國立臺灣大學理學院物理所 碩士論文
Department or Graduate Institute of Physics College of Science
National Taiwan University Master Thesis
以蒙地卡羅法研討Ｂ介子至質子Λ重子衰變
Monte Carlo Study of
→ ( ) Decay at Belle
林冠伯 Kuan-Po Lin
指導教授﹕王名儒 博士 Advisor: Min-Zu Wang, Ph.D.
中華民國 106 年 2 月 February 2017
Acknowledgements
In this long long journey,
sometimes it’s sunny, but other times stormy.
What beat me but didn’t defeat me
have made me stronger, physically and mentally.
In this long long journey,
I walk along, with all your company.
Please accept my sincere gratitude, my dearest family, and my bonnie.
Thanks to my enemies, if any,
and those who helped me, yet too many.
I believe beside me you will always be, in each of the upcoming long long journey.
摘要
本篇論文探討 B 介子經由 → Λ → ̅ 雙重子途徑之衰變。本實驗以蒙
地卡羅法模擬日本高能加速器 B 介子工廠(KEKB)中 Belle 偵測器收集來自Υ 4S
共振態衰變的7 億 7 千萬 介子對。本實驗並應用以神經網路演算法為基礎的
NeuroBayes 來提升訊噪比，以估計 → Λ → ̅ 分支比在90%信心水準
下的上限值為5.46 × 10−7。此模擬結果可對Belle 偵測器真實數據之分支比量測 所幫助。相關主題亦值得於新一代 B 介子工廠(SuperKEKB)進行後續研究。
Abstract
B+ → p ¯Λ1520 is one of the simplest two-body baryonic B decay modes. We reconstruct ¯Λ1520 from ¯p and K+, then search for the signal B+ → p ¯Λ1520 (→ p¯pK+). Simulating the data collected at the Υ(4S) resonance with the Belle detector at KEKB, we use Monte Carlo method to generate 772 million B ¯B pairs collider events. After background suppression, we extract the signal from the Monte Carlo generated data and estimate the branching fraction:
B(B+ → p ¯Λ1520) < 5.46 × 10−7 at the 90% confidence level. This study helps measuring the branching fraction of the real data at Belle detector, and further, data at Belle II detector.
1 Introduction 1
1.1 Standard Model . . . 1
1.1.1 Feynman Diagram . . . 3
1.1.2 The Cabibbo-Kobayashi-Maskawa(CKM) matrix . . . . 4
1.2 B Meson Physics . . . 5
1.3 Motivation . . . 6
2 Belle Experiment 9 2.1 KEKB accelerator . . . 10
2.2 Belle Detector . . . 12
2.2.1 Beam Pipe . . . 12
2.2.2 Silicon Vertex Detector (SVD) . . . 15
2.2.3 Extreme Forward Calorimeter (EFC) . . . 15
2.2.4 Central Drift Chamber (CDC) . . . 17
2.2.5 Aerogel Cherenkov Counter System (ACC) . . . 18
2.2.6 Time-of-Flight Counters (TOF) . . . 20
2.2.7 Electromagnetic Calorimeter (ECL) . . . 20
2.3 Trigger and Data Acquisition . . . 22
3 Analysis 25 3.1 Blind Analysis . . . 25
3.2 Analysis Tools . . . 25
3.2.1 BASF . . . 25
3.2.2 EvtGen and GSIM . . . 26
3.2.3 ROOT and RooFit . . . 27
3.3 B meson reconstruction . . . 27
3.3.1 Data Sample . . . 27
3.3.2 Tracking . . . 27
3.3.3 Particle Identification . . . 28
3.3.4 Selection Summary . . . 28
3.4 Background Suppression . . . 30
ii CONTENTS
3.4.1 NeuroBayes (NB) . . . 30
3.4.2 Continuum Background . . . 31
3.4.3 Figure of Merit (F .O.M.) . . . 33
3.4.4 B ¯B Suppression . . . 37
3.5 Signal Extraction . . . 37
3.5.1 Ensemble Test on Fitters . . . 39
3.5.2 Upper Limit Estimation . . . 39
4 Conclusion 41
Bibliography 43
1.1 The Standard Model . . . 2
1.2 The Standard Model Interactions . . . 2
1.3 The Standard Model vertices . . . 3
1.4 The lowest-order Feynman diagram for muon decay process . . 4
1.5 e+e−→ Υ(4S) → B ¯B process. . . 6
1.6 Feynman diagram of B+→ p ¯Λ1520 → p¯pK+ process. The tree diagram is on the left, and the dominated penguin diagram is on the right. . . 7
2.1 Bird’s eye view of KEKB. . . 9
2.2 Configuration of the KEKB accelerator: The Belle detector is located in the interaction point in Tsukuba hall. . . 10
2.3 Configuration of the Belle detector. . . 13
2.4 Configuration of SVD . . . 15
2.5 Graphical illustration of sub-detector SVD1 and SVD2. . . 16
2.6 Isometric view of the BGO crystals of the forward and back- ward EFC detectors. . . 17
2.7 Overview of the CDC structure. The lengths in the figure are in units of mm. . . 18
2.8 Scatter plot for dE/dx versus momentum. Expected results for π, K, p and e are shown by solid curves. The unit of the momentum is GeV/c. . . 19
2.9 Arrangement of ACC at the central part of the Belle detector. The index of refraction (n) is for each ACC module. . . 19
2.10 Schematic drawing of a typical ACC counter module: (a) bar- rel and (b) end-cap ACC. . . 20
2.11 Illustration of a TOF/TSC module. The unit of length is in mm. . . 21
2.12 Mass distribution from TOF measurements for particle mo- menta below 1.2 GeV/c. . . 21
2.13 Overall configuration of ECL. . . 22
iv LIST OF FIGURES 2.14 The Level-1 trigger system for the Belle detector. . . 24 2.15 Belle DAQ system overview. . . 24 3.1 Schematic view of the BASF framework . . . 26 3.2 Distributions of dr and dz, Mbc, de for signal MC. The Monte
Carlo true events are red-filled. . . 29 3.3 Work flow of NeuroBayes . . . 31 3.4 The NB output distribution ON B of signal (red), q ¯q(blue) and
B ¯B(black) . . . 33 3.5 The F .O.M. curve . . . 35 3.6 Distributions of de, MΛ1520 and Mbc. The remained events are
red-filled.(The remained are 99% Monte Carlo True for signal) 36 3.7 Mbc and de fit results by Roofit . . . 38 3.8 Mbc and de fit results for ensemble fit by Roofit 38 3.9 Chart of ensemble tests . . . 39 3.10 Likelihood fit result . . . 40
1.1 Properties of B mesons. . . 6
2.1 Parameters of the KEKB accelerator . . . 11
2.2 Performance parameters for the Belle detector. . . 14
2.3 Geometrical parameters of ECL. . . 23
3.1 Preliminary Selection . . . 28
3.2 ON B cut optimization. . . 35
3.3 Strict Selection . . . 36
Chapter 1 Introduction
1.1 Standard Model
The goal of particle physics is to study the fundamental constituents of matter and their interactions. The Standard Model (SM) is the currently dominant theory of particle physics, formulated as relativistic quantum field theory. In SM, there are 4 families of elementary particles: 6 flavor of spin-1/2 quarks, 6 types of spin-1/2 leptons, 4 spin-1 gauge bosons and Higgs boson (shown in Fig 1.1), with their anti-particles. Besides, the 3 funda- mental interactions within particles in SM are the strong interaction, the electromagnetic interaction and the weak interaction, leaving gravitational interaction excluded.
