Search for B → (ρ/ω)γ decays at BABAR Citation Piatenko, Timofei (2009) Search for B → (ρ/ω)γ decays at BABAR. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JFX3-1629. https://resolver.caltech.edu/CaltechETD:etd-08252008-225854 Abstract We present the results of the search for the decays ß (0/±) ? ? (0/±) ? (previously observed) and ß 0 ? ?? (for which currently only an upper limit exists). Together with ß ? K * ? decays, B ? (?/?)? allow us to measure the ratio of CKM-matrix elements |V td /V ts |. The analysis is based on the full BaBar dataset of 424.35 fb -1 , corresponding to 465 million BB pairs, and makes heavy use of multivariate classification techniques based on decision trees. We find ßR(B ? ? ± ? ) = (1.20+0.42-0.38±0.20)×10 -6 , ßR(B ? ? 0 ?)= (0.95+0.23-0.21 ±0.06)×10 -6 , and ßR(B ? ??)=(0.51+0.27-0.24±0.10)×10 -6 , where the first error is statistical and the second is systematic. We do not observe a statistically significant signal in the latter channel and set an upper limit at ßR(B ? ??) < 0.9 × 10 -6 (90% C.L.). We also measure the isospin and SU(3) F violating quantities ?(B + ? ? + ?)/2?(B 0 ? ? 0 ?) - 1 = -0.43+0.25-0.22±0.10 and ?(B ? ??)/?(B 0 ? ? 0 ?) - 1 = -0.49+0.30-0.27±0.10. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: bagged decision trees; radiative penguin B decays 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) Weinstein, Alan Jay Porter, Frank C. Wise, Mark B. Defense Date: 28 July 2008 Record Number: CaltechETD:etd-08252008-225854 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-08252008-225854 DOI: 10.7907/JFX3-1629 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3226 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 22 Oct 2008 Last Modified: 26 Nov 2019 20:24 Thesis Files Preview PDF (Thesis.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

Search for B → (ρ/ω)γ decays at BABAR Thesis by Timofei Piatenko In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2009 (Defended July 28, 2008) ii c 2009 Timofei Piatenko All Rights Reserved iii To my wife, Rimma, who helped me get through six years of graduate school, and my son, Andrei, whose arrival finally pushed me out of the door. iv Acknowledgements I would like to thank David Cassel and Lawrence Gibbons who got me interested in particle physics while I was an undergraduate at Cornell, Alan Weinstein for introducing me to the Caltech physics department, my adviser David Hitlin for his guidance through the years, Mark Wise and Frank Porter for teaching me particle physics, Ilya Narsky for sharing his deep knowledge of Statistics that made this work possible, Alex Samuel and Alexei Dvoretskii for their senior student wisdom, and all the folks who contributed to this analysis in the past couple of years. v Abstract We present the results of the search for the decays B 0/± → ρ0/± γ (previously observed) and B 0 → ωγ (for which currently only an upper limit exists). Together with B → K ∗ γ decays, B → (ρ/ω)γ allow us to measure the ratio of CKM-matrix elements |Vtd /Vts |. The analysis is based on the full BABAR dataset of 424.35 fb−1 corresponding to 465 million BB pairs, and makes heavy use of multivariate classification −6 techniques based on decision trees. We find B(B ± → ρ± γ) = (1.20+0.42 −0.38 ±0.20)×10 , +0.27 −6 0 −6 B(B 0 → ρ0 γ) = (0.95+0.23 −0.21 ±0.06)×10 , and B(B → ωγ) = (0.51−0.24 ±0.10)×10 , where the first error is statistical and the second is systematic. We do not observe a statistically significant signal in the latter channel and set an upper limit at B(B 0 → ωγ) < 0.9 × 10−6 (90% C.L.). We also measure the isospin and SU(3)F violating quantities Γ(B + → ρ+ γ)/2Γ(B 0 → ρ0 γ) − 1 = −0.43+0.25 −0.22 ± 0.10 and Γ(B 0 → ωγ)/Γ(B 0 → ρ0 γ) − 1 = −0.49+0.30 −0.27 ± 0.10. vi Contents Acknowledgements iv Abstract v Contents vi List of Figures ix List of Tables xiv 1 Introduction 1 2 Theoretical background and previous results 4 2.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Previous experimental results . . . . . . . . . . . . . . . . . . . . . . 7 3 PEP-II and the BABAR detector 11 3.1 PEP-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The BABAR detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 The silicon vertex tracker . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 The drift chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 The detector of Cherenkov light . . . . . . . . . . . . . . . . . . . . . 16 3.6 The electromagnetic calorimeter . . . . . . . . . . . . . . . . . . . . . 17 3.7 The instrumented flux return . . . . . . . . . . . . . . . . . . . . . . 18 3.8 Trigger and data acquisition . . . . . . . . . . . . . . . . . . . . . . . 19 vii 4 5 6 Initial event selection 22 4.1 Event reconstruction overview . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Datasets used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Skim selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.4 PID selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5 Precuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.6 Combined candidate selection . . . . . . . . . . . . . . . . . . . . . . 27 Classification of events 30 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Decision trees, boosting, and bagging . . . . . . . . . . . . . . . . . . 31 5.3 Multivariate classifier training . . . . . . . . . . . . . . . . . . . . . . 32 5.4 Study of NN vs AdaBoost performance . . . . . . . . . . . . . . . . . 33 5.5 Study of variable categories . . . . . . . . . . . . . . . . . . . . . . . 37 5.6 Study of potential gain from extending the input variable list . . . . . 40 5.7 AdaBoost vs Bagger . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.8 Variable selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.8.1 Photon quality precuts . . . . . . . . . . . . . . . . . . . . . . 48 5.8.2 Correlations with mES and ∆E . . . . . . . . . . . . . . . . . 48 5.8.3 Signal photon variables . . . . . . . . . . . . . . . . . . . . . . 49 5.8.4 Dalitz and helicity angles and π + π − displacement for B 0 → ωγ 49 5.8.5 Meson masses . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.8.6 Likelihood ratio vetoes for π 0 /η → γ γ decays . . . . . . . . . 51 5.8.7 Final list of classifier inputs . . . . . . . . . . . . . . . . . . . 51 The fit strategy 59 6.1 General fitter configuration . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 B 0 → ωγ fitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 B 0 → ρ0 γ fitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.4 B ± → ρ± γ fitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.5 B → K ∗ γ fitter for cross check . . . . . . . . . . . . . . . . . . . . . . 73 viii 6.6 Peaking BB component in B → ργ . . . . . . . . . . . . . . . . . . . 83 6.7 Simultaneous B → (ρ/ω)γ fitter . . . . . . . . . . . . . . . . . . . . . 93 7 Validation studies 98 8 Results 102 8.1 Branching fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.1.1 9 Study of B 0 → ωγ yields by BABAR runs . . . . . . . . . . . . 103 8.2 Upper limit for B 0 → ωγ mode . . . . . . . . . . . . . . . . . . . . . 107 8.3 Isospin and SU(3)F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.4 Determination of |Vtd /Vts | . . . . . . . . . . . . . . . . . . . . . . . . 108 Systematic errors 109 9.1 π 0 efficiency from Neutral AWG . . . . . . . . . . . . . . . . . . . . . 109 9.2 Photon quality cuts using e+ e− → µµγ control sample . . . . . . . . 109 9.3 Particle ID systematic . . . . . . . . . . . . . . . . . . . . . . . . . . 114 9.4 Bagger validation with B → K ∗ γ . . . . . . . . . . . . . . . . . . . . 118 9.5 Fit model systematics 9.6 Other systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10 Summary 129 Bibliography 130 ix List of Figures 2.1 CKM triangle relevant to BABAR physics . . . . . . . . . . . . . . . . . 4 2.2 Feynman diagram for a b → dγ transition . . . . . . . . . . . . . . . . 5 2.3 Comparison of previous branching fraction measurements with theory predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Previous constraints on |Vtd /Vts | . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Schematic representation of the linear accelerator and the PEP-II storage rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.6 Schematic drawing of the BABAR detector: side view . . . . . . . . . . 13 3.7 Schematic drawing of the BABAR detector: front view . . . . . . . . . . 13 3.8 The configuration of the SVT layers . . . . . . . . . . . . . . . . . . . 14 3.9 Schematic drawing of the silicon vertex tracker: side view . . . . . . . 15 3.10 Schematic drawing of the DCH: side view . . . . . . . . . . . . . . . . 15 3.11 Schematic drawing of the Cherenkov light detector: side view . . . . . 16 3.12 Schematic drawing of the electromagnetic calorimeter: side view . . . . 17 3.13 Schematic drawing of an EMC crystal . . . . . . . . . . . . . . . . . . 18 3.14 Schematic drawing of the instrumented flux return . . . . . . . . . . . 19 3.15 Flow chart representing Level 1 trigger operation . . . . . . . . . . . . 21 3.16 Schematic drawing of the BABAR online system . . . . . . . . . . . . . 21 4.17 Plots of signal vs stacked background MC for photon quality precuts . 29 5.18 B 0 → ωγ and B 0 → ρ0 γ Bagger validation curves . . . . . . . . . . . . 33 5.19 B 0 → ωγ AdaBoost with NN variables from previous BABAR search for B → (ρ/ω)γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 x 5.20 B ± → ρ± γ AdaBoost with NN variables from previous BABAR search for B → (ρ/ω)γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21 35 B 0 → ρ0 γ AdaBoost with NN variables from previous BABAR search for B → (ρ/ω)γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.22 B 0 → ωγ AdaBoost with ROE and B meson variables only . . . . . . . 38 5.23 B 0 → ωγ AdaBoost with ROE, B meson, and ω variables . . . . . . . 38 5.24 B 0 → ωγ AdaBoost with ROE, B meson, ω, and γ variables . . . . . . 39 5.25 B 0 → ωγ AdaBoost with ROE, B meson, ω, γ, and π 0 variables . . . . 39 5.26 B 0 → ωγ AdaBoost with 303 variables available in the full ROOT ntuples 41 5.27 B 0 → ρ0 γ AdaBoost with 296 variables available in the full ROOT ntuples 41 5.28 B ± → ρ± γ AdaBoost with 266 variables available in the full ROOT ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.29 B 0 → ωγ AdaBoost output for complete vs pruned list of inputs . . . . 43 5.30 B 0 → ωγ Bagger output for complete vs pruned list of inputs . . . . . 43 5.31 Raw output of AdaBoost and Bagger in B 0 → ωγ mode . . . . . . . . 45 5.32 Comparison of efficiency and signal significance curves for AdaBoost vs Bagger in B 0 → ωγ mode . . . . . . . . . . . . . . . . . . . . . . . . . 5.33 Comparison of efficiency and signal significance curves for AdaBoost vs Bagger in B 0 → ρ0 γ mode . . . . . . . . . . . . . . . . . . . . . . . . . 5.34 46 MC signal and background components for mES and ∆E for B 0 → ωγ mode after AdaBoost cut . . . . . . . . . . . . . . . . . . . . . . . . . 5.35 46 47 MC signal and background components for mES and ∆E for B 0 → ωγ mode after Bagger cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.36 Correlations of GammaECal vs ∆E in signal and uds continuum MC . . 49 5.37 Final MC output signal and background curves for B 0 → ρ0 γ Bagger . 52 6.38 PDF shapes determined from Monte Carlo for the B 0 → ωγ mode . . . 62 6.39 Luminosity-sampled, combined Monte Carlo fit for the B 0 → ωγ mode 63 6.40 Pure and signal-embedded toy MC for the B 0 → ωγ mode . . . . . . . 64 6.41 PDF shapes determined from Monte Carlo for the B 0 → ρ0 γ mode . . 66 6.42 Luminosity-sampled, combined Monte Carlo fit for the B 0 → ρ0 γ mode 67 xi 6.43 Pure and signal-embedded toy MC for the B 0 → ρ0 γ mode . . . . . . . 68 6.44 PDF shapes determined from Monte Carlo for the B ± → ρ± γ mode . . 70 6.45 Luminosity-sampled, combined Monte Carlo fit for the B ± → ρ± γ mode 71 6.46 Pure and signal-embedded toy MC for the B ± → ρ± γ mode . . . . . . 72 6.47 Signal MC mES and ∆E for B 0 → ρ0 γ vs B 0 → K ∗0 γ . . . . . . . . . . 74 6.48 Signal MC mES and ∆E for B ± → ρ± γ vs B ± → K ∗± γ . . . . . . . . . 74 6.49 Signal and generic MC Bagger output for B 0 → ρ0 γ vs B 0 → K ∗0 γ and B ± → ρ± γ vs B ± → K ∗± γ . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.50 PDF shapes determined from Monte Carlo for the B 0 → K ∗0 γ mode . 79 6.51 Luminosity-weighted, combined Monte Carlo and on-peak data fits for the B 0 → K ∗0 γ mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.52 PDF shapes determined from Monte Carlo for the B ± → K ∗± γ mode . 81 6.53 Luminosity-weighted, combined Monte Carlo and on-peak data fits for the B ± → K ∗± γ mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.54 82 Plots for peaking B-background studies (B → K ∗ γ and B → Xs γ only) in B → ργ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.54 Plots for peaking B-background studies continued . . . . . . . . . . . . 86 6.54 Plots for peaking B-background studies continued . . . . . . . . . . . . 87 6.54 Plots for peaking B-background studies continued . . . . . . . . . . . . 88 6.55 Plots for peaking B-background studies (adding B ± → ρ± η and B ± → ρ± π 0 ) in B ± → ρ± γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.56 Plots for peaking B-background studies continued . . . . . . . . . . . . 90 6.57 Plots for peaking B-background studies continued . . . . . . . . . . . . 91 6.58 Plots for peaking B-background studies continued . . . . . . . . . . . . 92 6.59 2-dimensional distribution of BB MC in B ± → ρ± γ mode . . . . . . . 94 6.60 B-embedded toy MC for the B ± → ρ± γ mode . . . . . . . . . . . . . . 94 6.61 Luminosity-sampled Monte Carlo fit for combined B → (ρ/ω)γ . . . . 96 6.62 Pure and signal-embedded toy MC for combined B → (ρ/ω)γ fit . . . . 96 6.63 Luminosity-sampled Monte Carlo fit for combined B → (ρ0 /ρ± )γ . . . 97 6.64 Pure and signal-embedded toy MC for combined B → (ρ/ρ± )γ fit . . . 97 xii 7.65 Distributions of combined continuum MC vs off-peak data in B 0 → ωγ and signal MC in B 0 → ρ0 γ vs B 0 → K ∗0 γ . . . . . . . . . . . . . . . 99 7.66 Plots for pre-unlbinding data-MC studies . . . . . . . . . . . . . . . . . 100 7.66 Pre-unblinding studies continued . . . . . . . . . . . . . . . . . . . . . 101 8.67 mES and ∆E projections for the on-peak data fit in the B 0 → ωγ mode 104 8.68 mES and ∆E projections for the on-peak data fit in the B 0 → ρ0 γ mode 104 8.69 mES and ∆E projections for the on-peak data fit in the B ± → ρ± γ mode 104 8.70 mES and ∆E projections for the on-peak data fit for the combined B → (ρ/ω)γ mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.71 mES and ∆E projections for the on-peak data fit for the combined B → ργ mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.72 On-peak data fit in the B 0 → ωγ mode using BABAR runs 1–4 only . . 106 8.73 On-peak data fit in the B 0 → ωγ mode using BABAR runs 1–5 only . . 106 8.74 On-peak data fit in the B 0 → ωγ mode using BABAR run 5 only . . . . 106 9.75 Distributions of photon variables for µµγ data and MC. . . . . . . . . 110 9.75 Distributions of photon variables for µµγ continued . . . . . . . . . . . 111 9.75 Distributions of photon variables for µµγ continued . . . . . . . . . . . 112 9.75 Distributions of photon variables for µµγ continued . . . . . . . . . . . 113 9.76 Profile histograms of the photon precut variables for B →(ρ/ω)γ signal MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.76 Profile histograms of the photon precut variables continued . . . . . . 116 9.77 Distribution of weights applied to the µµγ sample . . . . . . . . . . . . 120 9.78 Particle ID performance in MC vs D∗ decays data . . . . . . . . . . . . 120 9.79 PDF shapes determined from Monte Carlo for the B 0 → K ∗0 γ (K ∗0 → K + π − ) mode. No Bagger cut is applied. . . . . . . . . . . . . . . . . . 121 9.80 Luminosity-sampled, combined Monte Carlo and on-peak data fits for the B 0 → K ∗0 γ (K ∗0 → K + π − ) mode. No Bagger cut is applied. . . . . 122 9.81 Pure and signal-embedded toy MC for the B 0 → K ∗0 γ (K ∗0 → K + π − ) mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 xiii 9.82 PDF shapes determined from Monte Carlo for the B ± → K ∗± γ (K ∗± → K ± π 0 ) mode. No Bagger cut is applied. . . . . . . . . . . . . . . . . . 124 9.83 Luminosity-sampled, combined Monte Carlo and on-peak data fits for the B ± → K ∗± γ (K ∗± → K ± π 0 ) mode. No Bagger cut is applied. . . . 125 9.84 Pure and signal-embedded toy MC for the B ± → K ∗± γ (K ∗± → K ± π 0 ) mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 xiv List of Tables 2.1 Branching fraction predictions . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Summary of previous measurements . . . . . . . . . . . . . . . . . . . 10 4.3 B → (ρ/ω)γ MC datasets summary . . . . . . . . . . . . . . . . . . . . 24 4.4 B → K ∗ γ MC datasets summary . . . . . . . . . . . . . . . . . . . . . 24 4.5 Monte Carlo Skim Efficiencies . . . . . . . . . . . . . . . . . . . . . . . 28 4.6 Summary of quality cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.7 Summary of AdaBoost vs Bagger performance for B 0 → ρ0 γ . . . . . . 44 5.8 Definitions of classifier (Bagger) training input variables . . . . . . . . 57 5.9 Top 15 Bagger inputs for ρ0 γ and ρ± γ modes . . . . . . . . . . . . . . 58 6.10 Summary of combined MC fits for signal modes . . . . . . . . . . . . . 73 6.11 Summary of cross-check fits for B → K ∗ γ fits . . . . . . . . . . . . . . 77 8.12 Summary of on-peak data fits for all signal modes 8.13 Summary of fit results using only BABAR runs 1-5 data . . . . . . . . . 103 9.14 PID efficiency weighted momentum spectra . . . . . . . . . . . . . . . 117 9.15 Bagger cut signal MC efficiency in B → K ∗ γ 9.16 Bagger cut on-peak data efficiency in B → K ∗ γ . . . . . . . . . . . . . 119 9.17 Summary of systematic errors . . . . . . . . . . . . . . . . . . . . . . . 128 . . . . . . . . . . . 102 . . . . . . . . . . . . . . 118 1 1 Introduction At BABAR, which is a particle detector located in the PEP-II storage rings at the Stanford Linear Accelerator Center (SLAC), we study weak decays of the B meson by colliding electrons and positrons at the energy equivalent to the mass of the Υ (4S) resonance. Because the latter decays almost entirely into BB pairs, and because they are produced nearly at rest in the rest frame of the Υ (4S), we end up with a distinctively isotropic pattern of decay products, in sharp contrast to the jets produced by the hadronization of lighter quarks produced in the continuum. In addition, having two identical B mesons in each decay allows us to tag one of them, using a meson or a lepton decay signature to determine the flavor, and to infer information about the other, signal B. Furthermore, using different energies for the electron and the positron beams, we introduce a boost between the laboratory frame and the e+ e− center of mass (CM) frame, which allows us to easily find the separation between the two B mesons in the event. We also rely on the precise knowledge of the beam parameters to reconstruct two kinematic variables ∆E ≡ ∗ EB∗ − Ebeam mES ≡ q ∗2 Ebeam − p∗2 B where EB∗ and p∗B are the energy and momentum of the B candidate measured in the ∗ e+ e− CM frame, while Ebeam is the CM energy of each beam. We use mES in place of the meson mass because the energy of the beam is more precisely known. For a correctly reconstructed candidate, ∆E should be close to zero, while mES should be 2 close the nominal B mass. While the original purpose of the BABAR experiment was to study the phenomenon known as CP violation, possibly responsible for the matter-antimatter asymmetry we observe in nature, as the experiment comes to a close, attention is shifting towards rare decays of the B meson. In this particular analysis, we attempt to reconstruct exclusively some of the rarest decay channels, B → (ρ/ω)γ, which proceed via secondorder quantum loop processes known as Radiative Penguins. Apart from providing direct constraints on parameters of the Standard Model (SM) of elementary particle physics, penguin processes are also sensitive to new particles that may appear in the loop and alter the expected decay rates. As in all typical BABAR (and other high-energy physics) analyses, we proceed in several well-defined steps. First, data is collected by the detector, maintained and operated by various university and research laboratory groups that are part of this large collaboration. This raw data is processed to select only the “interesting” events that are most likely to come from B decays. Physical quantities, such as the momenta and energies of the particles, are reconstructed from primary information collected by the various parts of the detector, which comes in the form of voltage and current readings. Then, reconstructed quantities are combined to build a complete picture of an event, where tracks and energy deposits are matched to form particle candidates, which are in turn combined with certain kinematic assumptions to form composite particles that decayed inside the detector. Once lists of all particles in an event are formed, an individual analyst can use this information to select the events that fall within the topic of the analysis. For example, radiative penguin decays are identified by the presence of a highly energetic photon candidate that originates, along with a light meson, from the primary B candidate. The rest of the analysis relies heavily on statistical techniques to separate as cleanly as possible signal events from background and measure total yields or asymmetries to the highest precision possible, which implies maximizing statistical significance while minimizing the statistical and systematic uncertainties. Mathematically, an analysis consists of event classification, regression, and hypothesis testing. 