Ph. D. Thesis

Analysis of the First MEG Physics Data
to Search for the Decay µ+ → e+γ

byYusuke Uchiyama

Department of Physics, School of Science,
the Univeristy of Tokyo

December 2009

Abstract
A search for the lepton-ﬂavor-violating muon decay µ+ → e+ γ was performed on thebasis of the data from the initial three months of operation of the MEG experiment. We
observed 5.2×1011 muon decays and explored the decay with a 90%-conﬁdence sensitivityofS2008 = 1.3 × 10−11 .
This sensitivity is comparable with the current best experimental limit of 1.2 × 10−11 .
This search was done in a blind analysis. The number of the µ+ → e+ γ decay events inthe data sample was extracted by a maximum likelihood ﬁt. The best-estimated value isat Nsig = 4.3 and the 90%-conﬁdence interval is evaluated to be 0 ≤ Nsig ≤ 14.7 including
systematic uncertainties. We set an upper limit on the branching ratio,
B(µ+ → e+ γ) < 2.8 × 10−11(90 % C.L.).

The large value compared to the sensitivity is considered due to a statistical ﬂuctuation.
We do not give a new record of the experimental limit but an independent result from ameasurement with a comparable sensitivity.

iContents
AbstractiContents
iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction

12 µ → eγ Decay
2.1 Muon Decay in the Standard Model . . . . . . . . . . . . . . . . . . . . . .
2.2 Physics Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Supersymmetry and Lepton-Flavor Violation . . . . . . . . . . . . .
2.2.3 Right-Handed Neutrinos as a LFV source . . . . . . . . . . . . . . .
2.2.4 LFV in Grand-Uniﬁed Theories . . . . . . . . . . . . . . . . . . . .
2.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Experimental Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 History of µ+ → e+ γ Search . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Requirements of Muons . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Event Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Requirements of µ+ → e+ γ Search . . . . . . . . . . . . . . . . . . .

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3 Experimental Apparatus
3.1 µ+ Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 PSI Proton Ring Cyclotron . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Surface Muon from πE5 Channel . . . . . . . . . . . . . . . . . . .
3.1.3 Beam Transport System . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Stopping Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.5 Mode and Proﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The Positron Spectrometer . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 The Gamma-ray Detector . . . . . . . . . . . . . . . . . . . . . . .
3.3 Electronics and Data Acquisition . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Electronics Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3.4 Domino Ring Sampler . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Data Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Calibration Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 π − Beam and Hydrogen Target . . . . . . . . . . . . . . . . . . . .
3.4.2 NaI Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Cockcroft-Walton Proton Accelerator . . . . . . . . . . . . . . . . .
3.5 Simulation and Analysis Software . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Event Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Detector Geometry and Response . . . . . . . . . . . . . . . . . . .
3.5.3 Event Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.4 Electronics Simulation . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.5 Analysis Framework . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Run
4.1 Run 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Physics Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Dedicated Radiative Decay Run . . . . . . . . . . . . . . . . . . . .
4.1.3 π 0 Calibration Run . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Normal Calibration Runs . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Run Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Discharge Problem on Drift Chamber . . . . . . . . . . . . . . . . .
4.2.2 Helium Concentration inside the Magnet . . . . . . . . . . . . . . .
4.2.3 Problems on Timing-Counter APD . . . . . . . . . . . . . . . . . .
4.2.4 Purity and Light Yield of LXe . . . . . . . . . . . . . . . . . . . . .
4.2.5 Bad Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Event Reconstruction
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Waveform Analysis of Drift Chamber . . . . . . . . . . . . . . . . .
5.2.2 Waveform Analysis of Timing Counter . . . . . . . . . . . . . . . .
5.2.3 Waveform Analysis of Gamma-ray Detector . . . . . . . . . . . . .
5.3 Positron Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Hit Reconstruction and Track Finding . . . . . . . . . . . . . . . .
5.3.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Gamma-ray Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.4 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.5 Pileup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Combined Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Relative Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Relative Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTSv

6 Calibration
6.1 DRS Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Voltage Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Time Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Calibration of Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 z-coordinate Calibration . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Time Oﬀset Calibration . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Time-to-Distance Calibration . . . . . . . . . . . . . . . . . . . . .
6.3 Calibration of Timing Counter . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 z-coordinate Calibration . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Inter-Bar Time Oﬀset Calibration . . . . . . . . . . . . . . . . . . .
6.4 Calibration of Gamma-ray Detector . . . . . . . . . . . . . . . . . . . . . .
6.4.1 PMT Response Calibration . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Non-uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Energy Scale and Light Yield . . . . . . . . . . . . . . . . . . . . .
6.4.4 Timing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Calibration of Relative Time . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Performance99
7.1 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.1.1 Deﬁnition of Fiducial Volume and Acceptance . . . . . . . . . . . . 99
7.1.2 Positron Analysis Cut . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.1.3 Gamma-ray Analysis Cut . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Positron Energy, Ee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3 Gamma-ray Energy, Eγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.4 Relative Angle, θeγ and φeγ . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4.1 Positron Emission Angle . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4.2 Muon Decay Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4.3 Gamma-ray Position . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4.4 Combined Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.5 Relative Time, teγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.5.1 Positron Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.5.2 Gamma-ray Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5.3 Gamma-Positron Relative Time . . . . . . . . . . . . . . . . . . . . 120
7.6 Positron Detection Eﬃciency, e . . . . . . . . . . . . . . . . . . . . . . . . 121
7.6.1 Conditional Timing-Counter Eﬃciency . . . . . . . . . . . . . . . . 121
7.6.2 Drift-Chamber Tracking Eﬃciency . . . . . . . . . . . . . . . . . . 121
7.7 Gamma-ray Detection Eﬃciency, γ . . . . . . . . . . . . . . . . . . . . . . 122
7.8 Summary of Performance in 2008 . . . . . . . . . . . . . . . . . . . . . . . 124
8 µ+ → e+ γ Search Analysis
125
8.1 Overview of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.1.1 Pre-selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.1.2 Blinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.1.3 Deﬁnition of Analysis Window . . . . . . . . . . . . . . . . . . . . . 126
8.1.4 Deﬁnition of Signal Boxes . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Background Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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8.3

8.4

8.5
8.6

8.7

8.2.1 Rate of Gamma-ray Background . . . . . . . . . . . . . . . . . . . . 128
8.2.2 Estimation of Background Level . . . . . . . . . . . . . . . . . . . . 128
8.2.3 Distribution of Background Events from Sideband Data . . . . . . . 130
Maximum Likelihood Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.3.2 Probability Density Functions . . . . . . . . . . . . . . . . . . . . . 135
Conﬁdence Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4.1 Toy MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4.2 Construction of Conﬁdence Region . . . . . . . . . . . . . . . . . . 139
Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.6.1 Analysis on Sideband Data . . . . . . . . . . . . . . . . . . . . . . . 144
8.6.2 The Maximum Likelihood Fit on the Data . . . . . . . . . . . . . . 144
8.6.3 Conﬁdence Region from the Data . . . . . . . . . . . . . . . . . . . 144
8.6.4 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 146
8.6.5 Upper Limit on the Branching Ratio µ+ → e+ γ . . . . . . . . . . . 148
8.6.6 Sensitivity of Run 2008 . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.6.7 Cut Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.6.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9 Conclusion

159

A Radiative Decay Analysis
161
A.1 Eγ -Sideband Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2 Normalization by Radiative Decay . . . . . . . . . . . . . . . . . . . . . . . 162
A.3 Calculation of Expected Number of Radiative-Decay Events . . . . . . . . 164
A.4 Detail Description of Radiative-Decay Diﬀerential Branching Ratio . . . . 164
B Normalization with Background167
B.1 Overall Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.2 Normalization Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Acknowledgment

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Bibliography

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List of Figures
Example of Feynman diagram describing the µ → eγ decay in the SM witha neutrino-mass extension. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 cLFV generation mechanism in mSUGRA scenario. . . . . . . . . . . . . .
2.3 Dependence of the branching ratio of µ → eγ on right-handed neutrinomass in the MSSM with right-handed neutrinos. . . . . . . . . . . . . . . .
2.4 Feynman diagrams contributing to µ → eγ in SUSY SU (5) model. . . . . .
2.5 Branching ratio of µ → eγ as a function of right-handed selectron mass in
SUSY SU (5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Branching ratio of µ → eγ as a function of M1/2 in SUSY SO(10). . . . . .
2.7 Historical progress of µ+ → e+ γ search. . . . . . . . . . . . . . . . . . . . .
2.8 Schematics of µ+ → e+ γ event signature, and two types of backgrounds. . .
2.9 Positron energy spectrum of unpolarized µ+ → e+ νe ν̄µ decay (Michel spectrum). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Photon energy spectrum of unpolarized µ+ → e+ ν ν̄γ decay. . . . . . . . . .
2.1

3.1 Overview of the MEG experiment. . . . . . . . . . . . . . . . . . . . . . .
3.2 590 MeV proton ring cyclotron at PSI. . . . . . . . . . . . . . . . . . . . .
3.3 (a) πE5 beamline. (b) Muon and pion ﬂux at πE5 channel. . . . . . . . . .
3.4 Schematic and picture of beam transport system. . . . . . . . . . . . . . .
3.5 Pictures of MEG stopping target. . . . . . . . . . . . . . . . . . . . . . . .
3.6 Schematic view of MEG detector. . . . . . . . . . . . . . . . . . . . . . . .
3.7 Conceptual illustrations of the COBRA spectrometer. . . . . . . . . . . . .
3.8 Rate of Michel positrons per cm2 per second as a function of radius. . . . .
3.9 Picture of the COBRA magnet. . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Schematic of the COBRA magnet. . . . . . . . . . . . . . . . . . . . . . .
3.11 Proﬁle of the magnetic ﬁeld along the axis of the magnet. . . . . . . . . . .
3.12 Response of PMT used in the gamma-ray detector in magnetic ﬁeld. . . . .
3.13 Contour plot of fringe ﬁeld around the gamma-ray detector. . . . . . . . .
3.14 Picture of drift chamber system inside the COBRA magnet. . . . . . . . .
3.15 A module of drift chamber. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16 Cell conﬁguration of a drift chamber module. . . . . . . . . . . . . . . . . .
3.17 Schematic of vernier pattern on the cathode pads. . . . . . . . . . . . . . .
3.18 Field map and drift lines of drift chamber. . . . . . . . . . . . . . . . . . .
3.19 Picture and design of timing φ-counter. . . . . . . . . . . . . . . . . . . . .
3.20 Picture of timing z-counter and APDs. . . . . . . . . . . . . . . . . . . . .
3.21 Conceptual ﬁgure of LXe gamma-ray detector. . . . . . . . . . . . . . . . .
3.22 Photon cross section of xenon atoms. . . . . . . . . . . . . . . . . . . . . .
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LIST OF FIGURES
3.23 Xenon phase diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.24 Schematic view of LXe gamma-ray detector. . . . . . . . . . . . . . . . . .
3.25 Development view of LXe gamma-ray detector. . . . . . . . . . . . . . . .
3.26 Picture of 200 W pulse-tube refrigerator and its cooling power. . . . . . . .
3.27 Picture of LXe gamma-ray detector. . . . . . . . . . . . . . . . . . . . . . .
3.28 Pictures of xenon storage systems. . . . . . . . . . . . . . . . . . . . . . . .
3.29 Picture of PMT for the LXe gamma-ray detector (R9869). . . . . . . . . .
3.30 Micro-picture of a 241 Am source mounted on a 100 µm-diameter wire. . . .
3.31 Picture of LEDs and alpha sources. . . . . . . . . . . . . . . . . . . . . . .
3.32 Xenon scintillation absorption. . . . . . . . . . . . . . . . . . . . . . . . . .
3.33 Schematic of data ﬂow and electronics. . . . . . . . . . . . . . . . . . . . .
3.34 Schematic of DAQ system. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.35 Schematic of DRS principle. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.36 Picture of DRS chips and boards. . . . . . . . . . . . . . . . . . . . . . . .
3.37 π 0 decay kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.38 Picture of LH2 target cell. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.39 Picture of the NaI detector. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.40 Movable stage of the NaI detector. . . . . . . . . . . . . . . . . . . . . . .
3.41 1 MeV Cockcroft-Walton proton accelerator. . . . . . . . . . . . . . . . . .
3.42 Proton beamline from the CW accelerator to the detector center. . . . . .
3.43 Li2 B4 O7 target for the proton-nucleus reactions. . . . . . . . . . . . . . . .
3.44 Structure of MEG software. . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.1
4.2
4.3
4.4
4.5

Integrated amount of physics data taken in 2008. . . . . . . . . . . . . . .
Conﬁguration of patch and deﬁnition of patch number. . . . . . . . . . . .
Conﬁguration of lead bricks installation. . . . . . . . . . . . . . . . . . . .
Lead bricks for a study of position resolution. . . . . . . . . . . . . . . . .
History of HVs applied to the drift chambers. . . . . . . . . . . . . . . . .

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63

5.1 Flow chart of reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Typical waveforms of drift chamber. . . . . . . . . . . . . . . . . . . . . . .
5.3 Typical waveforms from a TIC PMT. . . . . . . . . . . . . . . . . . . . . .
5.4 Typical waveform of LXe interacting with a gamma ray. . . . . . . . . . .
5.5 Time extraction by the digital-constant-fraction method. . . . . . . . . . .
5.6 LXe waveform shaped with a high-pass ﬁlter. . . . . . . . . . . . . . . . . .
5.7 Reconstructed energy of pedestal events in normal data taking. . . . . . . .
5.8 Saturated signal and charge estimation with ToT. . . . . . . . . . . . . . .
5.9 z reconstruction of the drift-chamber hit. . . . . . . . . . . . . . . . . . . .
5.10 Time-to-distance functions for various incident angles. . . . . . . . . . . . .
5.11 An example of track ﬁtting using Kalman ﬁlter. . . . . . . . . . . . . . . .
5.12 Schematic view of connection between track and timing-counter hits. . . .
5.13 Examples of electromagnetic shower in LXe in response to 52.8 MeV gamma
simulated with MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 Schematic explanation of position ﬁtting. . . . . . . . . . . . . . . . . . . .
5.15 PMT selection used in the position ﬁtting. . . . . . . . . . . . . . . . . . .
5.16 Correlation between Ωin,max and sum of inner PMT Npho for wγ < 3 cm. .
5.17 Linearity plot of Eγ v.s. Nsum . . . . . . . . . . . . . . . . . . . . . . . . . .

65
67
67
68
68
68
69
70
71
72
73
75
77
79
79
80
80

LIST OF FIGURES
5.18 Illustration of time reconstruction. . . . . . . . . . . . . . . . . . . . . . . .
5.19 Distribution of normalized χ2 of the time ﬁtting. . . . . . . . . . . . . . . .
5.20 Light distribution before and after eliminating pileup contribution. . . . . .
5.21 Energy spectrum in the sideband data. . . . . . . . . . . . . . . . . . . . .

ix
81
82
83
83

6.1 Typical response curve of a DRS cell. . . . . . . . . . . . . . . . . . . . . . 85
6.2 Typical sampling intervals of a DRS channel. . . . . . . . . . . . . . . . . . 85
6.3 Examples of the global clock signal digitized by DRS. . . . . . . . . . . . . 87
6.4 An example of corrected hit time distribution. . . . . . . . . . . . . . . . . 88
6.5 An example of gain calculation with LED data. . . . . . . . . . . . . . . . 90
6.6 Distributions of gain calculation. . . . . . . . . . . . . . . . . . . . . . . . 90
6.7 Examples of evolution of PMT gains. . . . . . . . . . . . . . . . . . . . . . 91
6.8 Time dependence of the gain shift when counting rate is changed. . . . . . 92
6.9 Plot of total number of photoelectrons of LED events vs. date. . . . . . . . 92
6.10 Relative energy response map in (u, v)-plane. . . . . . . . . . . . . . . . . . 94
6.11 Evolution of the total amountof observed light. . . . . . . . . . . . . . . . 95√
6.12 tdelay as a function of 1/ Npe . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.13 tdelay as a function of cosine of PMT incident angle η after tprop is subtracted. 96
6.14 Dependence of reconstructed time on the total number of photoelectrons
with diﬀerent value of vef f used in the reconstruction. . . . . . . . . . . . . 97
6.15 Distributions of thit reconstructed by individual PMT after calibrated tdelay
is subtracted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.16 Change of LXe pulse shape. . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.17 Drift of teγ time reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.1 Deﬁnition of the ﬁducial volume. . . . . . . . . . . . . . . . . . . . . . . . 100
7.2 Distribution of track span v.s. number of multi-hit chambers. . . . . . . . 101
7.3 Track quality selections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.4 Two-dimensional distribution of muon decay vertex. . . . . . . . . . . . . . 102
7.5 Distributions of DCH-TIC matching quantities in r- and z- directions. . . . 103
7.6 Gamma energy spectrum with additional pileup cut. . . . . . . . . . . . . 104
7.7 Cosmic-ray spectrum with several cuts. . . . . . . . . . . . . . . . . . . . . 105
7.8 Fitting of Michel spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.9 Reconstructed energy distribution of 55 MeV gamma events from π 0 decays.107
7.10 Fitting function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.11 Distribution of the gamma-ray detector energy resolution. . . . . . . . . . 109
7.12 Gamma background spectrum in the sideband data. . . . . . . . . . . . . . 110
7.13 Diﬀerence of positron directions reconstructed with each turn. . . . . . . . 111
7.14 Target hole ﬁt to evaluate vertex resolutions. . . . . . . . . . . . . . . . . . 112
7.15 Event distribution of π 0 run with a Pb brick. . . . . . . . . . . . . . . . . . 113
7.16 Position response of the gamma-ray detector obtained from the MC simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.17 Position resolution as a function of the distance from the nearest inner
PMT center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.18 Time resolution of timing-counter bars. . . . . . . . . . . . . . . . . . . . . 115
7.19 Dependence of time resolution on |∆zDCH−T IC |. . . . . . . . . . . . . . . . 116
7.20 PMT grouping scheme for the intrinsic time resolution study. . . . . . . . . 117

x

LIST OF FIGURES
7.21 Distribution of half of time diﬀerence of two group reconstructed time. . . 117
7.22 Intrinsic time resolution at diﬀerent position. . . . . . . . . . . . . . . . . . 117
7.23 Intrinsic time resolution at diﬀerent position. . . . . . . . . . . . . . . . . . 118
7.24 Distribution of time diﬀerence between the gamma-ray detector and thereference counter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.25 Energy dependence of the gamma-ray detector timing resolution. . . . . . . 119
7.26 Distribution of tγ − te in physics run for 40 < Eγ < 45 MeV. . . . . . . . . 120
7.27 Conditional timing-counter eﬃciency curve. . . . . . . . . . . . . . . . . . 122
7.28 Evaluation of gamma-ray detection eﬃciency with the MC simulation. . . . 123
7.29 Evaluation of gamma-ray detection eﬃciency with π 0 data. . . . . . . . . . 124
8.1 Distribution of teγ v.s. Eγ after the pre-selection and blinding process. . . . 127
8.2 Absolute rate of gamma-ray background. . . . . . . . . . . . . . . . . . . . 129
8.3 Event distribution in the teγ -sideband. . . . . . . . . . . . . . . . . . . . . 130
8.4 teγ distribution in the sideband. . . . . . . . . . . . . . . . . . . . . . . . . 131
8.5 Scatter plot of teγ v.s. Eγ with analysis window cuts. . . . . . . . . . . . . 131
8.6 Positron background spectrum in the sideband data. . . . . . . . . . . . . 132
8.7 Gamma background spectrum in the sideband data. . . . . . . . . . . . . . 132
8.8 Angular distributions of background in the sideband. . . . . . . . . . . . . 132
8.9 PDFs for µ+ → e+ γ signal event. . . . . . . . . . . . . . . . . . . . . . . . 136
8.10 PDFs for µ+ → e+ νe ν µ γ RD event. . . . . . . . . . . . . . . . . . . . . . . 137
8.11 PDFs for accidental background event. . . . . . . . . . . . . . . . . . . . . 138
8.12 Illustration of conﬁdence region construction. . . . . . . . . . . . . . . . . . 140
8.13 Result of the maximum likelihood ﬁt. Projected event distribution with
ﬁtted PDFs and the likelihood function. . . . . . . . . . . . . . . . . . . . 145
best
8.14 Distribution of Nsig conﬁdence level on the NRD-line. . . . . . . . . . . . . 146
best
8.15 Distribution of NRD conﬁdence level on the Nsig -line. . . . . . . . . . . . . 146
8.16 Scan of the deviations of the Nsig best-ﬁt value. . . . . . . . . . . . . . . . 147
8.17 Distribution of upper limits of toy experiments. . . . . . . . . . . . . . . . 148
8.18 Event distributions in signal box B. . . . . . . . . . . . . . . . . . . . . . . 150
8.19 Ee vs Eγ two-dimensional plots of the signal boxes. . . . . . . . . . . . . . 151
8.20 Distribution of the ratio of event-type likelihoods, S/B . . . . . . . . . . . 153
8.21 Event display of the ﬁrst-rank event. It found to be a double-pileup event. 153
8.22 Change of the likelihood function by the change of the pileup search threshold.154
8.23 Two-dimensional PDFs with expected performance. . . . . . . . . . . . . . 158
8.24 Expected 90%-conﬁdence sensitivity as a function of muon stopping rate. . 158
8.25 90%-conﬁdence belt in Feldman-Cousins method with the Poisson statistics
in presence of backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.1 Eﬃciency curves applied to the calculation of RD branching ratio. . . . . . 162
A.2 Result of a maximum likelihood ﬁt on the Eγ -sideband data. . . . . . . . . 163

List of Tables
2.1
2.2
2.3

Decay modes and branching fractions of muon. . . . . . . . . . . . . . . . . 4
Particles content in the MSSM. . . . . . . . . . . . . . . . . . . . . . . . . 8
Historical progress of search for µ+ → e+ γ since the era of meson factories
with 90 % C.L. upper limits. . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1
3.2
3.3
3.4
3.5
3.6

Properties of LXe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of various scintillators. . . . . . . . . . . . . . . . . . . . . . .
Properties of the PMT (R9869). These are typical values. . . . . . . . . . .
List of trigger settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of MEG CW accelerator. . . . . . . . . . . . . . . . . . . . . . .
List of event type implemented in the MC simulation. . . . . . . . . . . . .

7.1
7.2
7.3
7.4

Results of the Michel ﬁt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Alternative parameter set estimated with analysis window cuts. . . . . . . 106
Ineﬃciency of gamma-ray measurement. . . . . . . . . . . . . . . . . . . . 123
Summary of detector performance. Resolutions are given in sigma. . . . . . 124

8.1
8.2
8.3
8.4
8.5
8.6
8.7

List of factors in normalization calculation. . . . . . . . . . . . . . . . . . . 143
Summary of normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Results of likelihood analysis in the sidebands. . . . . . . . . . . . . . . . . 144
Summary of systematic uncertainties. . . . . . . . . . . . . . . . . . . . . . 147
Summary of cut analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
List of events with large likelihood ratio S/B. . . . . . . . . . . . . . . . . 152
Summary of the upper limit and sensitivity. . . . . . . . . . . . . . . . . . 155

36
38
43
50
54
57

A.1 Eﬀective branching ratio of radiative decay in several boxes. . . . . . . . . 164

xiChapter 1
Introduction
Since when the muon was discovered, we have tried to understand why more than one
generations of fundamental fermions exist and how the generations are related. The
standard model (SM) of the elementary particle physics is one of the greatest successes ofmodern physics. Almost all kinds of phenomena can be explained by the SM. However,the SM is based on the three generations of leptons and quarks, but does not give thereason for the existence of the generations nor prediction of their masses and mixing
patterns.
Recently, oscillations between types of neutrinos have been established by various
experiments. The mixing patterns have been revealed progressively. Meanwhile, in thequark sector, the ﬂavor-changing neutral currents (FCNC) have been studied for many
years. The quark mixings are described by the quark-mixing matrix (known as CKM
matrix). Understanding the mixings in charged-lepton sector will obviously give us a new
insight for the generation mechanism of the fundamental particles and their interactions.
So far, no evidence of charged-lepton-ﬂavor violation (cLFV) is found. Within the SM, theconservation of lepton ﬂavor is considered accidental; there is no explicit gauge symmetry
for which lepton ﬂavor is the conserved quantity, but, in the absence of neutrino mass,
there is no mechanism for breaking this symmetry. Even if we introduce a tiny but
ﬁnite neutrino masses in the SM, the rate of cLFV is too small to reach with possible
experimental technique because of the GIM mechanism.
If there are either a new force mediated by a new gauge boson with non-diagonal
lepton couplings or a new class of heavy particles with lepton-ﬂavor mixings in this new
sector, the cLFV processes would be induced. The possibility of cLFV exists in essentiallyall extensions to the SM. Because cLFV processes are free from the SM background,searches for them are among the most sensitive ways to look for physics beyond the SM.
In particular, some models incorporating supersymmetry predict large branching ratiosthat can be reached with current or near-future experiments.
The cLFV processes have been investigated experimentally in various channels. The
muon system gives the most stringent constraint to the new physics, while the kaon andtau systems also give important results. The µ+ → e+ γ decay is the simplest and themost famous process out of them. The current best limit on the branching ratio of the
decay is given as B(µ+ → e+ γ) < 1.2 × 10−11 (90 % C.L.) by the MEGA experiment[1] in 1999. This sensitivity already entered into the phase space predicted by several
interesting models, and started to set constraint to them.
In 1999, a proposal for a new µ+ → e+ γ decay search [2] by a group of Japanese12

CHAPTER 1. Introduction

physicists was approved by the research committee of the Paul Scherrer Institut (PSI).
Since then, an experimental collaboration, which is now called the MEG (Muon to Electron and Gamma) collaboration, have been formed by approximately 60 physicists fromJapan, Switzerland, Italy, Russia, and the United States. MEG aims at searching forthe decay with a sensitivity two orders of magnitude below the current limit. The mainfeatures of the MEG experiment are the world’s most intense DC muon beam from the590 MeV ring cyclotron at PSI, a novel positron spectrometer with a specially graded magnetic ﬁeld, and an innovative 900 liter liquid xenon gamma-ray detector. MEG started
physics data taking in autumn 2008 after intense R&D and construction periods for 10
years. In 2008, it ran for three months and accumulated data of muon decay. We measured ∼ 1 × 1014 muons stopping on the target, the largest data set so far.
The theme of this thesis is a search for the µ+ → e+ γ decay using the ﬁrst three months
data of MEG taken in 2008. In Chapter 2, we summarize theoretical, phenomenological,
and experimental features of the µ+ → e+ γ search. In Chapter 3, we describe the experimental apparatus of MEG in detail. The run 2008 that is the full data set of this analysis
is summarized in Chapter 4. The event reconstruction and calibration that give bases of
the analysis are described in Chapter 5 and Chapter 6, respectively. Then the performanceof the detector is evaluated in Chapter 7. The analysis to search for the µ+ → e+ γ decay
is described in Chapter 8 and the result is discussed. Finally, we conclude this thesis in
Chapter 9.

Chapter 2µ → eγ Decay
We discuss theoretical and experimental features of the µ → eγ decay in this chapter.
First, we shortly summarize ‘standard’ muon decays. Theoretical frameworks and phenomenology of µ → eγ decay are brieﬂy reviewed in the second section. In this section,
we see that µ → eγ decay search can be a clear probe to new physics beyond the SM.
In the next section, experimental features as well as the history of µ → eγ search aresummarized.

2.1

Muon Decay in the Standard Model

The muon is the second-generation charged lepton which is a replication of the electronwith a heavier mass. It interacts through the electromagnetic and weak interactions. It
also couples to the Higgs boson. The Lagrangian for those interactions are given as
L = eµ̄γ µ µAµg− √ (ν̄µL γ µ µL WL+ + µ̄L γ µ νµL Wµ− )
2
[
(
)
]
√
1
µ
2
µ
2
2
02
− g + g µ̄L γ − + sin θW µL + µ̄R γ sin θW µR Zµ0
2
mµ
µ̄µH.
−
v

(2.1)

Muons decay through the charged-weak current interaction mediated by Wµ± guage bosons,the second line of the Eq.2.1 (V − A interaction). The decay modes and their branchingratios are summarized in Table 2.1. The dominant process, µ → eνµ ν e , is called “Michel
decay” [3].

2.2

Physics Motivations

2.2.1

Beyond the Standard Model

The SM is based on the fundamental fermions, quarks and leptons, and the gauge theory.
It describes the interactions of quarks and leptons that are the constituents of all matterthat we know about. The strong interactions are described by quantum chromodynamics(QCD) while the electromagnetic and the weak interactions have been uniﬁed into a single
3

4

CHAPTER 2. µ → eγ Decay
Table 2.1: Decay modes and branching fractions of muon (Listed in PDG table [4]).
Decay modeµ− → e− νµ ν eµ− → e− νµ ν e γµ− → e− νµ ν e e+ e−µ− → e− νe ν µµ− → e− γµ− → e− e+ e−µ− → e− γγBranching ratio∼ 100 %
(1.4 ± 0.4) %(3.4 ± 0.4) × 10−5
< 1.2 %
< 1.2 × 10−11< 1.0 × 10−12< 7.2 × 10−11

Reference
[5]
[6]
[7]
[1]
[8]
[9]

electroweak framework. They are all gauge interactions, which are based on the principleof gauge symmetry. Quarks and leptons consist of three generations, each of which has
identical gauge quantum numbers.
This theory has proven to be extremely successful in describing a tremendous variety
of experimental data over an energy range from a fraction of an electron volt to about
100 GeV, a range of over 12 orders of magnitude. Nevertheless, it is thought as aneﬀective theory in low-energy approximation, and more fundamental theory is thought toexist. The reason is that the SM contains some fundamental and theoretical problems.
In addition, recently a few experiments and observations have indicated results whichcannot be described in the SM. Clearly, it is desirable to have a more uniﬁed theory thatcan combine all these three interactions as components of a single force with one coupling
constant.
The SM does not account for the existence of three generations of quarks and leptons.
This problem is known as a ﬂavor puzzle of the SM. We lack any understanding of particle
masses and mixing patterns, which results in the large number of underlying parametersin the SM.
Another problem of the SM is the hierarchy problem. It is related to the huge diﬀerenceof energy scales between the week scale of O(100 GeV) and the reduced Planck scale ofO(1018 GeV), where quantum gravitational eﬀects become important. To understand thisproblem, let us think about the Higgs mass. The Higgs mass parameter receives enormous
quantum corrections from the virtual eﬀects of every particle which couples to the Higgs
ﬁeld,
m2HSM (phys) ≈ m2HSM +

c
Λ2 ,
2
16π

(2.2)

where Λ is a cut-oﬀ energy interpreted as the scale at which the SM ceases to be valid. In
the SM, incredible ﬁne tuning is required: if we assume the validity of the SM as a lowenergy eﬀective theory below the GUT scale, and take Λ = MGU T ∼ 1016 GeV, then the
Lagrangian mass parameter m2HSM will have to be ﬁne-tuned to 1 part in 1026 to provide
the needed cancellation that will maintain a physical Higgs mass below its unitarity limit.
To avoid excessive ﬁne tuning between the two terms on the right-hand side, we would
have to deduce that Λ ≤ O(TeV). Thus it indicates that new degrees of freedom that are
not included in the SM are there at TeV-scale. Those degrees of freedom could be a newforce mediated by a new gauge boson or a new class of heavy particles.

2.2. Physics Motivations

5

Incompleteness of the SM is also given by experimental arguments such as neutrino oscillations and existence of cold dark matter in the Universe for which there is no candidatein the SM. In addition, the anomalous magnetic moment (g − 2) of the muon and some
low-energy CP-violating observables measured at the B-factories could indicate the SM
incompleteness, whereas the hadronic uncertainties as well as the limited experimental
resolutions prevent any conclusive evidence of new physics in the quark sector.
Neutrino Mass and Mixing
In the (minimum) SM, neutrinos are strictly massless. No neutrino Dirac mass term canbe introduced because of the absence of right-handed neutrinos, and no Majorana massterms can be generated because of exact B − L conservation. Owing to the masslessneutrinos, there is no leptonic mixing in the SM, which in turn leads to separate leptonnumber (lepton-ﬂavor) conservation. Therefore, the observation of neutrino oscillations isevidence of physics beyond the SM. In another words, the lepton-ﬂavor violation (LFV)
is forbidden by an ‘accidental’ symmetry in the SM, and it can occur in all extensions of
the SM.
If neutrinos are not massless, their mass matrix, just as in the case for quarks, will be
non-diagonal and complex. The mass eigenstates are diﬀerent from ﬂavor ones
να =∑Uαi νi ,(2.3)

iwhere να = νe , νµ , ντ are weak-ﬂavor eigenstates and νi = ν1 , ν2 , ν3 are mass eigenstateswith mass eigen values m1 , m2 , m3 . U is a unitary matrix known as Pontecorvo-MakiNakagawa-Sakata (PMNS) matrix. A mass eigenstate of i after a time interval t is givenby|νi (t)) >= e−iEi t |νi (0)i.