Quarks, including up (u), down (d), charm (c), strange (s), bottom (b), and top (t) quarks, are the particles participating in strong interactions. On the other hand, leptons contain electrons (e), muons (µ), taus (τ ) and their corresponding neutrinos νe, νµand ντ. Each type of the 3 fundamental inter- actions is described in terms of the exchange of a spin 1 gauge boson. Gluons (g) mediate the strong interaction, while W± and Z0 mediate the weak in- teraction. For all charged particles (quarks are fractional charged, whereas others are integral), they are involved in the electromagnetic interaction, mediated by photons (γ). Finally, Higgs boson (H) is a spin-0 particle, postulated to explain the mechanism how elementary particles acquire their mass.
To sum up, a particle couples to a gauge boson only if it carries the charge of the interaction: electrically charged particles couple to γ; color-charged ones couple to g; and all couple to W± and Z0 due to weak charges. Fig 1.2 shows the diagram of the summary.
Figure 1.1: SM elementary particles classified as 4 kinds [1]
Figure 1.2: Diagram summarizing the interactions between elementary par- ticles
1.1. STANDARD MODEL 3
1.1.1 Feynman Diagram
Feynman diagram, invented by Richard Feynman, is not only a graphical representation but also a calculating tool used to evaluate probability of transitions between initial and final states.
A Feynman diagram consists of 3 basic elements: external lines, internal lines and vertices. All Feynman diagrams can be constructed from the Stan- dard Model vertices, partially shown in Fig 1.3. Given the initial and the final states, there are infinite Feynman diagrams associated with it. The diagram with the largest absolute value, the dominant term, is called the lowest-order diagram. However, no matter which order it is, once a Feynman diagram of a transition is drawn, the corresponding QM transition matrix elements can be calculated, applying F eynman Rules. Fig 1.4 shows an simple Feynman diagram example.
Figure 1.3: The SM vertices
Figure 1.4: A simple Feynman diagram for the neutron decay process:
n → pµ−ν¯µ.
1.1.2 The Cabibbo-Kobayashi-Maskawa(CKM) matrix
According to SM, leptons or quarks can change their flavor only through weak interactions. Moreover, the weak interaction allows quarks to change from one generation to another. Therefore, the d, s and b quarks are not pure mass eigenstates but mixtures with regard to weak interactions. This concept was proposed by M. Kobayashi and T. Maskawa in 1973 [2] based on Cabibbo mechanism: the mixture eigenstates of d, s and b quark can be represented by a 3 × 3 unitary matrix called Cabibbo-Kabayashi-Maskawa matrix (CKM matrix).
d0 s0 b0
=
Vud Vus Vub Vcd Vcs Vcb
Vtd Vts Vtb
d s b
(1.1)
is the form of the CKM matrix.
The entries can be parameterized by 3 mixing angles and 1 CP -violating phase as following, [3]
c12c13 s12c13 s13e−iδ
−s12c23− c12s23s13eiδ c12c23− s12s23s13eiδ s23c13 s12s23− c12c23s13eiδ −c12s23− s12c23s13eiδ c23c13
, (1.2) where sij = sin θji, cij = cos θji, and δ is the CP-violating phase.
A more convenient parameterization introduced by Lincoln Wolfenstein [4]
as an expansion in powers of λ is,
1.2. B MESON PHYSICS 5
1 −12λ2 λ Aλ3(ρ − iη + 2iηλ2)
−λ 1 − 12λ2− iηA2λ4 Aλ2(1 + iηλ2)
Aλ3(1 − ρ − iη) −Aλ2 1
+ O(λ4), (1.3) where λ = s12 = sin θc, Aλ2 = s23, and Aλ3(ρ − iη) = s13e−iδ. This parameterization implies the hierarchy
|Vub|2 << |Vcb|2 << |Vus|2 << 1, (1.4) such that b quark has an inclination to decay into c rather than u.
The magnitude of entries can be determined from adequate weak decays.
After measurements of the 4 real quantities in the Wolfenstein parametriza- tion,
λ = 0.2253 ± 0.0007, A = 0.808+0.022−0.015,
¯
ρ = 0.132+0.022−0.014, η = 0.341 ± 0.013,¯ , (1.5) the CKM elements are [6]
0.97428 ± 0.00015 0.2253 ± 0.0007 0.00347+0.00016−0.00012 0.2252 ± 0.0007 0.97345+0.00015−0.00016 0.0410+0.0011−0.0007
0.00862+0.00026−0.00020 0.0403+0.0011−0.0007 0.999152+0.000030−0.000045
. (1.6) Notice that the measured value is consistent with the unitary relations,
|Vud|2+ |Vus|2+ |Vub|2 = 1
|Vcd|2+ |Vcs|2+ |Vcb|2 = 1
|Vtd|2+ |Vts|2+ |Vtb|2 = 1
(1.7)
1.2 B Meson Physics
In 1977, the third generation bottom quark (b) was observed by the CFS1 E288 experiment headed by Leon Lederman at Fermilab [5]. B mesons are the particles composed of a bottom antiquark (¯b) and either a u quark or a d quark, leading to be charged (B+) or neutral (B0) respectively. Besides, a bottom antiquark and a strange quark (s) or a charm quark (c) form a strange B meson (Bs0) or a charmed B meson (Bc+). The properties of these B mesons are shown in Table 1.1.