3 Our main challenge is separating signal events from a very large amount of background from light quark decays, known as continuum. This includes e+ e− → udsc and is often combined with leptonic e+ e− → τ + τ − processes. The former are the main sources of our background, as they produce copious amounts of π 0 s and ηs, which decay to γγ pairs where one of the photons can be energetic enough to fake our signal signature. Along with continuum events, other B decays, in particular B → K ∗ γ where a kaon is misidentified as a pion, or B → ρπ 0 and B → ρη decays where one of the photons from π 0 or η decay is missed, also contribute. Therefore, we rely on the large number of variables that describe our events to carry out multivariate classification and remove much of the background. We then proceed to model the remaining distributions with signal and background components and perform maximum likelihood estimation of model parameters. Finally, we apply hypothesis testing methods, usually based on likelihood ratios, to determine the statistical significance of our measurements. In this analysis, we measure branching fractions, meaning that we are interested in the fraction of the total decay rate represented by our particular channels. Therefore, the measure of significance for us is simply the amount of deviation from zero in the total signal event yield. Finally, it’s important to note that we adopt the blind analysis methodology often used in physics experiments. We rely on precise modeling of the data in Monte Carlo (MC) simulations, which include a detailed physical description of all the detector components as well as all known information on elementary particle kinematics and their interactions with bulk matter. The analysis is carried out entirely on the MC simulated data, until all selection criteria and fit models are finalized.1 Only then does one “open the box” and perform that final measurement on the actual on-peak data collected by the detector. The intention is to minimize any potential bias in the choice of analysis procedure originating from the desire of the analyst to maximize the significance of the results. 1 A small amount of off-resonance data collected by the detector at a slightly lower energy is sometimes used for validation studies. 4 2 Theoretical background and previous results 2.1 Theoretical background The Standard Model of elementary particle physics attempts to describe the most fundamental constituents of matter and their interaction via the known forces, excluding gravity. Matter is postulated to consist of fermionic particles with fractional spin quantum numbers that interact via exchange of bosons with integer spins. Fermions 1.4 Violation in the Standard Model 21 further divide into two sectors: leptons (electrons, muons, taus, and corresponding & tb neutrinos) and quarks, which we are most interested in at B ABAVRtd.VQuarks are grouped $ into three generations with an up (positive charge) and&a down (negative charge) type # VudVub quark in each one. & Vcd Vcb " These particles interact via the strong (a) or the electroweak force by exchanging a spin-1 gluon, photon, W , or Z boson. A % & VudVub &| |Vcd Vcb Exchange of the latter two between two quarks allows them to transform between Vtd V&tb &| |Vcd Vcb $ up and down types and among the three generations. This mechanism is summarized concisely in the mathematical for- 0 0 7–92 # " ! 1 (b) 7204A5 Figure 2.1: CKM triangle relevant to malism of the SM in the form of Figure the 1-2. The rescaled Unitarity Triangle, all sides divided by B A B AR physics Cabbibo-Kobayashi-Maskawa (CKM) ma- . The rescaled Unitarity Triangle (Fig. 1-2) is derived from (1.82) by (a) choosing a phase convention ; (a) aligns one side of the triangle with the real axis, and (b) makes the length of this side 1. The form of the triangle graphically as a collection of triangles due to the unitarity constraint on its rows and is unchanged. Two vertices of the rescaled Unitarity Triangle are thus fixed at (0,0) and (1,0). The coordinates of the remaining vertex are denoted by . It is customary these days to express the CKM-matrix in terms of four Wolfenstein parameters with playing the role of an expansion parameter and representing the -violating phase [27]: trix [2], which in turn maysuch bethat represented is real, and (b) dividing the lengths of all sides by 5 columns. Figure 2.1 shows the CKM triangle most relevant to the physical processes studied at BABAR[1]. Within the SM, flavor-changing-neutral-current (FCNC) transitions among same type quarks of different generations are forbidden at tree level (lowest order processes described by Feynman diagrams without loops) because each CKM matrix element matches an up type quark to a down type quark. Thus, the leading order FCNC processes proceed via one-loop electroweak, so-called penguin diagrams where a weak boson is emitted and reabsorbed by a quark. The particular Feynman diagram that describes B → (ρ/ω)γ decays studied in this analysis is shown in Figure 2.2. Because the virtual top quark dominates the loop due to its high mass, b → W dγ decays are sensitive to the value of the CKM matrix element Vtd . In the context γ u, c, t b d of elementary particle theories beyond the SM, such as supersymmetry, new virtual d d particles may appear in the loop, leading to measurable effects on experimental observables such as branching fractions and Figure 2.2: Feynman diagram for a b → dγ transition CP (charge-parity) asymmetries [3]. Branching fraction results from recent next-to-leading order (NLO) SM theoretical calculations for the ρ isospin triplet and the ω singlet are listed in Table 2.1. The uncertainties of these predictions are rather large, mainly due to long-distance nonperturbative QCD effects, unaccounted for in the simplified approach to calculating the decay rates for these processes in the framework of the Heavy Quark Effective Theory (HQET). Normalizing to the branching fraction for the corresponding b → sγ decay, B → K ∗ γ, eliminates some of these uncertainties. This branching fraction ratio is related to the ratio of CKM matrix elements |Vtd /Vts | via [7]: Vtd 2 1 − m2ρ /MB2 B(B → (ρ/ω)γ) = S ρ/ω B(B → K ∗ γ) Vts 1 − m2K ∗ /MB2 !3 ζ 2 [1 + ∆R]. (2.1) 6 Here, the isospin factor Sρ/ω equals 1 and 0.5 for charged and neutral modes, respectively. ζ is the ratio of the ρ and K ∗ form factors computed using HQET approach, where the top quark and the W boson are integrated out of the effective interaction Hamiltonian. ∆R accounts for differences in the decay dynamics: the weak annihilation contribution for the decay B ± → ρ± γ and the W boson exchange contribution for the neutral channels [4, 6, 8, 9]. It is also worth noting that ∆R is itself dependent on Vtd /Vts and can be expressed as follows: ∆R = + i h 2A F1 + 2A (F12 + F22 ) 2 h 1− 2A(1)K (0)eff C7 ∗ ! − 2A(1)K ∗ (0)eff C7 ∗ (1)K Re A(1)ρ + F1 (Au + A A(1)t ) + A (F12 + F22 )Au sp − Asp (0)eff C7 i (2.2) where F1 = −|λu | cos α, F2 = −|λu | sin α, |λu | ≃ (1 − λ2 /2)|Vub /Vtd |, λ = Vus = 0.2265, and α is one of the angles of the CKM triangle [4]. Ci are known as Wilson Coefficients and represent strong interaction effects above the scale µ ∼ mb . They arise in HQET as the coefficients for the corresponding four-quark operators of various orders in the effective Hamiltonian that takes a B meson to the final ργ state. The most general form for the amplitude of the decay can be written in terms of the Fermi Constant, Wilson Coefficients, and CKM factors λi as [10]: GF X λi Ci (µ)hργ|Oi (µ)|Bi. A(B → ργ) = √ 2 i (2.3) From Equation 2.1, we see that radiative B decays provide the same CKM constraint as the ratio of Bd0 and Bs0 mixing frequencies, for which a precise measurement became recently available [11]. Comparing |Vtd /Vts | results obtained with these independent methods provides an important cross check of the SM, as significant inconsistencies would be a sign of new physics processes influencing decays of B mesons and B 0 oscillations in different ways. The other measurement that comes out of studying the branching ratios for the B → (ρ/ω)γ transitions (independent of the corresponding B → K ∗ γ decay modes) 7 is the isospin and SU(3)F breaking2 between the charged and the neutral channels, defined as: B(B ± → ρ± γ)τB 0 −1 2B(B 0 → ρ0 γ)τB + B(B 0 → ωγ) − 1. = B(B 0 → ρ0 γ) ∆ρ = (2.4) ∆ω (2.5) The amount by which these quantities differ from zero may signal evidence for new physics. In particular, for the ratio of the ρ modes, this deviation is expected to come from the weak annihilation contribution in the charged and W -exchange in the neutral channels. In the case of isospin violation between the ρ0 and the ω modes, the symmetry is broken by the different values of the form factors. However, these effects are all expected to be rather small, and observing an unexpectedly large deviation implies contribution from Feynman diagrams unaccounted for in the framework of the SM. The caveat to this is that theoretical uncertainties on second-order contributions are rather large, especially where the form factors are involved. Different theorists seem to agree on predictions for ∆ω and give values around −0.20 ± 0.09 [6]. There is a bit more variation when it comes to ∆ρ , which depends strongly on the value of the CKM angle γ. Ball, Jones, and Zwicky, for example, provide a range of predictions from −0.05 ± 0.07 at γ = 40◦ to 0.11 ± 0.03 at γ = 70◦ [6]. 2.2 Previous experimental results After earlier searches by CLEO (using 10 million BB events) [12] and BABAR (using 211 million BB events) [13], which had not yielded significant signals, b → dγ transitions were first observed by the Belle collaboration in the decay B 0 → ρ0 γ [14] using a sample of 386 million BB pairs. The latest BABAR measurement [15], which was based on 347 million BB events, confirmed Belle’s B 0 → ρ0 γ observation and found the first evidence of the decay 2 For the lack of a better name, we choose the convention used by Ali and Parkhomenko[4]. 8 B ± → ρ± γ; no significant signal in the B 0 → ωγ channel was seen, though. These results, which are detailed in Table 2.2, are in good agreement with the latest preliminary Belle measurement (based on a sample of 657 million BB pairs) [16] and theoretical predictions (see Figure 2.3). Using the most recent BABAR measurement of the B → K ∗ γ branching fraction [17] and Equation 2.1 with theory input from [6], the published BABAR B → (ρ/ω)γ branching fraction translates into |Vtd /Vts | = 0.200+0.021 −0.020 ± 0.015, where the first error is experimental and the second comes from theoretical uncertainties. As shown in Figure 2.4, this agrees well within errors with the recent Bs mixing results [11] and a global, independent CKM fit [18]. BABAR also measured the isospin and SU(3)F breaking ratios ∆ρ = −0.35 ± 0.27 and ∆ω = −0.49 ± 0.63. Although experimental measurements seem to indicate some evidence for significant isospin breaking between the charged and the neutral channels, for the purpose of event selection criteria optimizations and estimates of signal yields from simulated data (described in later chapters), we assume B(B 0 → ρ0 γ) = B(B 0 → ωγ) = 0.5 × 10−6 and B(B ± → ρ± γ) = 1.0 × 10−6 throughout this analysis. This is loosely based on the theoretical predictions by Ball, Jones, and Zwicky [6] as well as Ali and Parkhomenko [4]. 9 B(B ± → ρ± γ)/10−6 B(B 0 → ρ0 γ)/10−6 B(B 0 → ωγ)/10−6 1.37 ± 0.26 ± 0.09 0.65 ± 0.12 ± 0.03 0.53 ± 0.12 ± 0.02 Bosch et al [5] 1.58+0.53 −0.46 0.76+0.26 −0.23 – Ball et al 1.16 ± 0.26 0.55 ± 0.13 0.44 ± 0.10 Ali et al [4] [6] Table 2.1: NLO SM branching fraction predictions. Where two sets of errors are present, the first is theoretical, the second experimental. (ρ+, ρ0, ω) γ (combined) (ρ+, ρ0) γ (combined) -1 BaBar, 316 fb PRL 98, 151802 (2007) ρ+γ -1 Belle, 598 fb preliminary Ball, Jones, Zwicky ρ0γ PRD 75, 054004 (2007) derived from B. J .Z. assuming isospin ωγ 0 0.5 1 1.5 2 -6 x 10 2.5 3 3.5 4 Branching Fraction Figure 2.3: Comparison of previous branching fraction measurements with theory predictions 10 Mode nsig Σ B(10−6 ) (%) Belle B(10−6 ) Belle Σ B ± → ρ± γ 42.0+14.0 −12.7 3.8σ 11.0 1.10+0.37 −0.33 ± 0.09 +0.09 0.87+0.29 −0.27 −0.11 3.3σ B 0 → ρ0 γ 38.7+10.6 4.9σ −9.8 14.1 0.79+0.22 −0.20 ± 0.06 +0.09 0.78+0.17 −0.16 −0.10 5.0σ B 0 → ωγ 11.0+6.7 −5.6 2.2σ 7.9 0.40+0.24 −0.20 ± 0.05 0.40+0.19 −0.17 ± 0.13 2.6σ B → (ρ/ω)γ 6.4σ 1.25+0.25 −0.24 ± 0.09 1.14 ± 0.20+0.10 −0.12 6.2σ B → ργ 6.0σ 1.36+0.29 −0.27 ± 0.10 1.21+0.24 −0.22 ± 0.12 5.8σ η Table 2.2: The signal yield (nsig ), significance (Σ) including systematic errors, efficiency (), and branching fraction (B) for each mode. The second set of errors for (B) are systematic. Branching fraction results from Belle [16] are included for comparison. 1 95% prob. intervals BR(B→ρ/ωγ ) BR(B→K*γ ) ∆md ∆ms 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 ρ Figure 2.4: Contraints on |Vtd /Vts | from the BABAR run 1–5 measurement of B → (ρ/ω)γ (dark band), CDF Bs mixing results (light band), and a global CKM fit (ellipse). 11 3 PEP-II and the BABAR detector 3.1 PEP-II PEP-II [23] is an asymmetric energy e+ e− collider that uses separate storage rings for √ electrons and positrons before bringing them together at the energy of s = 10.58 GeV, corresponding to the peak production of the bb̄ resonance Υ (4s) with the crosssection of σ(e+ e− → Υ (4s)) = 1.05 nb. The high-energy ring (HER) stores electrons with the energy of 9.0 GeV, while the low-energy ring (LER) holds 3.1 GeV positrons. Both rings are located in the PEP tunnel with circumference of 2.2 km. Figure 3.5 shows a schematic representation of PEP-II. Figure 3.5: Schematic representation of the linear accelerator and the PEP-II storage rings Apart from the desired e+ e− → Υ (4s) reaction, processes that take place in the interaction region include elastic Bhabha scattering with σ(e+ e− → e+ e− ) ≈ 40 nb, muon production with σ(e+ e− → µ+ µ− ) = 1.16 nb, tau production with σ(e+ e− → τ + τ − ) = 0.94 nb, and light quark production with σ(e+ e− → uū) = 1.39 12 ¯ = 0.35 nb, σ(e+ e− → ss̄) = 0.35 nb, and σ(e+ e− → cc̄) = 1.30 nb, σ(e+ e− → dd) nb. To study these processes with real data, PEP-II is run 40 MeV below the Υ (4s) resonance for a fraction of the operating time. This off-peak data can be used, for example, to validate Monte Carlo simulations used for event selection optimization. The asymmetric nature of e+ e− collisions at PEP-II results in a boost of βγ = 0.56 between the center-of-mass (CM) and the laboratory frames. This translates to a typical separation between the two B meson vertices of about 250 µm, within the resolution of the BABAR silicon vertex tracker. This separation distance is a good discriminator between signal BB events and background light quark (or continuum) decays. The design luminosity for the PEP-II collider was 3.0 × 1033 cm−2 s−1 . However, the actual performance far exceeded this target, as the record peak luminosity reached 12.1 × 1033 cm−2 s−1 on August 16, 2006. Over the lifetime of the experiment, PEP-II delivered 557 fb−1 of data, of which 531 fb−1 were successfully recorded by the BABAR detector. 3.2 The BABAR detector The BABAR detector is a multi-system particle detector operating in the PEP-II collider interaction region. In order of increasing distance from the interaction point where electron and positron beams are brought head-on, it consists of the silicon vertex tracker (SVT), the drift chamber (DCH), the detector of internally reflected Cherenkov light (DIRC), the CsI(Tl) electromagnetic calorimeter (EMC), a 1.5 T superconducting magnet, and the instrumented flux return (IFR). Detector control and monitoring, event triggering, and data readout are performed by the data acquisition system (DAQ). The origin of the BABAR coordinate system coincides with the nominal interaction point. The z axis is pointing in the direction of the electron beam along the beam line. The y axis points upward, while the x axis points horizontally outward from the center of the storage rings. 13 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 1225 Magnetic Shield for DIRC Forward 3045 End Plug Bucking Coil 1375 Support Tube 810 e– e+ Q4 Q2 Q1 3500 B1 Floor 3-2001 8583A50 Figure 3.6: Schematic drawing of the BABAR detector: side view 0 IFR Barrel Superconducting Coil Scale 4m BABAR Coordinate System y Cutaway Section x z DIRC EMC IFR Cylindrical RPCs Earthquake Tie-down DCH SVT Corner Plates Gap Filler Plates 3500 Earthquake Isolator Floor 3-2001 8583A51 Figure 3.7: Schematic drawing of the BABAR detector: front view 14 3.3 The silicon vertex tracker The SVT is the innermost component of the BABAR detector responsible for precise vertexing and, in combination with the drift chamber, charged particle tracking and identification through dE/dx measurements. Many of the decay products of a B meson have low transverse momentum, and the SVT provides standalone tracking for particles with pt < 120 MeV/c, which is the cutoff for a reliable measurement using the DCH. The tracker consists of five unequally-spaced layers of double-sided silicon microstrip sensors, Beam Pipe 27.8mm radius Layer 5a where strips on one side are oriented parallel to Layer 5b the beam to provide angular information, while Layer 4b strips on the other side are oriented perpendic- Layer 4a ular to the beam to measure z. Each sensor is 300 µm thick. The innermost layer is only 33 Layer 3 Layer 2 mm from the interaction point, while the outer- Layer 1 most layer is at the distance of 146 mm. Figure 3.9 demonstrates the layout and dimensions of the SVT as well as its angular coverage, which Figure 3.8: The configuration of the SVT layers is constrained by the final beam focusing magnets located just outside the tracker. Figure 3.8 shows the layout of the five SVT layers from a cross-section perspective. The segments overlap slightly to provide better coverage at the edges. The helical trajectory of a charged track in a magnetic field is characterized by the following five parameters: the distance of closest approach to the z axis, d0 ; the corresponding azimuthal angle, φ0 ; the corresponding distance along the z axis, z0 ; the dip angle, λ, of the helix; and the curvature of the track, κ ∼ 1/pt . The precision of the first four of these is usually dominated by the uncertainties on the SVT measurements: σd0 = 23 µm, σφ0 = 0.43 mrad, σz0 = 29 µm, and σtan λ = 0.53 × 10−3 . 15 580 mm Space Frame Bkwd. support cone 520 mrad e- Fwd. support350 mrad cone e+ Front end electronics Beam Pipe Figure 3.9: Schematic drawing of the silicon vertex tracker: side view 3.4 The drift chamber The DCH consists of 40 layers (10 superlayers) and 7104 hexagonal drift cells. It is designed to provide precise charged particle tracking, such that a particle with pt > 180 MeV/c will be recorded in all the 40 layers. For particles with momenta of less than 700 MeV/c, the DCH provides dE/dx measurements necessary for particle identification. As shown in Figure 3.10, the subsystem is offset from the interaction point by 370 630 mm in the direction of the electron beam 1749 485 e– 809 27.4 464 1358 Be sitioned at angles of 45–76 mrad, alternat- 17.2 236 IP e+ 469 are axial, with wires parallel to the z axis. The wires in the other superlayers are po- 68 Elec– tronics to improve forward coverage based on the PEP-II boost. Superlayers 1, 4, 7, and 10 1015 1-2001 8583A13 Figure 3.10: Schematic drawing of the DCH: side view ing in sign, to enable z coordinate measurements. The DCH is filled with a 4:1 mixture of helium and isobutane gases. An incident particle ionizes the gas, and the resulting free electrons are accelerated toward the high-voltage sensor wires. Colliding with other molecules along the way, these elec- 16 trons produce further ionization electrons, resulting in an avalanche effect. The time of arrival of this avalanche determines the distance of closest approach of the particle to the wire, while the integrated charge provides dE/dx information with resolution of 7.5%. The DCH dominates the precision of momentum measurement for most tracks, parametrized by σpt /pt = (0.13 ± 0.01)% · pt + (0.45 ± 0.03)%. 3.5 The detector of Cherenkov light The DRC is a particle identification system built on the principles that a charged particle traveling faster than the speed of light in a refractive medium (v > c/n) emits a cone of radiation, and that the magnitudes of angles are preserved upon reflection from a flat surface. This radiation is called Cherenkov light, governed by cos θc = 1/nβ, where θc is the opening angle. PMT + Base 10,752 PMT's Light Catcher Purified Water Standoff Box 17.25 mm Thickness (35.00 mm Width) 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 3.11: Schematic drawing of the Cherenkov light detector: side view The detector (shown in Figure 3.11) consists of 144 synthetic quartz bars that trap Cherenkov light via total internal reflection. Photons that make it all the way 17 down to the front of a bar are reflected back by a mirror. On the back end, the photons are allowed to propagate into a standoff box filled with 6000 liters of purified water. Photons travel through the water, where the refractive index matches that of the quartz, preserving the conical pattern and the Cherenkov angle. The final image is detected by an array of 10,752 photomultiplier tubes (PMTs), and the arrival time is used to associate hits with charged tracks reconstructed in the DCH. The timing resolution measured with µ+ µ− events is 1.7 ns, only slightly worse than the intrinsic PMT spread of 1.5 ns. The corresponding angular resolution is 2.5 mrad. The DRC allows for a 4.2σ separation between kaons and pions at 3 GeV/c. This is particularly important in this analysis, where the signal modes produce only pions in the final state, while backgrounds from kinematically similar B → K ∗ γ decays produce pairs of kaons and pions. 