(2.4)

Thus a neutrino of the ﬂavor α after a time interval of t is given by|να (t) >=∑Uαi e−iEi t |νi (0)i.

(2.5)

i

The probability of ﬁnding ﬂavor νβ in να beam at a distance x from the source is givenbyPνα →νβ =∑i|Uαi | |Uβi | +
2

2

∑

∗
∗
Uαi UβiUαjUβj cos

i6=j

(

)

2πx
,
Lij

(2.6)

where Lij = 2π/(Ei −Ej ) ' 4πp/|m2i −m2j | called the oscillation length. To have neutrino
oscillation, we must have nonzero neutrino masses and mixing angles.
It is possible to accommodate the Dirac mass terms for the neutrinos if SU (2) singlet
right-handed neutrinos are added to the minimal SM
mD ν̄L νR .

(2.7)

If Yukawa coupling for neutrinos is very small, the small masses of neutrinos can be
explained. For example, the Yukawa coupling constant should be O(10−11 ) for a neutrinomass of 1 eV/c2 .

6

CHAPTER 2. µ → eγ Decay

A more natural explanation for the small neutrino masses is given by the see-saw
mechanism [10, 11]. Since neutrinos are electrically neutral, it is possible that their antiparticles are themselves. Then the Majorana mass term can be included in the Lagrangian,mL νL νLc + MR νR νRc .

(2.8)

The complete mass term for the neutrino is written as
mD ν̄L νR + mL νL νLc + MR νR νRc + (h.c.)
(
)(
)
mL mD
νL
c
= (ν̄L ν̄R )
.
mD MR
νR

(2.9)

In the see-saw model, the left-handed Majorana mass is set to zero (mL = 0) and the Diracmass is required to be much smaller than the right-handed Majorana mass (mD  MR ).
Then the neutrino physical masses are
mN ' MR ,

mν '

m2D
.
MR

(2.10)

One neutrino remains super heavy while the other gets a tiny mass. For example, if MR
is 1015 GeV and the Dirac mass is of the order of 100 GeV, then neutrino mass becomesnaturally O(10−2 ) eV. The Majorana mass terms violate lepton number by two units.
The neutrino mass can be written with Yukawa couplings Yνmν = −Yν MR−1 YνT hHu i2 ,(2.11)

where hHu i is the vacuum-expectation value (VEV) acquired by the up-sector Higgs ﬁeld.
The matrix Yν can be written in the general form√√Yν = UP M N S mν R MR /hHu i,(2.12)

where R is an arbitrary complex orthogonal matrix. A complete knowledge of Yν cannot
be achieved by only low-energy observables from the neutrino sector.
µ → eγ Decay through the Neutrino Oscillation
The neutrino oscillations show that lepton ﬂavor is not conserved in neutrino sector. It
also induces the transition between charged leptons at the one-loop level through diagrams
such as shown in Figure 2.1. However, the LFV processes in charged lepton (cLFV) areseverely suppressed because of the GIM mechanism. The µ → eγ rate in the minimum
extension of SM is given as
B(µ → eγ) =≈3α ∑ ∗∆m2 2
Uµi Uei 2 1i
32π i
mW

(

)

(

α
∆m221
2
sin 2θ12
2
2π
MW

)2

< 10−54 ,(2.13)

where sin2 2θ12 = 0.86 and ∆m221 ∼ 8 × 10−5 eV2 are used [4]. It is far from experimental
reach.

2.2. Physics Motivations

7

Figure 2.1: Example of Feynman diagram describing the µ → eγ decay in the SM witha neutrino-mass extension.

In conclusion, the cLFV processes are free from the SM background. Searches for
cLFV processes are among the most sensitive ways to investigate physics beyond theSM. In the following sub-sections, we review some models beyond the SM focused on
the supersymmetric (SUSY) scenarios. In those models, the branching ratio of µ →eγ decay is enhanced by the new physics and becomes accessible with present or nearfuture experimental techniques.

2.2.2

Supersymmetry and Lepton-Flavor Violation

The SUSY is a symmetry between bosons and fermions with the same quantum numbers
apart from their spins. A SUSY transformation turns a bosonic state into a fermionicstate, and vice versa. It predicts for every particle a supersymmetric partner with, inthe limit of non-broken SUSY, the same mass. Those two particles belong to a supermultiplet. The Higgs mass is kept under control by the cancellation between those twosince the contributions to the quantum correction of the two have opposite sign due tothe diﬀerence in Fermi-Bose statistics. In this way, the hierarchy problem in the SM issolved naturally in SUSY extensions.
Table 2.2 lists the super-partners in the minimum SUSY extension of the SM (MSSM).
The super-partners diﬀer by 1/2 unit of spin from the corresponding particles. Afterelectroweak symmetry breaking, the wino (W̃ ), the bino (B̃), and Higssino (H̃) mix one
another and form two charged Dirac fermions called charginos (χ̃±i ; i = 1, 2), and four0Majorana fermions called neutralinos (χ̃i ; i = 1 − 4). In general, SUSY models contain atleast two Higgs doublet ﬁelds to keep the SUSY invariance for three types of the Yukawa
coupling constants: one Higgs ﬁeld provides the mass terms for up-type quarks while theother provides mass terms for the down-type quarks and charged leptons. The ratio of
the VEVs of the two is called tan β.
SUSY breaking and SUSY ﬂavor problem If the symmetry is exact, a particle andits super-partner are degenerated and have the same mass. However, no superparticles
with the mass are not observed. Thus SUSY is broken. The LFV would originate fromthe misalignment between particle and superparticle mass eigenstates. In the basis where
the lepton mass matrix is diagonalized, the presence of nonzero oﬀ-diagonal matrix elements in the slepton mass matrix would induce LFV. However, constrains from LFV and8

CHAPTER 2. µ → eγ Decay
Table 2.2: Particles content in the MSSM.
Particlequark (q)lepton (l)gluon (g)W ±, Z 0, γ
Higgs boson (h, H, A, H ± )

Spin
1/2
1/2
1
1
0

SUSY particles
Spinsquark (q̃)
0˜slepton (l)0gluino (g̃)
1/2chargino (χ̃±
;
i
=
1,
2)
1/2
i
neutralino (χ̃0i ; i = 1 − 4) 1/2

Figure 2.2: cLFV generation mechanism in mSUGRA scenario. The oﬀ-diagonal terms
in the slepton mass matrix are induced by renormalization eﬀects due to GUT and/or
neutrino interactions.

FCNC in quark sector suggest the presence of a quite small amount of fermion-sfermion
misalignment (SUSY ﬂavor problem). There should be a special suppression mechanism
for the ﬂavor mixing of sfermions from the dynamics of SUSY breaking.
There are several scenarios that solve the SUSY ﬂavor problem. One of them is thesuper gravity model (SUGRA). In SUGRA, all sleptons and squarks have the same mass(the universal structure) and the mass matrix is diagonal. Thus in this model, there is no
cLFV at the Planck scale. However, if there are some interactions violating ﬂavor between
the Planck scale and the weak scale, cLFV can be induced by the slepton mass matrix
through the renormalization group (RG) evolution. Here, we introduce so-called the Mass
Insertion (MI) notation to denote the various ﬂavor-violating entries of the slepton matrix.
L/R
The MI parameters δij for the left/right-handed sleptons are deﬁned as
L/R

δij

=

(m2L̃/R̃ )ij
m2l̃

,(2.14)

where (m2L̃/R̃ ) is the left/right-handed slepton mass matrix, whose oﬀ-diagonal entries are2.2. Physics Motivations

9zero at the high scale but generated by RG evolution at the weak scale. This mechanism of cLFV generation is illustrated in Figure 2.2. Those interaction could be Yukawa
interaction of neutrino and/or that of grand uniﬁed theory.

2.2.3

Right-Handed Neutrinos as a LFV source

MSSM with Right-Handed Neutrino
In a MSSM with right-handed neutrino model, the neutrino Yukawa couplings becomelarge sources of LFV [12],(M )3m20 + A20
X
†
)
ln
(Y
)
(Y
,
ν ik
ν jk
8π 2
MRk

(m2L̃ )i6=j = −(2.15)

where MX denotes the scale of SUSY-breaking mediation; m0 and A0 stand for the universal SUSY-breaking scalar mass and trilinear coupling at MX , respectively. A complete
determination of (m2L̃ )i6=j would require a complete knowledge of the neutrino Yukawa
matrix.
In general, there is no direct relationship between the neutrino mixing and the slepton
mixing. If, however, we assume that the neutrino mixing mostly originates from theneutrino Yukawa coupling constants (R ≈ 1), the information from neutrino oscillations
can be related to the slepton mixing. Figure 2.3 shows the predicted branching ratio of
the µ → eγ decay as a function of a Majorana mass scale. A large fraction of the range
for MR is already excluded by the current experimental limit.
M2=130GeV,m~ eL=170GeV,mνt=0.07eV,mνµ=0.004eV

-8

10

-9

10

-10

eγ )

10

Br(µ

10

10

Experimental
bound

-11

-12

-13

10

-14

10

tanβ=3,10,30

-15

10

12

10

13

10

14

10

Mν2 (GeV)

Figure 2.3: Dependence of the branching ratio of µ → eγ on the second-generationright-handed neutrino Majorana mass Mν2 in the MSSM with right-handed neutrinos.
The assumptions of the SUSY parameters are described in the ﬁgure. The larger tan βcorresponds to the upper curve.

10

2.2.4

CHAPTER 2. µ → eγ Decay

LFV in Grand-Uniﬁed Theories

Grand-uniﬁed theories (GUT) try to unify SU (2) × U (1) electroweak interaction andSU (3) strong interaction in a single simple gauge group. The simplest GUT model isthe minimal SU (5) model. This model incorporates leptons and quarks into the same
multiplets. Therefore, the coupling constants for the strong and electroweak interactions
must be related to each other. As a consequence, there would be a new interaction in which
leptons and quarks transform one another mediated by heavy bosons. This interaction
brakes both baryon-number (B) and lepton-number (L) symmetries and leads to proton
decay.
The proton decay rate predicted by the minimal SU (5) is larger than the experimental
limit by, for example, KAMIOKANDE [17]. Furthermore, the minimal SU (5) fails tounify the gauge couplings of the SM at a single scale. Thus, the minimal SU (5) model is
excluded already. On the other hand, an extended version of SU (5) with SUSY is foundto unify them greatly.
It was ﬁrst realized by Hall and Barbieri, that LFV will occur at experimentally
accessible levels in a large class of SUSY-GUT models [13].
SUSY SU (5)The SU (5) running from the soft-braking scale MX to the GUT scale already breaks theuniversality by generating LFV entries at MGU T . Since both Q and ec are hosted in the
10, the CKM matrix mixing the left-handed quarks will give rise to oﬀ-diagonal entries
in the running of the right-handed slepton masses due to the colored Higgs,(m2R̃ )i6=j = −3

( M)
3m20 + A20
X
V
V
ln
.
ti tk
8π 2
MGU T

(2.16)

It is independent source of LFV from the neutrino Yukawa matrix. The LFV appears onlyin the right-handed slepton sector, in contrast to the neutrino Yukawa case, which has only
left-handed slepton sector. The Feynman diagrams which can enhance the µ → eγ rate
are shown in Figure 2.4.
However, except for few cases, the rate of µ → eγ turns out to be suppressed. The
reasons of the suppression are the following: the relevant sources of LFV are CKM suppressed; the amplitude involving only δ R do not have chargino contributions; and incertain regions of the parameter space, there could be cancellations between the bino andthe Higgsino-bino-Higgsino contributions. Only for light SUSY particles and for moderate
to large values of tan β and A0 , large values of µ → eγ branching ratio could be achieved.
The predicted branching ratio calculated in [14] is shown in Figure 2.5.
SUSY SU (5) with Right-handed Neutrinos
The main problems of pure SUSY SU (5) model are mass-less neutrinos, unless one breaks
R-parity. One way is going to SO(10), where the the see-saw mechanism can naturally
arise. The other way out includes adding singlets, right-handed neutrinos (SU (5)RN ).
The situation for LFV can drastically change in the SU (5)RN [15]. In this case,in addition to (m2R̃ )ij , we also have the (m2L̃ )ij MIs. In the SU (5)RN , the dominant
L L
Lδτ e ) through the loop exchange of(and δµτ
contributions to µ → eγ arise either from δµe
L Rcharginos/sneutrinos or from δµτ δτ e through the loop exchange of a pure bino.

2.2. Physics Motivations

11

Figure 2.4: Feynman diagrams contributing to µ → eγ in SUSY SU (5) model.

ft (M)=2.4

-10

µ >0

M 1 =50GeV

ft (M)=2.4

-10

10

µ <0

M 1 =50GeV

10
Experimental boundeγ )Experimental bound

Br( µ

-12

-12

10

10

tan β =30

tan β =30
tan β =10tan β =10

-14

10

-14

10

tan β =3tan β =3

-16

-16

10

10

-18

-18

10

10

-20

10

-20

100

120

140

160

180

200

220

240

260

280

300

10

100

120

140

160

180

200

(GeV)

m ~e

m ~e

(a)

(b)

R

220

240

260

280

300

(GeV)

R

Figure 2.5: Branching ratio of µ → eγ as a function of right-handed selectron mass in
SUSY SU (5).

SUSY SO(10)There are several features that make the SO(10) appealing. In SO(10) theories, thesee-saw mechanism is naturally present, and the neutrino Yukawa couplings are related

12

CHAPTER 2. µ → eγ Decay

(a)

(b)

Figure 2.6: Branching ratio of µ → eγ as a function of M1/2 in SUSY SO(10). Twocolored plots show two extremal case of the mixing of neutrino Yukawa. Green ones
show the case of maximal-mixing scenario where the mixing is given by UP M N S with
|Ue3 | = 0.07, and red ones show the case of minimal-mixing scenario by UCKM . Scatterplots are obtained by scanning SUSY parameter space in the LHC accessible region.

to those of the up quarks. Another interesting point is that the matter parity M =(−1)3(B−L) (equivalent to the R-parity) is a gauge transformation in SUSY SO(10), resulting in the lightest SUSY partner (LSP) can be a natural candidate for the dark matter.
Also in the SUSY SO(10), both (m2R̃ )ij and (m2L̃ )ij can contribute to the source of LFV.
The branching ratio of µ → eγ calculated in [16] is shown in Figure 2.6 as a function of
uniﬁed gaugino mass M1/2 . In these plots, two extremal case of the mixing of neutrino
Yukawa are shown as benchmark cases. The green ones show the case of maximal-mixing
scenario where the mixing is given by UP M N S . In this case, the LFV is dominated by thecontribution from neutrino Yukawa interaction. Since R = 1, the rate depends on theunknown neutrino mixing Ue3 . The red ones show the case of minimal-mixing scenario
by UCKM . The rate is suppressed by the small mixing of the CKM angles. The existenceof the large top Yukawa coupling would work to compensate such a suppression, but still
about two order smaller than that in the UP M N S -case. In this case, the contribution
L R
from the right-handed sleptons through δµτδτ e becomes sub-leading enhanced by a factor
2(mτ /mµ ) .

2.2.5

Summary

So far we saw some examples of SUSY extensions, but in general, cLFV could be induced
in all extensions to the SM (see, for example, [18, 19] as reviews). Searches for the LFVprocesses are very sensitive to high-energy physics beyond the SM such as SUSY and its
breaking mechanism, origin of the neutrino masses, and grand uniﬁcation. Pushing down
the sensitivity of µ → eγ search to 10−13 level can cover large part of the interestingregion. Undoubtedly through the search, we will get a large amount of information for
the new physics. A search for µ → eγ decay with better sensitivity than current limit
always has some possibility of discovering.

2.3. Experimental Search

2.3

Experimental Search

2.3.1

History of µ+ → e+ γ Search

13

The muon was discovered in 1937 by Neddermeyer and Anderson in cosmic rays [20]. Since
then for over a decade, the muon was thought to be the quantum mediating the strong
nuclear force, as predicted by Yukawa [21]. In 1947, however, an experiment [22] clearly
showed that the muon does not interact via the strong interaction, and thus the muoncannot be the π meson of Yukawa. Subsequent high-precision tests of the electromagnetic
and weak couplings of muons and electrons have found no signiﬁcant diﬀerences between
these two particles except for their masses. It was believed that the muon decays into
an electron and a neutral particle. In particular, the muon was thought to decay into aelectron and a gamma ray if it is simply a heavy electron.
The ﬁrst search for µ+ → e+ γ was made in 1947 using cosmic-ray muons [23]. Its
negative result set an upper limit on the branching ratio (B) of less than 10 %. This
was the beginning of the search for LFV. In 1948, the continuous spectrum of electrons
from muon decays was established [24]. This suggested a three-body decay with a ﬁnal
state of an electron accompanied by two neutral particles. The searches were signiﬁcantly
improved when muons became artiﬁcially produced at accelerators. In 1955, an upper
limit of B < 2 × 10−5 was set using the Nevis cyclotron at the Columbia University [25].
In the late 1950s, it was suggested that the weak interactions all arise from the couplingof a vector current with a heavy charged boson. One of the consequences of the existenceof such an intermediate boson was the occurrence of µ+ → e+ γ decay with a branchingratio of order of 10−4 [26], because at that time there was thought to be only one typeof neutrino. The puzzle of the absence of neutrino-less muon decays was phenomenologically solved by incorporating the separate electron-number (Le ) and muon-number (Lµ )conservation into the V − A theory of weak interactions. This conservation law requires
separate type of neutrinos for the muon and electron. The two-neutrino hypothesis was
veriﬁed experimentally in 1962 at the Brookhaven National Laboratory (BNL) by observing neutrinos emitted in pion decay do not produce inverse-β decay [27]. With this
discovery, interest in the search for neutrino-less decay modes of the muon waned andexperiments essentially ended for about 15 years.
Now our understanding of modern elementary-particle physics is based on the SM. In
the minimal version of the SM, where only an Higgs doublet is included and mass-lessneutrinos are assumed, lepton-number conservation is an automatic consequence of gauge
invariance and the renormalizability of the SM Lagrangian. It is the basis of a natural
explanation for the smallness of lepton-ﬂavor violation in charged-lepton processes.
In 1977, rumors circulated that an experiment at the Swiss Institute for Nuclear Research (SIN, currently PSI) had found a signal of the decay µ+ → e+ γ . This underscoredthe fact that conservation of muon number is only empirical and without a fundamental
basis. The rumors were later refuted, but motivated many theoretical and experimental
activities. On the experimental side, besides the search at SIN [28], two other experiments
were quickly assembled to search with improved sensitivity. One was located at the TriUniversity Meson Facility (TRIUMF) [29], and the other was at the Los Alamos Meson
Physics Facility (LAMPF) [30]. Results from these experiments showed no evidence forthe decay at a level of 1.7 × 10−10 .
Now the interest of the search moved to the physics beyond the SM. The motivations

14

CHAPTER 2. µ → eγ Decay

Upper limits of BR (90% C.L.)of the search have been provoked by the various theoretical models which predict large
values of the branching ratio reachable with current experimental techniques, as alreadydiscussed in the previous section.
In Figure 2.7, the historical progress of the upper limit on the branching ratio ofµ+ → e+ γ decay is shown. The sensitivities have been improved primarily as the numberof observed muons increases. Innovations in muon source have enabled breakthrough.
The usage of accelerator achieved the ﬁrst big progress in the middle of 1950s. Until
1964, muons were obtained from the π + beam stopped in a target. Next series of experiments from the middle of 1970s to present have used high-intensity µ+ beam frommeson factories. In particular in 1978, the experiment at LAMPF ﬁrst used the surface
muon [31] which was ﬁrst exploited at the Lawrence Berkeley Laboratory 184” cyclotron
in 1971. Owing to this technique, high-intensity µ+ beam can be stopped in a very thintarget. As the muon rate increases, the rate of background also increases. Therefore,µ+ → e+ γ search experiments have gradually improved detector resolutions. Table 2.3
lists the µ+ → e+ γ search experiments in the era of meson factories and their results withsome experimental parameters. The MEGA experiment at LAMP set the current best
limit of B < 1.2 × 10−11 in 1999 [1]. A new experiment, MEG, has just started physics
data taking since 2008 aiming at the sensitivity of two orders of magnitude below thecurrent limits.

1
10-1

Cosmic µ

10-2
10-3
10-4
10-5
10-6

Stopped π

10-7
10-8
10-9

µ beam

10-10
10-11
10-12
10-13
1940 1950 1960 1970 1980 1990 2000 2010
Year
Figure 2.7: Historical progress of µ+ → e+ γ search.

2.3. Experimental Search

15

Table 2.3: Historical progress of search for µ+ → e+ γ since the era of meson factories
with 90 % C.L. upper limits. The beam rates quoted are given as average beam intensities.
The resolutions are given as full width at half maximum (FWHM).
Year
Site
1977
SIN
1977 TRIUMF
1979 LAMPF
1986 LAMPF
1999 LAMPF

2.3.2

Beam rate
5 × 105 µ+ /s
2 × 105 π + /s
2.4 × 106 µ+ /s
4 × 105 µ+ /s
1.3 × 107 µ+ /s∆Ee∆Eγ ∆teγ
10 % 8.7 % 6.7 ns
8.7 % 9.3 % 1.4 ns
8.8 % 8 %1.9 ns8%8%
1.8 ns
1.2 % 4.5 % 1.6 ns∆Θeγ

Upper Limit
1.0 × 10−9
3.6 × 10−937 mrad 1.7 × 10−1087 mrad 4.9 × 10−11
15 mrad 1.2 × 10−11

Ref.
[28]
[29]
[30]
[9]
[1]

Requirements of Muons

To improve the sensitivity, we have to measure a huge number of muons. Let us thinkof observing 3 × 1013 muons to achieve a sensitivity of 10−13 in a realistic period of datataking, let’s say, 2 years (∼ 6 × 107 sec). Taking a reasonable detection eﬃciency (∼ 4 %)and eﬃciency of accelerator operation period (∼ 50 %) into account, we need a muon
intensity of ∼ 3 × 107 µ+ /sec.
Negative muons (µ− ) form muonic atoms when they are captured by nuclei in material.
It is not good for the µ → eγ search since bound states are formed and recoils of nuclei
make the event signature complex. Therefore, experimentally positive muons (µ+ ) havebeen used.

2.3.3

Event Signature

The µ+ → e+ γ decay is a simple two-body decay from a muon at rest as shown in
Figure 2.8(a), and the event signature has the following features:• The positron has a monochromatic energy of 52.8 MeV, which is a half of the muon
mass.
• The gamma ray has a monochromatic energy of 52.8 MeV.
• They are in time coincidence.
• They are emitted back-to-back from the same vertex.
Therefore, the following four variables are generally used in distinguishing signal eventsfrom a large amount of background events: the positron energy (Ee ), the gamma-ray
energy (Eγ ), the relative timing between the positron and gamma (teγ ), and the opening
angle between them (Θeγ ).

2.3.4

Backgrounds

There are two kinds of major backgrounds mimicking the signature of µ+ → e+ γ decay discussed above. One is a physics background from a radiative muon decay, µ+ → e+ νe ν µ γ .
It becomes a prompt background when the two neutrinos carry oﬀ little energy as shown

16

CHAPTER 2. µ → eγ Decay

Figure 2.8: Schematics of µ+ → e+ γ event signature (a), and two types of backgrounds(b)(c).

in Figure 2.8(b). The other is an accidental overlap of an uncorrelated Michel positron
and a high-energy gamma from any kinds of sources. We call this type of background
accidental background (Figure 2.8(c)). The rates of backgrounds crucially depend on thedetector performance. We discuss the details of those backgrounds in the following.
Prompt Background
Approximately 1 % of muon decays are accompanied by a radiation of photon Eγ >10 MeV. These radiative muon decay (RD) events can mimic a µ+ → e+ γ signal when
the two neutrinos carry oﬀ little energy.
The diﬀerential branching ratio was calculated by several authors [32, 33]. Within the
V −A interaction, the diﬀerential branching ratio, where the ﬁnal position and gamma areemitted at energy intervals of dx and dy with solid angles of dΩe and dΩγ , respectively,in the muon rest frame can be write down as,
dBRD (µ+ → e+ ν ν̄γ) =αdy
βdx dΩe dΩγ [F (x, y, d)364πy−β P~µ · p̂e G(x, y, d)−P~µ · p̂γ H(x, y, d)],(2.17)

where P~µ is the muon polarization vector; p̂e and p̂γ are the unit vectors of momentaof the positron and the gamma, respectively; β is deﬁned as β ≡ |~pe |/Ee ; d is given byd ≡ 1 − β p̂e · p̂γ ; and x and y are normalized positron and gamma energies, x = 2Ee /mµ
and y = 2Eγ /mµ . The formulas of F (x, y, d), G(x, y, d), and H(x, y, d) in the SM aregiven in Appendix A.4. In addition, we deﬁne here z as z ≡ π − Θeγ . In this notation, theRD looks similar to the µ+ → e+ γ decay when x ≈ 1, y ≈ 1, and z ≈ 0. Let us consider
here only unpolarized muons (Pµ = 0). Then it becomes,
dBRD (µ+ → e+ ν ν̄γ) =αdy
βdx dΩe dΩγ F (x, y, d)364πy2.3. Experimental Search=

α β
F (x, y, d)dxdyd(cos Θeγ ).
8π y17(2.18)

When exactly x = 1 and y = 1, the diﬀerential decay width vanishes. In reality, theﬁnite detector resolutions introduce background events. Given the detector resolutions,the sensitivity limitation from RD can be evaluated by integrating the diﬀerential branching ratio over the resolutions, more precisely, over the kinematic-box region of the signal
which is determined by the resolutions. Let us take δx, δy, and δz to be the kinematic
range of the signal region for positron energy (1 − δx ≤ x ≤ 1), that for gamma energy(1 − δy ≤ y ≤ 1), and that for the opening angle (0 ≤ z ≤ δz), respectively. The partial
branching ratio after the integration is given by [34],∫ 1∫ 1∫ min[δz,2√(1−x)(1−y)]dB(µ+ → e+ ν ν̄γ)++dBRD (µ → e ν ν̄γ) =dx
dydz
dxdydz
1−δx
1−δy
0
α
=
[J1 + J2 ],(2.19)
8π
where J1 and J2 are given by,8δzδz
1 1 δz 8(δx)3 (δy)( )2 − (δx)2 ( )4 +
( ),
3
2
2
3 (δy)2 2
δz
δz
8 δz
= 8(δx)2 (δy)2 ( )2 − 8(δx)(δy)( )4 + ( )6 ,
2
2
3 2

J1 =
J2

(2.20)√
when δz <√2 δxδy, which means an angular resolution better than the kinematic constraint of 2 δxδy, and this assumption ﬁts into our case.
To take some idea, let us take an example of MEGA detector resolutions. To keep 90 %of the signal coverage, we set here the width of the signal box to be 1.4 times FWHM.
From the FWHM resolutions in Table 2.3, the integration ranges become,
δx
δy
δz
δteγ

=
=
=
=

0.0084,0.023,
0.0075,
1.12 ns1 .

(2.21)

Put these number in the formula, then the eﬀective branching ratio of the prompt background is given as,
BRD (µ+ → e+ ν ν̄γ) ∼ 4.4 × 10−15 .

(2.22)The prompt background is not found to be a serious problem as long as such detector
resolutions are achieved.
Accidental BackgroundIn a high intensity environment, an accidental overlap of uncorrelated positrons andgamma rays can also mimic the µ+ → e+ γ signal with some probability. The higher
the muon intensity becomes, the more serious the accidental background becomes.
1Time resolution does not work for discrimination of the prompt background, but this number is used
for accidental background later.

18

CHAPTER 2. µ → eγ Decay
The eﬀective branching ratio of the accidental background can be estimated by
Bacc = Rµ · fe0 · fγ0 · (δωeγ) · (2δteγ ),4π(2.23)

where Rµ is an instant muon intensity; δteγ is a half width of the signal region for time
coincidence; δωeγ is that for back-to-back constraint. The terms fe0 and fγ0 are the fractionsof the spectrum within the signal box of positron in the Michel decay and that of gamma
from such as RD, respectively.
The positron energy spectrum of µ+ → e+ νe ν̄µ decay (Michel spectrum) is shown in
Figure 2.9. fe0 can be estimated by integrating the Michel spectrum over 1 − δx ≤ x ≤ 1,
yielding fe0 ≈ 2(δx) since it is almost ﬂat at x ≈ 1.
To estimate fγ0 , the diﬀerential branching ratio of RD (Eq.2.18) is integrated overpositron energy (x) and the angle between positron and gamma (Θeγ ). By neglecting theterms suppressed by me /mµ , the diﬀerential branching ratio is calculated to be [35]dBRD (µ+ → e+ ν ν̄γ) =2[J+ (y) + J− (y)]dy,y(2.24)

where J+ (y) and J− (y) are deﬁned by[()

1 − y 17
3 ln
−
r
2
(
)
1−y
+ − 3 ln
+ 7 (1 − y)r()]
1 − y 13+ 2 ln−(1 − y)2 ,
r
3
[(
)
α
1 − y 93J− (y) =(1 − y)2 3 ln
−
6π
r
12
(
)
1 − y 29
+ − 4 ln+
(1 − y)r
3()]
1 − y 55
2
+ 2 ln−(1 − y) ,r
12α
J+ (y) =(1 − y)6π(2.25)(2.26)

where r = (me /mµ )2 . It is shown in Figure 2.10. The partial branching ratio integrated
over the signal region (1 − δy ≤ y ≤ 1 − r) can be calculated with neglecting the higher
order terms of (1 − y),
fγ0 =

∫ 1−r

dy
1−δy

dBRD (µ+ → e+ ν ν̄γ)dyα≈(δy)2 [ln(δy) + 7.33].
2π(2.27)In this estimation, the other sources of high-energy gamma such as positron annihilationin ﬂight (AIF), external bremsstrahlung are ignored. Those contributions depend on
the actual design of the experiment. If we achieved extremely a good energy resolution,
the AIF contribution becomes more important. In addition, accidental pileups of thosegamma rays can be another source of background in high-energy region.
Given the angle resolution of δz, the size of signal box for back-to-back condition is
given by δωeγ = π(δz)2 .

2Differential Branching Ratio

Differential Branching Ratio

2.3. Experimental Search

1.8
1.6
1.4
1.2
1
0.8
0.6

19

1
10-1
10-2
10-3
10-4

0.4
0.2
0
0

0.2

0.4
0.6
0.8
1
Normalized Positron Energy (x)Figure 2.9: Positron energy spectrum of
unpolarized µ+ → e+ νe ν̄µ decay (Michel
spectrum). A radiative correction due tothe virtual photon emission and the inner
bremsstrahlung is applied in the spectrum
[36].

10-5
0

0.2

0.4
0.6
0.8
1
Normalized Photon Energy (y)Figure 2.10: Photon energy spectrum of
unpolarized µ+ → e+ ν ν̄γ decay. This isobtained by integrating over the positronenergy and the angle between a positron
and a photon.

From the above, the eﬀective branching ratio of accidental background is approximately given by
[

]

α
(δz)2
Bacc ≈ Rµ · (2δx) ·(δy)2 (ln(δy) + 7.33) ·· (2δteγ )2π
4(2.28)

Again, we here calculate an example of the eﬀective branching ratio of the accidental
background using numbers in Eq.2.21. The instant beam intensity was 2.6×108 in MEGA.
It is higher than the average intensity listed in Table 2.3 because they used a pulsed beamwith duty cycle 6 %. The eﬀective branching ratio is then given as
Bacc ∼ 1.2 × 10−12 .

(2.29)

This could be a serious problem. A new idea to suppress the background is necessary to
go into the sensitivity of 10−13 level.

2.3.5

Requirements of µ+ → e+ γ Search

By the naive calculations of backgrounds above, the accidental background is found to be
the dominant background source, and it will limit the experiment.
First, from Eq.2.23 we see the eﬀective branching ratio of the accidental backgroundis proportional to the instant muon beam intensity. Whereas we estimated that we needa > 107 /sec muon intensity to get enough statistics. To achieve such an intensity withminimizing the background, using a direct current (DC) muon beam is the best solution.
If a DC beam was used in MEGA, the background rate would be reduced by a factor 16
with same statistics.