1Columbia-Fermilab-Stony Brook collaboration
Table 1.1: Properties of B mesons.
Type Quark content I(JP) Rest mass (MeV/c2) Mean lifetime(ps) B+ u¯b 12(0−) 5279.17 ± 0.29 1.641 ± 0.008
B0 d¯b 12(0−) 5279.50 ± 0.30 1.519 ± 0.007 Bs0 s¯b 0(0−) 5366.3 ± 0.6 1.425 ± 0.041
Bc+ c¯b 0(0−) 6277 ± 6 0.453 ± 0.041
To study the physics of B mesons, namely B physics, 2 B factories, KEKB(Belle) and PEP-II(BaBar), were built in the 1990s. Since the mass of Υ(4S) is only 20 MeV above B ¯B threshold, and the branching fraction of Υ(4S) to B ¯B is larger than 96% [6], both of the B factories chose to utilize e+e− collider with asymmetry energy in order to produce B mesons via the process e+e−→ Υ(4S) → B ¯B(Fig 1.5).
Figure 1.5: e+e− → Υ(4S) → B ¯B process.
1.3 Motivation
B meson can decay into final states with charmed or charmless hadrons through quark fragmentation and hadronization. The branching ratio of B+ → p ¯Λ1520 decay, one of the simplest 2-body decay mode, has been mea- sured by LHCb to the order 10−7 − 10−6, at the boundary to be observed with the current data sets accumulated at B-factories. Since ¯Λ1520 decays to
¯
p and K+ and the lifetime of ¯Λ1520 is short enough to neglect, we believe it is a clear mode to study.
1.3. MOTIVATION 7 The leading Feynman Diagrams for B+ → p ¯Λ1520 → p¯pK+ are shown in Fig 1.6. The penguin diagram (¯b → ¯s) dominates the tree diagram (¯b → ¯u), due to the latter be CKM suppressed.
Figure 1.6: Feynman diagram of B+ → p ¯Λ1520 → p¯pK+ process. The tree diagram is on the left, and the dominated penguin diagram is on the right.
Chapter 2
Belle Experiment
The Belle experiment is designed to investigating the CP violation effects.
It is conducted by the Belle Collaboration, an international collaboration of more than 400 physicists and engineers from 16 countries, and located at the High Energy Accelerator Research Organization (KEK) in Tsukuba, Ibaraki Prefecture, Japan.(Fig 2.1) The Belle experiment includes 2 major facilities:
KEKB accelerator and the Belle detector.
Figure 2.1: Bird’s eye view of KEKB.
Figure 2.2: Configuration of the KEKB accelerator: The Belle detector is located in the interaction point in Tsukuba hall.
2.1 KEKB accelerator
KEKB [7], which stands for the KEK B-factory, is an asymmetric energy e− − e+ collider at KEK. The KEKB accelerator has 2 separate rings: a low-energy ring (LER) for positrons with 3.5 GeV beam energy while a high- energy ring (HER) for electrons with 8 GeV beam energy. Both rings are constructed side by side in the tunnel with a circumference of 3 km. To insure exactly the same circumference for the two rings, a cross-over design in Fuji area is built. (Fig 2.2) To reduce parasitic collisions near IP for reaching higher peak luminosities, the crossing angle of e−− e+ collision is set at ±11 mrad.
For convenience, we define ˆz to be the inverse direction of LER, and ˆx be perpendicular to z-axis which let x-z plane containing both HER and LER. The KEKB is designed to operate with a peak luminosity at the order of 1034 cm−2s−1, corresponding to 108 B ¯B pairs per year. Total integrated luminosity of 1052 fb−1 (including 772 × 106 B ¯B pairs running at Υ(4S)) was accumulated by Belle detector during the operation of KEKB, from Dec, 1998 to Jun, 2010. More parameters for the KEKB accelerator are listed in Table 2.1.
2.1. KEKB ACCELERATOR 11
Table 2.1: Parameters of the KEKB accelerator
Ring LER HER Unit
Energy E 3.5 8.0 GeV
Circumference C 3016.26 m
Luminosity L 1 × 1034 cm−2s−1
Crossing angle θx ±11 mrad
Tune shifts ξx/ξy 0.039/0.052
Beta function at IP βx∗/βy∗ 0.33/0.01 m
Beam current I 2.6 1.1 A
Natural bunch length σz 0.4 cm
Energy spread σε 7.1 × 10−4 6.7 × 10−4
Bunch spacing sb 0.59 m
Particles/bunch N 3.3 × 1010 1.4 × 1010 Emittance εx/εy 1.8 × 10−8/3.6 × 10−10
Synchrotron νs 0.01 ∼ 0.02
Betatron tune νx/νy 45.52/45.08 47.52/43.08
Energy loss/turn U0 0.81†/1.5†† 3.5 MeV
RF voltage Vc 5 ∼ 10 10 ∼ 20 MV
RF frequency fRF 508.887 MHz
Harmonic number h 5120
Longitudinal damping time τε 43†/23†† 23 ms
Total beam power Pb 2.7†/4.5†† 4.0 MW
Radiation power PSR 2.1†/4.0†† 3.8 MW
HOM power PHOM 0.57 0.15 MW
Bending radius ρ 16.3 104.5 m
Length of bending magnet lB 0.915 5.86 m
†: without wigglers, ††: with wigglers
2.2 Belle Detector
As shown in Fig 2.3, the Belle detector is mounted around the interaction point (IP). The beampipe and its surrounding 1.5T superconducting solenoid go through the sub-detectors:
B-meson decay vertices are measured by SVD situated just outside of a cylindrical beryllium beam pipe; charged particle tracking is provided by dE/dx measurement in CDC; particles are identified with the information measured by CDC, ACC, and TOF;(See section 3.3.3) electromagnetic show- ers are detected in ECL, whereas Muon and KLdetected in KLM. The above sub-detectors cover the θ region extending from 17◦ to 150◦, and a part of the uncovered small-angle region is instrumented with a pair of BGO crystal arrays (EFC). The performance of the detectors are summarized in Table 2.2 and the descriptions of each sub-detector are included in the following sec- tions. (Since electrons and muons are not involved in our study, the detail of KLM is skipped.)