3.6 The electromagnetic calorimeter The EMC is the most important component of BABAR hardware for this analysis, since it is responsible for detecting photons, which include the primary energetic photon from B → (ρ/ω)γ decay, as well as the softer photon pairs coming from π 0 and η decays. Thus, the EMC is essential both for detecting signal events and for distinguishing them from backgrounds. 2359 1555 1375 920 2295 1127 38.2˚ 1801 558 Interaction Point External Support 1979 26.8˚ 22.7˚ 15.8˚ 1-2001 8572A03 Figure 3.12: Schematic drawing of the electromagnetic calorimeter: side view 18 The EMC is a total absorption calorimeter comOutput Cable posed of 6580 thallium-doped CsI crystals, arranged in a central barrel and a forward endcap (illustrated Fiber Optical Cable to Light Pulser Preamplifier Board Diode Carrier Plate in Figure 3.12), all pointing toward the interaction point. The crystals are 4.7 × 4.7 cm in crosssection and vary in length from 29.6 cm (16 radiation lengths) to 32.4 cm (17.5 radiation lengths). Aluminum Frame Silicon Photo-diodes TYVEK (Reflector) Each crystal has a thin reflective coating to contain the scintillation light produced by an incident Aluminum Foil (R.F. Shield) CsI(Tl) Crystal particle. Two silicon photodiodes with quantum efficiency of 85% read out the signal from the back of each crystal (see Figure 3.13). The calorimeter is calibrated using 6.13 MeV Mylar (Electrical Insulation) CFC Compartments (Mechanical Support) 11-2000 8572A02 gamma rays from activated flourinet fluid, as well Figure 3.13: Schematic drawing as with e+ e− → e+ e− Bhabha scattering events of an EMC crystal that produce 3–9 GeV clusters for high-energy calibration. The energy and angular resolutions are parametrized as follows σE E = (2.32 ± 0.30)% q 4 ⊕ (1.85 ± 0.12)% E( GeV)  σθ = σφ = 3.7 3.87 ± 0.07  q E( GeV)  + 0.00 ± 0.04 mrad. The instrumented flux return The IFR serves as a flux return for the 1.5 T magnetic field and as a support structure for the entire BABAR detector. It’s also instrumented to detect muons, KL , and neutrons, none of which appear in this analysis. It is a hexagonal structure consisting of a barrel and two endcaps, as shown in Figure 3.14. The original IFR design made use of resistive plate chambers (RPCs), which 19 Barrel 342 RPC Modules 3200 FW 920 3200 19 Layers 1940 BW 18 Layers 1250 4-2001 8583A3 432 RPC Modules End Doors Figure 3.14: Schematic drawing of the instrumented flux return consist of an argon-freon-isobutane gas mixture sandwiched between two layers of graphite-coated bakelite, foam, and aluminum strips. An ionizing particle induces charge and changes the capacitance of the system, which can be read out by electronics. However, the performance of the RPCs slowly degraded over the years, prompting a replacement technology based on limited streamer tubes (LSTs) to be installed in 2004 and 2006. The LSTs are long and narrow rectangular cells (1.5 × 1.7 × 400 cm) of graphitecoated PVC and a silver-plated wire. Each cell is filled with a CO2 -isobutane-argon gas mixture. These detectors work on the principle of an electromagnetic avalanche produced by an ionizing particle traversing the gas-filled gap between the anode and the cathode. A “streamer” is the result of a cascade of secondary avalanches that occur after the primary avalanche grows enough to cancel out the external electric field and saturates. 3.8 Trigger and data acquisition The purpose of the BABAR trigger system is to pick out the events of interest and reduce the logging rate to a reasonable value that can be handled by modern electronic systems. The trigger consists of a hardware part (Level 1) and a software component 20 (Level 3, for historical reasons). The Level 1 trigger combines three sub-triggers from the DCH, EMC, and IFR and feeds the information to a global trigger (GLT). The drift chamber trigger uses both the timing and z coordinate information to reject background events. The GLT matches calorimeter clusters to drift chamber tracks for physics triggering, while also using IFR information for diagnostic triggering on µ+ µ− pairs and cosmic rays. The overall rate for Level 1 trigger is about 2.5 kHz at luminosity of 8 × 1033 cm−2 s−1 . The Level 3 trigger operates on an online Linux computer farm comprised of 28 Dell 1650 (Dual Pentium-III 1.4 GHz) capable of processing an individual event in about 4 ms. Here, quick reconstruction of DCH helices and EMC clusters is performed, and events are selected for logging. These events are then further grouped into runs and written to tape for long-term storage. Event rates are further reduced by about a factor of 10 by the Level 3 trigger, which also vetoes Bhabha scattering events. Apart from hosting the Level 3 trigger, the BABAR online event processing system is also responsible for general monitoring tasks related to data acquisition, real-time data quality monitoring, detector calibrations performed regularly during normal detector operation, and running an event display that provides visual event reconstruction information. Figure 3.15 shows a flow chart that illustrates Level 1 trigger operations and provides the design numbers for transmission rates among the various components. Figure 3.16 shows a very general overview of the BABAR online system. 21 3 bits/134 ns IFR FEE FastOR IFR Muon Trigger Synch Board (IFS) [1] Global Level 1 Trigger (GLT) 90 bits/ [1] 134 ns Calorimeter Trigger EMC ROM Processor Board (TPB) [10] 24 bits/67 ns 32 bits/134 ns DCH FEE 16 bits/134 ns Drift Chamber Track Segment Finder (TSF) [24] Fast Control 320 bits/134 ns Drift Chamber Binary Link Tracker (BLT) [1] 176x16 bits/269 ns Drift Chamber PT Discriminator (PTD) [8] Figure 3.15: Flow chart representing Level 1 trigger operation BABAR detector FEE VME dataflow [24] VME detector control [15] x24 logging server file and DB servers [5] event builder switch [1] network switches 740 GB x78 [1] 720 GB x16 x10 [2] application servers [10] x78 x10 farm nodes [78] 1.2 Gbps G−Links console servers [10] various analog/digital links ... 1 Gbps ethernet 100 Mbps ethernet computer center monitors [17] Figure 3.16: Schematic drawing of the BABAR online system 22 4 Initial event selection 4.1 Event reconstruction overview We aim to fully (exclusively) reconstruct three decay modes: B ± → ρ± γ , ρ± → π ± π 0 with B(ρ± → π ± π 0 ) ≈ 100%; B 0 → ρ0 γ , ρ0 → π + π − with B(ρ0 → π + π − ) ≈ 100%; B 0 → ωγ , ω → π+π−π0 with B(ω → π + π − π 0 ) ≈ 89%. Event reconstruction proceeds in several distinct stages. First, a “skim” is used to run over all the decays reconstructed in the BABAR detector and select only the ones that could potentially contain the signal we are looking for. This is extremely important, as the entire dataset of several TB is far too large to work with in practice, and there are many possible categories of B decays, whose signatures are very different from each other. Next, standard BABAR lists of particles reconstructed from primary event information collected by the detector using centralized software routines are used to combine individual candidates into composites, until one has a full picture of the decay. In our case, we combine low-energy photons into π 0 s, then combine pions (and kaons for B → K ∗ γ decays) into ρ/ω (and K ∗ ) mesons, and finally combine these with a highenergy photon to form the signal B meson candidate. At all stages, we require that the daughter particles originate from the same source (based on kinematic fits and geometric constraints) and that the mass of the resulting parent particle is consistent with the hypothesis assigned to it. Finally, various event shape quantities are computed and saved for the reconstructed event, including spatial distributions of the decay products and angular 23 relationships between individual (or groups of) particles. It is important to note that multiple reconstructions are possible for the same event, in which case we save all the possibilities and decide which particular one to choose at a later stage of the analysis. The main challenge, as is generally the case in all high-energy physics analyses, is to separate the events of interest from numerous background events that mimic our signal. Thus, it is possible to mis-reconstruct a kaon as a pion, allowing B → K ∗ γ background to leak in, or to use one of the photons originating from a π 0 or η decay as the primary photon candidate, after having lost the other photon during reconstruction. Since π 0 and η are light mesons, they are produced in large numbers in the continuum (i.e., decays that happen slightly off-resonance and do not proceed via the Υ (4S)). Because the ω(782) has a width of about 8 MeV, which is comparable to the reconstruction resolution in the final state π + π − π 0 , a tight cut on the invariant mass of the resonance is a powerful handle to reject combinatorial background. The ρ(770), however, has a width of about 150 MeV, which provides a far less stringent background rejection criterion. Due to the two-body kinematics and the low mass of the particles in the final state, the photon and the meson are produced with very high momentum, p∗ , in the B meson reference frame. The high momentum of this signal photon allows reduction of the combinatorial background due to π 0 and η decays. Nevertheless, the background from the continuum is overwhelming, and developing a dedicated classification tool for continuum suppression is crucial to achieve the sensitivity needed for the measurement. To leverage a larger set of event variables, we moved from an artificial neural network, which was used in previous searches for B → (ρ/ω)γ decays, to a decision-tree-based selection, described in detail in the next chapter. In addition to high levels of continuum backgrounds, signal events also need to be separated from BB decays, such as B → Xs γ (mostly B → K ∗ γ) and B → ρ(π 0 /η). By using a mixture of continuum and BB Monte Carlo for multivariate classifier training (see Section 4.6), we can reduce the problem of separating signal from background to event selection based on a single variable — classifier output. 24 4.2 Datasets used The analysis was performed using BABAR’s full runs 1–6 on-peak data sample of 424.35 fb−1 . Additionally, 41.37 fb−1 of off-peak data were used for validation studies. Monte Carlo samples used for optimization of selection criteria and for analysis technique validation are summarized in Table 4.3. Mode Signal Run B 0 B̄ 0 B+B− cc uds τ B ± → ρ± γ 587K 1 37200K 26072K 58900K 47180K 20378K B 0 → ρ0 γ 587K 2 103352K 103124K 168844K 130858K 55546K B 0 → ωγ 587K 3 48466K 49766K 83974K 66892K 27988K 4 167332K 167994K 252830K 206764K 90032K 5 241224K 244192K 366758K 317846K 132234K 6 105210K 100818K 156912K 127926K 56436K 702784K 691966K 1088218K 897466K 382614K 1338.6 1318.0 837.1 429.4 407.0 Total Luminosity in fb −1 : Table 4.3: Summary of B → (ρ/ω)γ MC datasets for Runs 1–6 For the B → K ∗ γ (K + π − and K ± π 0 modes) control sample, we used Runs 1–5 onpeak sample of 341.8 fb−1 , while run 6 was added for K ∗+ → K + π 0 mode, increasing the data to 376.24 fb−1 . The MC samples are summarized in Table 4.4. Mode Signal Run B 0 B̄ 0 B+B− cc uds τ K ∗0 → K + π − 1164K 1 37200K 36968K 54214K 21164K 20378K K ∗+ → K + π 0 195K 2 103498K 103124K 168844K 130858K 55606K 3 50556K 49766K 83974K 49182K 27988K 4 156446K 167994K 252830K 213380K 90032K 5 244812K 244322K 360578K 301758K 132234K 555312K 602174K 920440K 716342K 326238K 1128.6 1147.0 708.0 342.7 347.1 Total Luminosity in fb −1 : Table 4.4: Summary of B → K ∗ γ MC datasets for Runs 1–5 25 4.3 Skim selection We use the BtoRhoGamma skim in BABAR’s FilterTools package, which contains the skims for all possible BABAR analyses, to apply the following basic event selection criteria: 1. The event passes either the BGFMutliHadron (simply meaning we have identified at least three charged tracks in the event) or the BGFNeutralHadron (which selects photons with p∗ > 500 MeV/c and −0.75 < cos(θγ ) < 0.96) tag filter. 2. There are at least two tracks in the GoodTracksLoose list, which requires tracks to have at least 100 MeV/c of transverse momentum, with maximum momentum of 10 GeV/c, at least 12 hits recorded in the drift chamber, and imposes basic restrictions on the point of closest approach to the interaction point. 3. The event shape: the ratio of the 2nd and Oth Fox-Wolfram moments [21] is calculated using all tracks in the event, R2All < 0.9. 4. The primary photon energy is in the range [1.5, 3.5] GeV. 5. The meson candidate mass is in the range [0.5, 1.2] GeV/c2 and [0.5, 1.3] GeV/c2 for the ρ0 and ρ± modes, respectively, and within 0.05 GeV/c2 around the nominal ω mass. 6. Loose kinematic cuts are 5.1 < mES < 5.5 GeV/c2 and −0.6 < ∆E < 0.6 GeV. 7. The kinematics of the ρ0 daughters are compatible with a common vertex. Skim efficiencies for Monte-Carlo-generated events are listed in Table 4.5. Note that the skim selects events across all three signal modes, allowing cross-feed candidates to be counted. Further processing of the events by the RhoGammaTools analysis package (which performs the full candidate reconstruction described earlier, calculates and saves various event and candidate-level quantities, and applies further selection criteria, described in detail below) constrains events to be consistent with only one specific signal mode. 26 4.4 PID selection A very important part of this analysis is correctly identifying the pions coming from ρ/ω decay. Misidentifying kaons as pions allows B → K ∗ γ background to leak though into our signal region. The optimum selection, based on BABAR’s standard particle identification algorithms, was determined via a detailed study in the previous runs 1–4 analysis of B → (ρ/ω)γ decays. BABAR deploys various classification techniques, such as likelihood ratios and artificial neural networks, to correctly identify particles based on their attributes and place them into corresponding lists. There is a progression of selector algorithms with increasingly stringent selection criteria, which always introduce the mutually exclusive choice of better background rejection vs higher signal efficiency. These are labeled from VeryLoose to VeryTight in BABAR software. The best pion selector for the B → ργ modes was found to be the likelihood-based piLHVeryTight, combined with a minimal significance criteria of 0.001 on the Poisson probability of the number of the photons seen in the DIRC for a particular candidate being different from the expected number for the assigned pion hypothesis. Because the B → K ∗ γ background is not as important in the B 0 → ωγ (ω → πππ 0 ) decay mode, the piLHTight selector is the better choice there. Also, as enforcing the DIRC consistency criteria actually decreased signal significance, this cut was not applied to the B 0 → ωγ channel. 4.5 Precuts Table 4.6 summarizes fixed selection criteria applied to all modes before further optimization is carried out. Included are six precuts on photon quality, which are simultaneously optimized using the Bump Hunter (or PRIM) algorithm [24], although four of these are also included in bagged decision trees later, as described in Section 5.8.1. The photon quality precuts are meant to clean up the input to the decision tree classifier, reducing the size of the training sample and thus making it’s job much easier, as well as to simplify the estimation of systematic errors later on by cutting away 27 regions of poor agreement between data and simulated events. See Figure 4.17 for the comparison plots of normalized signal vs background distributions. The variables are as follows: • GammaECal : laboratory frame calibrated energy of the EMC cluster • Gammas9s25 : ratio of the sums of the energies of the central 9 EMC crystals to the central 25 crystals surrounding the centroid • GammaLat : lateral moment of the EMC cluster • GammasecMom : second moment of the EMC cluster • GammaZ20 : absolute value of the complex Zernike(2,0) moment [25] of the EMC cluster • GammaA42 : absolute value of the complex Zernike(4,2) moment of the EMC cluster. 4.6 Combined candidate selection On top of the selection criteria described so far, many variables are combined in a sophisticated classification algorithm based on decision trees and described in the next chapter. Final selection also includes choosing the best B meson candidate in events where multiple candidates were reconstructed. Such candidates are chosen so that the mass of the daughter meson (ρ or ω) is closest to the nominal particle mass, as found in [26]. The final signal efficiencies for the entire candidate selection process (i.e., the candidates that pass all cuts and end up in the final fit) are: 6.1% for B 0 → ωγ, 8.5% for B 0 → ρ0 γ, and 4.9% for B ± → ρ± γ. For comparison, the corresponding efficiencies for light quark (uds) continuum events are all near 10−7 . 28 data set pre-skim post-skim [%] B 0 B̄ 0 702784000 333791 0.0477 B+B− 702558000 453767 0.0647 cc 1088218000 16888023 1.5539 uds 897466000 36352462 4.0515 τ +τ − 382614000 4821779 1.2595 ρ0 γ 587000 423508 72.1479 ρ± γ 587000 366158 62.3779 ωγ 587000 356081 60.6612 Table 4.5: BtoRhoGamma skim efficiencies (runs 1–6 R22d) Description B 0 → ρ0 γ B ± → ρ± γ B 0 → ωγ −0.74 < cos(θγ ) < 0.93 Number of EMC crystals > 4 High-energy γ No problem crystal > 25 cm isolation Tracking PID π 0 selection R2All Converted γ veto GoodTracksLoose requirements for all charged tracks VeryTight π ID + DIRC photon consistency n/a tight π ID 0.115 < mπ0 < 0.150 ( GeV/c2 ) < 0.7 0.10 ≤ mconv ≤ 0.16 and 0.50 ≤ mconv ≤ 0.59 ( GeV/c2 ) η π0 GammaZ20 > 0.82 Gammas9s25 [0.93, 0.99] GammaLat [0.12, 0.51] GammaA42 < 0.08 GammasecMom < 0.002 GammaECal [1.5, 4.4] (relaxed from [1.89, 4.14]) GeV/c2 Table 4.6: Fixed cuts applied before optimization is carried out for the remaining criteria 29 signal bbbar bpbm ccbar uds tau GammaA42 : Zernike(4,2) moment 22000 signal bbbar bpbm ccbar uds tau GammaZ20 {GammaisFromPi0==0} : Zernike(2,0) moment 20000 20000 18000 18000 16000 16000 14000 14000 12000 12000 10000 10000 8000 8000 6000 6000 4000 4000 2000 2000 0 0 0.05 0.1 0.15 0.2 0.25 signal bbbar bpbm ccbar uds tau GammasecMom {GammaisFromPi0==0} : second moment 25000 0 0.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 signal bbbar bpbm ccbar uds tau Gammas9s25 {GammaisFromPi0==0} : crystal energy ratio 60000 40000 15000 30000 10000 20000 5000 10000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 signal bbbar bpbm ccbar uds tau GammaECal {GammaisFromPi0==0} : calibrated energy 7000 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 signal bbbar bpbm ccbar uds tau GammaLat {GammaisFromPi0==0} : lateral moment 12000 6000 10000 5000 4000 8000 3000 6000 2000 4000 1000 2000 0 1 0.6 50000 20000 0 0 0.55 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4.17: Plots of signal (arbitrarily scaled) vs stacked background MC (scaled to 316 fb−1 of data) for photon quality precuts optimized with Bump Hunter 30 5 Classification of events 5.1 Overview In the previous BABAR analysis of B → (ρ/ω)γ decays, the statistical significance of signal yield was slightly below 5σ in the ρ0 mode and 4σ in the ρ± mode. We (and the subsequent update from the Belle experiment[14]) also failed to find any significant evidence for signal in B 0 → ωγ. Therefore, we decided to try to apply a different classification algorithm to suppress our overwhelming background. The (artificial) neural network (NN) that was previously used was already at its performance limit due to poor performance scaling with the number of input dimensions, as well as poor ability to deal with correlated and/or redundant inputs. We decided to look into the possibility of using boosted or bagged decision trees, which perform much better in higher dimensions and deal gracefully with varied, correlated, irrelevant, and redundant inputs. Because the B 0 → ωγ mode produced the lowest and least significant yield in all previous analyses, we chose to concentrate on it for all of our initial tests of decision tree performance. A new C++-based package written by former BABAR collaborator Ilya Narsky was adopted as our primary tool. Now open-source software, StatPatternRecognition [27] is available for download from http://sourceforge.net/projects/statpatrec (documentation is also available in the corresponding README file). 