20

CHAPTER 2. µ → eγ Decay

We can see in Figure 2.10 that the background source of gamma ray is strongly
suppressed at the signal region. It is also the case for gamma rays from AIF. With agood gamma-ray energy resolution, we can reduce background rate most eﬃciently. It isalso important to reduce material which interacts with positrons to suppress additional
gamma-ray yield.
In contrast, there are abundant positrons in the signal region. It is diﬃcult to reduce
background eﬀectively by improving the positron energy measurement. It is more important to eﬃciently measure a huge amount of positrons because every muon of such a highintensity generates a positron.
Let us consider MEGA a little more. Brieﬂy, they measured gamma rays using pair
spectrometers, which consist of lead conversion foils, a multi-wire proportional chamber(MWPC), drift chambers, and plastic scintillators. They measured the e− + e+ generated
from the gamma-ray conversion. This method gave a good energy and position resolution
and also some information of direction. It, however, limited the gamma-ray detection
eﬃciency and the timing resolution. It was a trade oﬀ between energy resolution anddetection eﬃciency by the thickness of the conversion foils. If we develop a new type of
gamma-ray detector with a high eﬃciency and a high time resolution, it becomes a great
advantage of background suppression.
We can summarize the requirements of µ+ → e+ γ search in the following:• high intensity DC µ+ beam,• high rate tolerable positron detector,• high performance gamma-ray detector.

Chapter 3
Experimental Apparatus
This chapter gives detail descriptions of the experimental apparatus of MEG. In Figure 3.1, an overview of the experiment is shown. To summarize, the key elements of MEG
are:• The world’s most intense DC muon beam,• An innovative positron spectrometer,• A new type of gamma-ray detector with liquid xenon.
With this unique apparatus, we cope with the unexplored region of the sensitivity.
All these components were constructed and commissioned by 2007 and an engineering
run was conducted in 2007. We started full-scale physics data taking in 2008.

Coordinate System
Before entering the detail descriptions of the experiment, here we deﬁne the global coordinate system of MEG. It is used throughout this thesis. The origin is deﬁned as thecenter of the positron spectrometer; it is also the center of the muon stopping target.
The z-axis is assigned to the beam direction. The y-axis is vertical axis from bottom totop. Then the x-axis is deﬁned as the other axis of right-handed rectangular coordinate
system. The θ is deﬁned as the polar angle from the z-axis, while the φ is the azimuthal
one. In this coordinate system, the gamma-ray detector is located at the negative x sideand trajectories of positrons are negative φ direction as shown in Figure 3.1.
In addition, we deﬁne a local coordinate system of the gamma-ray detector. It is
used in the reconstruction of gamma rays. We deﬁne (u, v, w) coordinates. The u-axis
is identical to the z-axis. Its center is at the center of the stopping target which is also
the center of the gamma-ray detector. The v-axis is deﬁned as a direction of negative φalong the surface of the inner face of the gamma-ray detector. Its center is at y = 0. The
w-axis is deﬁned as the depth in the detector. The direction is the same as the radial
direction in cylindrical coordinates but the origin of the axis is at the surface of the innerface. The detailed design of the gamma-ray detector is described in Sec.3.2.2.
21

22

CHAPTER 3. Experimental Apparatus

Figure 3.1: Overview of the MEG experiment.

3.1µ+ Beam

The sensitivity of the experiment improves primarily as the number of observed muons.
Therefore, a high intensity muon beam is required. On the other hand, the accidental
background rate increases as the instant intensity of the beam. Hence, a DC beam ispreferable than a pulsed beam for µ+ → e+ γ search experiments.
MEG is conducted at the 590 MeV proton ring cyclotron facility of PSI in Switzerland,
which provides the world’s most intense DC muon beam.

3.1.1

PSI Proton Ring Cyclotron

PSI operates a 590 MeV proton ring cyclotron (Figure 3.2) with the maximum proton
current of 2 mA and beam power of 1.2 MW. It is operated at an accelerator frequencyof 51 MHz. Thus the primary proton beam has the RF pulse structure of 20 ns period.
However, compared to the muon lifetime of 2.2 µsec, it is short enough and the muon
decay rate is almost constant. It can oﬀer > 1.5 × 108 /sec DC muon beam.
Furthermore, this cyclotron is currently being upgraded: its beam current is planed
to reach 2.6 mA in a few years, and 3.0 mA some years thereafter [37]. Eventually, we
will be able to get > 2 × 108 µ+ /sec.

3.1. µ+ Beam

23

Figure 3.2: 590 MeV proton ring cyclotron at PSI.

3.1.2

Surface Muon from πE5 Channel

The πE5 channel is one of the secondary beamlines which provide low-energy (10-120 MeV/c)pion and muon beam. We use a surface muon beam from this channel. The surface muons[31] are muons originating from pions stopping near the surface of the pion production target and decaying at rest. Before the surface muon method was developed, muon beams
were generally produced by transporting a pion beam over a distance1 . Only a small
fraction of the decay muons were accepted for the beamline. To get higher intensity, amoderately high momentum (∼ 100 MeV/c) was usually used and the momentum spread
was large, which forced one to use a thick stopping target. In contrast, muons from thedecay of π + at rest have an unique momentum of 29.8 MeV/c. During penetrating a thinlayer of the surface of the target, it looses some energy, but still have low momentum
spread. By tuning the beamline to accept ∼28 MeV/c, we can collect high intensity ofpositive muons.
In the πE5 channel, a thick (4 cm) graphite π/µ production target (E-target) is placedin the primary proton beamline. The surface muons are extracted to the channel at anangle of 175◦ with respect to the primary proton beam. Figure 3.3(a) shows the πE5beamline consisting of bending magnets, quadrupoles, hexapoles, and slits. Three sets ofhorizontal and a set of vertical slits deﬁne the momentum and acceptance. Figure 3.3(b)shows the expected and measured rates of muons and pions at the ﬁnal focus of πE5
just behind the second bending magnet (AST). An enhancement due to surface muons
is seen at momentum around 28 MeV/c. Because of the low momentum and the sharp
momentum distribution, it is easy to degrade muon momentum and possible to stop themin a very thin target with a high eﬃciency.

3.1.3

Beam Transport System

The muon beam from the channel is transported to a stopping target in our detector
system through several elements of the beam transport system to achieve high stopping
It is still the case of negative muon beam because π − stopped in a target are captured by nuclei andthe surface muon method cannot be applied.
1

24

CHAPTER 3. Experimental Apparatus
10 11
10 10

-1 -1

N (mA s )

10 9
10 8
π+
πmeasured pointsµ+µmeasured points

10 7
10 6
10 5
10 4
0

20

40

60

80

100

120

140

P (MeV/c)

(a)

(b)

Figure 3.3: (a) πE5 beamline. (b) Muon and pion ﬂux at πE5 channel.

Figure 3.4: Schematic and picture of beam transport system. Surface muons enter fromthe left and go through the Wien ﬁlter, the BTS, and the COBRA magnet to reach the
stopping target at the center of the magnet.

rate on a thin target with minimum beam-related background. The system consists of aquadrupole triplet (Triplet I), a crossed-ﬁeld separator (Wien ﬁlter), a quadrupole triplet(Triplet II), and a beam transport solenoid (BTS) in the order of the beam ﬂow as shownin Figure 3.4. The Wien ﬁlter cleanly separates eight times higher positron contamination
by 7.5 σ by applying a horizontal magnetic ﬁeld of 133 Gauss and a vertical electric ﬁeldof 195 kV. The BTS adjusts the oscillation of the beam proﬁle to minimize the spot sizeon the stopping target. We put a degrader made of 300 µm Mylar in the BTS to reducemuon momentum.

160

3.1. µ+ Beam(a) Stopping target.

25(b) Installation. View from downstream.

Figure 3.5: Pictures of MEG stopping target.

3.1.4

Stopping Target

Material of the stopping target itself can be causes of both scattering of positrons resulting
in degradation of positron measurement and yielding undesired gamma rays from AIF
of positrons becoming a source of accidental background. To minimize the thickness
with keeping muon stopping power, we put the target with a slant angle with respect
to the beam axis. We adopt a sheet of polyethylene/polyester with thickness of 205 µm(18 mg/cm2 ) for the target material. Pictures of the target are shown in Figure 3.5. The
sheet is supported by a Rohacell [38] frame whose density is 0.895 g/cm3 . The dimensionsof ellipse are 79.8 and 200.5 mm for minor and major axises, respectively. The slant angle
is optimized to 20.5◦ . We made six holes (10 mm diameter) on the sheet to study vertex
reconstruction performance and to align the target position by using data.

3.1.5

Mode and ProﬁleWe prepared several settings of beam modes with diﬀerent intensities in 2008 run. Socalled “normal” mode is nominal one for the physics data taking, and “ultra-low” modeis dedicated one for the study and calibration with RD. The others are prepared for the
background studies. We directly measured the beam proﬁle with an APD counter. The
spot size at the center of the detector system is σx = 9.5, σy = 10.2 mm for the normal
mode. The muon intensity of the normal mode was measured to be (3.69 ± 0.08) ×
107 µ+ /sec at the center when the proton current is 2 mA. With stopping eﬃciency of 0.82
evaluated using MC simulation, the stopping rate of the normal mode is evaluated to be
7 +Rµ,stop = (3.0 −0.1
+0.2 ) × 10 µ /sec

(3.1)

at 2 mA proton current. The stopping rate of the ultra-low mode is evaluated to be
1.2 × 106 µ+ /sec.
Note that we set the beam rate lower than the maximum capability of the channel(∼ 1.5 × 108 ) by the limitation from detector performance. From the beam point of view,
there is a room for improvement.

26

3.2

CHAPTER 3. Experimental Apparatus

Detector

An overview of the MEG detector is given in Figure 3.6. Positrons from muon decays
are analyzed by the COBRA (COnstant-Bending-RAdius) spectrometer, which consistsof a thin-walled superconducting solenoid magnet, a tracking system of low-mass drift
chambers, and fast scintillator timing-counter arrays. Gamma rays are detected by aliquid xenon scintillation detector. It measures gamma-ray energy and ﬁrst interaction
point and time. We describe the concept and principle of each sub-detector in the following
sections together with its actual design.
COBRA Magnet
Drift Chamber+µ Beame+

Timing Counterγ

Stopping Targetx

Liquid Xenon
γ-ray Detector

z

1m

γ

e+

y
x

Figure 3.6: Schematic view of MEG detector.

3.2. Detector

3.2.1

27

The Positron Spectrometer

The requirements of the positron spectrometer are summarized in the following:i) to have to work in very high rate environment (up to 1 × 108 s−1 ),ii) to be as low material as possible,iii) to measure momentum, direction, and timing of positrons around 50 MeV with highresolutions.
We adopt a highly graded magnetic ﬁeld to achieve i). The concepts of the ﬁeld
are schematically shown in Figure 3.7. Owing to the gradient ﬁelds, positrons with thesame momentum follow trajectories with an almost constant projected bending radius
independently of their emission angles. This allows a preferential acceptance of higher
momentum positrons in tracking device as well as sweeping particles out of the detector
more eﬃciently. Figure 3.8 shows the rate of positrons as a function of radius. By placing
the drift chamber at a radius over 20 cm, we can reduce the hit rate to a level below thelimit of stable operation.
The spectrometer is designed to be very thin to avoid undesirable gamma-ray generation from bremsstrahlung or AIF of positron, to minimize the probability that a gamma
ray interacts with some material before reaching gamma-ray detector, and to suppress
Coulomb multiple scattering of positron. The last point is crucial for the precise measurement of low-energy positron.
Momentum and emission angle of positron is measured by the trajectory in the drift
chamber and the muon decay vertex is reconstructed by extrapolating the track back to
the target plane. The track is also extrapolated down to the timing counter and gives
us the length of the trajectory from the decay vertex. This track length is converted tothe positron time-of-ﬂight and, together with measurement on the timing counter, used
to reconstruct the muon decay time.
COBRA Magnet
We constructed a superconducting magnet specially designed to form a highly gradedmagnetic ﬁeld (Figure 3.9) [39]. A step-structure solenoid realizes such gradient ﬁeld. It
consists of ﬁve coils with three diﬀerent radii: one central coil, two gradient coils andtwo end coils. The design and dimensions are shown in Figure 3.10. Figure 3.11 shows a
proﬁle of the magnetic ﬁeld along the magnet axis. The magnetic ﬁeld ranges from 1.27
at the center to 0.49 T at the edge.
As shown in Figure 3.6, gamma rays from muon decay on the target go through themagnet wall to reach the gamma-ray detector. If they interact with some material infront of the detector, it causes the signal ineﬃciency. Hence, the cable and wall of themagnet are designed to be very thin. The superconducting cable is made from NbTi multiﬁlament embedded in copper matrix and high-strength aluminum stabilizer [40]. Nickelof 5000 ppm is added into the aluminum stabilizer to reinforce it mechanically. Owing to
the high-strength superconductor, we can minimize the thickness of the support structure
required for a given electromagnetic force acting on the coils. The total thickness of themagnet including its cryostat is suppressed to be 0.197 X0 . The transmission eﬃciency of
gamma rays is 85 %.

28

CHAPTER 3. Experimental Apparatus

rate(Hz/cm2 )Figure 3.7: Conceptual illustrations of the COBRA spectrometer compared with one
with a uniform magnetic ﬁeld. (a) and (c) show trajectories of positrons emitted at 88◦ .
The uniform ﬁeld makes many turns inside the detector, whereas the gradient ﬁeld sweep
the positron out of the detector much more quickly. (b) and (d) show trajectories ofmono-energetic positrons emitted at various angles. In the uniform ﬁeld, the bending
radius depends on the emission angle, whereas it is independent in the gradient ﬁeld.

5

10

104

3

10

Uniform B-field

102
COBRA B-field

10

1
0

DC region

5

10

15

20

25

30
radius(cm)Figure 3.8: Rate of Michel positrons per cm2 per second as a function of radius assuming
muon decay rate of 3 × 107 /sec.

3.2. Detector

29

Figure 3.9: Picture of the COBRA magnet.
Compensation coil

Central coil
φ92

φ81

φ70

Gradient coil
Inner end coil
Outer end coil

GM Refrigerator
1m

Figure 3.10: Schematic of the COBRA magnet.

The magnetic ﬁeld produced by the COBRA magnet can signiﬁcantly deteriorate theperformance of PMTs used in the gamma-ray detector because their outputs are sensitive
to magnetic ﬁeld as shown in Figure 3.12. The dependence is diﬀerent for diﬀerent
axes of the PMT because of the dynode structure. The PMTs are placed with diﬀerent
directions in accordance with the location in the detector. The requirement for tolerable
magnetic ﬁeld is estimated to be less than 50 Gauss. To reduce the fringe ﬁeld, we use
a pair of resistive coils placed at the both ends of the solenoid. The fringe ﬁeld withthe compensation coils around the gamma-ray detector is shown in Figure 3.13. Owing
to the compensation coils, we successfully reduce the fringe ﬁeld at the position of the30

CHAPTER 3. Experimental Apparatus

Bz (T)

gamma-ray detector.

1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0

0.2

0.4

0.6

0.8
1
1.2
Distance along axis (m)Relative output [%]Figure 3.11: Proﬁle of the magnetic ﬁeld along the axis of the magnet.

140

x-axisY

Compensation coilZy-axis120

z-axisX

100

80

End coil60[Tesla]40

Gradient coil

20

0
-300

Central coil
-200

-100

0

100
200
300
Magnetic flux density [ Gauss]

Figure 3.12: Response of PMT used in
the gamma-ray detector in magnetic ﬁeld.

Center of magnet

LXe Gamma-ray detector

Figure 3.13: Contour plot of fringe ﬁeld
around the gamma-ray detector.

3.2. Detector

31

Drift Chamber System
The drift chamber system (DCH) measures trajectories of positrons and reconstructs their
momentum, decay vertex, emission angle and time-of-ﬂight. A picture of the whole system
is shown in Figure 3.14. It consists of 16 radially aligned modules, spaced at 10.5◦ intervals
in φ direction forming a half-circle around the target. The radial position ranges from
19.3 to 27.9 cm to measure only high momentum positron (>40 MeV). Design of a moduleis shown in Figure 3.15 with its dimensions. The inside of COBRA magnet is ﬁlled withpure helium to minimize material along positron trajectories. The drift chamber system
is placed in the helium atmosphere.

Figure 3.14: Picture of drift chamber system inside the COBRA magnet.

Each module contains two staggered layers of anode wire planes. Each layer containing
nine drift cells is shifted by one-half cell each other to resolve left-right ambiguity. The
two layers are separated by two inner cathode foils and also enclosed by a outer one.
The cell conﬁguration of a module is shown in Figure 3.16. The sense wires are madeof Ni/Cr (80:20). Their diameter is 25 mm and resistance is 2200 Ω/m. Cathodes are
made of 12.5 µm thick polyimide with aluminum deposition. They have vernier-pattern
structure of 5 cm period for a precise z reconstruction. The principle of the vernier-pad
method is illustrated in Figure 3.17 schematically. Owing to the vernier pattern, thenumber of cathode readouts is suppressed relatively low compared to that of conventional
strip-pattern cathodes. It results in less amount of material of preampliﬁers and cables
which also take place in the region of positron tracks. All cathodes and potential wires are
grounded. Positive high voltages are applied to sense wires; the nominal value is 1850 V.

32

CHAPTER 3. Experimental Apparatus

The active gas of the chamber is composed of helium:ethane (50:50). We optimize itfrom the points of ionization loss in gas and multiple Coulomb scattering. The ionization
loss for the minimum ionization particles is ≈ 65 e− /cm. The ﬁeld map and drift lines
calculated by a GARFIELD simulation are shown in Figure 3.18.
A support frame made of carbon ﬁber supports the wires and foils. It is, however,designed to “open-frame” structure; there is no support frame at the target side (Figure 3.15). This structure helps to reduce amount of material in positron trajectories at a
cost of a diﬃculty in the construction. Shape of each cell is formed by the thin cathodefoil itself because of the open-structure frame. Hence, a precise pressure control betweenin (chamber gas) and out (helium atmosphere) of the chamber is necessary to maintainthe cell spacing. We developed a dedicated gas-ﬂow control system and it successfully
controls the pressure better than 0.005 Pa stability while the required stability is less than
1 Pa.
As a result of all eﬀorts of material reduction, we constructed a tracking system with
2.0 × 10−3 X0 in total along a positron trajectory. Detailed description about the design,
construction, and performance in 2007 engineering run are found in [41].

(a) Picture of a module.

(b) Design of a module.

Figure 3.15: A module of drift chamber.

Figure 3.16: Cell conﬁguration of a drift
chamber module.

Figure 3.17: Schematic of vernier pattern on the cathode pads.

33

y−Axis [cm]y−Axis [cm]3.2. Detector

x−Axis [cm](a) Contour plot of the potential.

x−Axis [cm](b) Drift lines with isochrone map.

Figure 3.18: Field map and drift lines of drift chamber calculated by the GARFIELD
simulation.

Timing Counters
A pair of timing counters (TIC) are placed at the both sides of the drift chamber system.
Positron tracked by the drift chamber ﬁnally hits the timing counter and the impact
timing and position are measured. A module of the timing counter consists of two layers of
scintillation counter arrays stacked orthogonally each other. The two layers have diﬀerent
structures and each has speciﬁc tasks. The outer layer is called φ-counter. Its main tasks
are precise measurement of positron impact time and fast information of positron φemission angle. The inner one is called z-counter. Its main tasks are precise measurementof z impact position and fast information of positron θ-emission angle.
The φ-counter consists of 15 plastic scintillation bars (4 × 4 × 80 cm3 , Bicron BC-404[42]). They are placed at 10.5◦ intervals in φ-direction at a radius of 32 cm and they cover160◦ in total (−150◦ < φ < 10◦ ). Two 2-inch ﬁne-mesh PMTs (Hamamatsu R5924 [43])are attached on both ends of each bar. A picture of the counter is shown in Figure 3.19(a),and its detail design is illustrated in (b).
The z-counter consists of 128 scintillating ﬁbers (6 × 6 mm2 , Saint-Gobain BCF-20[42]). Each ﬁber is separated optically at the center and read out independently at theboth ends by a 5 × 5 mm2 silicon avalanche photo-diode (APD) (Hamamatsu S8664-55[43]). The z-counters are mounted on the φ-counters at a radius of 29 cm. A picture of
the counter is shown in Figure 3.20.
Details of the design, construction, and performance in beam tests are described in[44][45].
In 2008 operation, we got some problems on the z-counter read out. We did not use
the z-counter in trigger. Instead we used φ-counter also to get the z impact position witha slightly worse resolution. Also in the analysis of this thesis, the z-counter is not used,even though the fundamental functions were studied and conﬁrmed with the data.

34

CHAPTER 3. Experimental Apparatus

(a)

(b)

Figure 3.19: Picture and design of timing φ-counter. The scintillation bars are mountedwith a slant angle at 20◦ so that the path lengths of positrons inside the bar become uniform. The bars are hexagonally shaped to avoid geometrical conﬂict due to the rotation.
PMTs are attached with a slant angle at 10◦ with respect to the z-axis (at ∼ 30◦ withrespect to the magnetic ﬁeld) to recover their operation in a magnetic ﬁeld [46].

Figure 3.20: Picture of timing z-counter and APDs. It is mounted on the φ-counter.

3.2. Detector

3.2.2

35

The Gamma-ray Detector

The gamma-ray detector is undoubtedly the most innovative and challenging part of theexperiment. Its performance is crucial for a successful search for the µ+ → e+ γ decay. We
use a gamma-ray detector of a 900 liter homogeneous volume of liquid xenon (LXe). It
is placed just outside of the COBRA magnet. Gamma rays that penetrated the COBRA
spectrometer enter the detector. They interact with LXe and generate scintillation light.
The scintillation light is collected by a number of photomultipliers (PMT) surrounding
the active volume of LXe to measure the total energy released by the incident gamma ray
as well as the position and time of its ﬁrst interaction. A conceptual ﬁgure of the gammaray detector is shown in Figure 3.21. Sometimes multiple gamma rays enter the detector
and are measured at the same time in a high rate of low-energy gamma-ray backgroundbecause the detector consists of a large volume without any segmentation. Nevertheless,
we can handle those pileup events correctly because the image of the light distribution
from a large number of PMTs enables us to identify and unfold those multiple events. In
addition, the time distribution and waveform can also be used to identify pileup events.
The R&D works, performance of prototype detector, design and construction of ﬁnal
detectors are described in detail in [47][48].

Figure 3.21: Conceptual ﬁgure of LXe gamma-ray detector.

36

CHAPTER 3. Experimental Apparatus
Table 3.1: Properties of LXe.
Material Properties
Atomic Number
Atomic Weight
Density at 161.4 K
Boiling point
Melting point
Triple point(temperature)
Triple point(pressure)
Radiation length
Critical Energy
Mollier radius
Scinti. wavelength (peak±FWHM)
Refractive index at 175 nm
Wph for electron
Wph for α particles
Decay time (recombination)
Decay time (fast components)
Decay time (slow components)
Absorption length
Scattering length

Value & Unit54
131.2932.978 g/cm3
165.1 K
161.4 K
161.3 K
0.805 atm
2.77 cm
14.5 MeV
4.2 cm(178 ± 14) nm
1.57 to 1.72
21.6 eV17.9 eV
45 ns
4.2 ns22 ns> 100 cm
29 cm to 50 cm

Ref.
[4]
[53]
[4]
[4]
[54]
[54]
[4]
[55]
[55]
[56]
[57, 58, 59]
[60]
[60]
[61]
[61]
[61]
[47]
[59, 62, 63, 64]

Liquid Xenon as a Scintillation Material
Xenon is widely used as a detector material in diﬀerent ﬁelds (see, for example, [49] fora recent review). It produces both charge carriers and scintillation photons in responseto radiation. Therefore, it can be used as a scintillation or ionization medium. In some
application, both of them are used simultaneously. In our detector, only the scintillation
light is used.
The photon cross section of xenon atoms are shown in Figure 3.22. Photon around
50 MeV interacts with xenon mainly through the pair production. The scintillation mechanisms are given by the following two processes [51, 52]. One is a self-trapping process of
excited xenon atom,
Xe∗ + Xe + Xe → Xe∗2 + Xe,
Xe∗2 → 2Xe + hν,

(3.2)
(3.3)

where hν is vacuum ultraviolet (VUV) scintillation photon. There are two states of theexcited dimer Xe∗2 , singlet and triplet, which correspond to fast and slow components,
respectively. The other process is a recombination process originating from a xenon ion,
Xe+ + Xe
Xe+
2 +e
Xe∗∗
Xe∗ + Xe + Xe

→
→
→
→

Xe+
2,
Xe∗∗ + Xe,
Xe∗ + heat,Xe∗2 + Xe,

(3.4)
(3.5)
(3.6)
(3.7)

3.2. Detector
2

Pressure [atm. absolute]2

Attenuation Coefficients [cm / g]

1037

Total Attenuation

Photoelectric Absorption

Scatteringliquid

10

solid

gas

1

10

−1

Pair
Production

120

140

160

Photon Energy [MeV]

Figure 3.22: Photon cross section of
xenon atoms [50].

180

200

220

240

260

Temperature [K]Figure 3.23: Xenon phase diagram.

Xe∗2 → 2Xe + hν,(3.8)

where the time dependence of the scintillation is dominated by the kinematics of therecombination process. In both processes, the scintillation photon comes from a deexcitation of a excited dimer to dissociative ground state.
Main properties of LXe and its scintillation are summarized in Table 3.1. Here, we cansee several fascinating properties of LXe for a gamma-ray detector. The large stopping
power for penetrating radiation due to its large atomic number and high density makes
it eﬃcient to detect gamma ray. It has large light yield which is necessary for the precise
energy measurement. Its fast response enables high timing resolution. It is also suitable
for the measurement in high rate environment to minimize pileup event. Those excellent
properties become clear by comparing LXe with various other scintillators in Table 3.2. In
addition, LXe has some other advantages. Since the scintillation light is not emitted by the
excited state of xenon atom (Xe∗ ) but the excited dimer (Xe∗2 ), xenon scintillation photons
cannot be re-absorbed by xenon atoms themselves. This is an advantage to achieve highenergy resolution with a large detector. We can make a single homogeneous detector
with arbitrary size and shape because it is liquid. It is free from aging or damage byradioactivity. We can purify it at any point, which is not possible for crystal scintillators
once they are constructed. We concluded LXe is the best scintillator for the gamma-ray
detector of MEG.
However, we have to overcome some diﬃculties in practical usage of LXe. The scintillation is in VUV light. Those light can be easily absorbed by some contaminants likewater or oxygen. The detector performance heavily depends on the purity of LXe. We
also have to use VUV sensitive photon sensors. Another diﬃculty comes from the lowtemperature. The phase diagram of xenon is shown in Figure 3.23. We can see a narrowtemperature range for liquid phase at around atmospheric pressure. Stable control of its
temperature around 165 K is required to keep it in liquid.

38

CHAPTER 3. Experimental Apparatus
Table 3.2: Comparison of various scintillators.

LXe
Density (g/cm )2.98
Radiation length (cm) 2.77
Mollier radius (cm)4.2
Decay time (ns)45
Emission peak (nm)178
Relative output
75
3

1

LAr NaI(Tl)
1.40
3.67
14
2.59
7.2
4.13
1620
230
127
410
90
100

CsI(Tl)
4.51
1.86
3.57
1300
560
165

BGO
7.13
1.12
2.23
300
480
21

LSO(Ce)
7.40
1.14
2.07
40
420
83

PbWO4
8.3
0.89
2.00
30/101425/4201
0.083/0.291

slow/fast component

Design of the MEG LXe Detector
The schematic view of MEG LXe gamma-ray detector is shown in Figure 3.24. It is Cshaped to ﬁt the outer radius of COBRA magnet. It consists of ∼900 liter LXe and 846
PMTs. The PMTs are placed on all of six faces of the detector and directly immersed in
LXe surrounding the active volume of LXe. The deﬁnition of the six faces (inner, outer,
upstream, downstream, top, and bottom) are also shown in the ﬁgure. The arrangementand density of PMTs are diﬀerent for each face. They are shown in the development viewof the detector in Figure 3.25. PMTs are most closely arranged on the inner face and its
coverage of active photo-cathode is about 35 %. Part of the outer face has dense part
whose density is the same as that of the inner face. This dense part is for the possibility ofcalibration with gamma ray impinging from the back of the detector. The active volumeof the detector is ∼ 800 liter and it covers 11 % of the solid angle from the stopping
target. The depth of the active volume is 38.5 cm, which corresponds to ∼ 14 X0 andkeeps the shower inside the detector.
Cryogenic System
A powerful and stable cryogenic system is indispensable to operate the large LXe detector since the temperature range of LXe is narrow as already shown. LXe is ﬁlled ina cryostat consisting of two layers of vacuum-tight vessels. Outer one makes a vacuum
layer for thermal insulation. The gamma-ray entrance window is designed to be as thinas possible to maximize the probability of gamma-ray penetration. The window of outer
vessel is made of a 0.7 mm thickness stainless steel plate, while that of inner vessel ismade of aluminum honeycomb panels covered with carbon ﬁber plates because it requires
mechanical strength up to ∼ 3 atm. The total thickness of the window is 0.075 X0 . A
turbo-molecular pump is directly attached to each vessel to evacuate with high conductance. In addition, a cryo-pump is installed to the inner vessel to eﬃciently remove water
vapor that is the main component of residual gas from the detector. A 200 W pulse-tube
refrigerator [65], which was developed for this LXe detector, is mounted on top of thecryostat and controls the temperature of LXe. Its picture and cooling power are shownin Figure 3.26. In addition, cooling pipes of LN2 are also available when a high coolingpower is necessary such as in liquefaction. Several Pt-100 sensors are placed at diﬀerent
positions inside the cryostat and measure the temperatures. The level of LXe is measured

3.2. Detector

39

Liquid xenon
Top face

Upstream face

Level meter

Outer face

PMT
γ

γ

Stopping target

Stopping target

67.85cm1m
125.5 deg.

Inner face
Inner face

Outer face
1m

Downstream
Bottom face(a) Side view.

(b) Top view.

Figure 3.24: Schematic view of LXe gamma-ray detector.
outer

top

100

downstream

upstream

80
60
40
20

v-axis148.6cm0

w-axisinnerzu-axis

cm
.5
38

58cm

bottom96cm

Figure 3.25: Development view of LXe gamma-ray detector. The local coordinate
system of the detector is also shown as red arrows. Each light blue circle shows a PMT.

40

CHAPTER 3. Experimental Apparatus

by a capacitance level meter. A picture of the detector is shown in Figure 3.27. This
picture was taken just before closing covers. All components were mounted.
Besides the detector itself, two kinds of xenon storage systems and two types of circulation systems build the xenon system of MEG. One of the storage systems consists ofeight high-pressure gas tanks (Figure 3.28(a)). The other is a dewar that can hold 1000
liter xenon in liquid phase [66] (Figure 3.28(b)). Both system can store the whole amountof xenon used in the experiment. The high-pressure gas tanks supply xenon directly to thedetector or to the 1000-liter dewar by the pressure diﬀerence. Normally, we liquefy xenon
in the 1000-liter dewar in parallel to the maintenance and preparation of the detector.
The 1000-liter dewar and the detector are connected by a liquid-transfer line. Once theliquefaction is completed in the 1000-liter dewar, we can speedily transfer xenon to thedetector in liquid phase. The recovery of LXe from the detector is also done in liquidusing the liquid-phase circulation pump in the puriﬁcation system.

Cooling power (W)Cooling power (PC150)
200
150
100
50
0
50

(a)

200
100
150
Cold end temperature (K)(b)

Figure 3.26: (a) Picture of 200 W pulse-tube refrigerator. (b) Its cooling power when
operated with 6.5 kW GM helium compressor.

PMT
We developed a new model PMT suited to MEG in cooperation with Hamamatsu Photonics. The PMT, R9869, is shown in Figure 3.29 and its properties are summarized inTable 3.3. The PMTs must work in low-temperature LXe and be sensitive to the VUVLXe scintillation light. A quartz window is used to transmit VUV light. We chose Bialkali(K-Cs-Sb) for the VUV sensitive photo-cathode. Aluminum strips are attached on thephoto-cathode to prevent from the increase in the sheet resistance at a low temperature.
Thus, we achieved high sensitivity of ∼15 % quantum eﬃciency. In 2008, PMT gains
were set to be about 1.7 × 106 with typical HV of 810 V.

3.2. Detector

41

Figure 3.27: Picture of LXe gamma-ray detector. It was just before closing covers. Allcomponents were mounted.

Calibration Devices
In the detector, several LEDs and radioactive sources are installed as constant light sources
for PMT calibrations. Blue LEDs are placed at six positions on each lateral face. We cancontrol the number and set of LEDs to be ﬂashed, their intensity, and their ﬂashing rateby remotely controlling LED drivers. Typically 10 LEDs are ﬂashed at the same time innormal calibration to illuminate all the PMTs uniformly. Each LED is covered with ahand-made ﬁlter made of aluminum foil with some small pin-holes. It works as a lightattenuator. Owing to the ﬁlter, we can apply higher voltage to the LED, resulting in lessnoise ﬂuctuation in emitted light and also the stable operation of LED. LEDs are mainly
used to calibrate PMT gains.
The LDE light has diﬀerent wavelength from that of LXe scintillation. Therefore, it
is not suitable for the absolute response calibration or quantum eﬃciency measurementof the PMT because the quantum eﬃciency depends on the wavelength. Instead, we can

42

CHAPTER 3. Experimental Apparatus

(a)

(b)

Figure 3.28: Pictures of xenon storage systems. (a) eight high-pressure gas storage
tanks. (b) 1000 liter liquid storage dewar.