2.2.1 Beam Pipe
The precise determination of decay vertices is an essential feature of the Belle experiment. Coulomb scattering in the beam pipe wall (and also the 1st layer of the silicon detector) affects the z-vertex position resolution. There- fore, minimizing the thickness of the beam pipe is necessary. The originally designed beam pipe has an inner / outer radius of 2.0 / 2.3 cm. The central part (−4.6 cm ≤ z ≤ 10.1 cm) of the beam pipe consists of double beryllium cylinders of 0.5 mm thickness. Moreover, the aperture of the beam pipe near IP is designed to avoid hitting by the synchrotron radiation from the final- focus quadrupole magnets (QCS) and other magnets going through. Besides, a 2.5 mm gap between the inner and outer walls of the cylinder provides a helium gas channel for cooling.
2.2. BELLE DETECTOR 13
m / KL detection
14/15 lyr. RPC+Fe
Central Drift Chamber
small cell +He/C2H6
CsI(Tl) 16X0
Aerogel Čherenkov Cnt.
n=1.015~1.030
Si Vtx. Det.
3/4 lyr. DSSD
TOF conter SC solenoid
1.5T
8 GeV
e
-3.5 GeV
e
+Belle Detector
(a) Cut view of the Belle detector.
(b) Side view of the Belle detector.
Figure 2.3: Configuration of the Belle detector.
Table 2.2: Performance parameters for the Belle detector. There were two configurations of inner detectors used to collect two data sets, DS-I and DS- II, corresponding to a 3-layer SVD1 and a 4-layer SVD2 with a smaller beam pipe, respectively.
Detector Type Configuration Readout Performance
Beam pipe Beryllium Cylindrical, r = 20mm, for DS-I double wall 0.5/2.5/0.5(mm) = Be/He/Be
w/ He gas cooled Beam pipe Beryllium Cylindrical, r = 15mm,
for DS-II double wall 0.5/2.5/0.5(mm) = Be/PF200/Be
EFC BGO Photodiode readout 160 × 2 Rms energy resolution:
Segmentation : 7.3% at 8 GeV
32 in φ; 5 in θ 5.8% at 2.5 GeV
SVD1 Double-sided 3-layers: 8/10/14 ladders φ: 40.96k σ(zCP) ∼ 78.0µm Si strip Strip pitch: 25(p)/50(n)µm z: 40.96k for B → φK02s
SVD2 Double-sided 4-layers: 6/12/18/18 ladders σ(zCP) ∼ 78.9µm
Si strip Strip pitch: φ: 55.29k for B → φKs0
75(p)/50(n)µm (layer1-3) z: 55.296k 73(p)/65(n)µm (layer4)
CDC Small cell Anode: 50 layers Anode: 8.4k σrφ= 130µm
drift Cathode: 3 layers Cathod: 1.8k σz= 200 ∼ 1400µm
chamber r = 8.3 - 86.3 cm σP t/P t = 0.3%
q p2t+ 1
−77 ≤ z ≤ 160 cm σdE/dx= 0.6%
ACC Silica 960 barrel/228 end-cap Np.e.≥ 6
aerogel FM-PMT readout K/π seperation:
1.2 < p < 3.5GeV/c
TOF Scintillator 128 φ segmentation 128 × 2 σt= 100 ps
r = 120 cm, 3-cm long K/π seperation:
TSC 64 φ segmentation 64 up to 1.2 GeV/c
ECL CsI Barrel: r = 125 - 162 cm 6624 σE/E = 1.3%/√
E
(Towered- End-cap: z = 1152(F) σpos= 0.5 cm/√
E
structure) -102 cm and +196cm 960(B) (E in GeV)
KLM Resistive 14 layers θ: 16k ∆φ = ∆θ = 30mr
plate (5 cm Fe + 4cm gap) φ: 16k for KL
counters 2 RPCs in each gap ∼ 1% hadron fake
Magnet Supercon. Inner radius = 170 cm B=1.5T
2.2. BELLE DETECTOR 15
2.2.2 Silicon Vertex Detector (SVD)
The ∼ 100 µ m position resolution of SVD is primarily for the measurement of the difference in z-vertex positions for B ¯B pairs. The SVD consists of 3 layers, 32 ladders of double-sided silicon strip detectors (DSSD) and covers most part of solid angle (23◦ < θ < 139◦). Each DSSD has 1280 sense strips and 640 readout pads on opposite sides. The designed configuration is shown in Fig 2.4. The z-strip / φ strip pitch is 42 / 25 µ m, and the active regions are 53.5 × 32.0 / 54.5 × 32.0mm2 on the z-side / φ -side.
To ensure that the SVD tracks match the tracks detected by CDC, higher SVD strip yields and better S/N ratios are needed. Therefore, in 2003, the SVD system was upgraded to the so-called SVD2, consisting of 4 layers of DSSDs with a larger coverage ( 17◦ < θ < 150◦, same as CDC). 2 types of DSSD are used for different layers. Basically, the SVD2 improves the detect- ing efficiency a bit and operates in a similar way to SVD1. The comparison is shown in Fig 2.5.
Figure 2.4: Configuration of SVD
2.2.3 Extreme Forward Calorimeter (EFC)
EFC is designed for improving the experimental sensitivity to some physics processes such as B → τ ν and two-photon physics. It also reduces back- grounds for CDC. Moreover, it is used as a beam monitor for the KEKB control and a luminosity monitor for the Belle experiment. The EFC detec- tor, surrounding the beam pipe, is attached to the front faces of the cryostats of the compensation solenoid magnets of the KEKB accelerator. It covers the angular range from 6.4◦ to 11.5◦ in the forward (e−) direction and 163.3◦
(a) Side view comparison of SVD1 and SVD2.
(b) End view comparison of SVD1 and SVD2.
Figure 2.5: Graphical illustration of sub-detector SVD1 and SVD2.
2.2. BELLE DETECTOR 17 to 171.2◦ in the backward (e+) direction. To endure such long-term and high level radiation, radiation-hard BGO (Bismuth Germanate, Bi4Ge3O12) crystal is chosen to construct EFC. There are 320 channels with photodiode readout in EFC, providing an energy resolution of 7.3%/5.8% for the forward / backward EFC. An isometric view of the EFC detector is shown in Fig 2.6.