31 5.2 Decision trees, boosting, and bagging Decision trees are a well-known tool in machine learning, widely used for classification problems (in our case, separating signal from background). In HEP, a decision tree can be built by choosing the most useful event variables and making decision splits on them. This is similar to making a cut on a kinematic variable, except that this is done over and over for each new tree node. In StatPatternRecognition, a decision tree is built by trying all possible splits on all possible variables and choosing the one that optimizes a given figure of merit (FOM). Although one can specify various FOMs at runtime, the usual criteria to optimize is either the Gini index [28] (used throughout this analysis) or cross-entropy [29]. In this case, one is concerned mostly with the fraction of correctly classified events and does not have to worry about applying correct luminosity weights to the different MC samples used in training and validation, as is required, for example, in the case of statistical significance optimization. Optimizing the Gini index is equivalent to maximizing the purity of child nodes in a tree, so that one node will contain almost entirely signal, while the other almost entirely background events. StatPatternRecognition also does not apply any pruning algorithms, which are often used in other implementations to remove nodes, which produce splits that are no better than a random choice, or to merge nodes to further improve the overall FOM. Because trees are simple to build and easy to visualize, they are a powerful algorithmic tool. However, a single tree cannot rival the power of the complexity built into a neural network. Thus, we extend the capabilities of tree-based algorithms by applying two modern statistical tools: Boosting (adaptive boosting or AdaBoost) and bootstrap aggregation (bagging). Boosting was introduced by Freund and Shapire in the form of the AdaBoost algorithm in 1999 [30]. It consists of combining weak classifiers (individual trees) into a weighted sum that becomes a new, strong classifier. In the process of building new trees to classify our events, we increase the weights of those that have been 32 misclassified so far, so as to force the algorithm to pay them more attention in the future. The final decision of whether to classify a certain event as signal or background is made by a collective vote of all the weak classifiers. In the framework of StatPatternRecognition, one can control the minimum size of a terminal tree node (one that is not split further) as well as the number of weak classifiers to combine. Generally, one wants to build O(100) trees with large terminal nodes for the algorithm to perform near optimally. Bagging (or bootstrap aggregating) is the brain child of Leo Breiman, dating back to 1996 [31]. Here, the idea is to use bootstrap replicas of data, i.e., resampling the data a certain number of times, each time selecting with replacement a subset of training events. Thus, one “averages” over the bootstrap samples to gain better accuracy than a single-shot approach. Bootstrap was originally introduced as a way to estimate certain parameters of a statistical distribution, such as correlations among variables, for which there is no close-form algebraic expression. In the case of decision trees, it’s a different way of producing a more accurate decision based on some collective vote. While the parameters of Bagger in StatPatternRecognition are the same as for AdaBoost, in this case, the user should make the terminal nodes small to pick up on the variation among the bootstrap replicas of training data. 5.3 Multivariate classifier training The actual training of a decision-tree-based classifier is done in the following way. Samples of signal and background Monte Carlo are produced by applying all preselection criteria. The samples are then split randomly in half, the first subset becoming the training and the second the testing/validation samples. In earlier studies, our training sample consisted of about 30,000 signal and 45,000 background events, while for the final classifier training, we roughly doubled these numbers. The training algorithm is fed a vector of values corresponding to each input variable for each candidate in the sample. Once the classification is complete, it is applied to the testing sample, and the validation curve (exponential loss for AdaBoost or quadratic loss for 33 Bagger3 ) is plotted (e.g., see Figure 5.18). From this graph, we can see whether the training has converged (i.e., the loss function has leveled off, and we’ve fixed all the misclassifications we could), or if we need to add more cycles (i.e., build more trees). If the loss function starts shooting back up, we have overtrained and need to use fewer cycles. From these validation curves, we can see that most of the work is done in under 100 cycles, the next 100 providing a small performance increase, and after about 200 cycles, each additional one provides only a small incremental contribution. 0.1 0.082 FOM vs training cycle FOM vs training cycle 0.08 0.095 0.078 0.076 0.09 0.074 0.072 0.085 0.07 0.068 0.08 0.066 0.064 0.075 0.062 0.07 0.06 0 50 100 150 200 250 0 50 100 150 200 250 Figure 5.18: B 0 → ωγ (left) and B 0 → ρ0 γ (right) Bagger validation curves (quadratic loss) as a function of training cycle (number of trees built) 5.4 Study of NN vs AdaBoost performance Because this analysis is an update of earlier searches for B → (ρ/ω)γ decays at BABAR, much of the initial work went into developing an improved analysis framework that would approach the goal of extracting the desired signal events from data in a substantially different manner from the previous iterations. Therefore, several studies were performed that compared fairly the background suppression performance of NNs used for the most recent BABAR measurement [15]. The first thing to investigate was how the performance of the existing NN compares to that of a boosted decision tree with the same set of input variables. Because 3 A loss function measures the difference between the target and the result of classification. Thus, for quadratic loss used in bagging, a signal event with classifier output of 0.5 corresponds to a loss of (1 − 0.5)2 = 0.25. 34 the previous BABAR analysis also relied on Bump Hunter [24] to optimize the cuts for remaining variables that were not included in the NN, as well as the cut on the NN output itself, we included these variables in the decision tree inputs to produce a fair comparison. In order to assess the performance, we compared the signal vs background efficiency curves produced by scanning the output of each algorithm, with the aim of achieving the lowest possible background efficiency for a given signal efficiency (or, alternatively, the highest signal efficiency for a given background efficiency). Figures 5.19 through 5.21 show the normalized distributions of signal and background for AdaBoost trained with StatPatternRecognition next to the NN, as well as the efficiency curves for both. In these plots, the NN output is shown with the rest of the cuts already applied, which accounts for the difference in the behavior of the efficiency curves. Thus, we do not see the NN efficiencies reach 1, as is the case with AdaBoost. The optimal cut value, based on maximizing signal significance, is shown on the plot of AdaBoost output. A vertical line is also drawn on the NN plot to show the nominal cut applied in the previous analysis. Horizontal lines on the efficiency curve plots show the background level for each of these cuts. By following these lines, one can directly compare the performance (in terms of gain or loss in signal efficiency for the given background efficiency) of both the old and the new background rejection methods. Efficiency curves for AdaBoost are shown for both the training and the testing samples. From this quick study, we determined that the B 0 → ωγ mode can easily be improved, while the B → ργ modes require more effort. This study also demonstrated that while the old NN shows a better ability to concentrate the signal events near 1, AdaBost does better at moving most of the background to 0 ([0, 1] being the standard output range for these multivariate classification algorithms). This makes AdaBoost the more desirable method, as the overall goal is to reduce the background, while keeping enough signal to extract a statistically significant measurement. However, both classifiers exhibit an unwanted feature — the tendency for the signal to peak near 0 and for background to peak near 1. 35 signal vs BG efficiency NN output ada signal BG eff SPR output NN signal NN 1 -- SPR best cut ada BG ada test -- NN precut NN BG ada train 10-1 10 -1 10-1 best cut = 0.996 10-2 10-2 10-2 -3 10 10 -3 -3 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.19: B 0 → ωγ AdaBoost (signal in red, combined backgrounds in blue) with NN variables from previous BABAR search for B → (ρ/ω)γ 1 signal vs BG efficiency NN output ada signal BG eff SPR output NN signal NN 1 -- SPR best cut ada BG 10-1 ada test -- NN precut NN BG ada train 10-1 10-1 10-2 best cut = 0.963 10-2 10-2 10 -3 -3 10 -3 10 10-4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.20: B ± → ρ± γ AdaBoost (signal in red, combined backgrounds in blue) with NN variables from previous BABAR search for B → (ρ/ω)γ 36 SPR output signal vs BG efficiency NN output ada signal BG eff 1 NN signal NN 1 -- SPR best cut ada BG 10 ada test -- NN precut NN BG ada train 10-1 -1 10-1 best cut = 0.996 10-2 10-2 10-2 10 -3 -3 10 -3 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.21: B 0 → ρ0 γ AdaBoost (signal in red, combined backgrounds in blue) with NN variables from previous BABAR search for B → (ρ/ω)γ 37 5.5 Study of variable categories The next study performed was to determine the contribution to overall classifier performance coming from different variable categories: B meson variables, daughter meson (ρ/ω) variables, γ variables, and π (0) variables. Here, the B 0 → ωγ mode is used for illustration, being the channel we concentrate mostly on. Figures 5.22 through 5.25 show the progression of AdaBoost performance, compared to the same NN shown in previous section, as these categories of variables are added to all restof-event (ROE) variables available in the n-tuples. This demonstrates several points: a) ROE variables by themselves do not provide enough separating power, b) even with signal B meson variables added, we still can’t beat the NN c) all categories of variables add something to the overall performance. 38 signal vs BG efficiency NN output ada signal BG eff SPR output NN signal NN 1 -- SPR best cut ada BG 10-1 ada train 10-1 10-1 best cut = 0.983 10-2 ada test -- NN precut NN BG 10-2 10-2 10 -3 -3 10 -3 10 10-4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.22: B 0 → ωγ AdaBoost (signal in red, combined backgrounds in blue) with ROE and B meson variables only signal vs BG efficiency NN output ada signal BG eff SPR output NN signal NN 1 -- SPR best cut ada BG 10-1 ada test -- NN precut NN BG ada train 10-1 10-1 best cut = 0.996 10-2 10-2 10-2 -3 10 10 -3 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.23: B 0 → ωγ AdaBoost (signal in red, combined backgrounds in blue) with ROE, B meson, and ω variables 39 signal vs BG efficiency NN output BG eff SPR output 1 ada signal NN signal NN 1 -- SPR best cut ada BG 10 ada test -- NN precut NN BG ada train 10-1 -1 10-1 best cut = 0.996 10-2 10-2 -3 10 10-2 10 -3 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -3 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.24: B 0 → ωγ AdaBoost (signal in red, combined backgrounds in blue) with ROE, B meson, ω, and γ variables signal vs BG efficiency NN output BG eff SPR output 1 ada signal NN signal NN 1 -- SPR best cut ada BG 10 ada test -- NN precut NN BG ada train 10-1 -1 10-1 best cut = 0.996 10-2 10-2 -3 10 10-2 10 -3 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.25: B 0 → ωγ AdaBoost (signal in red, combined backgrounds in blue) with ROE, B meson, ω, γ, and π 0 variables 40 5.6 Study of potential gain from extending the input variable list Once we determined that adding variables from different categories helps significantly boost classifier performance, it seemed clear that there was a lot of potential in utilizing a much larger variable set than the one deployed for NN inputs, where signal-side variables were avoided, except for the cosine of the polar angle of the B candidate in the center of mass frame. Therefore, we performed a study of AdaBoost performance with the full set of n-tuple variables (∼ 300) vs that of the original NN (∼ 30). The results, shown in Figures 5.26 through 5.28, demonstrate a very high potential for performance gain. One has to remember that in this order-of-magnitude study we include variables that are not well modeled in Monte Carlo simulations, as well as variables highly correlated with our primary kinematic variables mES and ∆E. However, even once the input list is pruned, one expects a performance gain over the old approach. We tested this hypothesis by removing obviously problematic variables from the list and re-training AdaBoost. Bagger was also trained for both cases. Figures 5.29 and 5.30 show that the differences in output were found to be very small, thus proving that there exist enough extra kinematic variables to warrant the expansion of the variable input list by about a factor of 2. However, it is important to note here that momenta of particles were retained in the pruned input variable list. Therefore, these plots demonstrate that a very high level of signal and background separation can be obtained with decision-treebased methods, but the results would only be appropriate for a simple cut-and-count analysis due to correlations of particle momenta with our primary fit variables, mES and ∆E. Because even with this level of separation, there still remains a large number of background events near the signal peak at AdaBoost or Bagger output value of 1, such a treatment is unrealistic. Therefore, we remove particle momenta from the list of classifier inputs in all of the following studies. 41 1 signal vs BG efficiency NN output ada signal BG eff SPR output NN signal NN 1 -- SPR best cut ada BG ada test -- NN precut NN BG ada train 10-1 10-1 10-1 10-2 best cut = 0.996 10-2 10-2 -3 10 -3 10 10 -3 10-4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.26: B 0 → ωγ AdaBoost (signal in red, combined backgrounds in blue) with 303 variables available in the full ROOT n-tuples 1 signal vs BG efficiency NN output ada signal BG eff SPR output NN signal NN 1 -- SPR best cut ada BG ada test -- NN precut NN BG ada train 10-1 10-1 10-1 10-2 best cut = 0.996 10-2 10-2 -3 10 10 -3 -3 10 10-4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.27: B 0 → ρ0 γ AdaBoost (signal in red, combined backgrounds in blue) with 296 variables available in the full ROOT n-tuples 42 ada signal 1 signal vs BG efficiency NN output BG eff SPR output NN signal NN 1 -- SPR best cut ada BG 10-1 ada train 10-1 10-2 ada test -- NN precut NN BG 10-1 10-2 best cut = 0.996 10-2 10 -3 10 -3 -3 10 10-4 10-4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 signal eff Figure 5.28: B ± → ρ± γ AdaBoost (signal in red, combined backgrounds in blue) with 266 variables available in the full ROOT n-tuples 43 AdaBoost all vars AdaBoost best vars 1 1 10-1 10-1 10-2 10-2 10-3 10-3 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 5.29: B 0 → ωγ AdaBoost normalized output (signal in blue, background in red) compared for the cases where all 303 available variables were used as inputs (left) vs a pruned list of only 70 inputs (right) Bagger Bagger all vars best vars 104 104 103 103 102 102 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 5.30: B 0 → ωγ Bagger normalized output (signal in red, background in blue) compared for the cases where all 303 available variables were used as inputs (left) vs a pruned list of only 70 inputs (right) The studies described above demonstrated the potential for improvement over the previous BABAR analysis of B → (ρ/ω)γ decays by switching to a decision-tree-based 44 mode classifier Nsignal Nbackground √ S/ S + B B 0 → ρ0 γ AdaBoost 11.0 17.4 2.06 B 0 → ρ0 γ Bagger 14.3 27.7 2.20 Table 5.7: Summary of AdaBoost vs Bagger performance comparison using cut-andcount analysis in the signal region for B 0 → ρ0 γ classification scheme leveraging the power of boosting or bagging techniques and expanding the input variable set considerably. This improvement is made possible by the ability of decision trees to deal with correlated and irrelevant inputs and to perform well in higher dimensions, both of which are the downfall of neural networks. 5.7 AdaBoost vs Bagger Based on the preliminary results described so far, we decided to pursue this new strategy and develop in parallel two classifiers based on boosted and bagged decision trees. As seen in Figures 5.29 and 5.30, while AdaBoost and Bagger perform similarly for signal MC events, Bagger produces better-behaved background distributions that fall off steadily as they approach 1, rather than peaking there (thus mimicking signal), as in the case of AdaBoost. However, to really be able to compare them, two studies had to be carried out. First, we looked at the signal vs background efficiency curves, similar to the ones that compared AdaBoost to the NN earlier. We also looked at the corresponding FOM plots to determine which classifier provides better signal sensi√ tivity (defined as S/ S + B). Then, we performed a simple cut-and-count analysis in the signal region based on a cut on AdaBoost or Bagger output and compared the plots of mES and ∆E in the fit region (see Table 5.7). Figures 5.32 and 5.33 compare efficiency and signal significance curves for AdaBoost and Bagger for B 0 → ωγ and B 0 → ρ0 γ modes, respectively. Figure 5.34 shows the distributions of signal and background components in mES and ∆E for the B 0 → ωγ mode using a cut at 0.96 on AdaBoost output, while Figure 5.35 shows 45 signal bbbar bpbm ccbar uds tau Bagger output for signal and generic MC bag {(BmESS>=5.22&&BmESS<=5.3&&abs(BdeltaE)<0.3)&&(GammaZ20>0.82&&Gammas9s25>0.93&&GammaLat<0.51&&GammaA42<0.08&&GammasecMom<0.002&&GammaECal>1.5&&GammaECal<4.4)} 104 signal bbbar bpbm ccbar uds tau Adaboost output for signal and generic MC ada {(BmESS>=5.22&&BmESS<=5.3&&abs(BdeltaE)<0.3)&&(GammaZ20>0.82&&Gammas9s25>0.93&&GammaLat<0.51&&GammaA42<0.08&&GammasecMom<0.002&&GammaECal>1.5&&GammaECal<4.4)} 104 103 103 102 102 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5.31: Raw output of Bagger (left) and AdaBoost (right) in B 0 → ωγ mode. Signal is plotted against stacked backgrounds on a log scale. the same distributions using the same cut on Bagger output. These plots clearly demonstrate the superiority of Bagger over AdaBoost. What’s more, AdaBoost seems to have a really unwanted feature in that it lacks an extremum in the signal significance curve, thus making it hard to justify a choice of cut on the output. We believe this is due to the previously mentioned problem with AdaBoost output for background events — namely the tendency of the background distribution to peak near 1, similar to the signal. In the case of Bagger, the background distribution drops off steadily near 1, allowing for a much cleaner cut on the output that rejects most of the background while retaining enough signal to produce peaks in mES and ∆E. See Figure 5.31 for a comparison of raw outputs for the two classifiers. It is clear that although AdaBoost produces more sharply peaking distributions for both signal and background events, Bagger has the better-behaved output of the two in the more-interesting-to-us region near 1. These studies of classifier performance led us to ultimately select bagged in favor of boosted decision trees. 5.8 Variable selection Several studies were carried out in order to prune the list of input variables to AdaBoost or Bagger to a more manageable set of well-modeled quantities that are not 46 S/sqrt(S+B) vs cut value ada_6000-90 1 bag_10-110 10-1 S/sqrt(S+B) BG eff signal vs BG efficiency ada_6000-90 bag_10-110 1 0.8 10-2 -3 0.6 10-4 0.4 10 -5 10 0.2 0 0.2 0.4 0.6 0.8 1 signal eff 0 0.2 0.4 0.6 0.8 1 cut value Figure 5.32: Signal vs background efficiency curves (left) and signal significance curves (right) for AdaBoost (red) and Bagger (blue) trained for B 0 → ωγ mode S/sqrt(S+B) vs cut value 10-2 0.8 BG eff 10-1 S/sqrt(S+B) signal vs BG efficiency ada_11000-70 1 bag_10-110 ada_11000-70 1.2 bag_10-110 1 -3 10 0.6 10-4 0.4 -5 10 0.2 -6 10 0 0.2 0.4 0.6 0.8 1 signal eff 0 0.2 0.4 0.6 0.8 1 cut value Figure 5.33: Signal vs background efficiency curves (left) and signal significance curves (right) for AdaBoost (red) and Bagger (blue) trained for B 0 → ρ0 γ mode 47 signal bbbar bpbm ccbar uds tau BmESS BmESS {((BmESS>=5.22&&BmESS<=5.3&&abs(BdeltaE)<0.3)&&(GammaZ20>0.82&&Gammas9s25>0.93&&GammaLat<0.51&&GammaA42<0.08&&GammasecMom<0.002&&GammaECal>1.5&&GammaECal<4.4))&&(ada>0.96)} 18 16 18 16 14 14 12 12 10 10 8 8 6 6 4 4 2 2 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 signal bbbar bpbm ccbar uds tau BdeltaE BdeltaE 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆E (GeV) mES (GeV/c2) Figure 5.34: Distributions of MC signal overlaid on top of stacked background components for mES (left) and ∆E (right) for B 0 → ωγ mode after a cut on AdaBoost output signal bbbar bpbm ccbar uds tau BmESS BmESS {((BmESS>=5.22&&BmESS<=5.3&&abs(BdeltaE)<0.3)&&(GammaZ20>0.82&&Gammas9s25>0.93&&GammaLat<0.51&&GammaA42<0.08&&GammasecMom<0.002&&GammaECal>1.5&&GammaECal<4.4))&&(bag>0.95)} 7 6 signal bbbar bpbm ccbar uds tau BdeltaE BdeltaE {((BmESS>=5.22&&BmESS<=5.3&&abs(BdeltaE)<0.3)&&(GammaZ20>0.82&&Gammas9s25>0.93&&GammaLat<0.51&&GammaA42<0.08&&GammasecMom<0.002&&GammaECal>1.5&&GammaECal<4.4))&&(bag>0.95)} 5 4 5 4 3 3 2 2 1 1 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 mES (GeV/c2) 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆E (GeV) Figure 5.35: Distributions of MC signal overlaid on top of stacked background components for mES (left) and ∆E (right) for B 0 → ωγ mode after a cut on Bagger output 48 strongly correlated with our primary fitting variables mES and ∆E. Before anything else, input variables were checked for compatibility between off-peak data and combined, luminosity-scaled generic Monte Carlo. If distributions exhibited visuallydetectable differences, the corresponding variables were removed from training. We’ll return to this study in Section 6.7. 