Figure 3.29: Picture of PMT for the LXe gamma-ray detector (R9869).

obtain the scintillation light using radioactive sources. We developed a calibration methodwith a radioactive point-source lattice [67]. We use 241 Am as an alpha source. It emits
alpha ray at 5.485 MeV (84.5 %) and 5.443 MeV (13.0 %). Since the range of the alpharay in LXe is as short as 40 µm, it can be used as a point-like light source from a known
position. The half life of 241 Am is suﬃciently long (432 years) so that we can regard it
to be constant. The activity of each source is ≈ 200 Bq. The alpha sources are attachedon thin (100 µm diameter) tungsten wires as shown in Figure 3.30. On each wire, alphasources are attached at ﬁve positions with a constant interval of 12.4 cm. Five wires are
installed in the detector. Therefore as a whole, a lattice of 25 alpha-source spots is formedin LXe active volume. In Figure 3.31, one of the wires installed in the detector can be3.2. Detector

43

Table 3.3: Properties of the PMT (R9869). These are typical values.
Size
Active area size
PMT length
Photo-cathode material
Dynode type
Number of dynode
Typical HV
Typical gain
Typical Q.E.
Rise time
Transit time
TTS

57 mm φ45 mm φ32 mm
K-Cs-Sb
Metal channel
12900 V
1 × 106
15 %2 nsec12.5 nsec Typ.
0.75 nsec Typ.

Figure 3.30: Micro-picture of a 241 Am source mounted on a 100 µm-diameter wire. The
longitudinal dimension is ∼2 mmseen together with LEDs mounted on a lateral face.
Puriﬁcation System
The performance of the gamma-ray detector severely depends on a purity of LXe. Owing
to the scintillation mechanism through the excited dimer Xe∗2 , LXe itself is transparent
to its own scintillation light. However, possible contaminants in LXe such as water andoxygen at ppm level considerably absorb the VUV light. Figure 3.32 shows absorption of
the LXe scintillation light by water vapor and oxygen. The absorption spectrum of water
vapor largely overlaps with the LXe scintillation spectrum. Water absorbs VUV light by
the photo-dissociation process
H2 O + hν → H2 + O∗ .

(3.9)

Water was found to be the dominant contaminant by investigating the residual gas in the
inner vessel of prototype detector. The residual water vapor originates at out-gas frommaterials inside the detector such as detector wall and PMT support structures because
we cannot heat the detector while evacuating the cryostat owing to the PMTs placed
inside. Thus, water is the most dangerous contaminant for the LXe scintillation in our
detector.

44

CHAPTER 3. Experimental Apparatus

Figure 3.31: Picture of LEDs and alpha sources. LEDs are shown with orange circles.
A wire of alpha source is indicated by a yellow arrow.

Besides the absorption, some contaminants cause quenching processes of LXe scintillation. Quenching processes of LXe scintillation are poorly known. In [69], the main
reaction of quenching process of liquid argon (LAr) by N2 contaminant was considered to
be a two-body collision of the excimer states with the contaminant,Ar∗2 + N2 → 2Ar + N2 .

(3.10)

Since it is a kinematic collision, the rate depends on both the concentration of the contaminant and the lifetime of the excimer. As a consequence, variation of the purity
changes the time dependence of the scintillation light. The article reported a suppressionof slow component of the LAr scintillation as nitrogen concentration increases. A similar
quenching process can occur also in LXe.
To remove impurities in LXe, we developed two types of puriﬁcation systems. One
is a liquid-phase puriﬁcation system consisting of a cryogenic centrifugal pump for aliquid circulation and a puriﬁer cartridge of molecular sieves. We developed this systemusing the prototype detector [70]. It is dedicated to remove water contaminant. In

3.2. Detector

45

Absorption coefficient (m-1 )addition, a puriﬁer using copper beads, which was developed for LAr puriﬁcation at
CERN, was installed at the outlet of the circulation pump to remove oxygen contaminant.
Owing to the circulation in liquid, the circulation speed is as high as ∼ 35 l/hour. The
low conductance of the copper puriﬁer limits the speed. We performed the liquid-phase
puriﬁcation before the data taking and during the beam-maintenance periods. We did
not during data taking since the pump generates noise on detectors.
The other puriﬁcation system is a gas-phase one using a metal-heated getter. A
diaphragm pump circulates gas xenon taken from the detector. During the circulation,
the getter removes H2 O, O2 , CO, CO2 , N2 , H2 , and hydro-carbon molecules from gas xenon
down to 1.0 ppb level. This system was also developed using the prototype detector andthe eﬀectivity was conﬁrmed [71]. The circulation speed is very slow compared to theliquid-phase puriﬁcation (< 100 cm3 liquid/hour). However, it can be used in parallelwith data acquisition.
Oxygen1

Xe scintillation spectrum

0.1

Water vapor

0.01
130

140

150

160

170

180

190

200

Wave length (nm)Absorption in O 2 -doped Xe
I / I0I / I0(a)Absorption in H 2 O-doped Xe
1

100 ppb

1

200 ppb
1 ppm

100 ppb

5 ppm

200 ppb
1010 ppm

−1

10

−1

1 ppm
5 ppm
10 ppm

−2

10

0

−2

10

20

30

40

50

60

10

Distance [cm]

(b)

0

10

20

30

40

50

Distance [cm]Figure 3.32: (a) Absorption coeﬃcient of VUV light in 1 ppm water vapor and oxygen. Xenon scintillation spectra is superimposed [68]. (b) Light absorption for various
concentrations of water and oxygen in LXe as a function of distance from a light source
[47].

46

CHAPTER 3. Experimental Apparatus

3.3

Electronics and Data Acquisition

3.3.1

Electronics Chain

The data ﬂow from detectors to digitizers is schematically summarized in Figure 3.33.
The outputs from the detectors pass through several devices with being processed adequately for each sub-detector, and are ﬁnally acquired as waveform digitized by a fast
waveform digitizer. Owing to the waveform digitizer and a ﬁeld programmable gate arrays(FPGA) trigger, the electronics chain becomes simple compared to that with conventional
ADC/TDC and logical trigger circuit.
PMT outputs from the gamma-ray detector are put in active splitters via patch-panels,
feed-through, and coaxial cables. The active splitter has three outputs: the ﬁrst one isa wide-band (1.9 GHz) fully diﬀerential output, the second one is a 320 MHz-bandwidth
diﬀerential output, and the third one is a four-to-one sum output. The ﬁrst one goes tothe waveform digitizer, the Domino Ring Sampler (DRS). The others go to the trigger.
For the inner PMTs, the one-to-one outputs are used for the trigger while the sum outputs
are used for the other PMTs. The splitter inverts negative PMT pulses to positive ones sothat the signal will match the digitizer’s dynamic ranges. The active splitter is connectedto the DRS by a 2 m long high-density (0.68 cm pitch) twisted-pair cable, and to thetrigger by a 2 m long lower-density (1.27 cm pitch) one.
The output from the timing φ-counter’s PMT ﬁrst goes to a passive splitter, and is

Figure 3.33: Schematic of data ﬂow and electronics.

3.3. Electronics and Data Acquisition

47

divided into three outputs with a fraction of 8:1:1. The largest output goes to a doublethreshold discriminator (DTD), which has two levels of thresholds for discriminating apulse of positron hit from noise or low-energy delta-ray hits. The lower level is used forthe timing determination, while the higher is for the validation of pulse which is not anoise hit and has suﬃcient energy loss in the bar. Owing to this mechanism, we can lower
the time-determining level as low as possible resulting in intrinsically good time resolution
and minimum time-walk eﬀect. In 2008, we set the lower level at 25 and the higher one
at 800 mV. The discriminator outputs standard NIM pulses2 of 50 ns width. The NIM
pulse goes to the active splitter and the ﬁrst output goes to the DRS. One of the smaller
outputs of the passive splitter goes to the active splitter and the ﬁrst output goes to the
DRS and the second one goes to the trigger. The other output of the passive splitter goesto current monitor to check the PMT lifetime.
Eight APDs of the timing z-counter are mounted on a frontend board together withpreampliﬁers. This frontend board outputs an analog sum signal of the eight channels
and a digital hit-pattern signal of individual channels. The analog output goes to thetrigger while the digital hit-pattern goes to a dedicated hit register module.
We get six waveforms from a drift cell of the drift chamber: two from the both ends
of a anode wire and four from the vernier-pattern cathode pads (two for outside pads andtwo for inside ones). The cathode pads are read out at one of the ends. Those outputs
are ﬁrst ampliﬁed by a preampliﬁer attached at the chamber frame structure. Here theanode output is inverted. At the patch-panel on the end-cap of the COBRA magnet, thewire signal is resistively divided into two outputs with a fraction of 9:1. The larger one
and the cathode signals go to the DRS directly by coaxial cables. The smaller one isampliﬁed and summed up over several wires. This sum signal goes to the trigger.

3.3.2

Data Acquisition System

An overview of the data acquisition (DAQ) system is shown in Figure 3.34. The MIDAS
system [72] controls the whole system. There are mainly nine frontend sub-systems: four
for trigger and ﬁve for DRS. Each of them, which corresponds to a VME crate, is controlled
by a Linux PC. Each frontend is running with multi-threading process, and each event isstored in a ring buﬀer. In the main DAQ PC, an event builder is running and each eventis re-built from the ring buﬀer of each frontend. The MIDAS system provides not only
the control of frontend processes but also the logging system, the online database system
for the parameters of DAQ and trigger, the slow control system, the alarm system, thehistory monitoring system, and web interfaces.
During the data taking, an online monitor is running and with its graphical display
the quality of the data is checked in parallel with the DAQ. In addition, after each run
is ﬁnished an oﬄine process is run automatically and more detail information can beavailable within a few dozens of minutes.
2

The fast-negative logic pulse of Nuclear Instrument Module (NIM). The voltage of 0 V for logic0 andthat of 0.8 V for logic1, where logic1 correspond to the presence of a hit here. The discriminator itself is
not a NIM standard module.

48

CHAPTER 3. Experimental Apparatus

Figure 3.34: Schematic of DAQ system.

3.3.3

Trigger

The trigger system is based on a coupled use of ﬂash analog to digital converters (FADC)
and FPGA. The input signals are sampled by FADCs and the digitized information isanalyzed by FPGAs.
The whole system is arranged in a tree structure on three layers with two diﬀerent
types of boards. The ﬁrst layer consists of the Type1 boards compliant with the 6U
VME standard. The Type1 board receives and digitizes analog input signals with FADCs(AD9218 [73]) at 100 MHz with 10 bits resolution , implements some reconstruction
algorithms on a large size FPGA (Xilinx Virtex-IIpro [74]), and sends the information
to the successive trigger layer. The two remaining trigger layers consist of a second typeof boards (Type2) compliant with the 9U VME standard. The second layer determines
trigger conditions of sub-systems, and ﬁnally the third layer makes a trigger decision.
In addition, an ancillary system was developed to ensure synchronous operation of thetree. It consists of ancillary boards of 9U VME boards, and distributes a reference clock
and controls signals such as start and stop of the DAQ system. A master board hosts
the reference clock oscillator (SaRonix SEL3935 [75]) generating a 19.44 MHz squared
clock, and receives control signals from the third layer of the trigger tree. Those signals
are fanned-out by three slave boards and distributed to all boards of the trigger and DRS.
The jitter of distributed clock is measured to be less than 30 ps.
The trigger for the µ+ → e+ γ event candidate (MEG trigger) is decided by the following three conditions: the gamma energy, positron-gamma time coincidence, and positrongamma direction match. Those kinematic conditions are checked with fast reconstruction
algorithms implemented in the FPGA. A requirement of global trigger latency to be lessthan ∼500 ns prevents us from using information of the drift chambers that have slower
signal due to the drift time. The pulse height of sum waveform of all PMTs gives anestimation of a gamma energy. A set of PMT-calibration factors can be incorporated

3.3. Electronics and Data Acquisition49

in the online reconstruction. The threshold was set to around 40 MeV to guarantee thefull eﬃciency at the signal energy. The online resolution is measured to be 13.8 % in
FWHM. The time coincidence is checked using the time diﬀerence between the PMTs
from the gamma-ray detector and the timing φ-counter. The PMT time is extracted bya parabolic interpolation of the waveform leading edge. We set the time window to 20 ns
taking the worse time resolution and the spread of the positron time-of-ﬂight into account.
The online time resolution is measured to be 3.4 ns in sigma. The position of the innerface PMT which observes the maximum light gives an estimation of the gamma direction.
There is a correlation between the emission angle of a positron on the target and the hit
position on the timing counter for the signal energy positron. This correlation is used
to estimate positron direction. In 2008, the charge ratio of φ-counter PMTs was used
for the z hit position reconstruction instead of the APD signal from the z-counter. The
resolution is measured to be 7.2 cm in sigma. The correlation is investigated using the MC
simulation, and a look-up table was formed in advance. Collinearity of the gamma andpositron directions is checked with this table and the index of the maximum output PMTin the gamma-ray detector. With these three conditions, the trigger rate was successfully
reduced to ∼ 6.5 Hz when the muon stopping rate was 3×107 /sec with the trigger latencyof less than 400 ns.
In addition to the MEG trigger, various kinds of trigger settings were prepared forthe calibration and the normalization. Table 3.4 lists those trigger settings. Multi-trigger
settings can be used simultaneously with a set of pre-scale factors. In the normal datataking, total 11 settings were used with the pre-scale factors listed in the table. The
trigger system also provides trigger rate scalers for each type of trigger setting which canbe used to calculate eﬃciencies and the normalization factor.

3.3.4

Domino Ring Sampler

The waveform image gives us useful information such as pileup and particle identiﬁcation,
noise properties, and event-by-event baseline, as well as the precise charge and time information. Thus the waveform digitizing can be a superior to the conventional ADC/TDC.
In high-energy physics, FADCs have been widely used as a waveform digitizer. They
are typically used in the range of 50 to 250 MHz sampling and 10 to 12 bits resolution.
However, with going to higher sampling frequencies, they suﬀer from the high power consumption and high cost. An alternative to the FADC is the usage of switched capacitor
arrays (SCA). The DRS is one of the SCAs developed at PSI.
A channel of DRS contains 1024 capacitive sampling cells fabricated in a 0.25 µm
CMOS and samples waveform at frequency raging from 0.5 to 5 GHz. The principleof the operation is illustrated in Figure 3.35. The sampling frequency is generated onchip by a series of inverters because it is very hard to generate and distribute clock
signals in the GHz range, The sampling frequency is freely and continuously running on
the inverter chain in circular fashion (domino wave). When an external trigger signal
comes, the domino wave is stopped and the waveform stored in the sampling cells is kept.
The stored signal is then read out by a shift resistor at lower frequency (33 MHz) anddigitized externally by a 12-bits commercial FADC. Several channels can be multiplexed
into a single FADC channels to reduce the cost. For the synchronization among chips, anexternal global clock can be sampled in each chip. A DRS chip consists of eight signal
channels, a trigger signal channel and a clock signal channel. Two chips are mounted on

50

CHAPTER 3. Experimental Apparatus
Table 3.4: List of trigger settings.

#

Name

0
1
2
3
4
5
6
7
8
9
10
14
16
18
22
24
27
31

MEG
MEG-Q
MEG-D
MEG-T
RD
RD-T
Pi0
Pi0-T
NaI
LXeHighQ
LXeLowQ
LED
Michel
DCH
TIC
TICPair
LXeCR
Pedestal

Prescale inphysics run
1
150
500
100
1000
8000
10
107
107
600
20000

Descriptionµ → eγ . (QLXe > Qhigh ) ∧ (|∆T | < TN )∧ direction-matchµ → eγ . low QLXe thresholdµ → eγ . wide direction-match windowµ → eγ . wide time-coincidence window
RD, π 0 -Dalitz. (QLXe > Qhigh ) ∧ (|∆T | < TN )RD, CW-B. wide time-coincidence window
π 0 . QLXe ∧ QN aI ∧ time-coincidence b/w LXe and NaI
π 0 . QLXe ∧ QN aI
NaI self. (QN aI )LXe self. (QLXe > Qhigh )LXe self, CW-Li, alpha. (QLXe > Qlow ).
LED. trigger signal from LED driver.
Michel. DCH hits ∧ TIC hits
Michel (DCH self).
Michel (TIC self), Cosmic-ray on TIC.
TIC test. A pair of TIC bar hits.
Cosmic-ray in LXe. (QLXe > QCR )Pedestal.

a mezzanine board, and two mezzanine boards are mounted on a VME board. A pictureof the boards and chips is shown in Figure 3.36.
We used mainly the second version of DRS (DRS2) [76] in 2008. However, since we
had a few boards of the succession version, DRS3 [77] which were found to posses bettertiming performance, we used them for the digitizing the DTD-output NIM signals of
the timing φ-counter. The DRS2 reads input signal in single-ended though the active
splitter outputs diﬀerential signal. On the other hand, the DRS3 reads diﬀerentially. The
dynamic range of DRS2 is from 0 to ∼0.5 V, limited by the non-linear response. To meetthis range, the PMT signal is inverted at the active splitter and the resistive attenuation
is mounted on the mezzanine board. We put the attenuator of factor 1.5 for the readoutof the gamma-ray detector. Together with the eﬀect of single-end readout, the signal isattenuated by a factor 3.
We set the sampling frequency to 1.6 GHz for the outputs from the gamma-ray detector
and the timing φ-counter. The total window of 640 ns works as a pipe-line for thetrigger latency and eliminates any delay cables. For the drift chambers, 0.5 GHz was
adopted since high time resolution is not necessary. Additional trigger delay of 500 ns
was artiﬁcially inserted to compensate the signal delay due to the drift time.

3.3.5

Data Size

Since waveform of each channel is recorded, the data size becomes huge. If no data
reduction is applied, the maximum data size is over 5 MB/event. Such large-size data

3.3. Electronics and Data Acquisition
Inverter "Domino" Chain

51

Rotating
signalInput

OutputShift Register

Figure 3.35: Schematic of DRS principle. The inverter chain generates the sampling
frequency (Domino wave) and the waveform is sampled in the series of sampling cells(switched-capacitor array). Using the shift register, the waveform stored in the cells isread out at slower speed.

Figure 3.36: Picture of DRS chips and boards. Right top boards is a DRS3 mezzanine
board with a USB adapter board. Right bottom board is a DRS2 mezzanine board. Eachmezzanine board mounts two DRS chips. Left board is a VME board which mounts two
mezzanine boards. In total, 32 channels can be digitized by a VME board.

cause several practical problems such as I/O speed and space of data storage. Thus, somedata reductions are applied. For the raw DRS waveform, the following algorithms are
implemented for each sub-detector:• zero-suppression (DCH, TIC)
• region-of-interest (TIC)• re-binning (DCH, LXe)Those reductions are applied in online frontend. Then in oﬄine, ﬁles are compressed withZIP by factor 2. The typical data size is about 1.5 MB/event. In 2008, we took 31 TB
data.

52

CHAPTER 3. Experimental Apparatus

3.4

Calibration Apparatus

3.4.1

π − Beam and Hydrogen Target

To calibrate the gamma-ray detector with high-energy gamma rays around and beyond
50 MeV, we use gamma rays from decays of neutral pion (π 0 → γγ). A neutral pion
is produced by a charge exchange reaction of a negative pion (π − p → π 0 n). It has amomentum of 28 MeV/c in laboratory frame and decays immediately into two gamma
rays with almost 100 % branching ratio. The spectrum of the gamma rays is continuous
between 54.9 and 82.9 MeV. However, because of the decay kinematics, the energies andopening angle of the two gamma rays have strong correlation as illustrated in Figure 3.37.
We can get almost monochromatic gamma rays at 54.9 and 82.9 MeV by selecting eventswith almost back-to-back gamma rays. For example, if we require the opening angle larger
than 175◦ (170◦ ), then the FWHM of the energy distribution becomes 0.5 % (2.3 %). In
addition, a 129 MeV gamma-line from the radiative capture reaction of negative pion
(π − p → γn) is available. The Dalitz decay of neutral pion (π 0 → γe+ e− ) is also aninteresting process. We can study the time synchronization and the time resolution of thedetector with a similar event topology to the µ+ → e+ γ decay.

k1= (k10,k1)

π0(E, p)

ψ

Opening angle ψ (degree)

180

175

170

165

160

k2= (k20,k2)
155

150

50

55

60

65

70

75

80

85

Gamma energy (MeV)

Figure 3.37: π 0 decay kinematics. The plot shows correlation between two gammas’
energies and their opening angle.

The πE5 channel can provide also pion beam. Because of the limitation of the BTS
current, momentum of the pion beam is set to 70.5 MeV/c, which is optimal within
the available momentum to separate pion from contaminants (beam electrons and cloud
muons). The spot size was measured to be σx ∼ 8.5 and σy ∼ 7.5 mm. The pion intensity
at the center of COBRA is calculated to be ∼ 1.5 MHz at the proton current of 1.8 mA.
We also prepared several settings with reduced beam intensity.
We use liquid hydrogen (LH2 ) as a target of the charge exchange reaction. Hydrogen isliqueﬁed in a cylindrical cell of 50 mm diameter, 75 mm length whose beam entrance side
is made of a thin 135 µm Mylar window. A picture of the cell is shown in Figure 3.38.
During the normal operation, the cell is ﬁlled with ∼150 cc liquid hydrogen and kept
lower than 20 K by a ﬂow of liquid helium.
It takes ∼5 days to set the beam and prepare the target.

3.4. Calibration Apparatus

53

Figure 3.38: Picture of LH2 target cell.

3.4.2

NaI Detector

To detect one of the gamma rays from π 0 decay, a NaI detector is placed at the opposite
side of the LXe gamma-ray detector. The NaI detector consists of nine crystals of NaI(62.5 × 62.5 × 300.5 mm3 ) each of which is viewed by an APD (10 × 10 mm2 , Hamamatsu
S8664-1010), two plastic scintillation counters (70 × 70 × 7 mm3 ) each of which is viewed
by two ﬁne-mesh PMTs (Hamamatsu H6152-70), and a lead plate (5 mm thickness)in front of the plastic counters as shown in Figure 3.39. This detector is mounted ona movable stage shown in Figure 3.40. We can obtain the back-to-back gamma rays atarbitrary regions of the LXe detector by moving the position of the NaI detector.

Figure 3.39: Picture of the NaI detector.
It consists of a lead converter, two plastic
scintillators, and nine NaI crystals.

Figure 3.40: Movable stage of the
NaI detector to be placed at the opposite side of the LXe gamma-ray detector.

54

3.4.3

CHAPTER 3. Experimental Apparatus

Cockcroft-Walton Proton Accelerator

The reaction 73 Li(p, γ)84 Be produces a 17.6 MeV gamma-line [78]. This reaction is excitablewith very low-energy protons, resonant at Ep = 440 keV with a resonance-width Γ ≈15 keV. It is almost unique in providing high-energy gamma rays with a large peakcross section (σpeak ≈ 5 mb). Therefore, it is very useful to monitor and calibrate the
gamma-ray detector although the energy is three times smaller than that of gamma fromµ+ → e+ γ decay. Actually, it was used in a previous µ+ → e+ γ search experiment [9]. In
addition, the reaction 115 B(p, γ)126 C excitable by protons, resonant at Ep = 163 keV is alsointeresting. It produces gamma-lines at 16.1, 11.7, 6.5, and 4.4 MeV. Since the 11.7 and
4.4 MeV gammas are produced in time coincidence with no angular correlation, they canbe used to calibrate timing between the gamma-ray detector and the timing counters.
We use a dedicated 1 MeV Cockcroft-Walton (CW) accelerator [79] (Figure 3.41) toproduce protons exciting those reactions. Table.3.5 summarizes the properties of theaccelerator. The accelerator is placed at the downstream side of the detector system
in a separate area. The proton beam goes to the center of COBRA in the opposite
direction of the muon beam. The proton beam intensity is 1012 /sec. We use a lithium
tetraborate (Li2 B4 O7 ) crystal disk target [80] which allows us to get the Li- and the Blines simultaneously. The rate of 17.6 MeV line is ≈ 106 /sec, isotropically. Part of the
proton beamline is made of bellows. This extendable beam pipe is remotely controlled andinserted into the COBRA volume to set the nuclear target at the center of COBRA withremoving the muon stopping target to the parking place. Owing to this target exchangesystem, we can move from normal muon mode to the CW mode in ∼20 minutes. The
details of MEG CW accelerator and its setup, operation, and calibration methods aredescribed in [81].
Table 3.5: Properties of MEG CW accelerator.
Nominal Measured at PSI
Terminal energy range (keV)
300 − 900
200 − 1100
Energy ripple(RMS eV)
< 500
< 50
Angular divergence (mrad×mrad)
5×5∼4×4
Spot size at 3 m (cm×cm)<3×3<1
Energy setting reproducibility (%)
0.1
OKEnergy stability (FWHM %)0.1
OK
Range of current (µA)
1 − 100
0.1 − 135
3
OK
Current stability (%)
10
OK
Current reproducibility (%)
Duty cycle (%)
100
OK
< 20
< 15
Start-up time (min)3.4. Calibration Apparatus

Figure 3.41: 1 MeV Cockcroft-Walton proton accelerator.

Figure 3.42: Proton beamline from theCW accelerator to the detector center.
The CW accelerator is placed at the downstream of the πE5 area.

Figure 3.43: Li2 B4 O7 target for the
proton-nucleus reactions. The target ismounted at the edge of the beamline andinserted to the center of the COBRA magnet with the bellow system.

55

56

3.5

CHAPTER 3. Experimental Apparatus

Simulation and Analysis Software

The MEG software is composed of three packages; MEGMC, MEGBartender, and MEGAnalyzer. The structure is shown in Figure 3.44. MEGMC and MEGBartender generate
simulation data and MEGAnalyzer analyzes data from both experiment and simulation.
A detailed simulation of the full apparatus was developed and used throughout theexperiment, from the design and optimization of all sub-systems to calculation of acceptances and eﬃciencies. The following sections describe a series of simulation processes.

Figure 3.44: Structure of MEG software.

3.5.1

Event Generation

MEGMC is a GEANT3.21 [82] based Monte Calro simulation (MC). It processes the event
generation and the detector simulation.
A muon decay on target can be generated both for normal SM decays and µ+ →+e γ decay. It is also possible to generate a muon in beamline and let it propagate anddecay. In addition to the event from muon decay, we prepared generations of several
calibration events. Table 3.6 lists the event types implemented in MEGMC. For each
event type, users can conﬁgure the kinematic conditions such as the phase space of the
decay, momentum ranges of particles.

3.5.2

Detector Geometry and Response

All apparatus components are described in MEGMC. Detailed geometry and materials
are implemented in the standard GEANT3 manner.
The transport of particles and their interactions with material are simulated by the
GEANT-tracking. All relevant electromagnetic processes (pair production, bremsstrahlung,Compton scattering, photo-electric eﬀect, ionization losses, multiple scattering and positronannihilation) are included, and the secondary particles are traced down to 10 keV. When

3.5. Simulation and Analysis Software

57

Table 3.6: List of event type implemented in the MC simulation.
µ+ → e+ γ signal event
Radiative decay (RD) event
Michel decay eventµ+ beame+ at 52.8 MeV
e+ from ﬂat spectrum
γ at 52.8 MeV
γ from RD
γ from e+ annihilation-in-ﬂight
γ from ﬂat spectrum
Cosmic-ray µ
α from 241 Am source on wirep+ from CW accelerator + LiF target
p+ from CW accelerator + B target
π 0 → γγ decay

particles interact with the active volume, the hit information such as space-time and energy deposit are recorded. The response of each sub-detector is then precisely simulated.
Simulation of drift cell The precise isochrone map of the drift-chamber cell is calculated with GARFIELD [83] which is widely used to simulate wire chambers in high-energy
experiments. It provides the ﬁeld map, drift lines, drift time tables and arrival time distributions of electrons and ions. The details of energy loss of fast charged particles ingases are computed by the HEED [84] program. The electron transport parameters areprovided by the MAGBOLTZ [85] program.
Scintillation photons in timing counter The propagation of scintillation photons in
scintillation bar is calculated analytically taking into account attenuation in the bar andreﬂection on the surface. The photo-electric eﬀect of PMT photo-cathode is simulated inaccordance with the Poisson statistics.
Scintillation photons in liquid xenon The transportation of scintillation photons
in LXe is simulated with our original code. The scintillation photons are isotropicallygenerated at each point of interaction with LXe in accordance with the deposit energy andscintillation eﬃciency. Then every photon is traced. During the tracking, several opticalprocesses are simulated such as absorption, Rayleigh scattering, reﬂection, transmittanceof PMT window and photo-electric eﬀect on photo-cathode.

3.5.3

Event Mixing

Simulating actual experiment requires to overlay several activities in a event. MEGBartender, which is the other simulation package developed by the MEG collaboration fromscratch, does this. It can read multiple events generated by MEGMC and mix them to58

CHAPTER 3. Experimental Apparatus

simulate accidental overlap. It is also possible to mix diﬀerent type of events in diﬀerent
output ﬁles of MEGMC. Events are randomly or sequentially picked up from the input
ﬁles and placed randomly in time in accordance with the Poisson distribution.

3.5.4

Electronics Simulation

Another task of MEGBartender is to simulate electronics and form digitized data with thesame format as experimental data, namely DRS waveform data. Signals from the detector
active volume pass through various devices such as PMT, wire/pad, cables, connectors,
splitter, and ﬁnally reach DRS. It is diﬃcult to simulate each element individually. However, if those devices are linear systems, we can regard the entire system as a device. Only
necessary information is the impulse response of the system.
For the drift chamber, we obtained the impulse response using the signal of 5.4 keV
soft X-ray from 54 Mn source. The soft X-ray usually generates a single-electron avalanche.
Signal from those events at the DRS represents the impulse response of the system. For
the gamma-ray detector, we used MC simulation for the input and the average waveformof real signal for the output. We can get the impulse response by deconvolution them.
Once we obtained the impulse response, we can synthesize waveform by convolving hit
information from MEGMC with the impulse response. We use a Fast Fourier Transform
technique to calculate the convolution.
For the timing counter, we synthesize waveform in analytical way since the execution
time is much faster than that by the convolution. We apply a series of digital ﬁlterswith some parameters: a randomization with a Gaussian for the transit time spread of
the PMT, a ﬁlter with a function t2 exp(−(t/s)2 ) as the single-electron response, and a
RC-ﬁlter and reﬂection for the electronics response. The parameters were adjusted sothat the output waveform can reproduce real data.
Before the digitization, electric noise is overlaid to the formed waveform. We generated noise with white and 1/f components, and applied a low-pass ﬁlter to simulate thefrequency property. Then, we digitize the waveform to simulate DRS both in samplingand quantizing.

3.5.5

Analysis Framework

We analyze all processes, from handling of raw data to the physics analysis, in MEGAnalyzer. It can handle both experimental and simulation data in the same manner. It, andalso MEGBartender, are organized by the analysis framework toolkit, ROME [86]. The
ROME-generated software is highly modular. The analysis processes and information(such as data, parameters and analysis results) are treated as objects, tasks and folders
respectively. A series of calculation codes is split up into tasks, which contain one orseveral calculation steps. We can exchange tasks arbitrarily as long as they access thesame data folders. This modularity makes it very ﬂexible to control the whole structure.
Any user can easily add and write their own code with simple knowledge of C++ and
ROOT [87].
Furthermore, the framework has features such as connection to DAQ systems, databaseaccess, socket connection, and Graphical User Interface (GUI) extension. Therefore,
MEGAnalyzer also works as an online monitor and an event display.

Chapter 4
Run
4.1

Run 2008

We started experiment in 2008 with a conditioning and calibrating Michel run in Junejust after completing detector installation. From end of July to beginning of September,
we conducted a π 0 calibration run. After a small period of trigger setup and backgroundstudy, ﬁnally we started a physics run on 12 September. It continued until 17 December.
Those data are the initial three month data of the MEG experiment and full data set ofthe analysis in this thesis. Run 2008 ﬁnished with another small π 0 calibration run and
it enters in annual accelerator-showdown period of PSI.

4.1.1

Physics Run

The normal run period consisted of the MEG runs taking physics data with the MEG
trigger, calibration runs (three times per week), dedicate RD runs (24 hours per week), andprogrammed beam-maintenance periods (three days per three weeks). In the MEG runs,
we mixed 11 diﬀerent types of trigger for the normalization, calibration, and monitoring(see Table 3.4). We took data with the normal mode of the muon beam (3 × 107 µ+ /sec,
Sec.3.1.5). Figure 4.1 shows the integrated amount of physics data as a function of date.
The livetime was 3.3 Msec with ∼85 % eﬃciency. In total, 9.5 × 1013 muons stopped on
the target.