Figure 2.6: Isometric view of the BGO crystals of the forward and backward EFC detectors.
2.2.4 Central Drift Chamber (CDC)
CDC is exploited to measure the transverse momentum (Pt) of charged parti- cles by the curvatures in the transverse plane. It also measure their momen- tum in the z-direction (Pz) from the helical track information and Pt. Beside Pt and Pz, CDC measures the energy loss (dE/dx) of charged particles as well.
The spatial resolution ranges from 120 to 150 µm, so precise that the resolution of Pt is smaller than 1% most of the time. Since gas with low atomic number minimize Coulomb scattering that effects the momentum resolution and ethane provides good dE/dx resolution, 50% He and 50%
ethane is used in CDC. Such high resolution makes particle identification more precise. (Fig 2.8)
The coverage of CDC is 17◦ < θ < 150◦, consisting of 50 layers and 3 cathode strip layers. It has a total of 8400 drift cells. The inner / outer radius of CDC is 103.5 / 874 mm, as shown in Fig 2.7.
747.0
790.0 1589.6
880
702.2 1501.8
BELLE Central Drift Chamber
5
10
r 2204
294
83
Cathode part
Inner part Main part Forward
Backward
e e
Interaction Point 150° 17°
y x 100mm y
x 100mm
- +
Figure 2.7: Overview of the CDC structure. The lengths in the figure are in units of mm.
2.2.5 Aerogel Cherenkov Counter System (ACC)
Although dE/dx measurement (by CDC) and time-of-flight measurement (by TOF) provide useful separation of K/π , an array of silica aerogel threshold Cherenkov counters is built to distinguish π from K in further extensive momentum coverage. (ACC covers 17◦ - 127◦ in polar angle.)
Fig 2.9 shows the arrangement of ACC, containing 960 / 228 counter modules in barrel / end-cap part, and Fig 2.10 shows the schematic drawing of the 2 types. These modules are segmented into 60 cells in the φ direction for the barrel ACC, and arranged in 5 concentric layers for the end-cap part.
The refractive indices of aerogels ranges from 1.01 to 1.03 depending on their spatial position, and the fine-mesh photomultiplier tubes (FM-PMTs) with corresponing diameters to obtain uniform response of Cherenkov light.
2.2. BELLE DETECTOR 19
0.5 1 1.5 2 2.5 3 3.5 4
-1.5 -1 -0.5 0 0.5 1
log10( p (GeV/c) )
dE/dx PK
e π
Figure 2.8: Scatter plot for dE/dx versus momentum. Expected results for π, K, p and e are shown by solid curves. The unit of the momentum is GeV/c.
Figure 2.9: Arrangement of ACC at the central part of the Belle detector.
The index of refraction (n) is for each ACC module.
Figure 2.10: Schematic drawing of a typical ACC counter module: (a) barrel and (b) end-cap ACC.
2.2.6 Time-of-Flight Counters (TOF)
TOF is used to identify charged particles with momenta below 1.2 GeV/c (almost 90% of the particles produced in Υ(4S) decays) and as a trigger due to fine time resolution (100 ps). The mass (Mtrack) distribution is shown in Fig 2.12.
The whole TOF system consists of 128 TOF counters and 64 trigger scin- tillation counters (TSC) with 70 kHz trigger rate preventing trigger queue from pile-up. A TOF / TSC module is constructed by 2 TOF and 1 TSC, as shown in Fig 2.11.
The length of the flight path is 1.2 m, and coverage of polar angle is 17◦ < θ < 127◦, same as that of CDC.
2.2.7 Electromagnetic Calorimeter (ECL)
ECL is designed to detect the energy and position of photons, and to identify electrons as well, by computing the ratio E/p (momentum p is provided by CDC).
The configuration is shown in Fig 2.13, and Table 2.3 lists some parame- ters. All crystals are shaped in a half-tower and point almost to the IP with a small tilt angle (1.3◦− 4◦ ) to avoid small leakage of photons from the gaps between the crystals. Moreover, the gaps between the end-cap and barrel parts are so small that only 3% of total acceptance is missing.
ECL use CsL(Tl) crystals as its scintillator. Since smaller size has bet- ter position resolution, while larger size having better energy resolution due
2.2. BELLE DETECTOR 21
TSC 0.5 t x 12.0 W x 263.0 L
PMT 122.0
182. 5 190. 5
R= 117. 5
R= 122. 0 R=120. 05
R= 117. 5 R=117. 5
PMT PMT - 72.5
- 80.5 - 91.5
Light guide
TOF 4.0 t x 6.0 W x 255.0 L
1. 0 PMT PMT
Forward Backward
4. 0
282. 0 287. 0
I.P (Z=0)
1. 5
Figure 2.11: Illustration of a TOF/TSC module. The unit of length is in mm.
Figure 2.12: Mass distribution from TOF measurements for particle mo- menta below 1.2 GeV/c.
to smaller crystal interfaces and lower cost, the cross-sectional area varies significantly from 44.52/542mm2 to 70.82/822mm2 for the front / rear faces.
By contrast, the length of crystals are all 30 cm (16.2X0) to avoid deteri- oration of energy resolution. After some calibrations, the energy resolution reached to 1.7% for the barrel ECL, and 1.74% and 2.85% for the forward and backward ECL respectively.
Figure 2.13: Overall configuration of ECL.
2.3 Trigger and Data Acquisition
At a luminosity of 1034 cm−2s−1, the trigger rate due to beam background is expected to be ∼ 100 Hz based on simulation studies. The trigger system is required to be robust against unexpectedly high beam background rates, and should be flexible so that background rates are kept within the tolerance of the data acquisition system (DAQ) 500 Hz, while the efficiency for events of interest is kept high. The redundant triggers to keep the efficiency high even for varying conditions is also considered.
2.3. TRIGGER AND DATA ACQUISITION 23
Table 2.3: Geometrical parameters of ECL.