5.8.1 Photon quality precuts As described in Section 4.5, photon quality cuts are applied to our samples before any classifier training is produced. While the original idea was to remove these six variables from the list of Bagger inputs, studies showed that it is beneficial to keep them even after rectangular cuts are applied. This is likely due to correlations among the variables, which are impossible to see from the plots, yet can be utilized by the flexible multidimensional classifier. However, the cut on GammaECal was already relaxed to avoid biases. The variable is also rather strongly correlated with ∆E (see Figure 5.36). Also, there was some concern regarding data-MC agreement for GammasecMom at lower values. Although the difference is not very dramatic, when we considered the fact that the optimal cut value for this variable never changed in any of our optimizations (using the Bump Hunter in both the old and the new analyses), we did not see the benefit in retaining it the list of inputs. Therefore, we decided that it would be best to remove the above two variables from the list of Bagger inputs. 5.8.2 Correlations with mES and ∆E Two different studies of variable correlations were carried out. First, we determined which of the potential Bagger inputs were correlated with ∆E at around 10% level or higher. One of the variables removed after this study (GammaECal) has already been mentioned. The other variables that were eliminated through this process are BcosHelB(Gamma/Meson), all correlated with ∆E at the 30% level in uds and with mES at the 20% level in signal Monte Carlo. These are the two calculations of the helicity angle of the B meson in the CM frame using either the meson or the photon 49 GammaECal vs ∆E (uds) ρ = 0.267137 GammaECal GammaECal GammaECal vs ∆ E (signal) ρ = 0.109104 4.5 5.5 5 4.5 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆E ∆E Figure 5.36: Correlations of GammaECal vs ∆E in signal and uds continuum MC daughter. 5.8.3 Signal photon variables Next, we studied the correlations between the signal photon variables and the rest of the Bagger inputs, demonstrating that the former can be treated as an independent sample, thus justifying the e+ e− → µµγ study described in Section 9.2 to evaluate the systematic error associated with the precuts. Due to the very large number of associated correlation plots, we decided not to include them here. 5.8.4 Dalitz and helicity angles and π + π − displacement for B 0 → ωγ The Dalitz angle (defined as the angle between the π + and π 0 momenta in the π + π − rest frame), the helicity angle (defined in Table 5.8), and the π + π − displacement (distance between the B and the ω vertices) were considered as inputs to the Bagger, as well as components of the final fit. However, the Dalitz angle and the π + π − displacement have no corresponding quantities in the B → K ∗ γ control sample, used 50 to validate the performance of the Bagger. On top of that, Bagger trained with the Dalitz angle as one of the inputs performed almost indistinguishably compared to the Dalitz-free configuration. The performance of the Bagger trained without π + π − displacement also did not suffer much (based on split counts, this could be explained by the substantial increase in contribution from the χ2 probability of the ω vertex). Therefore, it was decided to simplify the classifier architecture further by removing these two variables. After attempting to use the Dalitz angle as one of the components of the final fit and failing to find any improvement, the variable was completely left out of the analysis. While removing the helicity angles from Bagger configuration did not result in a significant performance drop for B 0 → ωγ and B ± → ρ± γ modes, we decided to keep this variable for all three signal modes to maintain uniformity. The reason for its lack of contribution in the case of these two modes, based again on the number of decision splits, is its strong correlation with the cosine of the angle between the two photons originating from the π 0 decay. As the latter variable is not present in the B 0 → ρ0 γ (ρ0 → π + π − ) mode, we chose to keep the helicity angle in all the signal modes, and split the load between it and θγγ for the modes containing a π 0 . 5.8.5 Meson masses Due to the extremely broad nature of the ρ resonance (∼ 150 MeV), it is impractical to include the meson mass as one of the components of the fit. However, including this variable in the Bagger configuration is problematic from the point of view of using the B → K ∗ γ control sample for validation for exactly the same reason. We therefore attempted to remove the meson mass from Bagger training and instead used simple cuts (optimized using the Bump Hunter algorithm in the last round of this analysis). Thus, the ranges of 633 < mπ+ π− < 957 MeV/c2 , 636 < mππ0 < 932 MeV/c2 , and 764 < mπ+ π− π0 < 795 MeV/c2 for ρ0 , ρ± , and ω, respectively, were used. These cuts in combination with a retrained Bagger recovered the performance of the original configuration. 51 5.8.6 Likelihood ratio vetoes for π 0 /η → γ γ decays As noted previously, π 0 /η → γ γ decays produce a very large amount of background where one of the daughter photons has a high enough momentum to mimic the signature of our signal events. These events are vetoed in our analysis using the Likelihood Ratio approach. The π 0 and η likelihood ratio functions are defined as LR(π 0 /η) = P (mγ1 γ2 , Eγ2 |π 0 /η) , P (mγ1 γ2 , Eγ2 |ρ/ω) + P (mγ1 γ2 , Eγ2 |π 0 /η) where P is the probability density function (PDF) given in terms of the invariant mass of the photon pair, M (γ1 γ2 ), and the energy of of the lower-energy photon in the lab frame, Eγ2 . We determine the PDFs by performing 2-dimensional kernel density estimation[32] using signal and continuum MC samples. For the signal PDF, γ1 is matched using MC truth information to have originated from the B meson, while for the π 0 /η PDFs, both γ1 and γ2 are truth-matched to have originated from the same mother particle. In previous searches for B → (ρ/ω)γ decays at BABAR, the likelihood ratios were not included in a multidimensional classifier, and their performance was assessed separately. Here, we choose to include these functions as inputs to the Bagger. Thus, their contribution to the overall performance can only be estimated. 5.8.7 Final list of classifier inputs The final list of Bagger inputs is given in Table 5.8 for B 0 → ωγ. The variables are sorted by their contribution to overall classifier performance, based on the number of decision splits. The inputs are basically identical in the other two modes, but the order of their importance is a bit different. You can see the top 15 variables for the B → ργ modes in Table 5.9. The final Monte Carlo output curves for the B 0 → ρ0 γ Bagger are shown in Figure 5.37. The signal efficiencies for the Bagger cuts alone are: 34.4% for B 0 → ωγ, 31.1% for B 0 → ρ0 γ, and 19.0% for B ± → ρ± γ. 52 BDT signal ouput for ρ0γ mode 0 1000 5000 2000 10000 5000 15000 10000 20000 20000 25000 BDT signal ouput for ρ0γ mode 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (note log scale) BDT background ouput for ρ0γ mode 0 1e+02 50000 5e+02 100000 5e+03 150000 5e+04 200000 250000 BDT background ouput for ρ0γ mode 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (note log scale) Figure 5.37: Final MC output signal (top) and background (bottom) curves for B 0 → ρ0 γ Bagger. Linear scale plots are shown on the left, logarithmic scale plots on the right The z component of the decay vertex of the signal B Likelihood Ratio based veto for η → γγ events The z component of the decay vertex of the tag B Missing mass in the rest of the event 36450 32969 30392 30265 24146 23109 22016 21892 R2All BcosThetaCM BroeGamL3 BzCP GammabestEtaLR BzTag BRoeMissMass into the CM frame with respect to the ROE thrust axis The third normalized Legendre moment of ROE boosted signal B Cosine of the polar angle of the CM momentum of the GoodNeutralLooseAcc moments for the event, using ChargedTracksAcc and The ratio of the second to the zeroth Fox-Wolfram the lab frame Cosine of the angle between the two γ daughters of π 0 in direction, divided by the error Pi0cosGammaAngle ∆z, the displacement between the two B vertices in the z Variable definition 48980 splits of decision Number BdeltaZFitOverErr Variable name 53 Likelihood Ratio based veto for π → γγ events Cosine of the polar angle of the CM momentum of the ω Rest of event momentum perpendicular to the beam di- Aplanarity of the rest of the event Kaon-based B-tag Invariant mass of the π 0 → γ γ system The second normalized Legendre moment of ROE boosted 21198 21131 20988 16867 16408 16024 15723 15178 15092 OmegacosthCM BRoePerpPBeam BrecoilR2prime50 BRoeAplanarity BKaonValueMor Pi0invMass BroeGamL2 BRoeDeltaE The ∆E of the tag B into the CM frame with respect to the ROE thrust axis γ recoil system R2 of EMC bumps of energy greater than 50 MeV in the rection in the ρ/K ∗ rest frame) daughter wrt the direction of the reconstructed B meson the (ρ/K ∗ )γ modes, it is the flight direction of a ρ/K ∗ the B flight direction, both in the ω rest frame. (For GammabestPi0LR The angle between the normal of the π + π − plane and Variable definition 21876 splits of decision Number OmegahelicityAngle Variable name 54 The centroid distance to the nearest neutral EMC bump Rest of event momentum perpendicular to the thrust axis R2 of charged tracks in the γ recoil system K-π B-tag The absolute value of the complex Zernike(2,0) moment The cosine of the π 0 helicity angle The cosine of the angle of the γ wrt the ROE thrust axis The absolute value of the complex Zernike(4,2) moment The ratio of the sums of the energies of the central 9 EMC The planarity of the rest of the event The spericity of the rest of the event 14508 14417 13397 13336 13063 12610 12250 12245 12154 12018 11891 11040 10473 10420 BRoePerpPThrust BrecoilR2primeCh BKaonSlowPionValue04 GammaZ20 Pi0cosHelicity BcosAngGamThrust BRoeKinLepTracks2P3CM BRoeKinLepTracks1CosThetaMissCM GammaA42 Gammas9s25 BRoeKinLepTracks2E90W BRoePlanarity BRoeSphericity crystals to the central 25 crystals surrounding the centroid of energy greater than 50 MeV GammadistNe The centroid distance to the nearest charged EMC bump Variable definition 14566 splits of decision Number GammadistCh50 Variable name 55 A lepton-based B-tag Maximum CM momentum B-tag 10132 9677 9643 8100 7972 7286 6477 5460 5315 5283 5175 5129 BRoeKinLepTracks1E90W BKinematicLeptonValue04 BMaxPstarValueMor BRoeKinLepTracks1P3CM BroeThrL2 BroeThrL1 BrecoilR2prime30 BRoeMissECM BRoeElectron1Pid BroeThrL3 BRoeElectron1P3CM the rest of the event The third normalized Legendre moment of the thrust of CM missing energy of the rest of the event γ recoil system R2 of EMC bumps of energy greater than 30 MeV in the rest of the event The first normalized Legendre moment of the thrust of the the rest of the event The second normalized Legendre moment of the thrust of The lateral moment of the photon shower shape into the CM frame with respect to the ROE thrust axis GammaLat The first normalized Legendre moment of ROE boosted Variable definition 10166 splits of decision Number BroeGamL1 Variable name 56 4557 4386 2732 2571 2541 1585 1464 BRoeKCharged1P3CM BRoeElectron1E90W BRoeMuon1E90W BRoeElectron1CosThetaMissCM BRoeMuon1P3CM BRoeKCharged2P3CM BRoeMuon1CosThetaMissCM Variable definition Variables are analogous for the B → ργ modes. Table 5.8: Definitions of classifier (Bagger) training input variables for B 0 → ωγ (some rest-of-event variables are self-explanatory). 4631 splits of decision Number BRoeKShort1P3CM Variable name 57 BdeltaZFitOverErr R2All BcosThetaCM BroeGamL3 Rho0helicityAngle BzCP BRoeMissMass GammabestEtaLR BRoePerpPBeam BzTag GammabestPi0LR Rho0chi2Prob BrecoilR2prime50 BKaonValueMor Rho0costhCM 79532 54972 51145 45449 40920 40622 35767 34995 34703 32832 32789 31739 29608 29588 27603 31459 32624 34322 34803 37764 38990 43646 43944 46173 48680 49988 51676 56497 69787 98566 Number of decision splits ρ± γ Table 5.9: Top 15 Bagger inputs for ρ0 γ and ρ± γ modes Variable name Number of decision splits ρ0 γ BRoePerpPThrust RhocosthCM BRoeAplanarity BrecoilR2prime50 GammabestPi0LR GammabestEtaLR BKaonValueMor RhohelicityAngle BRoeMissMass BRoePerpPBeam Pi0cosGammaAngle BcosThetaCM BroeGamL3 R2All BdeltaZFitOverErr Variable name 58 59 6 The fit strategy 6.1 General fitter configuration The fits for all modes, including the control sample B → K ∗ γ, were carried out using the RooFit[33] framework though the RooRarFit[34] configuration package. All fits employ the extended unbinned maximum likelihood technique, where the combined probability density function (PDF) for all the events, built from signal and background components, is augmented by a Poisson variation factor for the signal yield. Thus the yield becomes one of the parameters of the fit and can be extracted directly from the result. Various components were included in the fit procedure at different times (as will be described in subsequent sections) for different modes. In the end, we settled on a rather simple 2-dimensional fit using mES and ∆E only. In this configuration, the two dimensions are taken as statistically independent, resulting in a combined likelihood function that is a product of two uncorrelated PDFs for each fit model component. While the (transformed) output of the neural network was used as one of the fit components in the last round of this analysis[15], the shape of the Bagger output for signal is changing far too gradually in the fit region, and the overall number of signal events is too small. We therefore decided to impose a harder cut on the raw Bagger output, leaving the mES -∆E fit region rather clean. This cut was varied for combined, luminosity-weighted Monte Carlo fit for each signal mode, until the signal √ significance (S/ S + B ) was maximized. The values obtained using this procedure were bag > 0.94 and bag > 0.93 for the ρ and ω modes, respectively. In the previous analysis, the cut on the NN output was similarly optimized, but then subsequently 60 relaxed to allow more events in the fit region, since the output itself was one of the fit components. Meson helicity angles were also used in the previous analysis, but dropped this time around in favor of keeping this information as one of the inputs to the Bagger. While Bagger performance did not suffer significantly when helicity angles were removed, as mentioned earlier, including these angles as another fit dimension did not help the performance either. And, due to the similarity between the signal and background helicity shapes after the tight Bagger cut is imposed, it seems highly unlikely that this variable will be of any use in the fit. We also attempted to use the Dalitz angle as another fit dimension in the B 0 → ωγ mode, but the shape turned out to be too similar to that of the continuum background component, leading to zero gain in performance. Therefore, unlike the previous round of analysis, this variable was in the end dropped completely. In general, the setup is fundamentally different from the previous round of this analysis, where the approach was to cut loosely on the NN, leaving a (relatively) large number of events in the fit region, and to attempt to decompose the resulting distribution into signal and background components using a sophisticated multidimensional fit. We now go in the opposite direction by allowing the multidimensional classifier to do most of the work separating signal from background, resulting in a sparsely populated fit region and allowing for a simpler 2-dimensional fit. 6.2 B 0 → ωγ fitter As can be seen in Figure 6.38, the PDF shapes for the three components of the likelihood function (signal, BB, and continuum) were determined from luminosityweighted MC samples. The functions used were: Crystal Ball [35] (signal and BB) and ARGUS [36] (continuum) in mES , Cruiff (signal), exponential (BB), and a line (continuum) in ∆E.4 4 In all signal-mode ML fits, the only floating parameters are: signal yield, continuum yield, and parameters of the continuum PDFs in both mES and ∆E. In B → K ∗ γ fits with Bagger cut applied, we also float signal mean and width(s). 61 The Crystal Ball function is parametrized as:  exp − (x−µ) 2σ 2 ( f (x|µ, σ, α, n) = C ·  n n α  2 exp − α2  2  n − α + x−µ α σ >α , x−µ σ −n , x−µ ≤α σ (6.6) where α determines where the usual Gaussian turns into a power function with the tail parameter n, and C is the overall normalization. The ARGUS function is parametrized as: x2 f (x|b, Ebeam ) = C · 1 − 2 Ebeam !1 2 x2 exp −b 1 − 2 Ebeam " !# (6.7) where Ebeam , the beam energy, is the endpoint of the spectrum, and b is the shape parameter. The Cruiff function is used internally in the BABAR collaboration and is parametrized as: −(x − µ)2 f (x|µ, σL,R , αL,R ) = C · exp , 2 2σL,R + αL,R (x − µ)2 ! (6.8) where σL,R are the left and right-side width of this modified Gaussian, and αL,R are the measures of the corresponding tails. Figure 6.39 shows the result of a luminosity-sampled combined MC fit (i.e., each type of MC was randomly sampled to select the number of events expected in our onpeak datset). Signal and background components are also shown with dashed lines. The numbers are summarized in Table 6.10 for all three signal modes. Note that the number of BB events is fixed from PDF fit to BB MC. This is in fact done for all three modes by integrating the mES distribution of BB MC shown in Figures 6.38, 6.41, and 6.44. The fits were validated with both pure and signal-embedded toy MC (1000 experiments each). The results for these in B 0 → ωγ mode are shown in Figure 6.40 and display no significant biases. Toy MC experiments produce a distribution of signal event yields centered on the expected number, eliminating the need for any corrections. Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 5.22 0 2 4 6 8 10 12 14 0 5.22 0.5 1 1.5 2 2.5 3 3.5 0 5.22 1 2 3 4 5.23 5.23 5.23 5.24 5.24 5.24 5.25 5.25 5.25 5.26 5.26 5.26 5.27 5.27 5.27 6 5 α = 1.4 ± 1.6 2 5.29 2 5.28 ξ = -72.1 ± 14 χ 2 / ndf = 0.732 5.28 5.3 5.3 MES (GeV/c2 ) 5.29 MES (GeV/c2 ) 5.29 σ = 0.0053 ± 0.0046 GeV/c 2 n = 9.7 ± 8.3 5.3 MES (GeV/c2 ) µ = 5.2795 ± 0.0045 GeV/c α = 0.31 ± 0.24 χ 2 / ndf = 1.259 5.28 σ = 0.00294 ± 0.00070 GeV/c2 n = 4 ± 11 µ = 5.2798 ± 0.0011 GeV/c χ 2 / ndf = 4.463 0 -0.3 2 4 6 8 10 0 -0.3 0.5 1 1.5 2 2.5 3 3.5 0 -0.3 0.5 1 1.5 2 2.5 -0.2 -0.2 -0.2 -0.1 -0.1 -0.1 0 0 0 R α = 0.14 ± 0.18 = 0.23 ± 0.20 0.1 1 p = -0.145 ± 0.53 χ 2 / ndf = 0.693 0.1 c = -6.03 ± 2.3 χ 2 / ndf = 0.265 0.1 R σ = 0.039 ± 0.043 L σ = 0.070 ± 0.031 µ = -0.0061 ± 0.033 L α χ 2 / ndf = 1.400 0.2 0.2 0.2 0.3 ∆ E (GeV) ∆ E (GeV) 0.3 0.3 ∆ E (GeV) Figure 6.38: PDF shapes for mES (left) and ∆E (right) determined from Monte Carlo for the B 0 → ωγ mode. Signal on top, BB in the middle, continuum on the bottom 2 Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) 7 62 5.27 5.28 5.29 5.3 -0.2 -0.1 0 0.1 Figure 6.39: Luminosity-sampled, combined Monte Carlo fit for the B 0 → ωγ mode MES (GeV/c ) 2 0 -0.3 5.26 0 5.22 5.25 1 2 5.24 2 5.23 3 4 5 6 7 8 4 Events / ( 0.03 GeV ) 6 8 10 12 14 16 0.3 ∆ E (GeV) 0.2 63 Events / ( 0.004 GeV/c2 ) 5 -2 10 -1 15 0 20 80 90 -1 100 110 120 130 104.8 ± 0.4 140 11.13 ± 0.26 20 30 40 50 10 -2 0 1 Sigma Mean 81 ± 3.2 2 3 0.99 ± 0.02 -0.039 ± 0.032 Constant 5 -2 10 -1 15 0 20 68 ± 2.7 2 30 40 50 60 70 80 90 0 10 0 -3 80 -3 -2 90 nContinuum pull 20 1.1 ± 0.0 -0.12 ± 0.04 0 70 10 1 Sigma Mean Constant 30 20 40 60 80 20 30 40 50 60 70 80 nSignal pull 25 5.866 ± 0.158 100 nContinuum 100 -1 110 0 120 992 140 105.1 ± 0.4 78 ± 3.1 150 11.55 ± 0.27 1 Sigma 2 1 ± 0.0 Mean 0.0041 ± 0.0338 Constant 130 Sigma Mean Constant 97.17 ± 3.81 Entries Figure 6.40: Plots of signal and continuum yields (top) and their pulls (bottom) in pure (left four) and signal-embedded (right four) toy MC for the B 0 → ωγ mode 0 -3 10 20 2 20 30 1 30 40 -3 40 50 0 50 60 80 60 1 ± 0.0 90 nContinuum pull 70 Sigma 80 76 ± 3.0 -0.042 ± 0.034 70 Mean Constant 90 nSignal pull 0 70 Sigma Mean 12.61 ± 0.20 60 70 0 0 30 Sigma Mean 994 Constant 73.29 ± 2.93 Entries 80 nSignal 0 0 25 995 Constant 89.24 ± 3.49 Entries 10 20 40 60 80 100 nContinuum 10 20 30 40 50 13.03 ± 0.20 Sigma 60 5.924 ± 0.159 Mean 70 994 Constant 72.45 ± 2.89 Entries 80 nSignal 64 65 6.3 B 0 → ρ0γ fitter The fitter for this mode is almost analogous to the B 0 → ωγ one, except that the PDF for the BB component in mES is fit only in the range of [5.25, 5.3]. The very few events that appear at the lower range of mES can be absorbed by the continuum. This also affects the fixed BB yield, as the events in the tail are not counted. In addition, the PDF for the BB component in ∆E contains a peaking Crystal Ball contribution, which is taken from dedicated B → K ∗ γ MC (see Section 6.6). The PDF plots are shown in Figure 6.41 and the luminosity-sampled MC fit in Figure 6.42. The plots summarizing the results of toy MC studies are given in Figure 6.43. Again, no significant biases are observed, and no corrections are necessary. Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 5.26 5.26 5.26 5.27 5.27 5.27 5.28 5.3 0 -0.3 5.3 MES (GeV/c2 ) 2 0 5.22 6 8 10 12 14 16 18 0 -0.3 4 5.29 MES (GeV/c2 ) 0.5 1 1.5 2 2.5 0 -0.3 0.5 1 1.5 2 2.5 3 2 5.28 2 2 5.29 σ = 0.0037 ± 0.0022 GeV/c n = 100 ± 105 5.3 4 3.5 4 6 8 10 12 14 16 5.29 MES (GeV/c2 ) µ = 5.2794 ± 0.0036 GeV/c α = 0.62 ± 0.63 χ 2 / ndf = 0.944 5.28 ξ = -61.6 ± 12 5.25 5.25 5.25 χ 2 / ndf = 2.278 5.24 5.24 5.24 18 5.23 5.23 5.23 σ = 0.00257 ± 0.00094 GeV/c2 n = 3 ± 11 20 0 5.22 1 2 3 4 5 0 5.22 2 4 6 8 2 µ = 5.27975 ± 0.00068 GeV/c α = 1.9 ± 2.3 χ 2 / ndf = 16.022 -0.2 -0.2 -0.2 -0.1 -0.1 -0.1 0 0 0 R α = 0.12 ± 0.15 = 0.22 ± 0.21 0.2 0.1 1 p = 0.00 ± 0.50 χ 2 / ndf = 2.074 0.1 0.2 0.2 σ = 0.063 ± 0.049 GeV µ = -0.091 ± 0.12 GeV c = -6.56 ± 3.0 fracG = 0.39 ± 0.32 χ 2 / ndf = 0.329 0.1 R σ = 0.033 ± 0.022 L σ = 0.061 ± 0.054 µ = 0.006 ± 0.032 L α χ 2 / ndf = 2.967 0.3 ∆ E (GeV) ∆ E (GeV) 0.3 0.3 ∆ E (GeV) Figure 6.41: PDF shapes for mES (left) and ∆E (right) determined from Monte Carlo for the B 0 → ρ0 γ mode. Signal on top, BB in the middle, continuum on the bottom 2 Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) 10 66 0 5.22 5 10 15 20 25 5.23 5.25 5.26 5.27 5.29 5.3 2 MES (GeV/c ) 5.28 Events / ( 0.03 GeV ) 0 -0.3 -0.2 -0.1 0 0.1 Figure 6.42: Luminosity-sampled, combined Monte Carlo fit for the B 0 → ρ0 γ mode 5.24 2 4 6 8 10 12 14 0.3 ∆ E (GeV) 0.2 67 Events / ( 0.004 GeV/c2 ) 5 0 25 120 130 140 -1 150 160 20 10 0 20 10 0 -3 -2 0 92 ± 3.6 1 Sigma 2 3 0.98 ± 0.02 Mean -0.0018 ± 0.0311 Constant 200 5 2 3 0 -3 10 0 20 10 70 80 90 30 1 1.1 ± 0.0 120 130 -2 nContinuum pull 20 0 80 ± 3.2 -0.097 ± 0.034 0 110 10 40 -1 Sigma Mean 40 30 -2 35 Constant 30 50 25 40 20 20 30 40 50 60 70 80 50 15 6.303 ± 0.153 18.92 ± 0.20 60 -3 10 Sigma Mean 90 nContinuum 60 70 80 90 nSignal pull 0 988 Constant 83.68 ± 3.31 Entries -1 140 0 150 997 170 1 Sigma Mean 147.3 ± 0.4 180 80 ± 3.2 190 12.74 ± 0.30 2 3 0.99 ± 0.02 0.033 ± 0.032 Constant 160 Sigma Mean Constant 83.45 ± 3.28 Entries Figure 6.43: Plots of signal and continuum yields (top) and their pulls (bottom) in pure (left four) and signal-embedded (right four) toy MC for the B 0 → ρ0 γ mode 30 30 3 40 40 2 50 50 1 60 60 80 1.04 ± 0.02 70 90 nContinuum pull 190 10 10 180 20 20 170 30 30 0 40 40 -0.11 ± 0.03 80.8 ± 3.2 40 50 70 -3 35 Constant 30 Sigma 20 80 -1 15 60 50 70 80 90 nSignal 60 Sigma 12.77 ± 0.30 6.247 ± 0.146 Sigma Mean 152.8 ± 0.4 70 80 18.85 ± 0.20 994 Mean Entries Constant 88.12 ± 3.46 Mean -2 10 nContinuum 90 986 Constant 88.28 ± 3.47 Entries 90 nSignal pull 0 0 20 40 60 80 100 nSignal 68 69 6.4 B ± → ρ±γ fitter The fitter for this mode is a mix of the fitters for the neutral modes: the PDF for the BB component in mES is given by a Crystal Ball fit over the whole mES range, as in B 0 → ωγ, while the same component in ∆E is given by an exponential with a Crystal Ball contribution taken from dedicated B → K ∗ γ and B ± → ρ± π 0 MC (see Section 6.6), similar to B 0 → ρ0 γ. The PDF projection plots are shown in Figure 6.44 and the luminosity-sampled MC fit in Figure 6.45. The plots summarizing the results of toy MC studies are given in Figure 6.46. As in the other two modes, no significant biases are present, and no corrections are necessary. Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 2 5.3 2 5.3 10 12 14 0 -0.3 5.22 0 MES (GeV/c ) 2 0 -0.3 2 5.29 MES (GeV/c ) 5.29 1 2 3 4 5 0 -0.3 4 5.28 ξ = -67.0 ± 12 χ 2 / ndf = 1.003 5.28 σ = 0.0045 ± 0.0011 GeV/c2 n = 3.2 ± 8.7 5.3 2 MES (GeV/c ) 5.29 µ = 5.2800 ± 0.0064 GeV/c α = 0.36 ± 0.35 χ 2 / ndf = 1.227 5.28 2 5.27 5.27 5.27 6 5.26 5.26 5.26 4 5.25 5.25 5.25 8 5.24 5.24 5.24 0.5 1 1.5 2 -0.2 -0.1 0 -0.2 -0.2 -0.1 -0.1 0 0 0.2 0.1 1 χ 2 / ndf = 1.418 p = -0.206 ± 0.50 0.1 c = -9.45 ± 3.5 0.2 0.2 fracCB = 0.49 ± 0.19 χ 2 / ndf = 1.360 0.1 R σ = 0.046 ± 0.044 L σ = 0.070 ± 0.046 = 0.11 ± 0.16 2.5 R L µ = -0.0120 ± 0.031 α α = 0.25 ± 0.18 χ 2 / ndf = 2.617 3 3.5 6 5.23 5.23 5.23 σ = 0.00315 ± 0.00082 GeV/c2 n = 3.2 ± 8.0 µ = 5.2795 ± 0.0010 GeV/c 2 8 10 12 14 16 0 5.22 1 2 3 4 5 6 0 5.22 2 4 6 8 α = 1.3 ± 1.3 χ 2 / ndf = 6.169 0.3 ∆ E (GeV) 0.3 ∆ E (GeV) 0.3 ∆ E (GeV) Figure 6.44: PDF shapes for mES (left) and ∆E (right) determined from Monte Carlo for the B ± → ρ± γ mode. Signal on top, BB in the middle, continuum on the bottom 2 Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) 10 70 5.25 5.26 5.27 5.28 5.29 5.3 -0.2 -0.1 0 0.1 Figure 6.45: Luminosity-sampled, combined Monte Carlo fit for the B ± → ρ± γ mode MES (GeV/c ) 2 0 -0.3 5.24 0 5.22 5.23 2 2 6 8 10 12 14 16 4 Events / ( 0.03 GeV ) 4 6 8 10 12 14 16 18 0.3 ∆ E (GeV) 0.2 71 Events / ( 0.004 GeV/c2 ) 5 110 120 0 160 2 -3 -2 10 -1 20 0 30 991 1 Sigma Mean 2 21.99 ± 0.23 3 0.98 ± 0.02 0.064 ± 0.031 94 ± 3.7 50 7.307 ± 0.170 Constant 40 Sigma Mean Constant 93.91 ± 3.68 Entries 100 110 0 20 40 60 80 100 -3 -2 nContinuum pull 0 20 40 60 80 100 nContinuum 120 -1 130 0 140 150 998 1 Sigma Mean 2 136.4 ± 0.4 3 0.98 ± 0.02 -0.083 ± 0.031 95 ± 3.7 180 13.33 ± 0.31 170 Constant 160 Sigma Mean Constant 89.67 ± 3.50 Entries Figure 6.46: Plots of signal and continuum yields (top) and their pulls (bottom) in pure (left four) and signal-embedded (right four) toy MC for the B ± → ρ± γ mode 3 10 10 10 3 20 20 20 1 30 30 30 0 40 40 40 0 -3 50 50 0 -3 60 60 70 80 50 0.99 ± 0.02 90 nSignal pull 0 0 60 1 81 ± 3.2 180 20 40 60 80 70 2 170 -0.019 ± 0.032 Constant 150 Sigma -1 140 137.2 ± 0.4 13.48 ± 0.33 100 nSignal 70 -2 130 80 nContinuum pull 0 100 1 ± 0.0 77 ± 3.0 45 Sigma 0 40 Constant 35 80 30 90 -1 25 Sigma Mean Mean 20 10 20 30 40 997 Constant 78.99 ± 3.12 Entries 90 -2 15 60 7.524 ± 0.180 Sigma 50 70 21.08 ± 0.24 Mean 80 90 nContinuum -0.065 ± 0.033 10 993 Constant 84.47 ± 3.32 Entries Mean nSignal pull 0 0 20 40 60 80 100 nSignal 72 73 Nsignal NBB (fixed) Nudscτ Significance 104.9+12.1 −11.3 2.6σ Sampled MC B 0 → ωγ 12.3+6.2 −5.4 10.2 (13 expected) B 0 → ρ0 γ 19.1+6.8 −6.1 (105 expected) 10.5 (19 expected) B ± → ρ± γ 21.2+8.2 −7.4 150.6+13.7 −12.9 3.9σ (148 expected) 26.0 (21 expected) 135.6+14.2 −13.4 3.2σ (137 expected) Weighted MC B 0 → ωγ 15.4+0.0 −5.6 B 0 → ρ0 γ 21.0+6.8 −6.0 19.0+7.8 −7.0 B ± → ρ± γ 10.2 102.7+11.9 −11.2 3.2σ 10.5 148.6+13.5 −12.8 138.2+14.2 −13.4 4.4σ 26.0 3.1σ Table 6.10: Summary of luminosity-sampled and luminosity-weighted combined Monte Carlo fits for all three signal modes 6.5 B → K ∗γ fitter for cross check The purpose of performing these cross-check fits is twofold. First, we want to make sure we extract the correct branching fractions for these previously measured decays (B 0 → K ∗0 γ (K ∗0 → K + π − ) and B ± → K ∗± γ (K ∗± → K ± π 0 )) and make sure that on-peak data agrees well with our combined MC sample. Second, we want to extract the fit parameters (in particular the means and the widths of the signal distributions) from both MC and on-peak fits and use the differences as correction factors for the final on-peak data fits in our signal modes. The kinematics of B → K ∗ γ decays are almost identical to our signal modes. This allows us to use the B → K ∗ γ sample for several validation studies, including Bagger and fitter cross checks. Figures 6.47 and 6.48 show overlay plots of normalized signal distributions in mES and ∆E for B 0 → ρ0 γ vs B 0 → K ∗0 γ and B ± → ρ± γ vs B ± → K ∗± γ. Plots in Figure 6.49 show the distributions of Bagger output in signal and generic MC for the same four modes. These demonstrate that the B → K ∗ γ and 74 B → ργ signal shapes are very similar, as are the multivariate classifier outputs used for final event selection. ρ0 vs K π mES 0.12 0 ρ signal K π signal ρ0 vs K π ∆ E ρ signal 0 K π signal 0.08 0.07 0.1 0.06 0.08 0.05 0.06 0.04 0.03 0.04 0.02 0.02 0.01 0 5.25 5.255 5.26 5.265 5.27 5.275 5.28 5.285 5.29 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Figure 6.47: Signal MC mES and ∆E for B 0 → ρ0 γ vs B 0 → K ∗0 γ ρ+ vs K π0 mES + ρ signal ρ+ vs K π0 ∆ E ρ + signal 0.09 K π0 signal K π0 signal 0.08 0.05 0.07 0.06 0.04 0.05 0.03 0.04 0.03 0.02 0.02 0.01 0.01 0 5.25 5.255 5.26 5.265 5.27 5.275 5.28 5.285 5.29 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Figure 6.48: Signal MC mES and ∆E for B ± → ρ± γ vs B ± → K ∗± γ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K π backgorund K π signal ρ0 backgorund ρ0 signal 10-4 10-3 10-2 0 10-1 1 0 0.1 0.2 ρ vs K π bagger 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Kπ0 backgorund ρ backgorund Kπ0 signal ρ signal Figure 6.49: Signal and generic MC Bagger output for B 0 → ρ0 γ vs B 0 → K ∗0 γ and B ± → ρ± γ vs B ± → K ∗± γ 10 -4 10-3 10-2 10-1 ρ0 vs K π bagger 75 76 Since the statistics for B → K ∗ γ are much higher than for our signal modes, we did not use the full on-peak dataset for the cleaner K + π − mode, where we only used BABAR runs 1–5 data. We did, however, include run 6 data for K ± π 0 mode, which has a much lower signal-to-noise ratio. This is simply a result of trying to avoid potential technical problems associated with upgrading to a new BABAR dataset and the associated release version. The fitter configurations for the two B → K ∗ γ modes are again very similar to those described above. In K + π − mode, we have a small peaking Gaussian component for the BB PDF in ∆E, similar to B 0 → ρ0 γ. For K + π 0 mode, we perform two different sets of fits. The first uses the same PDFs as the B 0 → ωγ fit. The results of this are used to correct the signal shape parameters in B 0 → ωγ and B ± → ρ± γ modes, based on data-MC differences. The errors on the signal shape parameters obtained from this fit are used for systematic variations of fixed signal shape parameters in the above two signal modes. For the other fit, we split the signal into truth-matched and self-crossfeed parts. This is done because the two differ quite significantly if no cut on the Bagger output is made. The motivation for this split comes from the fact that mES -∆E correlation is much stronger for the self-crossfeed part, thus leading to a large bias in the fit if it is not handled separately. The separation cuts this bias in half. This becomes important when we extract signal yield from B → K ∗ γ to calculate Bagger efficiency systematic error (discussed in Section 9.4), where we need to perform the fit both with and without the Bagger cut applied. 124 ± 19 122 ± 17 34 ± 10 25 ± 7 987 ± 33 1036 ± 33 195 ± 17 239 ± 15 MC Data MC Data 5.2794 ± 0.0002 5.2798 ± 0.0003 5.2794 ± 0.00009 5.2797 ± 0.00009 mES µ 0.0028 ± 0.0002 0.0029 ± 0.0002 0.0026 ± 0.00007 0.0026 ± 0.00007 mES σ −0.0133 ± 0.0094 −0.0034 ± 0.0088 −0.0060 ± 0.0038 0.0064 ± 0.0040 ∆E µ 0.0534 ± 0.0081 0.0676 ± 0.0086 0.0469 ± 0.0033 0.0590 ± 0.0037 ∆E σL 0.0377 ± 0.0056 0.0382 ± 0.0061 0.0368 ± 0.0023 0.0324 ± 0.0026 ∆E σR Table 6.11: Summary of luminosity-weighted combined Monte Carlo and on-peak data fits for B → K ∗ γ (K ∗0 → K + π − and K ∗+ → K + π 0 ) cross check. K +π0 K +π− Nudscτ Nsignal 77 78 The PDF projection plots for B → K ∗ γ are shown in Figures 6.50 and 6.52, while the luminosity-weighted MC and on-peak data fits are shown in Figures 6.51 and 6.53. Table 6.11 summarizes the results (yields and PDF parameters). Note that the on-peak luminosity used for K ± π 0 mode was 376.2 fb−1 (run 6 data included), while MC was matched to only 341.8 fb−1 (runs 1–5) — the on-peak luminosity used for K + π − mode. With signal efficiencies of 10.6% and 4.2% and on-peak luminosities of 341.8 fb−1 and 376.2 fb−1 for the K + π − and K + π 0 modes respectively, we obtain: B(B 0 → K ∗0 γ (K ∗0 → K + π − )) = (4.11 ± 0.13) × 10−5 (Data) B(B 0 → K ∗0 γ (K ∗0 → K + π − )) = (3.89 ± 0.13) × 10−5 (M C) B(B ± → K ∗± γ (K ∗+ → K + π 0 )) = (4.34 ± 0.28) × 10−5 (Data) B(B ± → K ∗± γ (K ∗+ → K + π 0 )) = (3.88 ± 0.34) × 10−5 (M C) which compare favorably with the generated MC, based on world averages of 4.01 ± 0.20 × 10−5 and 4.03 ± 0.26 × 10−5 for B 0 → K ∗0 γ and B ± → K ∗± γ, respectively [26]. Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) χ 2 / ndf = 1.909 5.28 2 5.26 5.27 5.28 5.29 5.3 8 10 12 0 -0.3 0 5.22 2 4 MES (GeV/c2 ) 0 -0.3 2 4 5.25 5.3 6 5.24 5.29 MES (GeV/c2 ) 5 10 15 20 6 5.23 ξ = -67.4 ± 12 χ 2 / ndf = 1.479 5.28 σ = 0.00500 ± 0.00084 GeV/c2 25 0 -0.3 20 40 60 80 100 120 -0.2 -0.2 -0.1 -0.1 -0.1 0 0 0 µ = 0.0057 ± 0.0047 160 -0.2 R α 180 140 L α 200 0.1 1 p = -0.287 ± 0.49 χ 2 / ndf = 2.833 0.1 c = -8.161 ± 0.84 χ 2 / ndf = 2.262 0.1 R 0.2 0.2 0.2 σ = 0.0326 ± 0.0034 L σ = 0.0586 ± 0.0054 = 0.125 ± 0.020 = 0.224 ± 0.027 χ 2 / ndf = 6.842 220 8 10 12 14 16 0 5.22 2 4 6 8 10 12 5.3 2 MES (GeV/c ) 5.29 n = 17 ± 201 5.27 5.27 14 5.26 5.26 µ = 5.2804 ± 0.0011 GeV/c 5.25 5.25 16 5.24 5.24 α = 0.24 ± 0.11 5.23 5.23 2 σ = 0.002584 ± 0.000075 GeV/c2 n = 2.19 ± 0.92 µ = 5.279740 ± 0.000092 GeV/c 18 0 5.22 100 200 300 400 500 α = 2.24 ± 0.34 χ 2 / ndf = 34.664 0.3 ∆ E (GeV) ∆ E (GeV) 0.3 0.3 ∆ E (GeV) Figure 6.50: PDF shapes for mES (left) and ∆E (right) determined from Monte Carlo for the B 0 → K ∗0 γ mode. Signal on top, BB in the middle, continuum on the bottom 2 Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) 600 79 600 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 80 500 400 300 220 200 180 160 140 120 100 80 200 60 40 100 20 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 0 -0.3 5.3 -0.2 -0.1 0 0.1 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 600 500 400 0.2 0.3 ∆ E (GeV) 2 MES (GeV/c ) 250 200 150 300 100 200 50 100 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 2 MES (GeV/c ) 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Figure 6.51: Luminosity-weighted, combined Monte Carlo (top) and on-peak data (bottom) fits for the B 0 → K ∗0 γ mode Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 2 Events / ( 0.004 GeV/c ) 5.29 2 5.28 ξ = -35.6 ± 24 χ 2 / ndf = 2.437 5.28 5.3 5.29 5.3 MES (GeV/c2 ) 5.29 σ = 0.0041 ± 0.0011 GeV/c2 n = 1.1 ± 1.6 5.3 MES (GeV/c2 ) µ = 5.2820 ± 0.0033 GeV/c α = 0.53 ± 0.60 χ 2 / ndf = 0.999 5.28 5 6 7 0 -0.3 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 -0.3 -0.3 0 5.22 MES (GeV/c2 ) 1 0 1 2 5.27 5.27 5.27 3 5.26 5.26 5.26 2 5.25 5.25 5.25 4 5.24 5.24 5.24 5 10 15 20 25 30 35 3 5.23 5.23 5.23 σ = 0.00300 ± 0.00023 GeV/c2 n = 3.2 ± 1.6 µ = 5.27956 ± 0.00028 GeV/c 2 4 5 6 7 0 5.22 1 2 3 4 5 0 5.22 20 40 60 80 α = 1.43 ± 0.35 χ 2 / ndf = 2.343 -0.2 -0.2 -0.2 -0.1 -0.1 -0.1 0 0 0 R α = 0.142 ± 0.053 = 0.221 ± 0.088 0.1 1 p = -2.749 ± 0.92 χ 2 / ndf = 1.372 0.1 c = -5.44 ± 1.5 χ 2 / ndf = 0.527 0.1 R σ = 0.041 ± 0.010 L σ = 0.070 ± 0.017 µ = -0.0050 ± 0.013 L α χ 2 / ndf = 0.925 0.2 0.2 0.2 0.3 ∆ E (GeV) ∆ E (GeV) 0.3 0.3 ∆ E (GeV) Figure 6.52: PDF shapes for mES (left) and ∆E (right) determined from Monte Carlo for the B ± → K ∗± γ mode. True signal on top, self-crossfeed signal followed by BB in the middle, continuum on the bottom 2 Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) Events / ( 0.03 GeV ) 100 81 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 82 90 80 70 60 50 35 30 25 20 40 15 30 10 20 5 10 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 0 -0.3 5.3 -0.2 -0.1 0 0.1 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 120 100 80 0.2 0.3 ∆ E (GeV) 2 MES (GeV/c ) 50 40 30 60 20 40 10 20 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 MES (GeV/c2) 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Figure 6.53: Luminosity-weighted, combined Monte Carlo (top) and on-peak data (bottom) fits for the B ± → K ∗± γ mode 83 6.6 Peaking BB component in B → ργ We performed a detailed study of the composition of BB backgrounds in B → ργ modes using dedicated B → K ∗ γ, B → Xs γ, B ± → ρ± η, and B ± → ρ± π 0 MC samples. Figures 6.54 and 6.58 show distributions of all these various backgrounds after all selection criteria have been applied. We found that B → K ∗ γ describes completely the peak we observe in the generic BB distribution in B 0 → ρ0 γ. In B ± → ρ± γ, however, the story turned out to be a bit more complicated. Here, B → K ∗ γ contributes only about 2 events, whereas the overall number of BB candidates in generic MC is 26. Thus, we had to look at other potential sources of peaking backgrounds, including B ± → ρ± π 0 and B ± → ρ± η. The latter has a sizable contribution of about 7 events, but is not peaking as much around ∆E = −0.1. In fact, combined with B → Xs γ, it is described well by the exponential component of the BB PDF. B ± → ρ± π 0 , on the other hand, peaks in a similar fashion to B → K ∗ γ, and together they account for about 5 or 6 events. We therefore use the combined shape from these two backgrounds to describe the peak in generic BB distribution in Section 6.4. Second row of Figure 6.54 and bottom row of Figure 6.58 show the combined BB background distributions for B 0 → ρ0 γ (B → K ∗ γ and B → Xs γ) and B ± → ρ± γ (B → K ∗ γ, B ± → ρ± η, B ± → ρ± π 0 , and B → Xs γ). These agree well with the plots in Figures 6.41 and 6.44 in Sections 6.3 and 6.4, respectively. The only noticeable difference is the bin around ∆E = −0.09 in B ± → ρ± γ mode, which comes out low in these studies but high in generic BB MC. However, it is likely to be a fluctuation in the latter, where we have lower statistics. Otherwise, the fit is forced to produce a peak that seems too narrow, especially compared to B 0 → ρ0 γ mode. We performed an additional study using the B ± → ρ± γ mode to assess the stability of the BB-component contribution to the fit. The shift of the ∆E BB peaks in B → ργ modes with respect to the signal peak helps the fit to separate these components. However, there’s a possibility that the mES -∆E correlations are important enough that the nominal PDF, which is a simple product of two 1-dimensional PDFs, does not provide the correct model. This could potentially introduce a bias, because 84 the ∆E BB peak overlaps the signal, particularly in the B ± → ρ± γ mode. We studied the 2-dimensional mES -∆E distribution of all BB events in B ± → ρ± γ mode that pass our combined candidate selection process. This is shown in Figure 6.59, illustrating that, at least with the current statistics, the correlation is negligible. However, we also performed 500 toy MC experiments embedding random BB events, and redoing the fit. The results, shown in Figure 6.60, demonstrate that the fit remains stable, from which we conclude that the current fit model adequately describes the peaking BB component. * -0.1 0 0.1 1.6 0.2 0.3 ∆ E (GeV) 0 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0.2 0 0.05 0.1 ∆ E (MeV) n = 4.4 ± 3.1 σ = 0.057 ± 0.021 α = 0.87 ± 0.94 µ = -0.0881 ± 0.026 Figure 6.54: Plots for peaking B-background studies (B → K ∗ γ and B → Xs γ only) in B → ργ -0.3 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 0.2 0.4 0.6 0.4 0.8 0.6 1 0.8 1 1.2 8.431 K*γ peak in ρ0γ 1.2 -0.2 841 -0.09723 1.4 Integral Mean Entries 1.4 1.6 * K γ in ρ0γ 0 8.391 841 5.277 0 0.5 1 1.5 2 2.5 3 Entries Mean Integral Events / ( 0.03 MeV ) K γ in ρ0γ 85 * 3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Entries 875 Mean -0.1183 Integral 17.58 Figure 6.54: Plots for peaking B-background studies continued 0 0.5 1 1.5 2 2.5 * K γ and Xsγ in ρ0γ -0.3 17 875 5.274 0 Entries Mean Integral 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 1 2 3 4 5 K γ and Xsγ in ρ0γ 86 0 -0.3 0.05 0.1 0.1 0.2 0.3 ∆ E (GeV) 0 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 Figure 6.54: Plots for peaking B-background studies continued 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 0.1 0.2 0.15 0.2 0.3 0.2 0.35 0.25 0 1.86 K*γ peak in ρ+/-γ 0.25 -0.1 620 -0.1016 0.4 -0.2 Integral Mean Entries 0.3 0.35 K*+/-γ in ρ+/-γ 0.3 1.848 620 5.277 0.5 Entries Mean Integral Events / ( 0.03 MeV ) K*+/-γ in ρ+/-γ 0 0.05 0.1 ∆ E (MeV) n = 4 ± 26 σ = 0.067 ± 0.070 α = 1.0 ± 2.9 µ = -0.1014 ± 0.074 87 * 4 -0.2 -0.1 0 0.1 118 14.18 -0.1649 0.2 0.3 ∆ E (GeV) Integral Mean Entries Figure 6.54: Plots for peaking B-background studies continued 0 0.5 1 1.5 2 2.5 3 3.5 * K γ and Xsγ in ρ+/-γ -0.3 13.88 118 5.264 0 Entries Mean Integral 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 0.5 1 1.5 2 2.5 3 K γ and Xsγ in ρ+/-γ 88 -0.2 -0.1 0 0.1 392 7.448 -0.1565 0.2 0.3 ∆ E (GeV) Integral Mean Entries Figure 6.55: Plots for peaking B-background studies (adding B ± → ρ± η and B ± → ρ± π 0 ) in B ± → ρ± γ 0 -0.3 0 0.2 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 0.5 0.4 0.6 1 ρ+/-η in ρ+/-γ 1 7.41 392 5.276 0.8 Entries Mean Integral 1.5 2 ρ+/-η in ρ+/-γ 89 0.2 0 -0.3 0.05 0.1 0.15 -0.2 -0.1 0 0.1 2572 3.601 -0.0909 0.2 0.3 ∆ E (GeV) Integral Mean Entries 0 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0.1 0.2 0.3 0.4 0.5 ρπ0 peak in ρ+/-γ Events / ( 0.03 MeV ) Figure 6.56: Plots for peaking B-background studies continued 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 0.2 0.4 0.6 0.25 0.3 0.4 0.45 ρ+/-π0 in ρ+/-γ 0.8 3.573 2572 5.277 0.35 Entries Mean Integral 1 1.2 ρ+/-π0 in ρ+/-γ 0 0.05 0.1 ∆ E (MeV) n = 2 ± 42 σ = 0.060 ± 0.067 α = 0.4 ± 1.9 µ = -0.0519 ± 0.059 90 *+/- 0.2 0.3 ∆ E (GeV) 0 -0.3 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) *+/- +/- γ peak in ρ γ Figure 6.57: Plots for peaking B-background studies continued 0 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0.1 0.1 0.4 0.2 0.3 0.2 0.3 0.4 0.2 0.6 0.8 0.5 0.4 0.8 1 0.1 5.461 ρπ0 and K 0.5 0 3192 -0.09454 1.2 -0.1 Integral Mean Entries 0.6 -0.2 +/- γ and ρ π0 in ρ+/-γ 0.6 0.8 K 1.4 5.421 3192 5.277 0.7 Entries Mean Integral 0.7 +/- γ and ρ π0 in ρ+/-γ 1.6 1.8 *+/- Events / ( 0.03 MeV ) K 0 0.05 0.1 ∆ E (MeV) n = 1 ± 67 σ = 0.070 ± 0.034 α = 0.8 ± 1.8 µ = -0.0800 ± 0.077 91 * 3629 5.27 2 3 4 5 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Entries 3629 Mean -0.153 Integral 25.01 Figure 6.58: Plots for peaking B-background studies continued 0 -0.3 * K γ , Xsγ , ρ(π0/ η) in ρ+/-γ 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 24.67 1 Entries Mean Integral 1 2 3 4 5 6 K γ , Xsγ , ρ(π0/ η) in ρ+/-γ 92 93 6.7 Simultaneous B → (ρ/ω)γ fitter The individual decay modes can be combined together via the quark model prediction: Γ(B 0 → ρ0 γ) = Γ(B 0 → ωγ) = 1 · Γ(B ± → ρ± γ). 