4.1.2

Dedicated Radiative Decay Run

The RD events, which were introduced as one of the background sources before, can beused as a calibration source of relative timing between positron and gamma. Moreover, it
is very important to see the apparatus can detect RD events correctly to demonstrate thecapability of observing µ+ → e+ γ events. To clearly observe RD events, we conducted datataking with reduced-intensity beam by a factor 25 (the ultra-low mode). The reducedintensity beam suppresses the accidental background and makes signal-to-noise better.
We used the RD triggers (Trigger#4,5), which relax the condition of angular correlation
between positron and gamma. The energy threshold for gamma was also reduced toaround 25 MeV. The energy spectrum of gamma from RD and angular correlation arestudied as well as relative time. These special runs were carried out intermittently during
run 2008 for one day per week. They are indicated in Figure 4.1 as green arrows.
59

100
6000
5000
4000
3000

80
60

µ+ on target (× 1012)

CHAPTER 4. Run

Integrated proton current (C)

60

40

2000
1000

20

0
0
04/09 18/09 02/10 16/10 30/10 13/11 27/11 11/12 25/12
Date

Figure 4.1: Integrated amount of physics data taken in 2008. In total, data of 6300 C
proton current was acquired, which corresponds to 9.5 × 1013 µ+ on target. Gray arrows
show beam shutdown due to programmed beam maintenance, and green arrows show
radiative decay run periods.

4.1.3

π 0 Calibration Run

We conducted two π 0 runs, one at the beginning (August) and the other at the end(December) of the data taking period. The purposes of the π 0 runs are the following:• to evaluate energy, time, and position resolutions of the gamma-ray detector, andto get the responses,• to calibrate energy scale of the gamma-ray detector,• to study relative timing between positron and gamma and calibrate its t0 ,• to measure detection eﬃciency of the gamma-ray detector.
To achieve these purpose, we carried out several kinds of data taking.
π 0 → γγ patch scan To evaluate energy and time response at diﬀerent positions of thedetector, a full scan of detector inner face was performed with π 0 → γγ reaction. We
divided the inner face into 24 parts and prepared patches of trigger for each part. Each of
the patches contains 3 × 3 PMTs. The conﬁguration of the patches is shown in Figure 4.2.
Depending on the trigger patch, we moved the NaI detector to the opposite position of
the patch to collect back-to-back events. The trigger is ﬁred by a coincidence between
LXe patch and reference counter (Trigger#6) or NaI (Trigger#7). We use the former forthe time performance and the latter for the energy performance. The fraction of events
is about 3:7. We collected 200 ∼ 300k events for each patch.

4.2. Run Condition

61

π 0 → γγ reference run For the reference of the patch scan, we everyday took π 0 → γγdata at the same position (patch#8). We use them for corrections of light yield during
the scan. We determine the energy scale of the gamma-ray detector with those run.
Pb brick run To measure the position resolution, we took special data with lead bricks
mounted just in front of the gamma-ray detector entrance window. The lead bricks are
1.8 cm thick and have some slits of 1 cm width. Shadow of the bricks can be used
to estimate the position resolution. Figure 4.3 shows the conﬁguration of the bricks
installation. The design and picture of the lead bricks are shown in Figure 4.4. We usedthe LXe-self trigger (Trigger #9) and collected ∼ 1M events for each brick position.
π 0 → γe+ e− Dalitz decay To study relative time between positron gamma, we collected
π 0 Dalitz decay data. We used the RD trigger (Trigger#4) with gamma energy window
around 55 MeV. In total, we collected 500k events.
NaI single We collected NaI-self trigger data to evaluate the detection eﬃciency of
gamma-ray detector.

4.1.4

Normal Calibration Runs

We took a full set of calibration data three times per week. It takes about three hours totake one set. A set of the calibration consists of the following runs,• Pedestal run to check electronics and evaluate noise,• LED run to calibrate the PMT gains; it was taken everyday with beam on and oﬀ
conditions,• Alpha run to calibrate the PMT quantum eﬃciencies and monitor LXe (30k events),• Cosmic-ray run to monitor LXe (50k events),• CW-Li run to monitor and calibrate LXe light yield and to correct non-uniformityof the gamma-ray detector (30k events),• CW-B run to calibrate the timing counter and to monitor relative timing between
the gamma-ray detector and the timing counters (10k events).

4.2

Run Condition

4.2.1

Discharge Problem on Drift Chamber

We suﬀered from a severe problem on drift-chamber operation in 2008. Several chambers
frequently discharged. This problem is thought to be posed by helium gas permeated into
HV lines. As described in Sec.3.2.1, the drift chamber is placed in the helium atmosphere.
The outside of the chamber faced with pure-helium. To avoid connection of HV-to-GND
lines via helium gas, we molded the HV lines with glue as a dielectric. However, slowly

62

CHAPTER 4. Run

Figure 4.2: Conﬁguration ofpatch and deﬁnition of patch
number. The patch#8 wasused as a reference.

Figure 4.3: Conﬁguration of lead bricks installation.

(a) Design of a lead brick.

(b) Picture of lead bricks.

Figure 4.4: Lead bricks for a study of position resolution.

4.2. Run Condition

63

1850 (V)

30

1800
25

Plane ID

1750
20
1700
15
1650
10

1600

5

1550

0
18/09

02/10

16/10

30/10

13/11

27/11

1500

11/12
Date

Figure 4.5: History of HVs applied to the drift chambers. HV is applied independently
to each plane (in total 32 planes). The dark regions indicate planes to which we could
not apply HV because of the discharge problem.

helium permeated into the HV lines. Since it took a time for helium to permeate, thechambers suﬀering from the discharge gradually increased. Finally, out of 32 planes, 18
planes were operational, and only 12 planes worked at the nominal voltage. Because ofthis problem, the drift chamber system did not work at the nominal performance. In
particular, the eﬃciency was severely restricted in run 2008.

4.2.2

Helium Concentration inside the Magnet

To relax the potential of the discharge, we introduced small amount of air in the helium
atmosphere as a temporary treatment. On average, 6 % air was doped. The beam tuning
was done before the air-doping, and not corrected the eﬀect of this additional air.

4.2.3

Problems on Timing-Counter APD

There were some problems on the z-counter read out. In the digital output of APD
signal, 3/16 parts were out of control because of a hardware problem on the on-board
electronics in addition to the several hot channels. On the analog output to be used intrigger, the noise level was too high to be used in the trigger decision. Therefore, we
did not use the z-counter neither in trigger nor in the oﬄine analysis. Instead, in trigger
we reconstructed z-coordinate of hit position using the charge ratio of φ-counter PMTs,
while in oﬄine we do using the time diﬀerence between them. These alternative solutions
have worse resolutions. In addition, it was found later that there was a bias in the trigger
estimation of positron-emission angle from the z hit position reconstructed in this way.
Those eﬀects resulted in low trigger eﬃciency.

4.2.4

Purity and Light Yield of LXe

Since when we constructed the gamma-ray detector and ﬁrst liqueﬁed xenon in 2007,
we have progressively improved the purity of LXe by the puriﬁcation. Also during the

64

CHAPTER 4. Run

data taking period, we continued puriﬁcation, and the light yield of LXe scintillation was
improving.
At the beginning of the physics run, the absorption length was estimated to be sufﬁciently long, and it seemed the improvement of observed light amount got saturated
by the liquid-phase puriﬁcation. Therefore, we decided to stop the puriﬁcation and takedata with stable condition, even though the light yield was estimated to be lower thanthe expected value calculated using the data obtained with the prototype detector.
We took data with this stable condition until the end of October. However, we founda slow decrease of the light yield after a while. It is thought due to small leak in liquid
nitrogen cooling pipe in low temperature because there was correlation between the decrease of the light yield and the operation of liquid nitrogen. Or possible out-gas frommaterial in the detector. From when we discovered this correlation, we quited to use
liquid nitrogen pipe inside the detector for an additional cooling. To recover this small
decrease, we performed a liquid-phase puriﬁcation during a short beam-maintenance period. Then we found further improvement of the light yield, and newly found an eﬀectof gas-phase puriﬁcation which did not show a saturation of improvement. We thinkthat previously the gas-phase puriﬁcation had not worked eﬃciently because of a problem
on the gas-phase puriﬁcation pump, and after we ﬁxed the pump it started eﬀectively
working. The improvement of the light yield during the data taking causes an instability
in measurement point of view, but to understand the detector well and to extract a full
performance of LXe scintillation, continuous puriﬁcation was demanded. On the other
hand, we had established the monitoring and calibrating methods using the various kindsof periodic calibration data. Therefore, we decided to continue puriﬁcation and correct
the energy scale using the calibration data in oﬄine.
From the end of October to the end of the run, we continuously performed the gasphase puriﬁcation. In addition, we performed the liquid-phase in the periodic short beammaintenance periods. The light yield kept increasing until the end of the run. Roughly,
we observed 30 % of increase during the run. Yet at the end, it was still lower thanthe expectation by factor ∼1.5.1 Not only the light yield but also the pulse shape werechanged by the puriﬁcation. This change caused variation of the time measurement.
Monitoring and calibrating the light yield is described in Sec.6.4.3 and those of the timereference is in Sec.6.5.

4.2.5

Bad Channels

The timing φ-counter had a bad channel for the ﬁrst half data. All of the PMTs were
working ﬁne. There was, however, a broken channel of the DTD, and it was exchanged
at the end of October. After that, all channels were working.
There were three dead channels of the gamma-ray detector PMTs from the beginning.
One lost connection in signal line, and the other two had short in HV line. A channel
showed strange pulse shape and another channel showed unstable PMT gain. In addition,
two PMTs showed severe decrease of gains during the run. The point is that all the PMTs
on inner face worked. We did not lose any acceptance.

1

We performed further puriﬁcation during the long accelerator-shutdown period. At the beginning of2009 operation, the light yield became the same level of the expectation.

Chapter 5
Event Reconstruction
5.1

Overview

In this chapter, we describe the reconstruction of each kinematic variable focusing on themethods and algorithms. The reconstruction of MEG starts from the analysis of waveform
to extract physical quantities. Next, kinematic variables of each particle are reconstructed.
In this stage, we treat positron and gamma separately. Finally, we reconstruct the relative
angle and time between positron and gamma by combining each reconstruction. Figure 5.1
overviews the ﬂow of the reconstruction. To extract full performance of those methods,
precise calibrations are necessary in advance. They are described in the next chapter.
The performance of the reconstruction is evaluated in Chapter 7 using obtained data.

Figure 5.1: Flow chart of reconstruction.

65

66

CHAPTER 5. Event Reconstruction

5.2

Waveform Analysis

5.2.1

Waveform Analysis of Drift Chamber

We get six waveforms from a drift cell: two from both ends of an anode wire and four from
cathode pads (inner and outer pads with two vernier-pattern pads). Figure 5.2 shows anexample of six waveforms associated with a drift cell. We collectively treat those six.
First, using the waveforms of anode wire, we search for pulses. A simple peak-searchmethod cannot be used since the waveform of drift chamber has a multi-pulse structureof ionization clusters in single particle pass. Thus a pulse and its width is determined as
the following procedure. A pulse is identiﬁed by the maximum peak over a given threshold.
Then the width of the pulse is determined by pursuing the pulse in both directions until
it goes down under another threshold. This pulse range is masked, and the pulse searchis repeated.

Time extraction Time of each pulse is determined by a single-threshold crossing time.

Charge extraction The integration window of each pulse is determined by the anode
waveform and common to the six waveforms. The window width is set to 50 ns, which
was tuned to optimize the signal-to-noise ratio.

5.2.2

Waveform Analysis of Timing Counter

As described in Sec.3.3.1, the output of a φ-counter PMT is digitized in two ways. A
PMT output is divided passively into three outputs with fraction of 1:1:8. One of the 10%fraction outputs is digitized by DRS directly (PMT pulse). Together with the attenuation
factor two on the DRS board, the digitized signal totally becomes 5 % of the original
PMT output. The 80%-fraction one goes to the double-threshold discriminator (DTD)
and its output (NIM pulse) is digitized. We use DRS3 for the NIM-pulse digitization as
already mentioned in Sec.3.3.1. The other output of the splitter goes to a current monitor.
Figure 5.3 shows typical waveforms associated with a PMT.

Time extraction We use the NIM pulse to get a pulse time. We prepared in advancea template waveform by averaging many pulses channel-by-channel. It is used as a ﬁtting
function of the NIM pulse. The free parameters of the ﬁtting are baseline level andleading-edge time of the pulse.

Charge and amplitude A charge and an amplitude corresponding to the time-measured
NIM pulse is estimated by using the PMT pulse. The charge is estimated by integrating
pulse over 30 ns. The amplitude is measured as the voltage diﬀerence between the estimated baseline and the peak voltage of the pulse. We use the charge to measure the gains
to equalize them. We use the amplitude to correct the time-walk eﬀect of the NIM-pulse
time.

90
80
70

anode

80
70

60

60

50

50

40

40

30

30

20

20

10

10

0
-1000

-500

-10-20000(nsec)

35
30

25
20

15

15

10

10

5

5

0

0

-5

-5
-1000

-1000

-500

-1500

-1000

-500

-10
-2000

-500

(nse)

35

(mV)

(mV)

(nsec)

35
30

30

25

25

20

20

15

15

10

10

5

5

0

0

-5

-5

-10
-2000

-1500

-1000

0
(nsec)

30

20

-1500

-1500

35

25

-10
-2000

anode

0
-1500

(mV)

(mV)

-10-2000

67

90

(mV)

(mV)

5.2. Waveform Analysis

-10
-2000

-500

-1500

-1000

-500

(nsec)(nsec)

Amplitude (mV)

Amplitude (mV)

Figure 5.2: Typical waveforms of drift chamber. Those six belong to a drift cell. The
red horizontal line shows the calculated baseline, and vertical one shows the pulse timing.
The region between light blue dashed lines shows the expected region where triggeredsignal appear. Two sides of the cathode waveforms are re-binned for data reduction.

0
-100
-200

0
-20
-40

-300

-60
-400

-80

-500

-100

-600

-120

-700
-600

-500

-400

(a) Waveforms of a PMT-300
Time (nsec)

-500

-450

-400
Time (nsec)(b) Close-up view

Figure 5.3: Typical waveforms from a TIC PMT. Blue lines show waveform of attenuated
PMT signal (PMT pulse). Red lines show output pulse of double-threshold discriminator(NIM pulse) digitized by DRS3. A template waveform ﬁtted to the NIM pulse is overlaid
as black lines. The NIM pulses come with a delay of ∼ 20 ns.

Amplitude (mV)

CHAPTER 5. Event Reconstruction
Amplitude (mV)

68

0
-5
-10
-15

0
-500
-1000
-1500

-20

-2000

-25

-2500

-30

-3000

-35

-3500
-4000

-40
-600

-400

-200

-600

0

-400

-200

Time (nsec)(a) Waveform of a PMTTime (nsec)(b) Sum waveform over all PMTs

Figure 5.4: Typical waveform of LXe interacting with a gamma ray. (a) shows waveformof a PMT. (b) shows sum of waveform over all PMTs. The peak at the beginning of thewindow is related to the stop signal.

5.2.3

Waveform Analysis of Gamma-ray Detector

Typical waveform from LXe interacted with a gamma ray is shown in Figure 5.4. For thedata reduction, every eight sampling points are averaged and recorded as a point except
for the interesting regions; a leading-edge part and a baseline part just in front of theleading edge. The baseline is estimated by averaging the points in the baseline region
event-by-event for each channel.

Amplitude (mV)

Amplitude (mV)Time extraction For the time extraction, we do not apply any digital ﬁlters to retain
maximum information of pulse shape and band width. We extract the time of a pulse
with the digital-constant-fraction method. It determines the pulse time as a time at which
the signal reaches a given fraction (here 30 %) of the full pulse height (Figure 5.5). By
this method, we can determine the pulse time independently of the amplitude (no timewalk eﬀect). The pulse height is estimated by using the charge of the pulse not by the
direct measurement of the amplitude so that the noise and the photoelectron statisticalﬂuctuation in the pulse shape are eliminated. The measured pulse charge is converted toa voltage with a factor calculated by assuming the ﬁxed pulse shape. The level crossing
time is extracted with linear interpolation of the adjacent two points.

0
-5

30%

-10
-15
-20

50 ns

10

0

-10

-25
-20

-30
-35

-30

-40

Mov-Ave(high-pass) 89pnts
-600

-500

-400

-300

Time (nsec)

Figure 5.5: Time extraction by thedigital-constant-fraction method.
The
fraction is set to 30% of the full amplitude.

-600

-400

-200

0

Time (nsec)

Figure 5.6: LXe waveform shaped witha high-pass ﬁlter.

5.2. Waveform Analysis

69

Number of events

Charge estimation We estimate a pulse charge, which is later converted to the numberof photoelectrons observed by the PMT, by integrating the pulse. Since we have fullwaveform, digital ﬁlters can be used in oﬄine process to maximize the signal-to-noise
ratio. We observed some slow-component noise of ∼ 1 MHz. To eliminate this kind ofnoise, we apply a high-pass ﬁlter. On the other hand, we do not need ﬁltering the highfrequency components of noise since the integration itself works as a powerful low-passﬁlter, and charge estimation is not inﬂuenced so much. We form a high-pass ﬁlter basedon the moving-average method, which is usually used in a low-pass ﬁlter. The high-passﬁlter works as subtraction of waveforms between raw waveform and (low-pass) movingaveraged one. The number of averaging points is set to 89, which is corresponding to a
cut-oﬀ frequency of 11 MHz. It was adjusted with a trade oﬀ between gain and noise
reduction power. The ﬁltered waveform is shown in Figure 5.6. The gain is about 0.35.
The baseline becomes ﬂat and at zero. The pulse shape becomes sharper, resulting inshorter integration width. The integration range is determined by the sum waveform anda common window is used for all individual channels. It is determined as a window of48 ns width just in front of the zero-crossing time. By applying the ﬁlter, we successfully
reduce the noise contribution to the energy resolution from 0.6 % to 0.4 % of 52.8 MeV.
Figure 5.7 shows reconstructed energy of pedestal events with and without applying theﬁlter. The shorter window also works eﬀectively to reduce pileup events.

900

Constant

800

Mean

-0.1824 ± 0.0053

700

Sigma

0.3031 ± 0.0051

600

Constant

500

532.7 ± 11.3

822.2 ± 17.1

Mean

-0.02109 ± 0.00326

Sigma

0.2045 ± 0.0030

400
300
200
100
0
-4

-2

0

2

4
(MeV)

Figure 5.7: Reconstructed energy of pedestal events in normal data taking. The result
when we use normal waveform to estimate PMT charges is shown in ﬁlled histogram andthat with high-pass ﬁltered waveform is shown in hatched one.

When a gamma ray interacts at a very close point to a PMT, the PMT observes verylarge pulse and the signal gets saturated with a limited dynamic range of the electronics.
Figure 5.8(a) shows an example of saturated signal. To recover these saturated channels,
we use the time-over-threshold (ToT) method in estimating the charge. The ToT is aduration of the pulse over a given threshold. The threshold was set to 150 mV. We
know the average pulse shape of gamma-ray interaction events (shown in Figure 5.8(b)as a “template waveform”). Hence, we can convert the ToT value to the charge. The
recovering is important to achieve a high eﬃciency of gamma-ray detection because theprobability of gamma-ray interaction at shallow region is high; about 15 % events interact

70

CHAPTER 5. Event Reconstruction
Pulse height : ToT @ 150mV
Pusle height (V)Amplitude (a.u.)

0

-0.2
-0.4
-0.6

0
-0.5
-1
-1.5

-0.8

-2

-1

-2.5

-1.2

-100

-3
0

100

-200

0

200

50

100

150

200

ToT (nsec)

Time (nsec)

Charge : ToT @ 150mV-300

Charge (nC)

Amplitude (mV)

Template Waveform
Waveform
0

-400
-500

0
-0.5
-1

-600

-1.5

-700
-600

-400

-200

0

-2

Time (nsec)

-2.5
-3
0

(a) An example of saturated channel.

50

100

150

200
ToT (nsec)(b) ToT - charge relation.

Figure 5.8: Saturated signal and charge estimation with ToT. When a pulse saturates,
the ToT method is used for charge estimation instead of charge integration. Pulse shapeof template waveform is used to get the conversion factor from ToT to charge.

within 1 cm and most of them have at least one saturated channel.

5.3

Positron Reconstruction

5.3.1

Hit Reconstruction and Track Finding

Drift Chamber Hit Reconstruction
We measure the z-coordinate of drift-chamber hit by a combination of anode-chargedivision and cathode-vernier-pad. At ﬁrst, we roughly reconstructed it by the ratio of
charges measured at both ends of the hit wire. We deﬁne the charge-divisiona =Qu − Qd,
Qu + Qd(5.1)

where Qu(d) is charge on the upstream (downstream) end. With this charge-division, thez is reconstructed by
(

)

Z L
z=
+
· a ,
ρ2(5.2)

where Z is input impedance, L is wire length, and ρ is wire resistance per unit length.
With this method, we determine the turns of the vernier-pattern period (n).
Then, we reconstruct it more precisely using the vernier pattern. The charge-division
for the vernier pads are similarly deﬁned (1 , 2 ) for inner side cathode pad and outer one,respectively. Using the phase of the vernier pattern, α = tan−1 2 /1 , z is reconstructedbyl· α + n · l,(5.3)
2π
where l is the pattern pitch, equal to 5 cm. Figure 5.9 shows the vernier circle and relation
between anode charge-division and vernier circle.
z=5.3. Positron Reconstruction

(a)

71

(b)

Figure 5.9: z reconstruction of the drift-chamber hit. (a) shows the vernier circle. One
turn corresponds to a period of vernier pattern (5 cm). (b) shows relation of the vernierphase with anode-charge-division.

Track Finding
To ﬁnd tracks, we apply topological pattern recognition. First, using the z and time
information, we cluster hits associated with a positron pass within each module. Since thetwo layers of a module is staggered by a half cell, we can solve the “left-right ambiguity” inmost case. Next, we connect the clusters. If we ﬁnd a “seed” of track with three clusters,then we roughly reconstruct the trajectory with a circle. Using this swim function andhit coordinates, clusters associated with a positron are connected progressively, resulting
in a track candidate. During this process, we can reﬁne the left-right ambiguity. We canalso get information of incident angle to each cell.
For a found track candidate, we can estimate the time of the track, ttrack , by using
the timing of all hits belonging to the track. Using this drift-chamber self-contained ttrack
and the track direction, the timing-counter hit associated with the track is searched for,and the ttrack is reﬁned by the timing-counter hit time. The drift time of each hit is given
by twire,i − ttrack , where twire,i is a time extracted from the i-th wire waveforms.
Given drift time, incident angle, left-right solution, and B-ﬁeld strength, we can calculate the drift distance uniquely. Since the relation has strong angular dependence,
we prepared the time-to-distance functions for diﬀerent incident angle as shown in Figure 5.10. Using these functions, precise hit coordinate is determined.

5.3.2

Tracking

Track Fitting
Next, a trajectory of the track candidate is precisely analyzed by a track ﬁtting and thestate of positron on the target is reconstructed. We use the Kalman ﬁlter technique intrack ﬁtting. The Kalman ﬁlter [88] was originally developed as a linear estimation forthe state of a dynamic system from a series of incomplete and noisy measurements. In
this 20 years, it has been extensively used for track ﬁtting in high-energy physics (for

CHAPTER 5. Event Reconstruction
drift time (µ sec)

72

0.18

+60 deg

B=1.1T

0.16+30 deg

0.14
0 deg

0.12
-30 deg

0.1
0.08-60 deg

0.06
0.04
0.02
0
-0.5 -0.4 -0.3 -0.2 -0.1

0

0.1

0.2

0.3

0.4

0.5

drift distance (cm)Figure 5.10: Time-to-distance functions for various incident angles. In the practical usage, the functions are implemented as two-dimensional isochrone map for diﬀerent angles.

example [89]). It possesses the following features for eﬀective track ﬁtting:• multiple scattering and energy losses are included in natural way;
• a 3-dimensional trajectory is restored that approximates closely the real one;
• complex tracker geometries and non-uniform magnetic ﬁeld are handled in a simple
way;• N×N matrix inversion required in the global least squares ﬁt (where N is the total
number of measurements) is avoided;• control of error propagation is provided.
Those features are suitable for the tracking of the MEG positron spectrometer.
The track of a charged particle in a magnetic ﬁeld at each point can be uniquely
described by a vector of ﬁve parameters: two for the position, two for the direction and one
for the momentum. The trajectory is reconstructed recursively by adding measurements.
The operations of track ﬁtting with Kalman ﬁlter consist of the following three processes:• Prediction (xk−1k ) is estimation of ‘present’ k-th point state vector using all ‘past’
information. The extrapolation takes into account all possible eﬀects that aﬀects
the system (such as multiple scattering and energy loss).
• Filtering (xkk ) is estimation of ‘present’ k-th point state vector with all measurements including k-th measurement. The state vector is updated taking into account,with the appropriate weight, the present measurement and prediction by the previous step.
• Smoothing (xNk ) is backward estimation of state vector with all measurements.

Y (cm)Z (cm)

5.3. Positron Reconstruction

0

73

30
20
10

-10

0
-10

-20
-20
-30

-30
-30

-20

-10

0

10

20

30
X (cm)(a) x − y view

-30

-20

-10

0

10

20
30
X (cm)(b) x − z view

Figure 5.11: An example of track ﬁtting using Kalman ﬁlter.

The evolution of the state vectors (x) are described with a system equation.
xk = fk−1 (xk−1 ) + wk−1 ,(5.4)

where fk−1 is track propagator from ‘k-1’-th measurement to ‘k’-th measurement; w represent a random disturbance. The multiple scattering and energy loss are fetched in thesystem via this term and its covariance matrix. In the presence of a magnetic ﬁled thetrack propagator is nonlinear. It is approximated by a linear function, for example by thetwo terms of its Taylor expansion (extended Kalman ﬁlter) so that the concept of linear
ﬁltering can be applied. The system equation is then given byk−1
xk = fk−1 (xk−1k−1 ) + Fk−1 (xk−1 − xk−1 ) + wk−1 ,(5.5)

where F is the Jacobian matrix of f. The actual measurement (m) is brought in theevolution by converted to the state vector with linear relation of,
mk = Hk xk + k ,(5.6)

where matrix H is called projection matrix;  is the measurement uncertainty.
The accurate reconstruction of trajectory requires a precise track model for the trajectory of the charged particle in a magnetic ﬁeld, namely the deﬁnite description of the
track propagator. Because of the specially graded magnetic ﬁeld, the equation of motionof charged particles in the COBRA ﬁeld must be solved by numerical integration. We
use the ﬁfth order Cash-Karp method [90], which is one of the Runge-Kutta methods andcan maneuver the adaptive step-size eﬀectively to approximate the highly graded ﬁeld
suﬃciently. A measured ﬁeld map is incorporated in the system. A correct and eﬃcientdescription of the multiple scattering and energy loss eﬀects requires the knowledge bothof the detector resolutions and of the amount of material traversed by the particle.
In the process of ﬁltering, the local χ2k is calculated as a residual between the prediction
and measurement. It can be used as a powerful test of outlier hits. Thus the Kalmanﬁlter can correct the result of track ﬁnding by adaptive detection of outliers. If the χ2k islarger than a given threshold value, the hit is removed from the track candidate. It is also

74

CHAPTER 5. Event Reconstruction

possible to correct the left-right ambiguity in the process. On the other hand, a global
χ2track which is the sum of χ2k can be used as a test of ghost track or the track quality.
In the ﬁnal smoothing process, the state vector is back-propagated to the muon stopping target plane. An example of reconstructed trajectory is shown in Figure 5.11. The
state vector on the target gives the muon vertex point ~xµ = (xµ , yµ , zµ ), emission angle ofpositron p̂e = (θe , φe ) and its momentum |~pe | (thus its energy Ee ) with their covariancematrix. At the same time, the track is extrapolated to the timing counter. The total track
length from the target to the timing counter gives the positron time-of-ﬂight. The hit
information of the timing counters are not incorporated in the prediction of state vector
on the target because the extrapolation from the last measurement in the drift chamberto the ﬁrst timing-counter hit is so long that the it does not give a useful information.

5.3.3

Time

Timing Counter Hit Reconstruction
We reconstruct a hit time of a φ-counter by averaging
the two PMTs time after correcting√
the time-walk eﬀect as a linear function of 1/ Ai , where Ai is the measured amplitudeof i-th PMT. The hit position in z-coordinate is reconstructed by the time diﬀerence
between the two PMTs.
High momentum (∼ 50 MeV/c) positrons from muon decays on target often penetrate
a couple of bars like an example shown in Figure 5.12. For those events, hits associatedwith a positron track are clustered by their time and z hit position. Then the timingcounter hit time tT IC is given by the time of ﬁrst bar hit.
DCH-TIC Interconnection and Track Length
To reconstruct the time of muon decay, the time-of-ﬂight of positron teT oF has to be
subtracted from the timing-counter hit time. It is measured by the track length of thereconstructed Kalman track extrapolated to the surface plane of the φ-counter bar. The
connection is schematically explained in Figure 5.12. Matching between them is decided
by looking at the residual between√ 2track2extrapolation and timing-counter hit (∆z2DCH−TIC
and ∆rDCH−TIC in z and r = x + y directions respectively) and also the χDCH−TIC .
With the long projection from the last measurement of the drift chamber to the hit on thetiming counter, the projected track has large uncertainty due to the possible scattering.
To correct it, we use the value of |∆zDCH−TIC |. We extracted the correction function
from the π 0 Dalitz decay data. In addition to the multiple scattering, sometimes hard
scattering occurs on the materials placed along the trajectory like cables and preampliﬁer.
In such a case, the trajectory is completely changed and projection does not work, which
results in large value of |∆zDCH−TIC | and tail of time measurement.
Reﬁnement of the Timing-Counter Hit TimeFor positrons with several bar hits, we can improve the time measurement by combining
the hits. Even though we sometimes get more than three consecutive hits in a cluster,
we only use ﬁrst two hits since it is diﬃcult to estimate the pass of positron after hitting
a few bars. The propagation time between the two adjacent bars is estimated by the
extrapolated track, and by subtracting it we calculate the time of positron hitting the5.3. Positron Reconstruction

75

ﬁrst bar with another bar measurement. The time on the ﬁrst bar tT IC is modiﬁed as theaverage of the two measurements.
Finally, we reconstruct positron emission timete = tT IC − teT oF .

(5.7)

x-y viewyx

φ-z view

Figure 5.12: Schematic view of connection between track and timing-counter hits. Trackﬁtted to the drift-chamber hits is extrapolated to the surface plane of the ﬁrst hit bar. Bluecircles in the close-up views show the predicted hit position and green squares show timing
counter measurements. Because of the long distance projection, the predicted positionon the timing counter has large uncertainty especially in z direction. The ﬁrst two hits
associated with the track is combined by the track to improve timing measurement.

76

CHAPTER 5. Event Reconstruction

5.4

Gamma-ray Reconstruction

5.4.1

General Remarks

The response of LXe to a gamma ray around 50 MeV is in the transient region fromsingle process to electro-magnetic shower development. It can be understood noticingthat the xenon critical energy is ∼14 MeV. However, we call here this process a “shower”
even though the gamma ray losses its energy in a few steps. Some examples of theshower development in response to 52.8 MeV gamma in the MC simulation are shown in
Figure 5.13. Note that there is large ﬂuctuation in the shower development. The keys of
gamma-ray reconstruction are there in the treatments of the shower ﬂuctuation.
Before reconstructing gamma-ray hit, we convert the PMT charge to more physicalvariables, number of photoelectrons (Npe ) detected by the PMT,
Npe,i = Qi /(e · Gi ),(5.8)

where Qi is a pulse charge of the i-th PMT calculated by the charge integration of waveform, e is the elementary electric charge, and Gi is the PMT’s gain. Next, it is converted
to estimated value of the number of scintillation photons (Npho ) hitting the PMT photocathode,Npho,i = Npe,i /QEi ,(5.9)

where QEi is the PMT’s quantum eﬃciency including the collection eﬃciency of the ﬁrst
dynode. The calibration of PMTs is described in Sec.6.4.1. In reconstruction, we usually
use the calibrated charge, Npho , as an output of PMT. We use Npe when we estimate thestatistical ﬂuctuation in the observed number of photoelectrons.