Item θ coverage θ seg. φ seg. No. of crystals Forward end-cap 12.4◦− 31.4◦ 13 48−144 1152
Barrel 32.2◦− 128.7◦ 46 144 6624
Backward end-cap 130.7◦− 155.1◦ 10 64−144 960
The Belle trigger system consists of the level-1 hardware trigger and the level-3 software trigger. The latter has been designed to be implemented in the online computer farm. Figure 2.14 shows the schematic view of the level- 1 trigger system [8]. There are sub-detector trigger systems and a central trigger system called the Global Decision Logic (GDL). The sub-detector trigger systems are based on two categories: CDC and TOF are used to yield trigger signals for charged particles, while ECL trigger system provides triggers based on energy deposit and cluster counting of crystal hits. The KLM trigger gives additional information on muons and the EFC triggers are used for tagging two photon events and Bhabha events.
All the sub-trigger signals arrive at GDL within 1.85 µs after the event occurrence, and the level-1 final trigger signal is issued 2.2 µs after event crossing. The timing of the trigger is primarily determined by the TOF trigger which has the time jitter less than 10 ns. ECL trigger signals are also used as timing signals for events in which the TOF trigger is not available.
The DAQ system is designed to be tolerable to a trigger rate of up to 500 Hz. A distributed-parallel system has been devised such that it works at 500 Hz with a deadtime fraction of less than 10%. The global scheme of the system is illustrated in Figure 2.15. The DAQ system is segmented into 7 subsystems running in parallel, each handling the data from a sub-detector.
Data from each subsystem are combined into a single event record by an event builder. Its output is transferred to an online computer farm, where another level of event filtering is done after fast event reconstruction. The data are then sent to a mass storage system located at the computer center via optical fibers.
Figure 2.14: The Level-1 trigger system for the Belle detector.
Front-end elec.
Q-to-T
converter Master
VME
Master VME
Master VME Front-end
elec.
Q-to-T converter
Master VME Front-end
elec.
Q-to-T
converter Master
VME
Master VME Front-end
elec.
Q-to-T converter
Event builder
Online comp.
farm
Data storage system
Tape library Sequence control
Global trigger logic TDC
LRS1877S
TDC
LRS1877S
TDC
LRS1877S
TDC
LRS1877S
TDC
LRS1877S
TDC
LRS1877S
2km SVD
CDC
ACC
TOF
KLM ECL
Flash VME ADC Front-end
electronics
Subsystem trigger logics
Master VME TDC
LRS1877S
TRG
Belle Data Acquisition System
Front-end elec.
Q-to-T converter EFC
Hit multiplexer
Figure 2.15: Belle DAQ system overview.
Chapter 3 Analysis
3.1 Blind Analysis
This analysis employed the blind analysis which is completely based on the Monte Carlo simulation for signal selection. The blind analysis method is a good experimental way to reduce or eliminate experimental bias, for it being performed without looking at the answer, or the real data. Operating on the Monte Carlo instead of the real data prevents experimentalists from adjusting the parameters towards prior results or theoretical expectations.
Therefore, the blind analysis method, which is commonly used in particle physics, is employed in the study. The commonly used tools and methods involved in this analysis are shown in section 3.2 and 3.3 respectively.
3.2 Analysis Tools
3.2.1 BASF
Belle Analysis Framework (BASF) is the software framework to process data for Belle. BASF consists of 2 subsystems: the BASF kernel and the UI. The
”clients” connect to the ”server”, the BASF kernel, via the UI. (Fig3.1) The framework has the following features: module and path structure, dynamic linking of modules, multi-language support, unified data access method by Panther, parallel event-by-event processing, etc. An analysis code is written as a module, dynamically linked to BASF. The order of execution of modules is defined by creating a path. After then, Panther, a home-made memory manager, is utilized for data management.
doi: 10.6342/NTU201700568
AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA
AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA
modules
Module Pool parallel event processing
module1 module2
Panther path
Event Server
Output Server
Histo.
Server
module1 module2 shared objects dynamic link event process
FPDA
Input Data
Output Data
Histogram File
BASF Kernel BASF User Interface
kernel messages
dynamically linked
Graphical User Interface .etc.
Fig. 1. Schematic view of BASF architecture
Fig. 2. Dynamic linking by BASF
The execution sequence of modules is de ned by creating a path. A path is de ned as a chain of modules with a condition descriptor in which conditional branches to other paths can be de ned. Each path has a status variable which can be modi ed by each module in the path, and the conditional branches are de ned with respect to the status.
2.2 Dynamic link of modules
The modules are linked with BASF by the dynamic link. A module is made as a shared object and the BASF links the module when requested at run time using
dlopen()system call. Common utility libraries such as CERNLIB, Data Management Package, etc are also made as shared libraries and the external references from modules are resolve at run time.
The global initialization function and the user interface subsystem are also dynamically linked at run time so that the users can modify the initialization behavior and can choose their own user interface.
Figure 2 shows the structure of the dynamic linking by BASF.
Figure 3.1: Schematic view of the BASF framework
3.2.2 EvtGen and GSIM
The EvtGen package [12] is initialized by CLEO and developed by BaBar.
Written in C++ language, EvtGen implements many well-described decay models, including continuum q ¯q events, for B physics studying. In this frame- work, defining a new particle or a new decay model for event generation is allowed.
GSIM, an abbreviation for GEANT-based Detector Simulation, is used to simulate the behavior of detectors interacting with elementary particles
3.3. B MESON RECONSTRUCTION 27 passing through. GSIM was originally designed for HEP experiments, but now applies to other research areas as well.
EvtGen and GSIM are 2 important packages for the Monte Carlo simula- tion at Belle.
3.2.3 ROOT and RooFit
Although PAW (Physics Analysis Workstation) is still a popular software tool for data analysis in High Energy Physics, it could not be scaled up to the challenges offered by the Large Hadron Collider and SuperKEKB. Therefore, packages for data analysis, such as ROOT, were developed [16].
ROOT is an Object Oriented framework written mainly in C++, con- taining an efficient hierarchical OO database, a C++ interpreter, advanced statistical analysis (multi-dimensional histogram, fitting, minimization, clus- ter finding algorithms) and visualization tools.
RooFit, a part of ROOT packages, is a complete toolkit for curve fitting and data distribution modeling.
3.3 B meson reconstruction
3.3.1 Data Sample
The integrated luminosity collected by the Belle detector at KEKB e+e− collider is about 710 f b−1at the Υ(4S) resonance. The corresponding number of B ¯B events is 771.581 ± 10.566 million, and that is the number defined as 1 stream in the Monte Carlo simulation.