2 (6.9) Following [8], the combined branching fraction is defined as i 1 τB + h B(B → (ρ, ω)γ) = · B(B ± → ρ± γ) + · B(B 0 → ρ0 γ) + B(B 0 → ωγ) 2 τB 0 (6.10)   with the B meson lifetimes being τB + = 1.643 ± 0.010 and τB 0 = 1.528 ± 0.009. sim We define the combined “effective signal yield” as Neff = NBB × B(B → (ρ, ω)γ), so that it includes the efficiencies of all three individual modes. With the combined branching fraction defined above, and assuming the number of B 0B 0 decays is the same as the number of B +B − decays, each being half the total NBB , this becomes: → ρ± γ) τB + Nsig (B 0 → ρ0 γ) Nsig (B 0 → ωγ) + · + .  (B → ρ± γ) τB 0  (B 0 → ρ0 γ)  (B 0 → ωγ) Nsig (B sim Neff = ± ± " # (6.11) We determine this combined “effective signal yield” directly from the simultaneous maximum likelihood fit. Each component of this fit is identical to the one of the corresponding individual fit. The only difference is that the signal yield for each mode is given as a function of the combined “effective signal yield” using the assumption of Equation 6.9 above: Nsig (B ± → ρ± γ) = Nsig (B 0 → ρ0 γ) = Nsig (B 0 → ωγ) = 1 sim · Neff ·  (B ± → ρ± γ) 2 τ 1 sim · B0 · Neff ·  (B 0 → ρ0 γ) 4 τB + (6.12) τ 1 sim · B0 · Neff ·  (B 0 → ωγ) . 4 τB + Thus, we vary only one yield parameter in the combined fit, while the individual three yields are derived from it using the corresponding mode-by-mode signal efficiencies obtained from MC. These efficiency numbers also include systematic corrections and errors described in the next section. BmESSshift 94 BmESSshift BmESSshift:BdeltaE {Bagger > 0.94} 5.29 5.28 5.288 5.286 5.284 5.282 5.27 5.28 5.26 5.278 5.25 5.276 5.24 5.274 5.23 5.272 -0.14 5.22 -0.3 -0.2 -0.1 0 0.1 0.2 -0.12 -0.1 -0.08 -0.06 -0.04 0.3 BdeltaE -0.02 BdeltaE Figure 6.59: 2-dimensional distribution of BB MC in B ± → ρ± γ mode nSignal Entries 496 nContinuum Entries 499 45 50 Constant 46.16 ± 2.56 40 Mean 21.26 ± 0.34 Sigma 7.446 ± 0.249 30 Constant 37.35 ± 2.10 40 35 30 Mean 136.8 ± 0.7 Sigma 14.3 ± 0.5 25 20 20 15 10 10 5 0 0 10 20 30 nSignal pull 40 0 100 50 Constant 39 ± 2.2 45 120 130 140 150 nContinuum pull 160 170 180 35 ± 2.0 Constant 45 40 35 Mean -0.034 ± 0.046 Sigma 1 ± 0.0 40 35 30 30 25 25 20 20 15 15 10 10 5 5 0 -3 110 -2 -1 0 1 2 3 0 -3 -2 -1 0 Mean -0.036 ± 0.050 Sigma 1.1 ± 0.0 1 2 Figure 6.60: Plots of signal and continuum yields (top) and their pulls (bottom) in B-embedded toy MC for the B ± → ρ± γ mode 95 Figure 6.61 shows the results of the luminosity-sampled MC fit combining all three signal modes. The total effective yield is 905+213 −198 , consistent with the expected number of 865. The plots also show each individual PDF contribution; for clarity, BB components are omitted from the mES plots, and continuum components are omitted from the ∆E projections. Figure 6.62 shows results of pure and signal-embedded toy MC studies for this fit. Neither show any significant biases (there’s a slight positive pull in embedded toys). Figures 6.63 and 6.64 give the corresponding plots for the combined B → (ρ0 /ρ± )γ fit, which uses the exact same motivation as above, but ignores the B 0 → ωγ data. The relative sizes of individual contributions are again kept at 2 : 1 ratio, while the combined effective yield is equivalent to that obtained for the full fit above. Again, toy MC studies confirm that no significant biases are present. The one-shot luminositysampled MC fit gives the combined effective yield of 877+246 −225 , again consistent within errors with the expected yield of 865. Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 96 45 40 35 30 25 25 20 15 20 10 15 10 5 5 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) 2 MES (GeV/c ) Figure 6.61: Luminosity-sampled Monte Carlo fit for combined B → (ρ/ω)γ nSimSignal Entries 50 40 500 nSimSignal Constant 41.52 ± 2.30 45 Mean 870 ± 9.4 40 Sigma 208.6 ± 7.0 30 Entries 500 Constant 37.98 ± 2.13 Mean 855.6 ± 9.7 Sigma 211.2 ± 7.5 35 30 25 20 20 15 10 10 5 0 400 600 800 1000 nSimSignal pull 1200 0 1400 Entries 500 39 ± 2.2 Constant Mean -0.029 ± 0.047 45 40 1 ± 0.0 Sigma 35 30 400 600 800 1000 nSimSignal pull 1200 1400 Entries 500 35 ± 2.0 Constant Mean -0.08 ± 0.05 35 1.1 ± 0.0 Sigma 30 25 25 20 20 15 15 10 10 5 5 0 -3 -2 -1 0 1 2 3 0 -3 -2 -1 0 1 2 Figure 6.62: Pure (left) and signal-embedded (right) toy MC for combined B → (ρ/ω)γ fit. Effective signal yields on top, pulls on the bottom Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 97 35 30 25 20 24 22 20 18 16 14 12 10 15 8 10 6 4 5 2 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) 2 MES (GeV/c ) Figure 6.63: Luminosity-sampled Monte Carlo fit for combined B → (ρ0 /ρ± )γ nSimSignal Entries 495 nSimSignal 45 Constant 40.57 ± 2.27 45 40 Mean 864.8 ± 11.2 40 35 Sigma 244.2 ± 8.4 30 20 20 15 15 10 10 5 5 600 800 1000 nSimSignal pull 1200 1400 0 200 1600 Entries 495 38 ± 2.1 Constant Mean -0.06 ± 0.05 45 40 1.1 ± 0.0 Sigma Constant 41.69 ± 2.32 30 25 400 497 Mean 851.3 ± 10.1 Sigma 222.4 ± 7.5 35 25 0 200 Entries 400 600 800 1000 nSimSignal pull 1200 1400 1600 Entries 497 41 ± 2.3 Constant Mean -0.12 ± 0.04 45 40 0.97 ± 0.03 Sigma 35 35 30 30 25 25 20 20 15 15 10 10 5 5 0 -3 -2 -1 0 1 2 3 0 -3 -2 -1 0 1 2 3 Figure 6.64: Pure (left) and signal-embedded (right) toy MC for combined B → (ρ0 /ρ± )γ fit. Effective signal yields on top, pulls on the bottom 98 7 Validation studies Several studies were performed to validate the analysis technique. These fall into two different categories: validating the analysis against previous iterations attempted at BABAR, and performing studies that validate the use of MC-simulated samples in event selection, classification, and modeling. We will not cover the former category, as it is internal to BABAR analysis framework and depends on such details as the software release version used for event reconstruction. It suffices to say that the current analysis was validated against earlier results in the sense that distributions of the variables used for event selection (such as the Bagger inputs) were not significantly different from those seen in the past. Because we rely so heavily on MC simulated data samples, we need to find ways to cross-check that the simulation is indeed a good representation of our data. In order to achieve this goal, we combine the various simulated continuum MC datasets listed in Table 4.3 with appropriate luminosity weights that represent their expected proportional contribution in real data, and we compare the resulting distributions to those in the off-peak data sample. Only the variables for which the agreement is good are kept in the analysis. Because of the large number of variables used in this analysis, we do not include all the plots here, but a sample is shown in Figure 7.65. Another step in validating our analysis technique is to make sure that the control samples we use are indeed appropriate representations of our signal modes. Some of these cross-checks have already been described in Section 6.5, where we demonstrated that the distributions of the fit variables, mES and ∆E, as well as the outputs of the Bagger, agree very well between the B 0 → ρ0 γ and B ± → ρ± γ and the corresponding B → K ∗ γ modes. Another study that was carried out compared the individual 99 ccbar BdeltaZFitOverErr BdeltaZFitOverErr ρ 0γ signal 0.03 K*0γ signal uds tau 12000 off-peak data 10000 0.025 8000 0.02 6000 0.015 4000 0.01 2000 0.005 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Figure 7.65: Distributions of combined continuum MC vs off-peak data in B 0 → ωγ (left) and signal MC in B 0 → ρ0 γ vs B 0 → K ∗0 γ (right). The separation between the two B vertices is used as an example. Bagger inputs between B → ργ and corresponding B → K ∗ γ modes using signal MC samples. Again, we do not include plots for all the variables and provide a sample in Figure 7.65. Agreement was very good, indicating that B → K ∗ γ is indeed, as expected, an appropriate control sample to be used for validation and systematic effect evaluation for Bagger and for the fits. Finally, to get a sense for what differences in background levels we may expect in real data as compared to the combined MC fits, on which we base our expectations, we look at mES , ∆E, and Bagger distributions in off-peak data and in continuum MC. Figure 7.66 shows the results, which are mostly limited by the low off-peak statistics. We see on average higher background yields in data, which is not unexpected. In the last round of analysis, we observed about 25% more continuum background in the on-peak data fits than we expected from MC. The Bagger distributions here look reasonable within available off-peak statistics, leaving no strong reason to suspect a large systematic difference in classifier performance between data and MC. In fact, the distributions in on-peak data seem to follow the shape of the combined MC rather well (although there is again the tendency to have more background events there). 5.23 5.24 5.23 0.2 5.24 5.25 0 5.26 5.27 5.28 0.3 5.23 50 60 70 80 -0.2 -0.1 5.24 BdeltaE {bag>0.94} 0 5.22 10 20 30 40 50 BmESSshift {bag>0.94} 5.25 5.26 0.1 5.27 0.2 5.28 0.3 offPData tau uds ccbar 5.29 offPData tau uds ccbar 5.29 0 -0.3 10 5.24 5.23 40 50 60 0 -0.3 10 -0.2 -0.1 5.24 BdeltaE {bag>0.94} 0 5.22 10 20 30 40 50 5.25 5.25 0 5.26 5.26 0.1 5.27 5.27 0.2 5.28 5.28 0.3 offPData tau uds ccbar 5.29 offPData tau uds ccbar 5.29 offPData tau uds ccbar Figure 7.66: Plots for pre-unblinding data-MC studies: columns correspond to the individual modes (B 0 → ωγ, B 0 → ρ0 γ, and B ± → ρ± γ); rows correspond to pre-Bagger cut mES , post-Bagger cut mES and ∆E, and the high end of the Bagger output range (next page) in continuum MC vs off-peak and combined MC vs on-peak data. 0 -0.3 10 20 5.23 BmESSshift {bag>0.94} 0 5.22 1000 5.23 500 0 5.22 2000 1000 20 0.1 offPData tau uds ccbar 5.29 offPData tau uds ccbar 5.29 3000 1500 20 0 5.28 5.28 4000 2000 30 -0.2 5.27 5.27 5000 2500 30 5.26 5.26 6000 3000 40 5.25 5.25 7000 3500 9000 8000 offPData BmESSshift 4000 4500 tau offPData uds ccbar tau BmESSshift uds ccbar 30 40 50 60 -0.1 5.24 BdeltaE {bag>0.93} 0 5.22 10 20 30 40 50 60 BmESSshift {bag>0.93} 0 5.22 500 1000 1500 2000 2500 BmESSshift 100 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 signal bbbar bpbm ccbar uds tau onPData 1 60 70 80 90 bag 0 0.9 0 0.9 10 20 0 0.9 10 Figure 7.66: Pre-unblinding studies continued 0 0.9 10 20 60 70 80 90 bag 0 0.9 30 1 signal bbbar bpbm ccbar uds tau onPData 1 20 0.99 0.99 40 0.98 0.98 30 0.97 0.97 50 0.95 0.96 0.95 0.96 40 0.94 0.94 20 40 30 0.93 0.93 20 40 60 80 80 60 100 100 offPData 120 bag 50 0.91 0.92 0.91 0.92 offPData tau 120 tau ccbar uds bag uds ccbar 40 50 60 70 bag 0 0.9 20 40 60 80 100 bag 0.91 0.92 0.91 0.92 0.93 0.93 0.94 0.94 0.95 0.96 0.95 0.96 0.97 0.97 0.98 0.98 0.99 0.99 1 signal bbbar bpbm ccbar uds tau onPData 1 offPData tau uds ccbar 101 102 8 Results 8.1 Branching fractions Table 8.12 lists the results of unblinded on-peak data fits to all the signal modes, as well as the ρ0 /ρ± and ρ/ω combinations. Unfortunately, we do not observe a significant signal yield in the B 0 → ωγ mode, and we do not improve on the previous measurement of the B ± → ρ± γ channel either. Table 8.13 summarizes on-peak data fits to the three signal modes using only BABAR runs 1–5 data, corresponding to 383 million BB pairs. Compared to the previously published BABAR measurement using runs 1–5 data[15], our statistical significance is only slightly better for the neutral and slighly worse for the charged modes. The results are in good agreement. It appears that the increase in total integrated luminosity provided by run 6 (82 million BB pairs) benefits only the B 0 → ρ0 γ channel, while contributing mostly background events to the other two. More details are provided in Section 8.1.1 below. Our results directly supersede those previously published by the BABAR collaboration. Nsignal  Nudscτ Significance B (×106 ) B 0 → ωγ 12.4+6.6 −5.7 ± 1.6 6.1% 164.9+14.5 −13.8 2.2σ 0.50+0.27 −0.23 ± 0.09 B 0 → ρ0 γ 34.9+8.6 −7.9 ± 1.2 8.5% 271.4+18.0 −17.3 5.4σ 0.97+0.24 −0.22 ± 0.06 B ± → ρ± γ 23.3+8.1 −7.3 ± 4.2 4.9% 178.1+15.5 −14.8 3.2σ 1.20+0.42 −0.37 ± 0.20 B → (ρ/ω)γ 1509.8+276 −262 ± 104.4 — — 6.5σ 1.63+0.30 −0.28 ± 0.15 B → ργ 1604.6+317 −299 ± 131.5 — — 6.0σ 1.73+0.34 −0.32 ± 0.17 Table 8.12: Summary of on-peak data fits for all signal modes 103 Nsignal Significance B 0 → ωγ 11.4+6.0 −5.1 2.5σ B 0 → ρ0 γ 25.9+7.7 −7.0 4.7σ B ± → ρ± γ 23.6+8.2 −7.4 3.9σ Table 8.13: Summary of on-peak data fit results for all signal modes, using only BABAR runs 1–5 data Figures 8.67 through 8.69 show the mES and the ∆E projection plots for the on-peak data fits for all three signal modes. Here, signal region cuts are applied to the variable that is not being plotted to visually enhance the signal-to-noise ratio. Figures 8.70 and 8.71 do the same for the simultaneous, isospin-constrained fits to the ρ± and ρ/ω combinations. 8.1.1 Study of B 0 → ωγ yields by BABAR runs Failing to observe a significant signal yield in the B 0 → ωγ mode, and noting that excluding run 6 from the dataset does not significantly change the yield, we decided to perform additional fits to attempt to discover how the yield varies depending on the data-taking period. Figures 8.72, 8.73, and 8.74 show the mES projections for on-peak data fits in the B 0 → ωγ channel using BABAR runs 1–4 (corresponding to 236 million BB pairs), runs 1–5 (383 million BB pairs), and run 5 by itself (147 million BB pairs). We see that the first fit appears to contain mostly background events, the second fit looks very similar to our nominal result that uses the whole available BABAR dataset, and the latter has a very clean signal signature — virtually background free. Thus, it appears that the majority of our signal events come from run 5 alone, while the other 5 runs dilute these with a large amount of background. One can view this as an upward fluctuation in one third of our data, or as an unlucky downward fluctuation in the other two thirds. Based on the numbers in Table 8.13, it seems plausible that the B ± → ρ± γ mode suffers from a similarly uneven distribution of events in our data sample. Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 104 18 16 14 12 10 10 8 6 8 4 6 4 2 2 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 0 -0.3 5.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) mES (GeV/c2) Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) Figure 8.67: mES and ∆E projections for the on-peak data fit in the B 0 → ωγ mode 30 25 20 18 16 14 12 10 8 15 6 10 4 5 0 5.22 2 5.23 5.24 5.25 5.26 5.27 5.28 5.29 0 -0.3 5.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) mES (GeV/c2) Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) Figure 8.68: mES and ∆E projections for the on-peak data fit in the B 0 → ρ0 γ mode 25 20 15 18 16 14 12 10 8 10 6 4 5 2 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 mES (GeV/c2) 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Figure 8.69: mES and ∆E projections for the on-peak data fit in the B ± → ρ± γ mode Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 105 60 50 40 35 30 25 20 30 15 20 10 10 0 5.22 5 5.23 5.24 5.25 5.26 5.27 5.28 5.29 0 -0.3 5.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) MES (GeV/c2) Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) Figure 8.70: mES and ∆E projections for the on-peak data fit for the combined B → (ρ/ω)γ mode 50 40 30 25 20 15 20 10 10 0 5.22 30 5 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 MES (GeV/c2) 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Figure 8.71: mES and ∆E projections for the on-peak data fit for the combined B → ργ mode 106 projection of m Events / ( 0.004 GeV/c2 ) ES with lRatioFunc_mESshift>0 10 8 6 4 2 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 m 5.29 5.3 2 (GeV/c ) Figure 8.72: On-peak data fit in the B 0 → ωγ mode using BABAR runs 1–4 only projection of m Events / ( 0.004 GeV/c2 ) ES with lRatioFunc_mESshift>0 16 14 12 10 8 6 4 2 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 m 5.29 5.3 2 (GeV/c ) Figure 8.73: On-peak data fit in the B 0 → ωγ mode using BABAR runs 1–5 only projection of m Events / ( 0.004 GeV/c2 ) ES with lRatioFunc_mESshift>0 10 8 6 4 2 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 m 5.29 5.3 2 (GeV/c ) Figure 8.74: On-peak data fit in the B 0 → ωγ mode using BABAR run 5 only 107 8.2 Upper limit for B 0 → ωγ mode Because we do not observe the B 0 → ωγ signal above the 3σ level of statistical significance, we also set an upper limit on this branching fraction using a simple Bayesian technique. Bayes’ theorem applied to the extraction of signal yield via fitting a parametric model to a set of data points implies: L(parameters|data) = P (data|parameters) = P (parameters|data)P (data) P (parameters) where the left side of the equation is our likelihood function. Assuming flat priors P (data) and P (parameters), we readily obtain the probability distribution for the model parameters. Because we use an extended likelihood approach, where the event yield is incorporated into the fit model, setting an upper limit becomes as straightforward as integrating the likelihood curve with all other parameters held fixed.5 Thus, we find B(B 0 → ωγ) < 0.9 × 10−6 at the 90% confidence level. 8.3 Isospin and SU(3)F To test the hypothesis of isospin symmetry between the charged and neutral ρ channels and to find the degree of SU(3)F symmetry breaking between ρ0 and ω channels, we measure B(B + → ρ+ γ)τB 0 − 1 = −0.43+0.25 −0.22 (stat) ± 0.10 (syst), 2B(B 0 → ρ0 γ)τB + Γ(B 0 → ωγ) ∆ω = − 1 = −0.49+0.30 −0.27 (stat) ± 0.10 (syst). Γ(B 0 → ρ0 γ) ∆ρ = (8.13) (8.14) Though less consistent than previous results, this number is still in agreement with the theoretical expectation [8]. 5 Note that this procedure is not possible in the frequentist interpretation of statistics. 108 8.4 Determination of |Vtd/Vts| Using the current world average experimentally measured value of B(B → K ∗ γ) [26], we calculate B[B → (ρ/ω)γ] = 0.039 ± 0.008 B(B → K ∗ γ) B(B → ργ) = 0.042 ± 0.009. Rρ/ω = B(B → K ∗ γ) Rρ = (8.15) (8.16) This result is used to obtain the ratio of CKM elements |Vtd /Vts | by means of the following equation B(B → (ρ/ω)γ) Vtd 2 1 − m2ρ /MB2 = S ρ,ω B(B → K ∗ γ) Vts 1 − m2K ∗ /MB2 !3 ζ 2 [1 + ∆R]. (8.17) Following [6], we choose the values ζ = 0.85 ± 0.07, and ∆R = 0.1 ± 0.1. We find |Vtd /Vts |ρ = 0.235+0.026 −0.025 ± 0.018, (8.18) |Vtd /Vts |ρ/ω = 0.229+0.024 −0.023 ± 0.017, (8.19) where the first error is experimental and the second is theoretical. We would like to stress that we consider the individual branching fractions and the asymmetries to be the primary results of this analysis. While an important constraint on the Standard Model, the calculation of |Vtd /Vts | requires theoretical input, where assumptions differ from one theorist to another. 109 9 Systematic errors Various systematic effects have to be taken into account, including the modeling of reconstruction efficiency of charged and neutral particles, photon precut efficiencies, particle ID modeling, and the data-MC differences in signal efficiency of the cut on the multivariate classifier (Bagger). There are also systematic effects due to the choice of the fitting model. The errors are summarized in Table 9.17 and are described in detail below. 9.1 π 0 efficiency from Neutral AWG The efficiency for π 0 reconstruction is studied by the Neutral Reconstruction Analysis Working Group (AWG) in the BABAR collaboration. The standard approach is to use the double ratio of τ → ρν to τ → πν decays in data and MC. Based on these studies, we apply an efficiency correction of 0.971 to the B ± → ρ± γ mode and 0.968 to the B 0 → ωγ mode. The associated π 0 systematic error is 3%. 9.2 Photon quality cuts using e+e− → µµγ control sample Systematic uncertainties for the photon precuts on photon energy, number of crystals, GammaZ20, GammaA42, Gammas9s25, and lateral and second moments of the EMC cluster are determined using the e+ e− → µµγ samples published by Neutral Reconstruction AWG. Figure 9.75 shows comparison between data and MC for the relevant photon variables. 110 0.045 0.05 0.04 0.035 0.04 0.03 0.025 0.03 0.02 0.02 0.015 0.01 0.01 0.005 0 0 0.5 1 1.5 2 2.5 3 ECal (GeV) 3.5 4 4.5 5 0 -1 -0.5 0 cosTheta 0.5 1 4 5 0.03 0.03 0.02 0.02 0.01 0 0 0.01 1 2 3 ECal, Barrel 4 5 0 0 1 2 3 ECal, Endcap Figure 9.75: Distributions of photon variables for µµγ data (points with error bars) and MC (histograms) 111 0.18 0.14 0.16 0.12 0.14 0.1 0.12 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0 0.6 0.02 0.7 0.8 Z20, Barrel 0.9 1 0 0.6 0.7 0.8 0.9 Z20, Endcap 1 20 30 40 nCrystal, Endcap 50 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 0 10 20 30 40 nCrystal, Barrel 50 0 0 10 Figure 9.75: Distributions of photon variables for µµγ continued 112 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0.8 0.85 0.9 0.95 s9s25, Barrel 1 0 0.8 0.85 0.9 0.95 s9s25, Endcap 1 0.2 0.4 0.6 0.8 lateral moment, Endcap 1 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0 0 0.01 0.2 0.4 0.6 0.8 lateral moment, Barrel 1 0 0 Figure 9.75: Distributions of photon variables for µµγ continued 113 0.3 0.04 0.25 0.2 0.03 0.15 0.02 0.1 0.01 0 0 0.05 0.001 0.002 0.003 0.004 second moment, Barrel 0.005 0 0 0.001 0.002 0.003 0.004 second moment, Endcap 0.005 0.02 0.1 0.04 0.03 0.03 0.02 0.02 0.01 0 0 0.01 0.02 0.04 0.06 Z42, Barrel 0.08 0.1 0 0 0.04 0.06 Z42, Endcap 0.08 Figure 9.75: Distributions of photon variables for µµγ continued 114 Figure 9.76 shows the dependency of the photon variables on photon energy (ECal) and acceptance angle (θ) for B →(ρ/ω)γ signal MC. The µµγ events are weighted so that the ECal and cos(θ) distributions match those of B →(ρ/ω)γ signal MC. Figure 9.77 shows the weights in bins of ECal and cos(θ). The data-to-MC efficiency ratio for the photon precuts is calculated to be 0.996 ± 0.020. We don’t apply signal efficiency corrections and quote a 2% systematic error. EMC bump isolation cut of 25 cm was also studied in detail by the Neutral Reconstruction AWG, and a 2% discrepancy between data and MC was found. These errors are added in quadrature to become the photon selection systematic error. 