5.4.2

Position

A gamma-ray interaction point is reconstructed by ﬁtting the PMT-output distribution.
The method is schematically shown in Figure 5.14. We calculate an expected light distribution so that each PMT output is proportional to the solid angle of the photo-cathodeviewed from the reconstructed position. The solid angle of a photo-cathode (a circle) froman arbitrary point, Ωi (x, y, z), is calculated numerically. The three dimensional positionof gamma-ray interaction, (xγ , yγ , zγ ), is calculated by minimizing
χ2pos =∑ Npho,i − c × Ωi (xγ , yγ , zγ )iσpho,i (Npho,i ),(5.10)

where c is a constant factor and one of the free parameters of the ﬁtting, and σpho,i (Npho,i )is statistical uncertainty of the PMT’s charge calculated by,1
× σpe,i (Npe,i )QEi√
1× Npe,i
=
QEi
√
Npe,i
=
.
QEi

σpho,i (Npho,i ) =(5.11)(5.12)(5.13)

5.4. Gamma-ray Reconstruction
EVENT

1

EVENT

77

2

000 0
0
0000
00 0
000 000
0000
00000
000000000
00000000000000 000
00
00
0

00
0000000000000000000000
00000
0
00

00 00

0
0000

0

EVENT

EVENT

3

4

00

0
00
00

0
0 0000000000000000 000
0

0

0 00
0
00000000000000000000 0
0 00 0 00 000
0 00

0 0 000

Figure 5.13: Examples of electromagnetic shower in LXe in response to 52.8 MeV gamma
simulated with MC. Each small circle shows an energy deposit.

This ﬁtting estimates three-dimensional position. Therefore we can reconstruct not only
the two-dimensional position projected to the inner face (uγ and vγ ) but also the depthof the interaction (wγ ).

78

CHAPTER 5. Event Reconstruction

The precision of this method will be limited by the ﬂuctuation in shower developing
behind the ﬁrst interaction point since we assume here the scintillation light comes froma point-like source. To minimize the eﬀect, we perform the ﬁtting twice only using PMTs
in restricted region on the inner face. First ﬁtting uses typically 45 PMTs around one
observed maximum light. Second ﬁtting is then performed with fewer, typically 15 PMTs,
around the result of the ﬁrst ﬁtting. The way of PMT selection is explained in the
Figure 5.15.
Nevertheless, the reconstructed position by the ﬁtting has bias in u- and w-coordinates,
while there is no bias in v-coordinate. In u-coordinate, the gamma ray comes into thedetector with incident angle respect to the PMT direction. This oblique incidence causes
a bias of reconstruction to outside direction because the shower usually develops in theoriginal gamma ray direction. In w-coordinate, the bias is in deeper direction by the same
reason. These biases are studied with MC simulation, and corrections are applied.
In addition, the event-by-event diﬀerence of shower shape can be estimated by the difference between the two ﬁtting results. It gives us the direction of the shower development.
It is also used to correct the remaining inﬂuence of the shower ﬂuctuation.

5.4.3

EnergyGamma rays around 50 MeV converted in LXe usually deposit all of their energy in the
LXe active volume. Almost all of the deposit energy is converted to the emission of
scintillation. Our basic idea of energy reconstruction is to collect as much scintillation
photons as possible. If we can collect all of the photons, of course it is not possible, andif the absorption length is suﬃciently long, then the total number of collected photons
is proportional to the original gamma energy regardless of the position and shape of theshower.
On the basis of the idea, we reconstruct gamma energy by summing up outputs of all
PMTs. In summation, we put a weight for each PMT to take into account the diﬀerent
coverage of photo-cathode for diﬀerent location of the PMT. We deﬁne the following
quantity,
Nsum =∑

wi × Npho,i ,(5.14)iwhere a weight factor wi is the inverse of photo-cathode coverage of i-th PMT.
This quantity can be a good estimator of the deposit energy when the ﬁrst interaction
point in LXe (wγ ) is deeper than a couple of centimeter regardless of the interaction point
and the event-by-event ﬂuctuation in shower development. However, when the gamma
conversion occurs very close to the inner surface, the Nsum becomes very sensitive tothe position of conversion point relative to the PMT alignment. Eﬃciency of photon
collection changes event-by-event. For those events, a solid angle of photo-cathode froma conversion point can be a better variable to correct the collection eﬃciency than the
coverage. We investigated the dependence of Nsum on conversion depth and solid angle
for the response to the 55 MeV gamma ray, and found a good correlation of sum of Npho
over inner face PMTs with the solid angle of the maximum-output PMT on inner face(Ωin,max ) as shown in Figure 5.16. We apply a correction of Nsum with Ωin,max for eventswith wγ < 3 cm. The correction function is given as a linear function, whose coeﬃcientsare obtained by the 55 MeV-gamma events in the π 0 calibration.

5.4. Gamma-ray Reconstruction

Figure 5.14: Schematic explanation of position ﬁtting. Red star marker shows the ﬁrst
interaction point of an incident gamma. Orange ellipses show PMT photo-cathodes, andtheir color indicate the amount of observed
light (the deeper color, the larger amount oflight). The solid angle of each PMT from theinteraction point can be estimated by the relative amount of light observed by the PMT.

79

Figure 5.15: PMT selection usedin the position ﬁtting.
The blue
cross marker shows the initial estimate of interaction point calculated by
the weighted mean around the maximum output PMT shown as red color.
The PMTs used in the ﬁrst ﬁtting arethose within 3.5-PMT distance fromthe point. The second ﬁtting is done inmore restricted region, 2-PMT radius
shown as green PMTs from the resultof the ﬁrst ﬁtting shown as black cross.

The remaining position dependence is corrected globally after the Nsum reconstruction.
The detail of the calibration of the position dependence and the correction are describedin Section 6.4.2.
The linearity of Nsum to gamma energy Eγ was checked using several energy points
available from our calibration methods as shown in Figure 5.17. A conversion factor from
Nsum to Eγ is given from the 55 MeV peak.

5.4.4

Time

With the reconstructed interaction point, we can reconstruct the time of ﬁrst interaction
by individual PMT,
thit,i = tP M T,i − tdelay,i − tof f set,i ,(5.15)

where tP M T,i is time of i-th PMT measured by the constant-fraction method describedin Sec.5.2.3; tdelay,i is time delay during the scintillation light propagation in LXe; and80

CHAPTER 5. Event Reconstruction

The number of scintillation photons

Figure 5.16: Correlation between Ωin,max and sum of inner PMT Npho for wγ < 3 cm.

×103
400
350
300
250
200
150
100
50
20

40

60
80
Energy (MeV)

Figure 5.17: Linearity plot of Eγ v.s. Nsum . The ﬁve points are given from 4.4 and11.7 MeV from CW-B run, 17.6 MeV from CW-Li run, and 54.9 and 82.9 MeV from π 0
run. Red line is the best-ﬁt linear function whose intercept is ﬁxed to zero.

5.4. Gamma-ray Reconstruction81

tof f set,i is a constant time oﬀset of the channel. Let us think about the second term. We
can think of possible cause of the delay as the following three: time of scintillation lightpropagating in LXe from light source to the PMT, tprop ; that of light which reaches the
PMT indirectly from the light source, tindir ; and the time-walk eﬀect of time-extraction
method, twalk . The tprop can be a function of distance between light source and PMT dand eﬀective light speed in LXe vef f . We could model tindir as a function of incident angle
to the PMT η because as η gets larger, the fraction of scattered or reﬂected photons in
the observed light increases. The twalk would be a function of Npe . Those are illustrated
in Figure 5.18. Thus the tdelay can be write down astdelay = tprop (d, vef f ) + tindir (η) + twalk (Npe ).

(5.16)

Details of the extraction of those function forms and coeﬃcients, and calibration of time
oﬀset are described in Sec 6.4.4.

Figure 5.18: Illustration of time reconstruction.

Then, we reconstruct the hit time, tLXe , by combining those measurements. The χ2time
is calculated with PMTs which collect more than 50 photoelectrons,
χ2time =∑ (thit,i − tLXe )2i

σt,i (Npe )2,(5.17)

where σt,i (Npe )2 is time resolution of each PMT as a function of the number of photoelectrons. The tLXe is determined so that the χ2time becomes minimum. Typically about 150
PMTs are used for around 50 MeV gamma. During this ﬁtting process, large χ2 channels
are rejected to remove pileup eﬀect and the minimization is iterated.

5.4.5

PileupPileup Identiﬁcation
We identify pileup events byi) light distribution in the inner surface PMTs,ii) time distribution in all PMTs.

82

CHAPTER 5. Event Reconstructiona.u.

The ﬁrst item separates multiple gamma rays spatially. We perform a peak search on
the light distribution of inner surface PMTs. We can ﬁnd the position of each gamma
ray. The power and mis-identiﬁcation probability depend on the threshold for the peaksearch. In 2008 analysis, the threshold was set to 500 photons, which corresponds to ∼80
photoelectrons and ∼8 mV of pulse-height.
The second item uses temporal separation of diﬀerent gamma rays. For this, we canuse the normalized χ2 value of the time ﬁtting, χ̂2time . For a single gamma event, all PMTs
have hits in coincidence. On the other hand when pileup occurs by the accidental overlap,
the two gammas have diﬀerent conversion time. Even though the overlapping gamma ray
has small energy, the closest PMT observe relatively large amount of light and the timeis measured as the overlapping hit time. Those events results in large χ2 values. The
distribution of χ̂2time of signal MC and data are shown in Figure 5.19. We set a thresholdfor the identiﬁcation at 3.
These two methods are complementary each other.

10-1

10-2

10-3

10-4

0

5

10

15
χ2/NDF

Figure 5.19: Distribution of normalized χ2 of the time ﬁtting. Dotted plot shows thatof data and solid line shows that of signal MC.

Pileup Elimination
By the MC study, in 3 × 107 /sec beam rate, about 8.5 % of events suﬀer from pileups, andit causes an ineﬃciency for the signal detection. Even if an event is overlapped by anothergamma ray, we can use those events by unfolding or eliminating the overlapping one. We
developed an algorithm for the pileup elimination to recover the eﬃciency. Eliminationof contribution of energy by pileup gamma ray is done by the following steps.
i) Estimate energy by ﬁtting PMT charges except for those around the pileup gamma.
The ﬁtting is done on the basis of a table obtained from CW data. In the table,
average outputs of each PMT for each position mesh(1.55 × 1.55 × 1.55 cm) are
written.

5.4. Gamma-ray Reconstruction
100

83

21999

80

100

60

80

40

60

20

40

0

20

6304

1806

517

0

148

-20
42

-40
-60

12

-80
3

-100
-100
z

-50

0

50

100

150

200

250

300
1

(a)
100

21999

80

100

60

80

40

60

20

40

0

20

6304

1806

517

0

148

-20
42

-40
-60

12

-80
3

-100
-100
z

-50

0

50

100

150

200

250

300
1

(b)
Figure 5.20: Light distribution before(a) and after(b) eliminating pileup contribution.
The color axis shows the amount of observed light on each PMT.

105

104

103

102

10
40

50

60

70
80
Energy [MeV]

Figure 5.21: Energy spectrum in the sideband data. Red line is that after subtracting
pileup events identiﬁed by time but not by light. Blue line is that after pileup elimination(the same event set as the red line).

84

CHAPTER 5. Event Reconstructionii) Calculate the expectation of outputs of PMTs around the pileup gamma ray using
the table, and replace them with the expectation.

iii) Reconstruct the energy in the usual way using the replaced PMT charges.
Figure 5.20 shows PMT output distribution “before” and “after” eliminating contributionof a pileup gamma ray.
The left plot of Figure 5.21 shows gamma energy spectrum of the sideband in 2008
muon runs with various cuts. For the blue histogram, the same cuts of gamma ray forphysics analysis are applied.

5.5

Combined Analysis

5.5.1

Relative Angle

The gamma-ray direction p̂γ (θγ , φγ ) is reconstructed as the direction from the muon decay
vertex ~xµ reconstructed by the positron tracking to the reconstructed position of the
gamma ray ~xγ ; p̂γ = (~xγ − ~xµ )/|~xγ − ~xµ |. Then the opening angle between positron andgamma ray Θeγ is given bycos Θeγ = p̂e · p̂γ .

(5.18)The relative angles in θ -and φ-directions are reconstructed as
θeγ = θγ − (π − θe ),φeγ = φγ − (π − φe ).

5.5.2(5.19)(5.20)

Relative TimeThe time-of-ﬂight of gamma tγT oF is calculated by a line segment between the reconstructed interaction point and muon decay vertex reconstructed by the positron tracking.
Then the gamma emission time tγ is reconstructed as,tγ = tLXe − tγT oF .

(5.21)

Then the relative time diﬀerence between positron and gamma ray is given byteγ = tγ − te .

(5.22)

Chapter 6
Calibration
Needless to say, calibrations of detectors are crucial in a precise measurement. In addition, for a long-term experiment, continuous monitoring of detectors becomes a key toachieve high performance and to validate the results. We developed various calibrationand monitoring methods. Some of them are complementary and some are duplicative
for the cross check. Some instabilities and deviations of observables are corrected by thecalibration data.

6.1

DRS Calibration

6.1.1

Voltage Calibration

Time interval (nsec)

Output [ADC ch]

Figure 6.1 shows a typical response curve of a DRS cell. The DRS has non-linear response.
We see worse resolution over 0.5 V input from the curve. To use good resolution range,
we adjusted inputs from MEG detectors to be smaller than 0.5 V. We prepared these
response curves in advance by looking at the response for diﬀerent input voltages frominternal DC voltage generator on DRS boards. The online frontend load and apply the
prepared calibration tables. Therefore, the data written in the disk is already calibrated
and in unit of volt.
4000
3500
3000
2500
2000

0.7

0.65

0.6

1500
1000

0.55

500
0

0

0.5

1

1.5

0.5
0

2

Input [V]Figure 6.1: Typical response curve of a DRS
cell.

85

200

400

600

800
1000
Cell number

Figure 6.2: Typical sampling intervalsof a DRS channel.

86

6.1.2

CHAPTER 6. Calibration

Time Calibration

A sampling frequency is generated by a series of inverters in a DRS chip. The speedof this “domino wave” is controlled by an analog voltage. Thus the speed varies if thevoltage varies, for example, because of a change of temperature. The speed is regulated
once at the beginning of each run but there is no feed-back system to regulate it during
the run. The sampling interval between each point reﬂects a characteristic of individual
inverter. As a result, each sampling point has diﬀerent sampling interval. It is, however,a hardware-related characteristic, and thus stable over time relatively to the speed ofdomino wave.
Therefore we ﬁrst apply the time interval calibration for the time of each sample point.
A dynamic event-by-event calibration is then performed by analyzing a global clock signal.
During this clock analysis, the phases of domino wave of all chips are synchronized to get
a common time zero.
Static Calibration of Sampling Intervals
A sampling interval corresponding to a given inverter was calculated using a periodic
wave from an external function generator. Sine waves at 240 (60) MHz were used forchips used at 1.6 (0.5) GHz sampling. We took data of the sine waves with random
trigger and analyzed them in oﬄine. On each event, periods between zero-crossing pointsof the sine waves were measured. Sampling intervals of cells between the zero-crossing
period were adjusted so that the measured zero-crossing period become that of input sine
wave. This regulations were repeated until sampling intervals of all cells got converged.
The calculated sampling intervals of all cells of all channels are stored in a database, andthey are applied when the waveform data are read in oﬄine analysis. Figure 6.2 shows
an example of sampling intervals of a chip calculated by this procedure.
Clock Analysis
A global common clock (square wave of 19.44 MHz) is distributed to all DRS boards fromthe trigger system. It is used as the time reference of the experiment. The clock signal isdigitized and recorded in an additional channel of DRS and analyzed i) to calibrate thedomino wave event-by-event, and ii) to synchronize the phase among chips. The domino
wave of each chip is running freely and independently. It stops when an external trigger
signal comes. Although the trigger signal from the trigger system is also common, becauseof the independent circulation of domino wave on each chip, actual stop timing of eachchip ﬂuctuates as large as one sampling interval, namely ∼600 ps for 1.6 GHz sampling.
Therefore, the synchronization of chip timings is quite important to achieve the time
resolution of <100 ps. On the other hand, for the DCH analysis, we do not need such ahigh time resolution. Thus we analyze the clock only for the data of 1.6 GHz sampling.
The clock signal taken by DRS2 and DRS3 are shown in Figure 6.3. It is clear thatDRS3 has better quality of the signal: higher bandwidth and much less distortion. It
is also known that DRS3 has less cross talk of the clock signal onto the normal datachannels. Taking those better timing performance and the available number of channels
into account, we adopted DRS3 to digitize timing-counter NIM pulses, which are one of
the most important parts of timing analysis. While DRS2 chips were used for all of LXe

Amplitude (mV)

Amplitude (mV)

6.1. DRS Calibration

600
550

87

120
100
80

500

60
450

40
400

20
350

0

300
-400

-200

-600

0
Time (nsec)

Amplitude (mV)

Amplitude (mV)

-600

300
250
200

-400

-200

0
Time (nsec)

-400

-200

0
Time (nsec)

120
100
80
60

150

40

100

20

50

0

0
-600

-400

(a) Raw data.

-200

0Time (nsec)-600(b) After analysis.

Figure 6.3: Examples of the global clock signal digitized by DRS2 (top) and DRS3
(bottom). You can see the clock signals are out of phase before the clock analysis (a).
We analyze clock signal so that they get in phase (b).

channels, which are also important but whose timing performance can be recovered byusing many channels on many chips.
The procedure of clock analysis is the following:i) Measure roughly the domino speed by a peak search with diﬀerential signal of theclock, and correct the speed.
ii) Search for a edge of the clock closest to the expected timing where the pulse oftriggered signal should be around. A reference time of the channel is deﬁned as
the edge time and a global time zero is deﬁned as the reference time of a reference
channel deﬁned in advance.
iii) Fit several edges around the reference time with a template clock waveform.
iv) Adjust the domino wave locally around the reference time using the results of ﬁttingsin step iii).
v) Perform a ﬁtting with wider range and determine the reference time precisely.
vi) Shift the time of all sampling points of the channel to synchronize the reference timewith the global time zero.
We estimate the precision of the synchronization of chips by the diﬀerence of time
resolutions of the timing counter evaluated by looking at the time diﬀerence of two consecutive bar’s hits between the case that two bars are connected to a common DRS chipand the case using diﬀerent two chips. Average values are estimated to be σclock ∼47 and∼37 ps for DRS2 and DRS3, respectively.

88

CHAPTER 6. Calibration

6.2

Calibration of Drift Chamber

6.2.1

z-coordinate Calibration

The z-coordinate calibration of drift-chamber hit, which means the relative gain correctionof each ends, is done using the known pattern of vernier pad. We calibrate the anoderelative gains so that the reconstructed length corresponding to a turn of vernier circle
becomes the pitch length of 5 cm. We calibrate the cathode gains using the fact that themean charge collected on each pad should be a quarter of total charge.

6.2.2

Time Oﬀset Calibration

We calibrate the time oﬀset of each channel by ﬁnding the leading edge of hit-time distribution corrected by ttrack . We ﬁt the distribution with a polynomial function, empiricallygiven as ﬁfth-order one, and the oﬀset is determined by the time that the height of the
distribution reaches 15 % of the maximum. An example of the ﬁtted distribution is shownin Figure 6.4.

Figure 6.4: An example of corrected hit time distribution.

6.2.3

Time-to-Distance Calibration

The time-to-distance functions are calculated by GARFIELD program. If deviation fromdata is observed, the function is modiﬁed so that the deviation disappear. This calibration
is done with simple iterative procedure.
i) Calculate the residual between obtained drift distance and the closest approach fromthe wire to the reconstructed track.
ii) Put a corrective oﬀset to the drift distance so that the residual is minimized.
iii) Perform above for many events with similar incident angle and closed ﬁeld strength,and build a new time-to-distance plot.
iv) Fit the plot and form a new time-to-distance function.
Repeat this procedure until the functions are converged.

6.3. Calibration of Timing Counter

6.3

Calibration of Timing Counter6.3.1

z-coordinate Calibration

89

The z-coordinate calibration means adjustment of time oﬀset between the two PMTs
because z hit position is reconstructed by the time diﬀerence between the two. We use
the reconstructed positron tracks as a reference of z hit position. The relative time oﬀsets
were calibrated so that the distribution of ∆zDCH−T IC has mean at zero. This calibration
was conﬁrmed by looking at the distribution of z hit position diﬀerences between adjacentbars hits for multiple-bar hits.

6.3.2

Inter-Bar Time Oﬀset Calibration

We use CW-B data to calibrate the relative time oﬀsets among bars. The 115 B(p, γ)126 C
reaction produces simultaneous gamma rays at 11.7 and 4.4 MeV with no angular correlation. We use the 4.4 MeV gamma ray at the gamma-ray detector as reference time, andmeasure the 11.7 MeV one on each bar. After the time-of-ﬂights of two gamma rays arecorrected, the distributions of time diﬀerence of the two give the relative oﬀsets.

6.4

Calibration of Gamma-ray Detector

6.4.1

PMT Response Calibration

PMT Gain
We calibrate the PMT gains using LEDs. We use LEDs as stable light sources to monitor
PMT outputs and also to calculate absolute value of PMT gains.
Gain Calculation We calculate the gain by means of the photoelectron statistics. Here
we use a simple model. We assume LED output is constant and statistics of Npe obeys
the Poisson distribution. Let Npho be the number of photon from LED to the PMT,
which is constant in our assumption. Then the average number√ of photoelectrons will
be N pe = Npho · QE, and its standard deviation will be σpe = N pe , while the observed
charge is given by Q = Ge · Npe . If we take many LED events, the mean and variance of22
respectively. Therefore= (Ge · σpe )2 + σnoise
the distribution become Q = Ge · N pe and σQ
we obtain the relation,
22
σQ= Ge · Q + σnoise
.

(6.1)

By taking LED events at diﬀerent intensities, we can extract the gain, G, as a slope
2.
of the variance-mean plot. This method is insensitive to the noise contribution, σnoise
Figure 6.5(a) shows the charge distribution of LED runs. In the normal calibration, we
took data at 10 diﬀerent LED intensities as one set of gain calibration, and each step
contains 3000 events. The number of photoelectrons in each steps is suﬃciently large
to regard the distribution to be a Gaussian. We ﬁt the charge distribution of each stepwith a Gaussian and estimate the mean and variance. Figure 6.5(b) shows the obtained
mean-variance plot. It shows that those plots are well on a line. By ﬁtting them with alinear function, we get the PMT’s gain. The χ2 distribution of the mean-variance ﬁt is

90

CHAPTER 6. Calibration

Counts

LED gain calibration : PMT103 (I103)

(a)
250

200

150

100

50

Variance

0

0.45

0
×10

0.2

0.4

0.6

6

× 10

9

× 10

9

0.8
1
-Charge (e+)

(b)
0.4
0.35
Gain = 1.71 x 10 6

0.3
0.25
0.2

2

χ / ndf
3.234 / 8
p05.759e+03 ± 2.133e+03
p1±
0.0005714 1.042e-05

0.15
0.1
0.05
0

0

0.2

0.4

0.6

0.8

1
mean

Figure 6.5: An example of gain calculation with LED data. (a) shows charge spectra of diﬀerent intensity LED runs. (b) is
the mean-variance plot of (a). The gain isextracted from the slope of the ﬁtted line.

Figure 6.6: Distributions of gain calculation. (a) χ2 distribution of mean-varianceﬁt. (b) Uncertainties of gain calculation.
(c) Gain distribution.

shown in Figure 6.6(a). By monitoring these χ2 values, we checked bad channels. The
precision of this method is about 1.8 % as shown in Figure 6.6(b). Gains were adjusted
around 1.7 × 106 as shown in Figure 6.6(c).
Stability and Correction We checked the long-term stability of PMT gains by looking
at the LED event peak. Figure 6.7 shows two typical examples of time evolutions of
PMT gains. We found several points from those plots. First, we observed two kinds of
instabilities of PMT outputs. One is a gradual gain decrease. The gradual gain decreases
were severe particularly in the π 0 run (in August and last few points in December), whilein physics run gains were stable. These decreases are considered to be an aging eﬀectof dynode surface material due to the load from large current. When PMT sees a large
amount of light, a huge amount of electrons hit dynodes (especially one at the last stage),and cause local heat and material change. It causes the change of the second-emissionratio of the dynode resulting in the change of gain.
The other instability is a rate-dependent gain shift. For example, in Figure 6.7(b)we see two bands of gain plots. Those two correspond to the gain values measured inbeam on and oﬀ. Figure 6.8 clearly shows the shift of gain value after opening the beam
blocker. The shift values are diﬀerent for each PMT, and dependent on the beam rate.

Gain

6.4. Calibration of Gamma-ray Detector

91

×103
1900
1800
1700
1600

(a)

1500
1400
1300
1200

Gain

1100
09/08

08/09

08/10

07/11

07/12

09/08

08/09

08/10

07/11

07/12

Date

×103
2100
2000
1900
1800

(b)

1700
1600
1500
1400
1300
1200
Date

Figure 6.7: Examples of evolution of PMT gains. Red blank circle plots show absolute
gain values calculated by the photoelectron statistical method. Black square plots show
relative gain traced with the LED peak. (a) typical example of a stable PMT. (b) typicalexample showing rate dependent gain shift.

Typical shift value is ∼ 2 % to higher gain in the normal beam. In [91], similar kind oftime dependence of rate-dependent gain shift was explained by a hole-trapping model on
the dynode surface.
To correct these eﬀects, we took the LED run every day. In particular, to measure
the response under the conditions of physics run, we took LED data under the same
beam conditions, while normal calibration sets including LED runs described previous
paragraph are taken without beam. From Figure 6.7, we found that the light intensities
of LEDs are quite stable and we can use them as constant light sources over time. Owing
to this stability, the relative gain correction can be done much precisely by using theabsolute measured peak value of the LED events, instead of measuring absolute gain
value each time with the statistical method. Thus, we correct the gains relatively byusing the variation of the peak value of LED events.
We checked the stability of the correction using LED events mixed in the physics run
taken at 0.5 Hz. Figure 6.9 shows the total number of photoelectrons of the LED events
as a function of time. After the gain correction, LED peaks are quite stable over timewithin ± 0.5 %.
PMT Quantum Eﬃciency
The PMTs’ quantum eﬃciencies, or in another words equalization factors, can be calibrated using alpha events and 17.6 MeV line from 73 Li(p, γ)84 Be reaction. The LED is notsuitable for this purpose because wavelength of the light is diﬀerent from that of xenon
scintillation, and the response of the PMT is sensitive to the wavelength. The overall
PMT response factor can be decomposed into
f = 1/(G × QE × CE),

(6.2)

CHAPTER 6. Calibration
Total number of photoelectrons

92

×103
650
645
640
635
630
open beam blocker
625
10h26

10h28

10h30

10h32

10h34

10h36
time

Total number of photoelectrons

Figure 6.8: Time dependence of the gain shift when counting rate is changed. Red
dot-dashed line shows the time when the beam blocker was opened.
×103
680
670
660
650
640

1%

630
620
610
600
590
16/10

30/10

13/11

27/11

11/12
Date

Figure 6.9: Plot of total number of photoelectrons of LED events vs. date. LED eventswere taken during the physics run. Red plots show peak value for each slice. After thegain correction, LED peaks are quite stable over time.

where CE is a collection eﬃciency of photoelectrons at the ﬁrst dynode. The PMT gain
G is calculated in the previous sub-section. Here, we want to calculate remaining part.
According to our convention, in this thesis we call the multiplication of the quantum eﬃciency and the correction eﬃciency as “QE” because we do not calculate them separately.
The point-like alpha source on wires are mounted in the detector as described in
Section 3.2.2. Since we know the positions of alpha sources and spectrum of the alphaparticles, we can estimate the amount of light observed by given PMT. We use the MC
simulation for the estimation. The QE is measured by comparing the observed charge
spectrum with that of the MC estimation. In this process, we can use two types of alphadata. One is alpha data taken in LXe in normal calibration. The other is data taken in gas
xenon (GXe). We took those data after the physics data taking while we were recovering
xenon from the detector to the storage tank. These two kinds of data have advantages
and disadvantages, and they are complementary. The LXe-alpha data are real-time data
under the same conditions, while GXe-alpha data are less sensitive to the parameters of
scintillation optics used in the MC simulation.

6.4. Calibration of Gamma-ray Detector

93

We took the LXe-alpha data in normal calibration run three times per week. For every
calibration set, we can calculate a QE of each PMT. The precision of the measurements
was estimated to be 3 %. We monitored the stability throughout the data taking, andconﬁrmed the stability better than the each measurement precision. Therefore, we calculated average value using seven calibration sets taken when the gains and light yield of
LXe were stable. In this way, we can reduce statistical uncertainty of the measurement.
In GXe data, we observed upstream-downstream asymmetry of the measured QEs,
which can be explained by the absence of BTS magnetic ﬁeld in the measurement. Thisasymmetry is corrected with respect to the QEs measured with LXe data which does notshow such asymmetry. On the other hand, there is systematic dependence of calculated
QEs in LXe data. PMTs placed close to the corner of the detector have higher QEs,
while this kind of dependence is not seen in ones measured with GXe data. The reason
can be thought by the mismatch of optical parameters like scatter length and reﬂection
coeﬃcients. Therefore, the dependence is corrected globally, and systematic uncertainty
due to the mismatch of MC parameters can be reduced. Finally, the measurement of bothare averaged to reduce statistical uncertainty of measurement.
If the QEs measured in this way are free from systematic error, then the response to
the 17.6 MeV peak becomes symmetry to the detector geometry. However, we observedsome asymmetry: top-bottom asymmetry and a little upstream-downstream asymmetry.
The reason is not fully understood yet. Displacement of PMTs and alpha wires is apossible cause. We can correct these asymmetries, for example, by applying a global
position correction after reconstructing the energy. However, it is better to correct the
PMT responses in advance. Thus, we corrected QE values to cancel out the asymmetry
in 17.6 MeV line. Note that we do not correct them so that the response to 17.6 MeV
become uniform, but correct only the asymmetry in order not to introduce systematic
uncertainty again by the unknown optical parameters. A set of QEs calculated in thisway is used throughout the data in 2008.
Another set of QEs was calculated by another analysis group of our collaboration withindependent method. We carried out a study to estimate the eﬀect of QE-measurement
error in the reconstruction. The diﬀerence of the two sets of QEs was about 8 % in
RMS. This diﬀerence was used as possible systematic errors of the measurement. The
impact of wrong QEs on the reconstruction performance was investigated using the MC
simulation with wrong QE values. This study shows that this level of error has little
eﬀect on the energy resolution if the position dependence is corrected. More eﬀect canbe seen in position resolution, but still suﬃciently small respect to the current-achieved
position resolution.

6.4.2

Non-uniformity

The gamma-ray detector has small non-uniformity of energy response with the Nsum reconstruction described in Section 5.4.3, even though we try to cancel out coverage
diﬀerence. The non-uniformity stems from a geometrical eﬀect of diﬀerent eﬀective coverage and also the ﬁnite absorption length. To correct it, we use the actual response to
the 17.6 MeV-monochromatic gamma rays. The data are divided into a mesh of (uγ , vγ )plane, and at each point, we measure the peak of the spectrum. Figure 6.10 shows the
response maps. We observed a change of the dependence during data taking. The change
is due to the change of optical properties by the puriﬁcation. Therefore, we prepared two

1.04

60

1.02

40

1

20

[cm]

CHAPTER 6. Calibration

[cm]

94

1.02

40

1

20

0.98

0

1.04

60

0.98

0

0.96
-20

0.96
-20

0.94
-40

0.94
-40

0.92
-60

0.92
-60

0.9
-20

-10

0

10

20

[cm]

(a)

0.9
-20

-10

0

10

20

[cm]

(b)

Figure 6.10: Relative energy response map in (u, v)-plane. These maps were made using
the response to the 17.6 MeV line. The position dependence changed before (a) and after(b) the puriﬁcation. The correction factors are extracted as inverse of those maps.

sets of correction tables for the data before and after the large jump of the LXe light yield(see the next sub-section).
For the w-coordinate, we calculated the correction factor using the central part of thedetector. The w-correction is applied independently of the (u, v)-correction.

6.4.3

Energy Scale and Light Yield

We calibrate the energy scale using the 55 MeV gamma ray from π 0 decay. The uncertaintyof the peak determination is evaluated to be 0.8 % by the error of the ﬁtting.
As described in Sec 4.2.4, during the physics run in 2008, the light yield was changing
as a result of puriﬁcation. Therefore, we had to monitor the light yield continuously andcorrect the energy scale. We use several calibration sources to follow the variation of thelight yield. Figure 6.11 shows the time evolution of amount of the observed light relative
to the point when we determined the energy scale. We use mainly 17.6 MeV line during
the physics run, and use 55 MeV peak during the π 0 run. We use cosmic-ray Landau peakfor the connections between the π 0 run and the physics run, because we cannot use CW
nor π − beam during the exchange of beamline. Several jumps in the plot were causedby the liquid-phase puriﬁcation during the beam-maintenance periods. The continuous
increase after 30 October was due to the gas-phase puriﬁcation.
This history is used for the correction of energy scale. The uncertainty of energy scaleTotal amount of observed light (a.u.)