3.3.2 Tracking
The tracks of charged particles are reconstructed with the information from SVD, CDC and TOF. TOF provides the event timing, and CDC reconstructs the tracks, also extrapolated by SVD detector. These detectors make the momenta and the trajectories of the charged particles accurate, leading to the precision of vertex reconstruction.
In order to rule out the secondary tracks generated by hadronic interac- tion, |dr| < 0.3 cm and |dz| < 3 cm are required, where dr and dz defined as the distance from IP (interaction point) in the transverse (x − y) plane and the z−direction respectively.
3.3.3 Particle Identification
According to the information from SVD, CDC and TOF, the likelihoods of charged particles, Lp, LK, Lπ, Leand Lµ, are determined. Since only protons and Kaons are considered in this analysis, particles with Le> 0.95 and those with Lµ> 0.95, i.e., lepton-like, are excluded.
To distinguish p and K from π, we define likelihood ratio LRX/Y as LX/(LX + LY). A proton is identified with the conditions LRp/K > 0.6 and LRp/π > 0.6. Similarly, a Kaon is identified with the conditions LRK/π> 0.6 from the remaining charged particles.
3.3.4 Selection Summary
Besides the tracking constraints and particle identification, we also trimmed the events out of the range of energy difference (de) and the range of con- strained mass of B meson (Mbc). The certain 2-dimension range of ∆E-Mbc is called the ”Sample Box”.
The principle of selection is to keep most signal events (up to 3 σ ≈ 99.7%
see Fig 3.2), therefore the preliminary selections are very loose. The summary of the preliminary selection range is shown in Table 3.1.
Table 3.1: Preliminary Selection Track Impact Parameter |dr| < 0.3 cm
|dz| < 3 cm Particle Identification Le < 0.95
Lµ< 0.95 (Both required) p± LRp/K > 0.6
LRp/π > 0.6 K± LRK/π > 0.6
(LRp/K < 0.6 or LRp/π < 0.6) Sample Box
Mbc 5.20 < Mbc< 5.29 GeV de |de| < 0.24 GeV
3.3. B MESON RECONSTRUCTION 29
(a) dr for p(left), ¯p(middle) and K(right)
(b) dz for p(left), ¯p(middle) and K(right)
(c) M bc(left), de(middle) and de − M bc(right)
Figure 3.2: Distributions of dr and dz, Mbc, de for signal MC. The Monte Carlo true events are red-filled.
3.4 Background Suppression
Continuum background (e+e− → q ¯q, where q denotes a quark: u, d, s or c) and generic B background (b → c transition including mixed and charged decay) are the main background of B meson analysis. In this section, the methods for background suppression will be introduced.
We study the backgrounds with Monte Carlo data based on the new track- ing algorithm, with 5 stream of events for each.
3.4.1 NeuroBayes (NB)
NeuroBayes method is a multivariate analysis tool that can help us opti- mize the purity and efficiency of the signal. NB is not just a kind of neural network but also involving weight decay, Bayesian approaches and other al- gorithms. To begin with, some samples in both signal and background events are used to train the model. Only part of variables (better independent of each other) of events are chosen in the training. NB first transforms the input variables nonlinearly into flat distributions then into Gaussians, so the variables can be decorrelated and the insignificant ones are pruned. After that, NB finds another set of variables, where only one variable is correlated with the target variable. This new variable, called NB output (ON B) in our study and ranging from 1 (signal-like) to -1 (background-like), has all the separating information from all other variables. After training, each event of the data can be told by the model whether it is more like a signal or a background, by given an number ON B. We choose the ON B cut to maximize the signal-to-noise ratio, by F .O.M., mentioned in Section 3.4.3.
3.4. BACKGROUND SUPPRESSION 31
Figure 3.3: Work flow of NeuroBayes
3.4.2 Continuum Background
The continuum background dominates generic B background by 3 times, thus the suppression is important. Since q ¯q events are more jet-like than the spherical B ¯B ones (including the signal), shape variables are employed to NeuroBayes to help distinguishing continuum background from B ¯B events.
We use the following shape variables:
Kakuno SFW (KSFW)
KSFW is one of the modified Super Fox-Wolfram (SFW), defined as:
KSF W ≡
4
X
l=0
Rsol +
4
X
l=0
Rloo+ γ
Nt
X
l=0
|(Pt)l| (3.1)
and the terms of KSFW is like
Rlso= αc(Hc)sol + αn(Hn)sol + αm(Hm)sol
Ebeam− ∆E (3.2)
We can see that KSFW involves the term of total transverse momen- tum (P Pt) and (Hc)sol , (Hn)sol , (Hm)sol . The last 3 terms are the ra- tio of nth-order to zeroth-order Super Fox-Wolfram (SFW) moments:
charged particles from B candidate and the remaining charged ones ( (Hc)sol , 5 variables for l = 0−4); the neutral particles from B candidate and the remaining neutral ones ( (Hn)sol , 3 variables for l = 0, 2, 4); the neutral particles from B candidate and the total missing momentum ( (Hm)sol , 3 variables for l = 0, 2, 4); the charged and neutral particles excluding the particles from B candidate ( (Hm)ool to calculate Rool , 5 variables for l = 0 − 4).
Missing mass counts for NeuroBayes method because it is related to KSFW.
Thrust Angle (cos θT) and Sphericity
The thrust angel θT is defined as the angle between the thrust axis ~nT of the B candidate and that of the remaining particles. The thrust axis
~nT is defined as the direction that maximize T (~n)
T (~n) =
N
P
i=1
P~i· ~n
N
P
i=1
P~i
, (3.3)
where N is the number of daughter particles used to reconstruct B candidates and ~Pi is the 3-momentum of i-th daughter particle of B candidates.
The distribution of cos θT has peaks near ±1 for q ¯q due to jet-like event shape, whereas for B ¯B is much flatter.
Sphericity is the quantity to describe how ”spher-like” it is. Sphericity can be obtained by calculating the ratio of magnitude of total trans- verse momentum to total momentum. In q ¯q events, most are along the thrust axis, leading to small transverse momentum then low sphericity.
B Flight Direction (cos θB) and Vertex Difference ∆Z
The θBis the angle between the flight direction of B and the beam direc- tion in the Υ(4S) rest frame. The distribution of cos θBobeys 1−cos2θB (based on angular momentum conservation quantum physics) for B ¯B events and is roughly flat for non-B ¯B events.