9.3 Particle ID systematic Since we use a particle identification criteria that is different from those defined by the BABAR PID group for B → ργ, we need to validate its performance ourselves. We do this using the standard PID sample of D∗ decays, which, after background subtraction, provide clean kinematically selected samples of pions and kaons. There are two aspects of the PID that need to be checked: the K ± mis-identification rate, and the π ± efficiency. The first is the more critical because K ± could cause a bias in the fit yield (an additive systematic), whereas π ± efficiency is simply a multiplicative systematic. Figure 9.78 shows the performance of our PID criteria for the full generic BB and cc MC and D∗ decay data (883033 events). Kaon mis-identification rates are low (≈ 1%), and there is generally a good agreement between data and MC. However, at high momentum (above 3 GeV/c), the agreement between D∗ data and BB MC is poor. Since the K ± momenta in BB decays are generally low (below 3 GeV/c), the high momentum K ± s are produced and boosted, almost exclusively, in the forward direction. These high momentum kaons enjoy better DIRC resolution, leading to lower mis-identification rates. The agreement between D∗ data and cc MC at high momentum is much better; high momentum K ± s are present throughout the entire detector in both samples. Therefore, the cc MC PID rates were used to weight the π ± (B → (ρ/ω)γ sample) and K ± 115 Profile of secMom versus eCal Profile of lat versus eCal 0.45 0.4 0.35 0.3 0.001 0.25 0.2 0.15 0.1 0.05 01 1.5 2 2.5 3 3.5 4 4.5 5 Profile of nCrys versus eCal 01 1.5 2 2.5 3 3.5 4 4.5 5 3.5 4 4.5 5 3.5 4 4.5 5 Profile of s9s25 versus eCal 1 35 0.8 30 25 0.6 20 15 0.4 10 0.2 5 0 1 1.5 2 2.5 3 3.5 4 4.5 5 Profile of Z20 versus eCal 0 1 0.045 0.8 0.04 0.7 0.035 0.6 0.03 0.5 0.025 0.4 0.02 0.3 0.015 0.2 0.01 0.1 0.005 1.5 2 2.5 3 2 2.5 3 Profile of Z42 versus eCal 0.9 01 1.5 3.5 4 4.5 5 01 1.5 2 2.5 3 Figure 9.76: Profile histograms of the photon precut variables for B →(ρ/ω)γ signal MC 116 Profile of secMom versus cosTheta Profile of lat versus cosTheta 0.4 0.35 0.3 0.25 0.001 0.2 0.15 0.1 0.05 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Profile of nCrys versus cosTheta Profile of s9s25 versus cosTheta 35 1 30 1 0.8 25 0.6 20 15 0.4 10 0.2 5 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Profile of Z20 versus cosTheta 0 0.045 0.8 0.04 0.7 0.035 0.6 0.03 0.5 0.025 0.4 0.02 0.3 0.015 0.2 0.01 0.1 0.005 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 Profile of Z42 versus cosTheta 0.9 0 -0.8 -0.6 -0.4 -0.2 -0.8 -0.6 -0.4 -0.2 1 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 9.76: Profile histograms of the photon precut variables continued 117 (B 0 → K ∗0 γ sample) momentum spectra. Weighting the momentum spectrum of π ± s from B → (ρ/ω)γ decays (kaons from B 0 → K ∗0 γ decays) with the D∗ data π ± efficiencies (D∗ data K ± mis-identification rates), integrating, then dividing by the same momenum spectrum weighted by cc MC π ± efficiencies (cc MC K ± mis-identification rates) gives the relative difference in the mis-identification rates predicted by the MC to those projected using the D∗ decays data. The results of the PID K ± mis-identification and π ± efficiency weighted momentum spectra study are summarized in Table 9.14. Uncertainties in π ± efficiency affect the overall signal reconstruction efficiency, while K ± mis-identification rates are directly reflected in the overall normalization of the B → K ∗ γ peaking background component included (and fixed) in the fits. Thus the former error is multiplicative, while the latter is additive. Based on this cross check, we found a relative uncertainty of 23% in the overall B → K ∗ γ background normalization attributed to kaon misidentification rates. We choose to vary the peaking BB contribution by a slightly more conservative 30% in the fits. Agreement between data and MC is also quite good for π ± efficiency. Based on this, we assign a 1% per track multiplicative systematic in B 0 → ρ0 γ and B ± → ρ± γ modes. data set BB cc data (BB − data)/BB (cc − data)/cc B 0 → K ∗0 γ 3147.88 4421.95 5434.32 0.726341 0.228941 B 0 → ρ0 γ 328236 331301 330395 0.00657558 0.00273468 ± B →ρ γ 159574 161431 161101 0.00957159 0.00204576 B 0 → ωγ 251909 262206 259542 0.030304 0.0101593 ± Table 9.14: Signal π ± and K ± momentum spectrum integrals after weighting with PID π ± efficiency (B → (ρ/ω)γ data samples) and K ± mis-identification (B 0 → K ∗0 γ data sample) rates. The last two columns give the overall rates after averaging the PID weighted momentum bins using the BB and cc rates, respectively. The cc weighted overall rates were used in this analysis. 118 9.4 Bagger validation with B → K ∗γ Together with the cross-check fits described in Section 6.5, the fits described here give us a handle on the systematic uncertainty associated with how well we understand the efficiency of the Bagger cut. We carry out the same 2-dimensional mES -∆E fits for the K + π − and K ± π 0 modes as before, but remove the Bagger cut altogether. Thus we have the difference in the fraction of events passing this cut in Monte Carlo vs on-peak data, which we can then use to apply a correction and assign a systematic error to our B → (ρ/ω)γ branching fraction measurements. We begin by simply looking at signal MC numbers before and after the Bagger cut is applied, and we obtain the results shown in Table 9.15. Mode Before cut After cut  K +π− 423570 122813 29.0% K ±π0 38186 7159 18.7% Table 9.15: Bagger cut signal MC efficiency in B → K ∗ γ To do the same exercise in on-peak data, we need to perform maximum likelihood fits to separate the signal from background. Results for these fits with the Bagger cut applied have already been presented in Table 6.11. The same fits performed with no Bagger applied (PDFs shown in Figures 9.79 and 9.82, MC and on-peak data fits in Figures 9.80 an 9.83) give us signal yields of 3986 ± 96 for K + π − and 1321 ± 68 for K ± π 0 modes. However, after performing pure and signal-embedded toy MC studies (500 experiments each, shown in Figures 9.81 and 9.84), we find significant biases in the embedded toys that require correction. The mean of the signal yield distribution is shifted upward by 48 (from 3389, corresponding to 1.4%) in K + π − and by 38 (from 916, corresponding to 4.1%) in K ± π 0 . Applying these corrections, we end up with the numbers listed in Table 9.16. Thus, in order to correct our data efficiency, we need to multiply it by 0.91 in B 0 → ρ0 γ and 0.88 in B ± → ρ± γ mode. For B 0 → ωγ channel, we use the average of the two, because, even though there’s a π 0 in the final state similar to the B ± → ρ± γ, 119 Mode Before cut (corrected) After cut  K +π− 3931 ± 96 1037 ± 33 26.4% K ±π0 1269 ± 68 210 ± 14 16.5% Table 9.16: Bagger cut on-peak data efficiency in B → K ∗ γ the output of the Bagger is almost identical to B 0 → ρ0 γ and the efficiency of the cut is exactly the same for the same cut value. There remains the question of statistical errors on the on-peak data yields. We believe that a correction is justified by the fact that Bagger inputs agree well in B → ργ and B → K ∗ γ (which is kinematically very similar to our signal). But the correction is obtained using central values for the data yields, while these also carry statistical errors themselves. Therefore, we apply the efficiency correction and take these errors as systematic errors. We obtain 3.2% error on the yield with the Bagger cut applied and 2.7% without it (taking this number from signal-embedded toy MC studies) for K + π − mode. Similarly, the numbers are 6.7% and 6.4% for K ± π 0 . Now we use the following facts: f (x) = x =⇒ df dx = f x and f (x) = 1 df dx =⇒ =− . x f x Therefore, an error of 1% on either yield will translate to the same fractional error after cut on the efficiency of the Bagger cut (since cut = NN before ). And because the two cut sets of fits are independent, we can treat these errors as uncorrelated (i.e., an upward fluctuation in one yield does not need to correspond to a downward fluctuation in another). Then, adding the two sets of errors in quadrature, we end up with a 4.2% error in K + π − to be applied to B 0 → ρ0 γ, and 9.3% error in K + π 0 to be applied to B ± → ρ± γ. Because we took the average of the two efficiency corrections for the ρ channels and applied that to the ω, we take half the quadrature sum of the corresponding errors, which comes out to be 5.1%, as the error to be applied to the B 0 → ωγ mode. 120 cosTheta eCal vs cosTheta 9 0.8 8 0.6 7 0.4 6 0.2 5 -0 4 -0.2 3 -0.4 2 -0.6 1 1 1.5 2 2.5 3 3.5 4 0 4.5 5 5.5 ECal (GeV) Figure 9.77: Distributions of weights applied to the µµγ sample 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.095 0.09 0.085 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 0.5 1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 1.5 2 2.5 3 3.5 4 0.1 0.095 0.09 0.085 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0.1 0.095 0.09 0.085 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 Figure 9.78: Particle ID performance in generic BB MC (left), generic cc MC (right), and Run 6 D∗ data (bottom). Filled circles are for π ± efficiency and use left-hand scale. Open circles are for K ± mis-identification and use the right-hand scale. Events / ( 0.00266667 GeV/c ) 2 Events / ( 0.00266667 GeV/c ) 2 Events / ( 0.00266667 GeV/c ) 0 5.22 500 1000 1500 2000 2500 3000 3500 0 5.22 20 40 60 80 100 0 5.22 200 400 600 800 1000 5.23 5.23 5.23 5.24 5.24 5.24 5.25 5.25 5.25 5.26 5.26 5.26 5.27 5.27 5.27 α = 2.29 ± 0.13 2 2 5.28 ξ = -24.680 ± 0.52 χ 2 / ndf = 0.815 5.28 5.3 2 5.3 MES (GeV/c2 ) 5.29 MES (GeV/c ) 5.29 σ = 0.005000 ± 0.000077 GeV/c2 n = 0.79 ± 0.84 5.3 2 MES (GeV/c ) 5.29 µ = 5.27949 ± 0.00037 GeV/c α = 0.091 ± 0.046 χ 2 / ndf = 1.525 5.28 σ = 0.002623 ± 0.000039 GeV/c2 n = 1.17 ± 0.19 µ = 5.279680 ± 0.000049 GeV/c χ 2 / ndf = 63.748 1200 -0.2 -0.1 0 -0.1 0 0 -0.3 500 1000 1500 2000 2500 3000 -0.2 -0.1 0 0.1 2 0.2 p = 1.30 ± 0.14 1 p = -1.8460 ± 0.024 0.2 3500 χ 2 / ndf = 0.951 0.1 σ = 0.094 ± 0.077 GeV 4000 0 -0.3 20 40 60 80 100 µ = -0.2680 ± 0.039 GeV c = -3.088 ± 0.51 120 fracG = 0.190 ± 0.090 0.2 140 χ 2 / ndf = 0.546 0.1 R σ = 0.0328 ± 0.0019 L σ = 0.0610 ± 0.0032 = 0.144 ± 0.011 160 0 -0.3 50 100 150 200 250 -0.2 µ = 0.0060 ± 0.0027 350 300 R α = 0.252 ± 0.016 L α 400 χ 2 / ndf = 10.573 450 0.3 ∆ E (GeV) ∆ E (GeV) 0.3 0.3 ∆ E (GeV) Figure 9.79: PDF shapes for mES (left) and ∆E (right) determined from Monte Carlo for the B 0 → K ∗0 γ (K ∗0 → K + π − ) mode. Signal on top, BB in the middle, continuum on the bottom. No Bagger cut is applied. 2 Events / ( 0.02 GeV ) Events / ( 0.02 GeV ) Events / ( 0.02 GeV ) 1400 121 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 122 3500 3000 2500 2000 1400 1200 1000 800 1500 600 1000 400 500 200 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 0 -0.3 5.3 -0.2 -0.1 0 0.1 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 4000 3500 3000 2500 1400 1200 1000 800 1500 600 1000 400 500 200 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 MES (GeV/c2) 0.3 1600 2000 0 5.22 0.2 ∆ E (GeV) MES (GeV/c2) 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Figure 9.80: Luminosity-sampled, combined Monte Carlo (top) and on-peak data (bottom) fits for the B 0 → K ∗0 γ (K ∗0 → K + π − ) mode. No Bagger cut is applied. 3200 3300 3400 -2 -1 0 1 Sigma 2 3 0.98 ± 0.03 -2 -1 0 1 2 3 5 5 0 10 10 0 -3 15 20 20 15 25 25 30 40 30 1 ± 0.0 45 -2 nSignal pull 35 Sigma 500 39 ± 2.2 Constant Mean -0.13 ± 0.05 Entries 35 40 45 nContinuum pull 0 -1 0 1 3600 Sigma 2 Constant Mean Entries 3500 3 1 ± 0.0 0.51 ± 0.05 498 37 ± 2.1 3700 77000 77200 0 -3 5 10 15 20 25 30 35 40 45 -2 nContinuum pull 0 5 10 3400 15 10 5 20 30 35 15 3300 3440 ± 4.3 90.62 ± 3.78 20 Sigma Mean 40 45 nContinuum 25 3200 498 Constant 39.85 ± 2.39 Entries 25 30 35 40 45 nSignal -1 77400 77600 0 77800 498 1 Sigma 2 0.96 ± 0.03 498 39 ± 2.2 Constant Mean -0.28 ± 0.04 78400 297.5 ± 10.8 78200 Entries 78000 Sigma Mean 7.769e+04 ± 14 Constant 35.87 ± 2.02 Entries Figure 9.81: Plots of signal and continuum yields (top) and their pulls (bottom) in pure (left four) and signal-embedded (right four) toy MC for the B 0 → K ∗0 γ (K ∗0 → K + π − ) mode. No Bagger cut is applied. Expected signal and continuum yields are 3389 and 77762, respectively. 0 -3 10 20 30 40 50 499 41 ± 2.3 Constant Mean 0.0039 ± 0.0445 0 76800 77000 77200 77400 77600 77800 78000 78200 78400 78600 78800 0 Entries 5 5 nSignal pull 10 10 3600 15 3500 20 312.6 ± 10.4 15 Sigma Mean 7.774e+04 ± 14 20 30 35 40 500 Constant 42.61 ± 2.36 Entries 25 91.67 ± 3.44 45 nContinuum 25 30 Sigma Mean 3388 ± 4.5 35 40 499 Constant 37.35 ± 2.16 Entries 45 nSignal 123 Events / ( 0.00266667 GeV/c ) 2 Events / ( 0.00266667 GeV/c ) 2 Events / ( 0.00266667 GeV/c ) 2 Events / ( 0.00266667 GeV/c ) 5.23 5.24 5.25 5.26 5.27 5.23 5.24 5.25 5.26 5.27 0 5.22 500 1000 1500 2000 2500 0 5.22 10 20 30 40 5.24 5.25 5.26 5.27 5.23 5.24 5.25 5.26 5.27 5.28 χ 2 / ndf = 1.661 ξ = -25.187 ± 0.61 5.28 5.29 5.29 2 50 σ = 0.00337 ± 0.00019 GeV/c 60 2 µ = 5.28200 ± 0.00043 GeV/c2 n = 100 ± 181 5.29 χ / ndf = 1.402 α = 0.0923 ± 0.0077 5.28 70 0 5.22 10 20 5.23 40 σ = 0.00373 ± 0.00038 GeV/c2 µ = 5.27981 ± 0.00056 GeV/c n = 1.31 ± 0.40 30 5.29 2 2 χ 2 / ndf = 1.822 α = 0.76 ± 0.21 5.28 50 0 5.22 50 100 150 200 σ = 0.00294 ± 0.00011 GeV/c µ = 5.27946 ± 0.00014 GeV/c2 n = 5.2 ± 1.9 2 χ / ndf = 4.652 α = 1.37 ± 0.18 250 MES (GeV/c 2 ) 5.3 MES (GeV/c 2 ) 5.3 MES (GeV/c 2 ) 5.3 MES (GeV/c 2 ) 5.3 R 0.1 0 -0.3 500 1000 1500 2000 2500 3000 0 -0.3 20 40 60 80 100 0 -0.3 2 4 6 8 10 12 14 18 -0.2 -0.2 -0.1 -0.1 0 0 R 1 0.1 2 p = 2.19 ± 0.17 χ 2 / ndf = 0.739 p = -1.9323 ± 0.029 0.1 2 = 0.056 ± 0.017 χ / ndf = 1.619 c = -5.184 ± 0.24 0.1 σ µ = -0.0233 ± 0.022 σ L = 0.18 ± 0.11 0 0 = 0.0422 ± 0.0050 16 -0.1 -0.1 R L χ 2 / ndf = 1.103 α L = 1.1 ± 1.5 α = 0.235 ± 0.072 -0.2 -0.2 σ µ = -0.00435 ± 0.0068 σ = 0.0707 ± 0.0082 α R = 0.126 ± 0.025 L 2 χ / ndf = 1.358 α = 0.228 ± 0.042 20 0 -0.3 20 40 60 80 100 0.2 0.2 0.2 0.2 ∆ E (GeV) 0.3 0.3 ∆ E (GeV) ∆ E (GeV) 0.3 0.3 ∆ E (GeV) Figure 9.82: PDF shapes for mES (left) and ∆E (right) determined from Monte Carlo for the B ± → K ∗± γ (K ∗± → K ± π 0 ) mode. True signal on top, self-crossfeed signal followed by BB in the middle, continuum on the bottom. No Bagger cut is applied. 2 Events / ( 0.02 GeV ) Events / ( 0.02 GeV ) Events / ( 0.02 GeV ) Events / ( 0.02 GeV ) 300 124 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 125 2200 1000 2000 1800 1600 1400 1200 800 600 1000 400 800 600 200 400 200 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 0 -0.3 5.3 -0.2 -0.1 0 0.1 Events / ( 0.03 GeV ) Events / ( 0.004 GeV/c2 ) 2400 0.2 0.3 ∆ E (GeV) 2 MES (GeV/c ) 1000 2200 2000 1800 1600 1400 800 600 1200 1000 400 800 600 200 400 200 0 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3 2 MES (GeV/c ) 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆ E (GeV) Figure 9.83: Luminosity-sampled, combined Monte Carlo (top) and on-peak data (bottom) fits for the B ± → K ∗± γ (K ∗± → K ± π 0 ) mode. No Bagger cut is applied. 750 0 950 Entries 1000 1 499 36 ± 2.0 1100 2 1 ± 0.0 0.02 ± 0.05 1050 56600 56800 57000 57200 57400 498 20 30 40 800 850 -1 900 0 950 1000 1050 Sigma Mean 949.7 ± 2.9 1100 495 38 ± 2.2 61.45 ± 2.32 20 25 30 35 40 -1 0 3 0 -2 1 2 3 0 -3 5 2 10 5 5 0 -3 15 20 25 30 35 40 10 1 0.52 ± 0.04 0.95 ± 0.03 56200 -2 nContinuum pull 10 Sigma Constant Mean Entries 5 0 55800 15 20 25 30 35 40 45 nSignal pull 0 15 20 25 30 -2 0.99 ± 0.03 Sigma 40 35 40 ± 2.2 Constant Mean 0.018 ± 0.045 Entries 45 nContinuum pull 0 55800 5 10 56000 nContinuum 10 56400 277.2 ± 9.4 495 Constant 39.69 ± 2.31 Entries 15 56200 Sigma Mean 5.669e+04 ± 13 50 nSignal 10 56000 498 Constant 40.71 ± 2.27 Entries 15 20 25 30 35 40 45 nContinuum -1 56400 0 56600 497 57200 57400 279.2 ± 9.8 1 Sigma 2 1 ± 0.0 497 36 ± 2.0 Constant Mean -0.28 ± 0.05 57000 Entries 56800 Sigma Mean 5.661e+04 ± 13 Constant 38.04 ± 2.13 Entries Figure 9.84: Plots of signal and continuum yields (top) and their pulls (bottom) in pure (left four) and signal-embedded (right four) toy MC for the B ± → K ∗± γ (K ∗± → K ± π 0 ) mode. No Bagger cut is applied. Expected signal and continuum yields are 916 and 56682, respectively. 0 -3 10 20 30 Sigma 900 40 -1 850 Constant Mean -2 800 64.08 ± 2.31 918.1 ± 3.1 Sigma Mean 499 38.1 ± 2.2 Constant Entries 50 nSignal pull 0 10 20 30 40 50 nSignal 126 127 9.5 Fit model systematics A maximum likelihood fit finds the best parameters for the chosen model, such that the model describes the data as closely as possible. The regression algorithm also returns the statistical errors for each parameter, based on the amount and distribution of the data sample. However, one can always ask the question of whether the choice of the model itself is the optimal one. Maybe a slightly different model could fit the data event better? This becomes especially important for those model parameters that are fixed in the fit and, therefore, have no associated statistical errors. In order to account for possible model variations in these parameters, we have to vary them explicitly by a specified amount and perform the fitting procedure again. The typical thing to do is to vary each fixed parameter by the error one obtains from the individual component PDF fits, where all parameters are floating (before they are fixed for the final multidimensional maximum likelihood fit). However, we can do a bit better than that for the signal model parameters by taking those errors from the B → K ∗ γ control sample fits instead. In order to match the signal mode fits as closely as possible, we use the B ± → K ∗± γ (K ∗± → K ± π 0 ) fit that does not split the signal into truth-matched and cross-feed components. We also vary parameters of the BB background components in mES and ∆E. The fraction attributed to the peaking B → K ∗ γ backgrounds is varied by ±30% based on the K ± mis-identification rate studied in Section 9.3 above. The overall contribution of the BB background to the likelihood function (the BB yield) is then varied by ±15% because the peaking component in ∆E accounts for about 30% of the entire PDF, and we choose to be a bit more conservative. Table 9.17 summarizes all systematic errors and includes a breakdown of the fit model systematics into those originating from signal and those coming from the background components. The former is dominant for the B 0 → ωγ channel, where we are limited by the low purity of the signal. In the cleanest B 0 → ρ0 γ channel, model errors from both sources are reasonably small. For the B ± → ρ± γ mode, both the shape of the continuum background component, which rises as we go from 128 the low to the high end of the fit region, and the relatively larger BB contribution combine to produce a rather high background model error. The simultaneous fits are driven mostly by the high yield in the B 0 → ρ0 γ channel, and are therefore much less sensitive to the variations in backgrounds in the other two modes. 9.6 Other systematics The other systematic effects we need to account for include charged particle reconstruction and the uncertainty on the overall number of BB pairs used in the analysis. The BABAR Tracking Efficiency Task Force provides standard recipes for all BABAR analyses dealing with charged tracks. From the tables they provide internally, we get a 0.43% error for B 0 → ωγ and B 0 → ρ0 γ modes, and 0.36% for B ± → ρ± γ, since all our tracks are taken from the GoodTracksLoose list. As for the uncertainty on the size of the BB sample, all analyses in the BABAR collaboration use a centrally computed uncertainty on the recorded number of Υ (4s) decays. For the exact sample used in this analysis, we obtain a 1.1% systematic error. B ± → ρ± γ B 0 → ρ0 γ B 0 → ωγ B → (ρ, ω)γ B → ργ Tracking efficiency 0.4% 0.4% 0.4% 0.4% 0.4% PID 1.0% 2.0% 1.0% 1.2% 1.4% Photon selection 2.8% 2.8% 2.8% 2.8% 2.8% π 0 reconstruction 3.0% - 3.0% 2.0% 1.7% Bagger cut efficiency 9.3% 4.2% 5.1% 7.5% 7.0% Signal model 7.1 2.1 16.3 3.0 3.0 Background model 10.9 2.8 2.7 3.6 4.3 BB sample 1.1% 1.1% 1.1% 1.1% 1.1% - - 0.8% 0.1% - 16.7% 6.6% 17.9% 9.5% 9.5% Source of error B(ω → π + π − π 0 ) Combined Table 9.17: Fractional systematic errors in % of the measured branching fractions 129 10 Summary In conclusion, we have performed an analysis of B → (ρ/ω)γ decays using the full dataset of 465 million BB pairs collected by the BABAR experiment, which ran from 1998 to 2008. We confirmed earlier observations of the B 0 → ρ0 γ channel, as well as evidence for B ± → ρ± γ. We do not observe a statistically significant signal in B 0 → ωγ channel, and a study of signal yields in this mode performed for different periods of data-taking suggests that there’s a great deal of fluctuation in the data. −6 We measure the branching fractions to be B(B ± → ρ± γ) = (1.20+0.42 −0.37 ± 0.20) × 10 , +0.27 −6 0 −6 B(B 0 → ρ0 γ) = (0.97+0.24 −0.22 ±0.06)×10 , and B(B → ωγ) = (0.50−0.23 ±0.09)×10 , and we set an upper limit for the B 0 → ωγ channel at B(B 0 → ωγ) < 0.9 × 10−6 at 90% confidence level using a simple Bayesian approach. Decay rates for these channels probe isospin and SU(3)F violation between the charged and neutral modes. 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