6.4. Calibration of Gamma-ray Detector

95

1.4
1.3
1.2
1.1
1
0.9
0.8

physics data taking

π0 runπ0 run0.7
09/08/08

08/09/08

08/10/08

07/11/08

07/12/08
Date

Figure 6.11: Evolution of the total amount of observed light. In this plot, results fromseveral calibration sources are combined. The y-axis is relative to the point when we
determined energy scale.

from the correction is estimated to be 0.18 %, mainly stemming from the uncertainty of
the cosmic-ray Landau peak ﬁtting.

6.4.4

Timing Parameters

Here, we calibrate parameters for the gamma-ray detector time reconstruction. The timereconstruction is based on the hit time reconstructed by individual PMT given by Eq.5.15
andEq.5.16. The tdelay extracted by the π 0 data is shown in Figure 6.12 as a function of√
1/ Npe . Here, the time reference is given by the reference counter in the NaI detector.
What we have to do are to extract function form of each term and to calibrate coeﬃcientsof it. The diﬃculty stems from the fact that those three terms are correlated one another.
For example, if we look at the delay time as a function of Npe , then d is also eﬀectively
changed. We cannot disentangle the eﬀect of light speed and time walk easily.
To disentangle each eﬀect, we apply the following steps. First, we determine vef f , andassume tprop is equal to d/vef f . By subtracting d/vef f , we can remove the contribution of
tprop at least in ﬁrst order. Then, we look at the dependence of tdelay as a function of η on
the data in narrow region of Npe as shown in Figure 6.13. We found that the dependence
is well modeled with a function of (1 − cos η)2 . We adopt the empirical parametrization oftindir and subtract it. Then, we look at the remaining dependence of Npe , and we consider
this dependence is the time-walk eﬀect. In this way, hit times by individual PMT are
calibrated. Now, we can reconstruct the hit time, tLXe . Finally, we see tLXe as a functionof the total number of photoelectrons Npe,sum . If the extracted twalk is wrong, it would
show some dependence. In this way, we disentangle the eﬀect of d and Npe .
We tried this procedure with vef f = 10 cm/ns, which is a group velocity in LXe

CHAPTER 6. Calibration
12

tPMT - tref (nsec)

tPMT - tref (nsec)

96

10
8
6

12
10
8
6

4
4

2
2

0
0

-2
-2

-4
-4

-6

-6

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.2

0.4

0.6

0.8

1

1/sqrt(Npe)cos(η)

Figure6.12: tdelay as a function of√
1/ Npe . It is not clear this dependenceshows√ time-walk eﬀect as a function of
1/ Npe or light propagation time as afunction of d.

Figure 6.13: tdelay as a function of cosine of PMT incident angle η after tprop is
subtracted.

expected for the scintillation spectrum with peak value at 178 nm.1 Then, we foundstrong correlation between tLXe and Npe,sum shown in Figure 6.14(a). This indicates slower
velocity of scintillation-photon in LXe. Then we tuned vef f so that the dependence on
Npe,sum disappear (Figure 6.14(b)). We found that vef f around 8 cm/ns, the dependence
disappear. At the same time, we found there is no time-walk eﬀect in the tP M T . The
fact means that the digital-constant-fraction method described in Sec.5.2.3 works welland eliminates time-walk eﬀect. This measured vef f would partly include the eﬀect ofindirect photons, resulting in slower value. Or it may indicating shorter wavelength of
LXe scintillation, possibly by the deformation of spectrum due to absorption by some
impurities.
Once we calibrated the tdelay and subtract it from the PMT time, we can reconstruct
tLXe by individual PMT. Therefore, the systematic diﬀerences of reconstructed time byeach PMT, thit,i , among the channels show the time oﬀsets of channels. We calibrate thetime oﬀset so that thit,i are synchronized. To avoid possible bias of position reconstruction
and time parameter calibration, we use all the scanned data over the inner face to get
uniform event distribution. Figure 6.15 shows thit distribution for all inner face PMTs
after the calibrated tdelay and tof f set are subtracted.
Finally, the time resolutions of a single PMT are evaluated√ by looking at the time
diﬀerence between the reference counter as a function of 1/ Npe . They are well linear
to that quantity as shown in Figure 6.15(d). Those resolutions are used in the chi-square
ﬁtting.
1

The propagation of light in media would obey the group velocity vg instead of the phase velocity of
vp = c/n = 18 cm/ns (for LXe). In case of the known wavelength spectrum of LXe around 178 nm, vg
become 10 cm/ns.

6.4. Calibration of Gamma-ray Detector(a) vef f = 10 cm/ns

97(b) vef f = 8 cm/ns

Figure 6.14: Dependence of reconstructed time on the total number of photoelectrons
with diﬀerent value of vef f used in the reconstruction.

dt VS nphe

×10

-9

12

thit - tref (sec)thit - tref (sec)dt VS distance

(a)

10
8

12

6
4

2

2

0

0

-2

-2

-4

-4
20

30

40

50

10

0.1

0.15
1/ Npe

0.05

0.1

0.15
1/ Npe

× 10

-9

(c)

8
6

2
1.8

(d)

1.6
1.4
1.2

4

1

2

0.8

0

0.6

-2

0.4

-4
-6
0

0.05

1 PMT time resolution-9

Sigma (sec)

12

-6
0

60
d (cm)

dt VS cos(angle)

×10

(b)

8

4

10

-9

10

6

-6
0

thit - tref (sec)

×10

0.2
0.2

0.4

0.6

0.8

1
cos( η)00

Figure 6.15: Distributions of thit reconstructedby individual PMT after calibrated tdelay√
is subtracted as a function of d (a), 1/ Npe (b), and cos(η) (c). Calibration works ﬁne
and no dependence is seen. (d) shows the spread of the distribution (b).

98

CHAPTER 6. Calibration

6.5

Calibration of Relative Timeteγ (nsec)

Amplitude (a.u.)

We discussed the variation of light yield as a result of the puriﬁcation and its correction
in Section 6.4.3. During the puriﬁcation, not only the light yield but also the pulse shapeof LXe scintillation were changing. The average pulse shape of LXe scintillation pulse
in response to a gamma-ray interaction at the beginning and at the end of run 2008
are shown in Figure 6.16. It was getting wider. As a result, the reference time of the
gamma-ray detector was also changed because the constant-fraction method is based on
the constant pulse shape.
To monitor and correct the eﬀect, we used the RD peak in the dedicated RD run. The
dedicated RD run, which is described in Section 4.1.2, were taken in reduced-intensity
beam. Therefore, we can calibrate the timing precisely with a good signal-to-noise ratio.
The run were taken once per week for about 24 hours. We can calibrate time reference (t0 )of teγ with each set. Figure 6.17 shows the plot of the t0 as a function of time. Before we
started the continuous puriﬁcation, it was stable within 15 ps. However after we started
the puriﬁcation, it started drifting. We used a ﬁtted linear function to correct the t0 .
After the correction, the stability of t0 over the whole period is evaluated to be better
than 20 ps by looking at the RD peak in normal MEG run.

0
-0.2

0.3
0.25
0.2

-0.4

0.15
0.1

-0.6

0.05

-0.8

χ2 / ndf
0.07846 / 3
p0-69.39 ± 10.45
p1
5.669e-08 ± 8.516e-09

0
-0.05

-1
0

50

100

150
200
Time (sec)Figure 6.16: Change of LXe pulse
shape. Blue line shows the averagepulse shape at the beginning of run
2008 (in September), and red one
shows that at the end of the run (inDecember). Those two are normalized by the pulse heights.

08/09/08

08/10/08

07/11/08

07/12/08
Date

Figure 6.17: Drift of teγ time reference. Eachpoint shows the t0 evaluated by a set of RD run.
After we started the continuous puriﬁcation, thetime reference started drifting. This time drift wascorrected by the ﬁtted linear function shown inthis ﬁgure.

Chapter 7
Performance
We evaluate performance of the detector in this chapter. First, we describe selection
criteria for the performance evaluation. These criteria are also applied in the ﬁnal analysis.
Note that the selection is not so critical in our analysis because ﬁnally we perform thelikelihood analysis with event-by-event PDFs. This method naturally takes the quality ofeach event into account. In an extreme case, we can use all the reconstructed events as longas we know the response to those events. Thus, our policy of event selection is to select
events with minimum quality cuts and evaluate performance in detail. The resolutionsof detector are evaluated using actual data, and response functions are extracted. Those
responses are used directly or with a small correction as a probability density functions
for the signal event in the likelihood analysis that will be described in Chapter 8. The
detection eﬃciencies of both particles are also evaluated.

7.1

Event Selection

7.1.1

Deﬁnition of Fiducial Volume and AcceptanceFirst, we deﬁne the ﬁducial volume of the gamma-ray detector. At the very edge of thedetector, the performance is extremely low in energy and position measurements andeﬃciency. Therefore, the detector was designed so that the half region of the outer-mostPMTs are out of the acceptance. We deﬁne the ﬁducial volume with this original design,
and we use full volume in the depth direction:|uγ | < 25,|vγ | < 71,

0 < wγ < 38.5 cm.

(7.1)

The positron acceptance is then deﬁned so that the inverse of the positron direction
goes into the gamma-ray detector ﬁducial volume. With this deﬁnition, the acceptanceof angular range is not deﬁned uniquely because of the spread of muon decay vertex on
the target. The acceptance is shown in Figure 7.1. In the original design, the center partin θ direction (| cos θ| < 0.08) was out of acceptance because a positron emitted in thisdirection turns many times in the spectrometer resulting in the diﬃculty of tracking andalso the low probability that the positron reaches the timing counter. Actually, the event
distribution shows low eﬃciency in this region. However, we do not intentionally exclude
these events.
99

100

CHAPTER 7. Performance

inner face (u-v plane)

148.6
cm
24.3cm

142.0
cm
23cm

38.5
cm
6.32cmtop / bottom face (u-w plane)

50.0 cm
7.88cm58.0
cm
9.43cmFigure 7.1: Deﬁnition of the ﬁducial volume. The red shaded part is the ﬁducial volume.
The corresponding positron acceptance is deﬁned so that the inverse of the direction goesto the ﬁducial volume. The light shaded part around center is positron low eﬃciency partdue to the multiple turns, but it is also in the acceptance.

7.1.2

Positron Analysis Cut

Number of Hits
To select positrons reconstructed with reliable tracking, we required the following: thetotal number of hits ≥ 7; the number of chambers that have hits > 3; the span of the
track > 4 chambers; the number of multi-hit chambers > 1. These are really minimum
criteria, just to exclude abnormal tracks. The ﬁrst and second criteria are actually fulleﬃcient. The third and fourth selections are shown in Figure 7.2.

Track-Fitting Quality Cut
We require the following criteria on the ﬁtted tracks to select events with good quality of
the ﬁtting: δEe ≤ 0.7 MeV ∧ δθe ≤ 0.6◦ ∧ δφe ≤ 1.5◦ , where δ means the uncertainty
evaluated by the covariance matrix of the Kalman ﬁlter. In addition, we require thenormalized χ2 of the Kalman ﬁlter to be χ̂2track ≤ 12.

span of the track

7.1. Event Selection

101

12
10
8
6
4
2
0
0

2

4
6
8
10
number of multi hits chamber

Number of events

40000
35000
30000
25000
20000

90000
80000
70000
60000
50000
40000

15000

30000

10000

20000

5000

10000

0
0

5

10

15

0
0

20

χ2 / NDF(a) Distribution of normalized χ2 .
θ uncertainty (deg)Number of events

Figure 7.2: Distribution of track span v.s. number of multi-hit chambers.

0.5

1
1.5
2
Positron energy uncertainty (MeV)(b) Distribution of Ee uncertainty.

1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2
2.5
3
φ uncertainty (deg)(c) Distribution of uncertainties of angular measurements.
Figure 7.3: Track quality selections.

102

CHAPTER 7. Performance

Vertex Cuty (cm)

To guarantee that the positron comes from a muon decay on target, we require thereconstructed muon decay vertex is on the target. Because of the additional air doped
inside the spectrometer, the beam distribution on the target was slightly oﬀ center to
upstream. The two-dimensional distribution of muon decay vertex and the selection
criteria are shown in Figure 7.4.

6
4
2
0
-2
-4
-6
-15

-10

-5

0

5

10

15
z (cm)

Figure 7.4: Two-dimensional distribution of muon decay vertex. Magenta dot-dashedline ellipse shows the target position selection and white dashed line ellipse shows beam
distribution selection. The beam distribution was slightly oﬀ center to upstream side
because of the additional air doping inside the spectrometer. A shadow of hole is clearlyseen at z = −3 cm. Events inside of both ellipses are selected.

DCH-TIC Matching Cut
To select events well matched between the drift-chamber track and the timing-counter
hit, we apply cuts on the quantities |∆zDCH−T IC | and |∆rDCH−T IC |. As will be shown in
Sec.7.5, the time resolution becomes drastically worse as these values increase. Figure 7.5
shows the distributions of those values. There are large tail components, which consistof two kinds of events. One is events with uncorrelated positrons for track and timingcounter hit. It makes a ﬂat distribution in these plots because it is accidental. The otheris positrons hardly scattered by some materials between the drift chamber and the timing
counter. These hard scatterings cannot be predicted in the tracking and the projection
becomes worse, resulting in the worse timing resolution. We required |∆zDCH−T IC −0.1| <10 and |∆rDCH−T IC +2.2| < 3 cm. The systematic bias in |∆rDCH−T IC | distribution stems
from that the r-position of timing-counter hit is just given as the center of the bar.

103

Number of events

Number of events

7.1. Event Selection

50000

45000
40000

40000

35000
30000

30000

25000
20000

20000

15000
10000

10000

5000
0
-10

-5

0

5
∆rDCH-TIC (cm)

0
-20

(a)

-10

0

10
20
∆zDCH-TIC (cm)(b)

Figure 7.5: Distributions of DCH-TIC matching quantities in (a) r- and (b) z- directions.

Ghost-Track Selection
Sometimes diﬀerent tracks share some hits in the track ﬁnding process in high rate environment. The Kalman ﬁlter attempts to ﬁt them separately. On the other hand,sometimes track ﬁnding process recognizes a track as diﬀerent multiple tracks with somecombinations of the hits connections. Even in such a case, the Kalman ﬁlter tries to ﬁtthem as diﬀerent tracks. We call these tracks associating with a common positron ghost
tracks.
Now we get two separate problems. One is that we have to decide the tracks whichshare some hits are really diﬀerent tracks or ghosts. The other is if they are ghosts, thenwe have to choose the best measured track among them.
We put the following conditions on ghost tracks:• ghosts must share a common timing-counter hit;
• the number of chambers which has at least one hit in common must be more thanthe half number of hit chambers on shorter of the two tracks.
If two tracks satisfy these conditions, then those two are recognized as ghosts, otherwise
as diﬀerent tracks.
We then rank those ghost tracks with the number of chambers spanned (S) and thenormalized χ2 of track ﬁtting, χ̂2track . We deﬁned the following quantity which represents
the measurement quality;
Grank =

1+ αχ̂2track ,
S

(7.2)

with which tracks are ranked. The smaller value a track has, the higher position it isranked in. The coeﬃcient α was tuned to 0.025 so that the second term dominates the
ranking when ghosts have very diﬀerent χ̂2track and the ﬁrst term dominates when theghosts have similar χ̂2track . We select the highest rank track out of the ghosts passing theother selections.

104

7.1.3

CHAPTER 7. Performance

Gamma-ray Analysis Cut

Pileup Cut
As described in Sec 5.4.5, we do not discard but use pileup events after eliminating theoverlapping gamma. In the following two case, however, we reject the events:• event on which the elimination does not seem working correctly;• event identiﬁed as a pileup by timing distribution but not by light distribution, thus
the elimination is not applied.
For the ﬁrst item, we require 0 ≤ (Eγnormal − Eγeliminate )/Eγnormal < 0.1 , where Eγnormal
gamma energy by normal reconstruction without pileup elimination and Eγeliminate is thatafter pileup elimination. Figure 7.6 shows the eﬀect of the cut. Ineﬃciency due to thislight
> 1. Ineﬃciency due to
cut is 1.7 %. For the second item, we require χ̂2time < 3 ∨ Npeak
this cut is 3.9 %. In total, ineﬃciency by pileup rejection is (4.5 ± 2.6) %.

Figure 7.6: Gamma energy spectrum with additional pileup cut. Black line is without
any cut, red is after cutting events for which the elimination works to increase energy,
green is after cutting large elimination events additionally.

Cosmic-ray Cut
To cut cosmic-ray events, we use the event topology. Since cosmic-ray events deposit
large energy in deep part of the detector, the ratio of inner PMTs sum to outer ones sum
becomes diﬀerent from that of gamma-ray events originating on the target. We require
Ninner /Nouter > 0.3. Figure 7.7 shows cosmic-ray spectrum taken with the LXe-self trigger
without beam. With the cut, the rate of cosmic-ray background is successfully reduced
well below that from gamma from muon decays. The rate in 51 < Eγ < 54.6 MeV is0.35 Hz, which is about 1 % of gamma-ray background from muon decays. This cut also
rejects gamma rays from the target when their interactions occur very at deep position.
The signal ineﬃciency by this cut is estimated to be 1.1 % by using the MC simulation.

Rate (Hz/2.0MeV)

7.2. Positron Energy, Ee

105

CR spectrum
Acceptance cut

2.5

timecut
CR cut

2
1.5
1
0.5

0

50

100

150
200
Energy (MeV)

Figure 7.7: Cosmic-ray spectrum with several cuts. Black line shows measured spectrum,blue one shows that in ﬁducial volume, green one requires χ̂2time < 3, and red one requires
also Ninner /Nouter > 0.3.

7.2

Positron Energy, Ee

We evaluated the positron energy scale and resolution by ﬁtting the kinematic edge ofthe measured Michel-positron energy spectrum. We formed the ﬁtting function by folding
the theoretical Michel spectrum with the energy-dependent detector eﬃciency, and the
response function for mono-energetic positrons. The former is empirically given by anerror function acc . The latter is well described in the MC simulation by a triple-Gaussian
gtri . Thus ﬁtting function is given by
f (x) = (ftheo (x) × acc (x)) ⊗ gtri (x),(7.3)

where ftheo (x) is the theoretical Michel spectrum. Figure 7.8 shows the ﬁtting on reconstructed Michel spectrum for a range of 51 < Ee < 55 MeV. The extracted response
function is also shown in the ﬁgure. The resolutions extracted from the data are 0.374,
1.06, and 2.00 MeV in sigma for the core component and two tails, with corresponding
fractions of 60, 33, and 7 %, respectively. The complete set of the spectrum parameters
is given in Table 7.1.
To suppress statistical uncertainty, the response function was estimated on a large
data sample: no cut on gamma energy, relative angles, and relative time. We evaluate
the impact of the cuts deﬁning the analysis window by repeating the Michel ﬁtting on
the teγ -sideband with gamma-energy and relative-angle cuts. The results are given in
Table 7.2. Parameters not listed in the table were ﬁxed to the nominal value. These
diﬀerences indicate the systematic uncertainty of the positron response function.

CHAPTER 7. Performance
Events / (0.05 MeV)

106

6000

5000

4000

3000

2000

1000

0

50

51

52

53

54

55

E e (MeV)

Figure 7.8: Fitting of Michel spectrum. Fitting range is 51 < Ee < 55 MeV. Dashedline shows obtained response function.

Table 7.1: Results of the Michel ﬁt.
ParameterValueµacc (MeV)
51.3
σacc (MeV)
3.32
fcore
0.607
µcore (MeV) 0.206
σcore (MeV) 0.371µtail (MeV)
0.11
σtail (MeV)
1.07
fout
0.07
µout (MeV)
0.0
σout (MeV)
2.0
2χ /NDF
88/95

7.3

Table 7.2: Alternative parameter set estimated with analysis window cuts.
Parameter
Value
fcore
0.81
µcore (MeV) 0.23
σcore (MeV)
0.4µtail (MeV)
1.4
σtail (MeV)
0.4

Gamma-ray Energy, Eγ

We measured the energy resolution of the gamma-ray detector using a response to the
55 MeV gamma from π 0 decay. To get 55 MeV gamma ray, we selected the π 0 decay events
where 83 MeV gamma ray were detected in the NaI detector with an opening angle larger
than 170◦ . A correction with the measured opening angle was then applied. Figure 7.9
shows the reconstructed energy spectrum at a point around center of the detector. The
lower tail comes from mainly two reasons. One is the interaction of gamma ray on somematerial in front of the LXe active volume. The other is the shower escape from the innerface. You also see higher tail in the distribution. It is particular to the π 0 data sincemuch more backgrounds from electrons in beam and high-energy gammas from π 0 decays
are there in the environment with π − beam.

Number of events /(0.64 MeV)

7.3. Gamma-ray Energy, Eγ

107

1600
1400
1200
sigma = 1.54 ± 0.06 %FWHM = 4.55 ± 0.20 %

1000
800
600
400
200
0
0

20

40

60
Eγ (MeV)a.u.

Figure 7.9: Reconstructed energy distribution of 55 MeV gamma events from π 0 decays.
The numbers quoted in the ﬁgure is the intrinsic resolution after the pedestal distribution
is unfolded.

ResponseFunction

0.03

0.09

pedestal distribution

0.08

0.025
0.07

0.02

0.06

⊗

0.05

0.015

0.04
0.03

0.01
0.02

0.005

0.01
0

0
0

-50000050000

100000200000300000Total number of photon

Figure 7.10: Fitting function.

We ﬁtted the distribution with an asymmetric function with lower-side tail, f0 (x),convolved with the pedestal distribution in the π 0 run, hπ (x) (Figure 7.10),
f0 (x) ⊗ hπ (x),and f0 (x) is given as
f0 (x) =


(
{
})

t
t

 A exp σ2− (x − x0 )
2
)
( up0

(x−x )2 A exp −2σ20
up0

(7.4)

x ≤ x0 + t,

(7.5)

x > x0 + t,

where A is a scale parameter; x0 is a peak-position parameter; t is a transition parameter;and σup0 is a resolution parameter that indicates the spread of the distribution in higher
side. For this function f0 (x), the FWHM is given as√ln 2 2t
FWHM = 2 ln 2σup0 −σup0 − .
(7.6)
t
2

108

CHAPTER 7. Performance

By deconvoluving the pedestal contribution in the spectrum, we can extract the intrinsic
resolution of the detector that is not aﬀected by a particular condition. To evaluate thepractical resolution in normal muon beam, we then convolved the response function f0 (x)with the pedestal distribution in normal muon run, hµ (x). For the resolution, we quote
here the eﬀective sigma of higher energy side, σup , of the response function f0 (x) ⊗ hµ (x)because it is the most important quantity for the background suppression, while the
FWHM value is important when we think about the eﬃciency.
Since the resolution is position dependent, the response was extracted for each position.
Figure 7.11 shows resolution maps for various positions of the gamma-ray interaction
for wγ ≥ 2 cm. The maps show better resolution around the center of the detector,
and the σup -distribution shows a peak at around σup = 1.8 %. However, since actual
event distribution shows fewer events around the center because of the low eﬃciencyof accompanied positron, the averaged resolution over the actual event distribution isσup = 2.0 % for wγ ≥ 2 cm. The relative uncertainties of resolution evaluation are
estimated to be 10 ∼ 14 % from the errors of ﬁtting and the variation of the resolutions inclose positions. The resolutions are measured separately for the events with 0 ≤ wγ < 1,
and 1 ≤ wγ < 2 cm because the resolution heavily depends on the depth of interaction
when interaction occurs in shallow region. The average resolutions are 3.0 and 4.2 % for
1 ≤ wγ < 2 and 0 ≤ wγ < 1 cm, respectively. Note that those numbers are representative
ones and in the likelihood analysis position-dependent responses are used.
Validation Check of the Analysis with Spectrum from Muon Decays
We can also use the gamma-ray background spectrum to evaluate the energy resolution
and the energy scale. The gamma-ray background comes mainly from RD and frompositron AIF especially in high-energy region. Those rates are rapidly suppressed as theirenergy increase as discussed in Sec 2.3.4. The steepness of the background suppression
indicates the energy resolution. The background spectrum was calculated with the MC
simulation. It was formed with detector resolution, pileup component, and a scale factor.
The measured spectrum was ﬁtted with the MC spectrum with taking the resolution,
pileup component fraction, and scale parameters as free parameters. We performed thisﬁtting separately for the diﬀerent positions. Figure 7.12(a) shows an example of theﬁtting. Comparisons between the parameters estimated with the π 0 data and those withthe ﬁtting are shown in Figure 7.12(b)(c). The consistent results validate the analysis ofenergy measurement. The ﬁt results of energy-scale parameter shown in (c) shows small
bias of ∼0.5 %. When the scale estimated in this way is plotted as a function of the time,
we observe slight increase of 0.25 % for whole period. This bias is used when we consider
the systematic uncertainty of the energy scale.
Systematic Uncertainty on Energy Scale
The factors of the systematic uncertainties on Eγ scale are listed as the following:• uncertainty of 55 MeV-peak estimation in the π 0 run: 0.08 %;• uncertainty of the gain shift during the the π 0 run: ∼ 0.2 %;• uncertainty of the light-yield connection between π 0 run and physics run, mainly
stemming from one of cosmic-ray peak estimation: 0.18 %;

4.5

109

10
60
9

40

FWHM (%)

60

v (cm)

5

σup (%)v (cm)

7.3. Gamma-ray Energy, Eγ

40

4

8
20

3.5

20

0

3

0

-20

2.5

-20

7

6
2

-40

-40
5

1.5

-60

-60

1
-10

0

10

4

20
u (cm)
Entries

Entries

-20
45
40

-20

-10

2

4

0

10

20
u (cm)

35
30

35
25
30
25

20

20

15

15

10

10
5

5
0
0

1

2

3

(a)

4

5
σup (%)

0
0

6

8

10
12
FWHM (%)(b)

Figure 7.11: Distribution of the gamma-ray detector energy resolution for wγ ≥ 2 cm,(a) high energy side sigma and (b) FWHM. Upper two-dimensional plots show the resolution map as a function of interaction position. White boxes in the maps indicate theacceptance. Lower histograms shows the distribution of resolution at each point inside
the acceptance. One entry corresponds to an area of one PMT.

CHAPTER 7. PerformanceσEγ (%)

110

6
5

3
2

103
1
0

10

102

10

1
40

20

30
Position ID

20

30
Position ID(b)
Energy Scale Factor

Number of events / (0.5MeV)

4

1.03
1.02
1.01
1

45

50

55

60
Eγ (MeV)

(a)

0.99
0.98
0.97
10

(c)
Figure 7.12: Gamma background spectrum ﬁt. Fitting function is formed by folding MC
spectrum (green line for single gamma and blue line for pileup component) and detector
resolution. Energy scale, resolution and pileup fraction are free parameters and estimatedby the ﬁtting. (b) and (c) show the results. (b) Comparison of energy resolution estimatedby π 0 data (red ﬁlled squares) and background ﬁtting (blue blank squares). (c) Energy
scale relative to that determined by the 55 MeV peak in π 0 run.
• variation of energy scale estimated by the ﬁtting of sideband data: 0.25 %;In total, we assign 0.4 % uncertainty. This estimation is consistent with the systematic
diﬀerence in the sideband ﬁt result of ∼ 0.5 %. The systematic error of the uniformity corrections with three times lower energy gamma (17.6 MeV) and the possible time variation
due to the puriﬁcation are included in those systematic uncertainties.

7.4

Relative Angle, θeγ and φeγ

7.4.1

Positron Emission Angle

The positron angular resolution was evaluated by exploiting tracks that make two turns
in the spectrometer, where each turn is treated as an independent track. The θ- andφ-resolutions were extracted separately from the diﬀerences of the two track-segments at
the point of closest approach to the beam-axis. The distributions of the diﬀerences are
7.13. The resolutions were estimated with the sigmas of ﬁtted Gaussiansplotted in Figure√
divided by 2. They are σθe = 18, σφe = 10 mrad. Those estimations do not include theeﬀect of scattering in the target. However, it is found to be negligible compared to the450
397.6 ± 9.3

Constant

400

Mean

-0.1284 ± 0.0241

Sigma

1.451 ± 0.024

350
300

Number of events

Number of events

7.4. Relative Angle, θeγ and φeγ

111

600
571.6 ± 14.0

Constant

500

Mean

0.1153 ± 0.0151

Sigma

0.8196 ± 0.0133

400

250
300

200
150

200

100
100
50
0
-10

-5

0

5

10
∆θ (degree)

(a)

0
-10

-5

0

5

10
∆φ (degree)(b)

Figure 7.13: Diﬀerence of positron directions reconstructed with each turn in (a) θ and(b) φ directions.

obtained resolutions. Uncertainty of the resolutions are estimated to be ∼ 1 mrad by thecomparison with the MC simulation and variation of the results by changing the cuts.

7.4.2

Muon Decay Vertex

The vertex position resolutions were measured by looking at the reconstructed edges ofholes on the target. Figure 7.14 shows distribution of reconstructed vertex. We selected
the region including a hole in horizontal and vertical directions as shown in the ﬁgureby the magenta and white lines. Projections of the slices are shown in (b) and (c) forhorizontal and vertical directions, respectively. The shape of the hole image was ﬁtted
with a function formed with beam proﬁle and vertex resolution. The vertex position
resolutions are measured to be ∼ 3.2 and ∼ 4.5 mm in the vertical and the horizontal
directions on the target plane, respectively. Those results were validated by anotherestimation using the two turn events in the same way as for the angular resolutions.

7.4.3

Gamma-ray Position

We measured the gamma-ray position resolution in the dedicated π 0 run with lead bricks
placed just in front of the gamma-ray entrance window of the detector. The details of
the brick are shown in Figure 4.4. Figure 7.15(a) shows the distribution of reconstructed
positions in the run. It shows the image of the lead brick. We used peaks of the slits dueto the shadow of the brick to evaluate the resolution. The projection of the distribution
on v-axis was ﬁtted with superimposition of three Gaussian (for slits peaks) , two Error
functions (for brick edges), and a ﬂoor component (for the events penetrated the brick)as shown in (b). The average of the sigmas is σdata = 6.9 mm. The width of the ﬁtted
Gaussian contains the width of slit itself and the contribution from the spread of π 0 decay
point. A dedicated MC simulation with the same conﬁguration was conducted to evaluate
these eﬀect. The same ﬁt on the MC is shown√ in (c). The average of the sigmas in MC is
2
2
− σMC
= 1.8 mm gives an additional
σMC = 6.5 mm. We consider the diﬀerence σdata

CHAPTER 7. Performance
Vertical axis on target (cm)

112

2
1
0
-1
-2
-5

0

5
Horizontal axis on target (cm)Number of events

Number of events

(a)
2200
2000
1800
1600

700

600

500

1400

400
1200

1000

300

800

200
600

400

100

200
-8

-6

-4

-2

0

2

4

6

8

Horizontal axis on target (cm)(b) Projection on horizontal direction

0

-4

-3

-2

-1

0

1

2

3

4

Vertical axis on target (cm)(c) Projection on vertical direction

Figure 7.14: Target hole ﬁt to evaluate vertex resolutions. (a) shows the distributionof reconstructed vertex. (b) and (c) show the projections of slices given by magenta andwhite lines in (a). Solid lines show the ﬁtted function formed with beam proﬁle and vertex
resolution.

contribution to the resolution stemming from errors of the PMT QEs’ measurements.
The bias of reconstructed position was estimated by looking at the absolute positionsof reconstructed slits. The bias is 0.7 mm in standard deviation. It also stems from theerrors of QEs. We take this bias into the response as an additional smearing because we
could not measure the bias for all position and thus we cannot correct them.
We investigated detail response of the position measurement to the signal events using
MC simulation with taking into account the diﬀerence from the data. Responses obtained
from the MC are shown in Figure 7.16. In u- and v-direction, the responses can be
expressed by double-Gaussians. The tail component comes from events with large spreadof shower or back scattered events. The core component has sigma of ∼ 4 mm with about
70% fraction, while the tail component has sigma of ∼ 8 mm with about 30% fraction.
The resolutions depend on the relative position to the PMT alignment. Figure 7.17 shows
the dependence of the resolutions on distance from the center position of the nearest PMT.
We can measure the position with better resolution when the interaction occurs at the
mid of PMTs, because the light is eﬀectively distributed to several PMTs. The eﬀective
resolutions are ∼ 5 cm for both u- and v-directions on average. The uncertainties are7.4. Relative Angle, θeγ and φeγ

113estimated to be 0.3 mm for both directions.
The response in w-direction has asymmetric shape. We quote here the half width ofthe distribution which contains 68 % of events as a resolution. The eﬀective resolution is∼ 6 mm. The uncertainty is estimated to be 0.7 mm.