The vertex difference ∆Z is defined by that between the B candi- date and the accompanying B. The distribution of ∆Z of q ¯q events
3.4. BACKGROUND SUPPRESSION 33 is narrower because quarks cannot be isolated due to color confinement.
To sum up, the above variables are used for training in NeuroBayes method. Fig 3.4 shows the NB outputs. We will choose the cut in the next subsection.
Figure 3.4: The NB output distribution ON B of signal (red), q ¯q(blue) and B ¯B(black)
3.4.3 Figure of Merit (F .O.M.)
To optimize the ON B cut, i.e., to maximize the statistical significance of signal and to maximize signal-to-noise ratio, the quantity called Figure of Merit (F .O.M.) is considered:
F .O.M. = Nsig
pNsig+ Nbg (3.4)
where the expected value Nsig is calculated by
Nsig = NB ¯B× B × M C,sig, (3.5) and
M C,sig = Nremainaf tercut
Ngenerated
(3.6) is the efficiency of MC.
Figure 3.5: The F .O.M. curve
Table 3.2: ON B cut optimization.
ON B cut Nsig Nqq NBB FOM
0.5 16 945 45 0.503
0.6 15 707 42 0.542
0.7 14 487 38 0.509
0.8 12 268 32 0.666
0.81 12 247 31 0.676
0.82 11 231 30 0.681
0.83 11 213 29 0.690
0.84 11 196 28 0.696
0.85 10 174 27 0.711
0.86 10 155 26 0.722
0.87 10 136 25 0.738
0.88 9 120 24 0.742
0.89 9 102 23 0.751
0.9 8 85 22 0.767
We choose ON B cut = 0.87 instead of 0.9 to keep more Nsig. Fig 3.6 shows some variables before and after ON B cut. To suppress more q ¯q background, we observe 2 variables de and MΛ1520. As in section 3.3.4, the principle of
3.4. BACKGROUND SUPPRESSION 35 selection is to keep most signal events (up to 99% ≈ 3σ). The summary of the strict selection range is showed in Table 3.3.
(a) signal MC
(b) qq MC
(c) generic B MC
Figure 3.6: Distributions of de, MΛ1520 and Mbc. The remained events are red-filled.(The remained are 99% Monte Carlo True for signal)
Table 3.3: Strict Selection ON B > 0.87
|de| < 0.12 GeV
1.45 < MΛ1520 < 1.69 GeV
3.4.4 B ¯ B Suppression
After the strict cut, the number of B ¯B is so small (≈ 10) that no further suppression is applied, also in order to keep the efficiency. In other words, we merge B ¯B events with q ¯q as ”background”.
We will show that B ¯B indeed does little effect to signal extraction and ensemble test in the following section (section 3.5).
3.5 Signal Extraction
To extract signal from background, the unbinned extended likelihood fit is used, which maximizes the likelihood function:
L = e−N N !
N
Y
i=1
[NsigPsigi (Mbc) + (NbgPbgi (Mbc)] (3.7)
where i denotes the ith event, Psig/bg denotes the signal/background prob- ability density function (PDF), and N(sig/bg) denotes the number of total (signal/background) events yield.
In this study, we use Gaussian function to describe Psig and Argus function for Pbg for Mbc variable; as for de, double Gaussian function fits Psig whereas 2nd-order Chebyshev polynomial function is used for Pbg. Fig 3.7 shows the fit result of pure signal/background events yield (5 stream of q ¯q events).
When applied to the ensemble (or real data in the future), the parameters of Mbc and de of signal, means and sigmas of Gaussians, are fixed (as in Fig 3.8). We will explain more for ensemble data in the next subsection.
One thing to mention is that it fits well to ensemble no matter we add B ¯B events or not.
3.5. SIGNAL EXTRACTION 37
(a) Mbcfit for pure signal (left) and q ¯q background (right)
(b) de fit for pure signal (left) and q ¯q background (right)
Figure 3.7: Mbc and de fit results by Roofit
Figure 3.8: Mbc and de fit results for ensemble fit by Roofit The ensemble includes signal, q ¯q and B ¯B.
3.5.1 Ensemble Test on Fitters
To verify the unbiasedness and stability of the above fitting model, the so- called ensemble test is applied. The model fits to plenty Monte Carlo data samples to see if the bias is small enough. In this ensemble test, 3000 MC samples are generated by mixture of Poisson-distributed numbers of signal and background, related to the expected numbers of signal and background calculated in Section 3.5.
The result is shown in Fig 3.9, where
P ull = Bias
F itting error = Nf it result− Nexpected
F itting error . (3.8) According to the fitting result of the ensemble test, pull and bias are well Gaussian-distributed and almost 0-mean, thus the fitter is reliable.
Figure 3.9: Chart of 3000 ensemble tests(Gaussian-distributed)
3.5.2 Upper Limit Estimation
Since the theoretical branching fraction of B → p ¯Λ1520 is too small, we can only estimate the upper limit (U.L.) of the branching fraction. The 90%
confidence level (C.L.) Bayesian upper limit is calculated by the following formula:
3.5. SIGNAL EXTRACTION 39
Z N 0
L(n)dn = 0.9 Z ∞
0
L(n)dn, (3.9)
where L(n) denotes the likelihood of the fit result, given the number of signal events n. As shown in Fig 3.10, the boundary of the 90% area is Nsig = 18.2, and the corresponding branching ratio B = 5.46 × 10−7.
With likelihood function, another method to compute the significance of signal is provided:
σ = r
−2 × ln L0
Lmax. (3.10)
The calculated result is 2.565 σ, indeed a little too small to detect B → p ¯Λ1520 decay signal.
Figure 3.10: Likelihood fit result
Chapter 4 Conclusion
In conclusion, we have performed a search for the 2-body baryonic B decay B+ → p ¯Λ1520 using some techniques including Monte Carlo method, Neu- roBayes, likelihood fit, and etc. Although we cannot find significant signal due to lack of data in Belle, nor compare the branching fraction measured by LHCb, we still set an expected upper limit on B(B+ → p ¯Λ1520) < 5.46 × 10−7 at the 90% C.L.
Since the KEKB accelerator and the Belle detector are now upgrading to SuperKEKB and Belle II, increasing the luminosity by a factor of 40, we are glad to see the larger integrated luminosity for a search of the decay mode, and hopefully some new physics can be found.
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