(a)
1800
500

1600

450

1400

400

1200

350

1000

300

800

250

600

200

400

150

200

100

0
-15

-10

-5

0

(b) data

5

10

15
v(cm)

50
-15

-10

-5

0

5

10

15
v(cm)

(c) MC simulation

Figure 7.15: Event distribution of π 0 run with a Pb brick. (a) shows scatter plot of thereconstructed position of the data. (b) shows the projection on v-axis for the data, and(c) shows that of the MC simulation.

6000

Simulation

5000
4000

7000
6000

Simulation

5000

60
Simulation
50
40

4000

30

3000

3000

2000

2000

1000

1000

0

Number of events

CHAPTER 7. Performance
Number of events

Number of events

114

-4

0

-2

2

4
urec - utrue (cm)

0

20
10

-4

(a)

0

-2

2

4
vrec - vtrue (cm)

0

-4

(b)

-2

0

2

4
wrec - wtrue (cm)(c)

Position resolution in sigma (cm)Figure 7.16: Position response of the gamma-ray detector obtained from the MC simulation in (a) u-, (b) v-, and (c) w-directions.
0.8
0.7

Simulation

0.6
0.5
0.4
0.3
0.2
0.1
0
0

1
2
3
Distance from the nearest inner PMT (cm)Figure 7.17: Position resolution as a function of the distance from the nearest inner
PMT center in u (red square), and v (blue circle) directions. Resolutions quoted here arevalue of sigma of core component Gaussian.

7.4.4

Combined Resolution

The angular resolutions of the gamma direction were evaluated to be σθγ = 9.9 and
σφγ = 9.2 mrad on average including the uncertainty of vertex position. We see here
worse resolution in θ-direction because both u and w measurements contribute to theθ-direction whereas only v measurement does for φ-direction. The resolutions of relative
angle in θ- and φ-directions were then obtained by combining the angular resolutions of
the two particles. They are evaluated to beσθeγ = 21,

σφeγ = 14 mrad.

(7.7)

There is a signiﬁcant diﬀerence in the responses between the two directions. Therefore,
we separately treat the two directions in the likelihood analysis.

7.5. Relative Time, teγ

7.5

Relative Time, teγ

7.5.1

Positron Time

115

A

σ(t - tB) / 2 (nsec)

We measured the resolutions of the timing-counter bars by looking at the distributions oftime diﬀerence between two consecutive bar’s hits. The results are shown in Figure 7.18.
Resolutions indicated by arrow markers were estimated using the two bars read out with
diﬀerent DRS chips. Thus those resolutions include the chip-synchronization accuracy.
On the other hand, the others were read out with a single chip for the two bars so that
we can measure the bar’s intrinsic resolutions. The resolutions were measured at two
diﬀerent period of the run, and found to be stable. One point with worse resolution on
bar 15 is due to the defect of a DTD channel described in Sec 4.2.5. After exchanging theDTD channel, the resolution was recovered (from black marker to red one). The average
resolution of single bar is estimated to be σ1bar = 67 ps, where the contribution of clock
synchronization (σclock(DRS3) = 37ps) is subtracted (intrinsic resolution). If it is included,
the resolution becomes 76 ps. For the positron with multiple hits we can improve theintrinsic resolution to 54 ps by combining the hits.
0.12
0.1
0.08
0.06
0.04

downstream

upstream

0.02
0

0

5

10

15

20

25
Bar #

Figure 7.18: Time resolution of timing-counter bars. They are evaluated at two diﬀerent
period of the run; September (black square) and November (red blank circle).

The timing resolution strongly depends on the quality of the DCH-TIC interconnection. We investigated the dependence using the π 0 Dalitz decay data. Figure 7.19 shows
the result. As the value of |∆zDCH−T IC | increases, the resolution becomes worse. Positrons
with large values of |∆zDCH−T IC | were probably scattered between the drift chamber andthe timing counter. Thus the measurements of the track length were screwed up. The
Dalitz decay has a similar topology to the µ+ → e+ γ decay but the conditions such as
target material, vertex distribution, positron and gamma-ray energy, and angular correlation are diﬀerent. Here, we assume those diﬀerences contribute independently asa quadratic sum in the resolution, and the dependence on |∆zDCH−T IC | is common topositrons from muon decays in the physics data taking. With this assumption, we extracted the |∆zDCH−T IC | dependence of positron time resolution. It is shown as a redplot in the ﬁgure. At the edge of |∆zDCH−T IC | = 0, the positron time resolution is

CHAPTER 7. Performance
Time resolution (σ, nsec)

116

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

5

10

15
|∆zDCH-TIC| (cm)Figure 7.19: Dependence of time resolution on |∆zDCH−T IC |. Black plot shows themeasured dependence of the Dalitz data. Red one shows estimated resolution for thesignal positrons. Those plots are ﬁtted with fourth order polynomial functions.

(100 ± 13) ps.
The accuracy of the time-of-ﬂight measurement cannot be measured independently.
Hence, we estimated it with the MC simulation tuned to reproduce the conditions of thedrift chamber. It is evaluated to be σeT oF = 88 ps for signal positron with |∆zDCH−T IC | <
6 cm.

7.5.2

Gamma-ray Time

Intrinsic Resolution
To investigate ‘intrinsic’ time resolution of the gamma-ray detector, we performed a divided analysis. We deﬁne two PMT groups, even- and odd-groups. PMTs are assigned
to the two alternately, and each group contains half number of PMTs as shown in Figure 7.20. The gamma-ray interaction time was reconstructed independently for each grouponly using PMTs belonging to the group. The intrinsic time resolution was evaluated by
the distribution of half of time diﬀerence between the two reconstructed time,∆tint
todd − teven
=
,
2
2

(7.8)

whose uncertainty is statistically same as that of average of the two, (todd +teven )/2, whichis equivalent to tLXe . The distribution for 55 MeV gamma events in August π 0 run isint= 45 ps at 55 MeV.
shown in Figure 7.21. It shows the intrinsic time resolution of σtγ
The position dependence of the resolution was investigated along u, v, and w. The
results are shown in Figure 7.22. The resolution is gradually improved as the interaction
occurs closer to lateral faces and suddenly becomes worse. It can be understood that as
closing in one of lateral faces, the total number of photoelectrons increases owing to the
PMTs on the lateral face. When it gets too close to the edge, then the shower escapedominantly works to reduce the number of photoelectrons resulting in worse resolution.

7.5. Relative Time, teγ

117

Number of events

Part of the worse resolution at the shallow part is also explained by the decrease of thephotoelectrons. In addition, there is another contribution. When interaction occurs near
the inner face, then the observed number of photoelectrons in each group heavily depends
on the interaction position. This disproportion of the observed number of photoelectrons
deteriorates the evaluation by the distribution of (todd − teven )/2 because this evaluation
assumes equal resolution of the two groups.
The intrinsic resolution becomes 36 ps at 83 MeV. Figure 7.23 shows the dependence of
the intrinsic resolution on the total number of photoelectrons and incident gamma energy.
They shows improvement as the number of photoelectrons increases. The intrinsic time
resolution is completely dominated by the photoelectron statistics.

4553 ± 36.5

4500

Constant

4000

Mean

0.002524 ± 0.000280

3500

Sigma

0.04474 ± 0.00022

(t

0.5
1
- teven) / 2 (nsec)

3000
2500
2000

z

1500
1000
500
0
-1

0.12
0.1
0.08
0.06

0.1
0.08
0.06
0.04

0.02

0.02
5

10

(a)

15

20

25
w (cm)

odd

0.12

0.04

0
0

0

Figure 7.21: Distribution of half oftime diﬀerence of two group reconstructed
time. The intrinsic time is estimated to be
45 ps in sigma at 55 MeV.

Sigma of ∆t/2 (nsec)

Sigma of ∆t/2 (nsec)

Figure 7.20: PMT grouping scheme forthe intrinsic time resolution study. PMTs
indicated with blue circles belong to odd
group, and those with orange circles belong to even group.

-0.5

0

-50

0

50
u or v (cm)(b)

Figure 7.22: Intrinsic time resolution at diﬀerent position, (a) as a function of wγ and(b) as a function of uγ (black circle) and vγ (blank square).

CHAPTER 7. Performance

Sigma of ∆t/2 (nsec)

Sigma of ∆t/2 (sec)

118

0.14
0.12
0.1
0.08

0.055
0.05
σ = 0.33 / E

0.045
0.04

0.06
0.04

0.035
0.02
0
0

20000

40000
60000
80000
Total number of photoelectrons

0.03
40

(a)

50

60

70

80

90
100
Eγ (MeV)(b)

Figure 7.23: Intrinsic time resolution as a function of total number of photoelectrons(a) and energy (b). The red cross mark in (a) shows the resolution estimated in the CW
run at 17.6 MeV.

Practical Resolution
We measured the practical time resolution of the gamma-ray detector using π 0 → γγ
data. We used events where one of the two gamma rays was measured by the plastic
scintillation counters mounted on the NaI detector as a reference counter.
Figure 7.24 shows the distribution of the time diﬀerence between the 55 MeV gamma
ray at the gamma-ray detector and the 83 MeV one at the reference counter,∆tabs = tγ − tref ,(7.9)

where the time-of-ﬂight of the two gamma rays are already subtracted with an assumptionof the π 0 decay vertex at the origin. The spread of the distribution includes contributions
from the uncertainty of the vertex position and the time resolution of the reference counter.
The contribution from the vertex uncertainty is evaluated to be 58 ± 2 ps from the spreadof the π − beam on the target. The reference counter resolution is evaluated to be 93±7 psby looking at the time diﬀerence between the two PMTs on a plate. By subtracting theseabs
contributions, the practical time resolution is evaluated to be σtγ= 135 58 93 = 78 ps
at 55 MeV.
At 83 MeV, the sigma of ∆tabs distribution is 127 ps. In the second π 0 run in December, the sigma of distribution for 55 MeV gamma ray is improved to 130 ps owing to theincrease in the LXe light yield. Using these measurement points, the energy (more precisely photoelectron) dependence is extracted. Figure 7.25 shows the plot and extracted
dependence-curve. Here the December data is plotted at an energy to which the mean
number of photoelectrons corresponds in August. The dashed curve shows the practical time resolution after subtracting the other contributions. From this curve, the time
resolution for the signal is evaluated to beabs= 80 ± 6 ps.
σtγ(7.10)

We consider the diﬀerence from the intrinsic resolution stems from errors of position
reconstruction and the spread of shower which can be eﬀectively canceled out in the7.5. Relative Time, teγ

119

Number of events /(0.03 nsec)

evaluation of the intrinsic resolution. In particular, the remaining constant term of 24 ps
would be contribution of the depth reconstruction error which directly deteriorates thetime-of-ﬂight measurement.
2000
χ2 / ndf

276.4 / 71
Constant
637.9 ± 4.4
Mean0.006298 ± 0.000925
Sigma
0.1346 ± 0.0008

1800
1600
1400
1200
1000
800
600
400
200
0
-1.5

-1

-0.5

0

0.5

1
1.5
tLXe - tref (nsec)

Figure 7.24: Distribution of time diﬀerence between the gamma-ray detector and thereference counter.

Sigma of ∆ t/2 (nsec)

0.16
0.14
0.12
0.1

0.08
0.06

π0 August

0.04

π0 December

0.02

0.5522 / E + 0.1122
0.5522 / E + 0.0242

0
40

50

60

70

80

90
100
Eγ (MeV)Figure 7.25: Energy dependence of the gamma-ray detector timing resolution. Plot withblank circle and blue ﬁlled square show the sigma of the time diﬀerence distribution in
August and December π 0 runs, respectively. Dashed line shows the gamma-ray detector
resolution after the other contributions are subtracted. Dot-dashed vertical line shows
the signal energy.

120

7.5.3

CHAPTER 7. Performance

Gamma-Positron Relative Time

The overall time resolution between positron and gamma ray was measured with the RD
events. Since the RDs are time coincident events, they make a peak in time distribution
on top of the ﬂat distribution of accidental background. We used the Eγ -sideband data ofphysics run. The event topology is very close to that of µ+ → e+ γ event, and the analysis
and run conditions are common to those for signal. The time distribution is shown in
Figure 7.26. It was ﬁtted with a Gaussian plus ﬂat ﬂoor. With taking into account thediﬀerent gamma-ray energy, the time resolution for the signal is evaluated to beσteγ = 148 ± 17 ps,(7.11)

Events /(0.080 nsec)

where the uncertainty includes the errors of ﬁtting and extrapolation to the signal energy.
This resolution is consistent with the combination of individual components of resolutions.

500
400
300
200

mean = 6 ± 16 ps
sigma = 152 ± 16 ps

100
0

-1

0

1
teγ (nsec)

Figure 7.26: Distribution of tγ − te in physics run for 40 < Eγ < 45 MeV. It shows clear
radiative decay peak on top of the accidental background ﬂoor.

7.6. Positron Detection Efficiency, e

7.6

121

Positron Detection Eﬃciency, e

The probability that a positron from µ+ → e+ γ decay would be measured is given by thefollowing joint probabilityp(acceptance, trigger, tracking, TIC hit, DCH − TIC match, selection).

(7.12)

Here, we investigate the positron detection eﬃciency e which is deﬁned as the probabilityof measuring a positron emitted in the acceptance deﬁned in Sec.7.1.1. The positrondetection eﬃciency can be decomposed into the two component; the tracking eﬃciency inthe drift chamber and the probability of getting a hit on the timing counter. We deﬁnethem more precisely with conditional probabilities,
e = p(good track | acceptance) × p(matched TIC hit | good track, acceptance)= DCH × AT IC ,(7.13)
where ‘good track’ requires the selection criteria for the ﬁnal analysis; ‘matched TIC hit’requires at least one hit on the timing counter ﬁred by the trigger and matched with areconstructed good track.
We use Michel positrons taken in the physics run as a control sample. Only thediﬀerence from positrons from µ+ → e+ γ decays is their energy. By selecting high-energy
positrons (Ee > 50 MeV), we can reduce the diﬀerence, and ﬁnally a correction will beapplied using the measured energy dependence.

7.6.1

Conditional Timing-Counter Eﬃciency

The conditional timing-counter eﬃciency AT IC was measured with data sample of theDCH-self trigger. We selected the good-track events in the sample without any reference
to the timing-counter information. Here, we only applied a ttrack cut to remove tracks
which are out of timing-counter digitization time because the digitization window of thedrift chamber is wider than that of the timing counter. Then the ratio between the numberof matched good tracks and the number of good tracks gives the probability. This ratio
is plotted in Figure 7.27 as a function of the positron energy. The average probability
for Michel positrons with Ee > 50 MeV is 34 %. By estimating the end-point of theprobability curve, the probability for the signal positron is evaluated to be AT IC = 38 %.
The ineﬃciency stems from positrons which do not hit the timing counter by changing
their trajectories from the expectations because of the energy loss, multiple scattering,
hard scattering, or annihilation by interacting with material in the spectrometer such aschamber frame, preampliﬁers, and cables.

7.6.2

Drift-Chamber Tracking Eﬃciency

The drift-chamber tracking eﬃciency DCH was evaluated with data sample of the TIC-self
trigger. The number of observed Michel positrons can be written as
Nobs = Nµ × f (Ee > 50 MeV) ×1
4π
× DCH × AT IC × ,Ω
P(7.14)

where Nµ is the number of muons stopping on the target, and we can calculate the number
with the stopping rate Rµ,stop given by Eq.3.1 and the total amount of proton current inCHAPTER 7. Performance
Efficiency

122

0.4
0.35
0.3

χ2 / ndf
p0
p1
p2

7.436 / 13
48.36 ± 0.15
2.373 ± 0.378
0.3853 ± 0.0140

0.25
0.2
0.15
0.1
0.05
0

46

48

50

52
Ee (MeV)Figure 7.27: Conditional timing-counter eﬃciency curve. The eﬃciency for the signal
is estimated by the end-point of the curve. Blue solid line shows the ﬁtted error function,and red dashed one shows the ﬁtted 4-th polynomial function.

the livetime; f (Ee > 50 MeV) is a fraction of Michel spectrum over 50 MeV calculated tobe 0.101; the solid angle of acceptance is 4π/Ω = 0.1; and P is the pre-scale factor of the
TIC-self trigger of 107 . With this relation, the positron detection eﬃciency is evaluatedto be 13.5 % for Michel positrons over 50 MeV. The energy dependence is almost ﬂat at
the high-energy region and the correction factor is extracted to 1.02 for the signal energy.
Thus the eﬃciency of signal positron is evaluated to bee ≈ 14 %.

(7.15)

With the evaluated AT IC , the drift-chamber tracking eﬃciency is evaluated to be DCH ≈37%. The ineﬃciency stems from the dead layers and low voltages due to the discharge
problem.

7.7

Gamma-ray Detection Eﬃciency, γ

The gamma-ray detection eﬃciency was estimated using the MC simulation, and conﬁrmed by the measurement of π 0 events and gamma-ray background rate in physics run.
Figure 7.28(a)(b) shows the distribution of energy deposit in LXe for 52.8 MeV
monochromatic gamma ray. The tail comes from interactions with material in frontof the active volume and shower escape from the inner face. Figure 7.28(c) shows theradius distribution of ﬁrst interaction point of the gamma rays. The contribution of theineﬃciency is summarized in Table 7.3. The detection eﬃciency is given in Figure 7.28(d)a function of the energy cut threshold. For the analysis window (Eγ > 46 MeV), theeﬃciency becomes 67 %. The eﬃciency was evaluated at each position of the detector
and average eﬃciency was calculated with a weight of actual event distribution of positron
direction. Then it is evaluated to be 66 %.
The eﬃciency was measured using NaI-self trigger data in the π 0 run. By tagging a
gamma around 83 MeV from π 0 decay, we can get the condition that a gamma around
55 MeV is emitted to the detector. We can calculate the eﬃciency by looking at themeasured spectrum shown in Figure 7.29. The eﬃciency is estimated to be (68 ± 2.5) %for Eγ > 46 MeV, where the uncertainty comes from the evaluation of the radiativeNumber of events

Number of events

7.7. Gamma-ray Detection Efficiency, γ

12000

12000

(a)

10000

8000

6000

6000

4000

4000

2000

2000

7000

(c)

0

40
50
Energy deposit (MeV)
Efficiency

30

LXe

20

COBRA

10

WindowPMT

00
Number of events

(b)

10000

8000

6000

123

46

48

50

52
54
Energy deposit (MeV)

0.7
0.6

(d)
0.5

5000
0.4
4000
0.3

3000

0.2

2000

0.1

1000
0
0

20

40

60

80
100
radius (cm)

0

46

48

50
52
54
Energy cut threshold (MeV)

Figure 7.28: Evaluation of gamma-ray detection eﬃciency with the MC simulation.
(a) Distribution of the energy deposit of 52.8 MeV gamma generated in the acceptancedirection. (b) Close-up view for the analysis region. (c) Distribution of radius position of
the ﬁrst interaction. Blue ﬁlled histogram shows that for the event Edep < 46 MeV, andblank one for all events. (d) Eﬃciency curve as a function of the energy cut thresholdvalue.

Table 7.3: Ineﬃciency of gamma-ray measurement.
Ineﬃciency (%))
Drift-chamber frame, support structure
2.5
14.2
COBRA magnet
Entrance window, PMT, support holder
7.3
4.0
Other materials
6.0
Shower escapeSelection
5.6 1
Total
37
1

For the detected events.

124

CHAPTER 7. Performance

Figure 7.29: Evaluation of gamma-ray detection eﬃciency with π 0 data. Diﬀerent
spectra at low energy show diﬀerent estimations of neutron events.

capture reaction, π − p → γn, which emit a 129 MeV gamma. This consistent results
validates the eﬃciency evaluation with the MC above.
In addition, the gamma-ray background rate was measured and compared to the expectation from the MC simulation. As will be discussed in Sec.8.2.1, the rate was agreedin ∼5 % relatively, while the expectation has a relative uncertainty of ∼7 % coming fromthe beam rate uncertainty and evaluation of the contribution from the muon decay oﬀ
target.
The cuts of pileup and cosmic-ray events give 5.6 % ineﬃciency. Thus in total, thegamma-ray measurement eﬃciency is given as
γ = (63 ± 4) %7.8

(7.16)

Summary of Performance in 2008

The summary of the detector performance is given in Table 7.4.
Table 7.4: Summary of detector performance. Resolutions are given in sigma.
Resolution (σ)
Positron energy
0.70 %
Positron emission angle (θ, φ) (18, 10) mrad
Muon decay vertex
3-4.5 mmTiming-counter time
55-70 ps
2.0 %Gamma energy
Gamma position (u, v and w) (5, 5, 6) mm
80 ps
Gamma timingteγ
148 ps(21, 14) mrad(θeγ , φeγ )eﬃciency
Positron eﬃciency
14 %Gamma eﬃciency
63 %Chapter 8µ+ → e+γ Search Analysis
8.1

Overview of Analysis

The goals of this chapter are• to estimate the best value of the number of µ+ → e+ γ decay events in the data;• to estimate the interval of that value;• to calculate the observed number of muon decays;• and ﬁnally to calculate the best-estimated value of the branching ratio and its
interval.
To measure the branching ratio on µ+ → e+ γ decay, we need to extract the numberof µ+ → e+ γ events in the data sample and the total number of muon decays that we
observed. The latter is measured by counting Michel decays in the data sample. For theformer, we perform a likelihood analysis. We actively detect the signal events instead of
passively counting the remaining events by cutting background candidates. A conﬁdence
interval of the measured branching ratio is then assigned with a frequentist approach.
In addition to the likelihood analysis, we also perform cut-counting analyses to understand background and to check the analysis. It is diﬃcult to see the background levelin the likelihood analysis because the analysis window is set relatively wide so that thebackground distribution is ﬁtted. Therefore, we conveniently deﬁne signal boxes to see
the background level.
To avoid introducing bias in analysis, we adopted a blind analysis. It is often used inrare-decay searches in high-energy physics (for example, see a review [92]). We blindedourselves by a simple hidden-box method. We masked events in the region where µ+ →e+ γ events might appear in (teγ , Eγ ) two-dimensional plane. We calibrated the detectors,
selected events, evaluated the performance, and estimated the background without looking
at events inside the hidden box.

8.1.1

Pre-selection

To reduce the data size and to analyze eﬃciently, ﬁrst we applied a pre-selection for all
events of the MEG trigger data. Full reconstruction was not done in this stage, but some
125

126

CHAPTER 8. µ+ → e+ γ Search Analysis

fundamental kinematics were reconstructed. The point of pre-selection is not to discard
any good events.
The selection criteria are the following,i) −6.9 < tγ − tT IC < 4.4 nsecii) |ttrack − tT IC | < 50 nsec,
where tγ is the gamma emission time reconstructed with an assumption that the muon
decay vertex is at the origin; tT IC is timing-counter hit timing and the time-of-ﬂight is
not subtracted; Thus we did not use precise tracking information for the pre-selection.
The reason of the asymmetric window for i) is to acquire the multi-turn events; becauseof the absence of the tracking, we do not know the number of turns before the hit on
the timing counter. The second criterion requires at least one track associated with thetrigger is found. Reduction power of the pre-selection is about factor 6.

8.1.2

Blinding

For the pre-selected event, we apply a blinding process, namely masking events in a predeﬁned hidden box for the blind analysis. We deﬁned the hidden parameters as a set of(teγ , Eγ ), and the hidden box asi) 48 < Eγ < 57.6 MeVii) |teγ | < 1 nsec,
Full reconstruction is required to eﬃciently hide the possible signal events. The box size
was determined with the detector resolutions to fully cover the signal region. Selected
events go to a separated data stream, and user cannot touch them. Only the unmasked
events are opened to the user. We call these unmasked events sideband data. Thus, thefollowing event selection criteria were tuned without looking at the data in the signal
region but only with the sideband data. Event distributions on (teγ , Eγ ) plane after the
pre-selection and blinding process is shown in Figure 8.1. No cuts are applied for eventsin the plot.

8.1.3

Deﬁnition of Analysis Window

We deﬁned the analysis window to which the likelihood analysis is applied as the following:• 50 < Ee < 56 MeV,
• 46 < Eγ < 60 MeV,• |θeγ | < 100 and |φeγ | < 100 mrad,• |teγ | < 1 nsec,
This window is not a so-called signal window. It includes background events to estimate
the background distribution in a ﬁtting. It is set wider than the hidden box so that we
can study the ﬁtting and the background before opening the box.

Eγ (MeV)

8.1. Overview of Analysis

127

70
65
60
55
50
45
40

-5

0

5
teγ (nsec)

Figure 8.1: Distribution of teγ v.s. Eγ after the pre-selection and blinding process. No
cuts are applied. Events outside the pre-selection time window come from accidental
overlap positrons. Analysis window for the likelihood analysis and signal box (A) are also
shown as blue and red boxes, respectively.

8.1.4

Deﬁnition of Signal Boxes

To know our status and to understand backgrounds, we need to estimate the level ofbackground in our data. For that purpose, here we deﬁne two signal boxes for convenience.
One is a box based on the evaluated resolutions; the half width of each variable is basicallygiven by 1.64 σ 1 .
• 52.2 < Ee < 53.8 MeV,
• 51.0 < Eγ < 54.6 MeV,
• |teγ | < 242 ps,• π − Θeγ < 33 mrad,
where we use the quantity of π − Θeγ for the back-to-back condition instead of separated
θeγ and φeγ for an eﬃcient cut. We call this box “signal box A”.
1

For a Gaussian distribution, this width retains 90% eﬃciency.

128

CHAPTER 8. µ+ → e+ γ Search Analysis

The other is more optimized to the sensitivity,• 52.0 < Ee < 56.0 MeV,
• 51.0 < Eγ < 56.0 MeV,
• |teγ | < 273 ps,• π − Θeγ < 42 mrad.
Asymmetric windows on Ee and Eγ are adopted because the background distributions
have large asymmetries for the signal energy. The window for time and angles are thenbroadened so that the sensitivity becomes the highest considering the signal eﬃciencies
and the number of expected backgrounds. We call this box “signal box B”.

8.2

Background Estimation

8.2.1

Rate of Gamma-ray Background

In Sec 2.3.4, we roughly estimated the background and found the accidental backgroundwill be a dominant source. The rate of the accidental background depends on the beam
intensity, time resolution, angular resolution, positron background spectrum, and gammaray background spectrum. In particular, the gamma-ray spectrum is the most importantpart for the suppression of the background. The theoretical spectrum of gamma from
RD was used in the previous estimation. In reality, it depends on the gamma-ray yielddepending on the actual design of the apparatus and its energy resolution.
A LXe-self trigger was mixed in the data taking. Using this data, we can measure theactual rate of the gamma-ray background in the detector. It is shown in Figure 8.2. The
rate in signal box A is 34 Hz.
The expected spectrum by the MC simulation is superimposed in this plot. For theabsolute rate, the MC spectrum has about 7 % uncertainty mainly from the uncertainty
of beam rate and contribution from moun decay oﬀ target. In this plot, we used the muon
beam rate measured at the center of COBRA, Rµ,center = (3.69 ± 0.08) × 107 µ+ /sec. Data
and MC match well both in shape and in absolute rate. AIF is found to be a large source ofhigh-energy gamma yield comparable with RD. For example, in signal box A, the fractionof RD is 51 %, that of AIF is 42 %, and that of pileup component is 8 %. The cosmic-ray
contribution is measured to be 1 % from the data with no beam (see Figure 7.7). The
origins of the AIF process are mainly the drift-chamber material (> 60 %) and that of
the target.

8.2.2

Estimation of Background Level

It is diﬃcult to deﬁne the expected number of backgrounds in the signal region in thelikelihood analysis because we use the background distribution positively in the ﬁtting and
we use also worse measured events, that is, the event selection is not optimized to eliminate
background events. However, to know our status and to understand backgrounds, we needto estimate the level of background in our data. We estimated the number of background
events in the signal boxes deﬁned in Sec.8.1.4 using the sideband data.

Rate (Hz /0.50 MeV)

8.2. Background Estimation

129

102
MC AllAIF
RD
Pileup
Data

10

1
2.4
2.2
2
1.8
1.6

10-1 1.4
1.2
1
0.8

400.6
0.4

45
40

45

50

55

60

50

55

60

65
65

70
Eγ (MeV)Figure 8.2: Absolute rate of gamma-ray background. It is compared with the MC expectation (green: gamma from radiative decay, magenta: that from positron annihilationin-ﬂight, blue: pileup component, and red: total). Discrepancy at the low-energy is due
to a trigger eﬀect.

Accidental Background Events
We use the teγ -sideband data to estimate the accidental background. The range is selected
to 1 < |teγ | < 4 ns not to be aﬀected by the trigger. In addition, to increase statistics, we
also use a wider range of π − Θeγ of 60 mrad with a correction of small trigger eﬀect.
The event distribution in the sideband is shown in Figure 8.3. The number of events
observed in the sideband window is 38 events. Taking the wider windows into account, thenumber of the accidental backgrounds in signal box A is estimated to be 0.95±0.15, where
the error is statistical uncertainty. That in signal box B is estimated to be 2.08 ± 0.03 inthe same way.
The background level was estimated in another way. A maximum likelihood ﬁt was
performed on another sideband window that includes lower range of Eγ (> 46 MeV). Fromthe probability ratio between the signal box and the sideband window, the expected
number of the accidental backgrounds is estimated to be 0.8 ± 0.02, where statisticaluncertainty is smaller owing to larger amount of statistics in low-energy region.

Radiative Decay Events
The expected number of RD events in signal box A (B) is estimated to be 0.02 ± 0.004(0.03 ± 0.006) by calculating the partial branching ratios in the boxes and scaling theobserved number of RD events in the Eγ -sideband by the ratio of the branching ratios.
The detail description is given in Appendix A. The RD events is not a severe backgroundsource in this analysis.

CHAPTER 8. µ+ → e+ γ Search Analysis

Eγ (MeV)

130

60
58
56
54
19 events

19 events

52
50
48
46
-4

-2

0

2

4
teγ (nsec)

Figure 8.3: Event distribution in the teγ -sideband. A looser cut on π − Θeγ than thatfor the signal box is applied. Green boxes show the windows used for the backgroundestimation. Red one shows the signal box. Hatched one shows the hidden box.

8.2.3

Distribution of Background Events from Sideband Data

Since the background is dominated by the accidental background, we can estimate distributions of the background in the analysis box by looking at those in the teγ -sideband.
Figure 8.4 shows the teγ distribution when we applied the analysis window cuts on variables except for teγ . The scatter plot of teγ v.s. Eγ is shown in Figure 8.5. The time
distribution is ﬂat within the statistical ﬂuctuation in this range. For a wider range, thetime distribution is deformed by the trigger resolution. From this plot, we estimated thenumber of events in the analysis box to be 1192 events.
We obtained the distribution of Ee and Eγ from the sideband data where the cuts on
the other energy and angles are relaxed. The modelings of the distributions are alreadydescribed in Sec.7.2 and Sec.7.3 for Ee and Eγ , respectively.
We would expect ﬂat distributions also for the angular distributions. However, acceptance and trigger eﬀects deform the distributions. Figure 8.8 shows the observed angle
distributions in the sideband. In particular, there was a bias in trigger conditions in φdirection. It is considered due to the mismatch in the look up table, which was preparedusing the timing z-counter while the φ-counter was actually used. We lost some trigger
eﬃciency because of this bias.

Number of events / (0.2 nsec)

8.2. Background Estimation

131

140
120
100
80
60

EntriesMeanχ2 / ndf

3577
28.52 / 29

p0

118.3 ± 2.0

0.0183

40
20
0
-4

-2

0

2

4
teγ (nsec)

Eγ (MeV)Figure 8.4: teγ distribution in the sideband. The analysis window cuts are applied on
variables except for teγ . Fewer events at the center part is due to the blinding.

60
58
56
54
52
50
48
46
-4

-2

0

2

4
teγ (nsec)

Figure 8.5: Scatter plot of teγ v.s. Eγ . The analysis window cuts are applied on variables
except for the two.

Number of events / (0.2MeV)

Number of events / (0.5MeV)

CHAPTER 8. µ+ → e+ γ Search Analysis

132

10000

8000

6000

4000

4500
4000
3500
3000
2500
2000
1500
1000

2000

500
45

50

55

0
40

60
Ee (MeV)
Number of events / (0.5MeV)

Number of events / (0.2MeV)

0
40

104

103

102

10

45

50

55

60
Eγ (MeV)

45

50

55

60
Eγ (MeV)

103

102

10
1
40

45

50

55

1
40

60
Ee (MeV)

Number of events

2200

Number of events

Figure 8.7: Gamma background spectrum. Black circle plot shows data spectrum measured in the sideband data. Red
solid line shows the MC spectrum foldedwith detector resolution ﬁtted to the data.
The MC spectrum is composed of singlegamma component (green dotted line) andpileup component (blue dashed line).

Number of events

Figure 8.6: Positron background spectrum. Black circle plot shows data spectrum measured in the sideband. Red solidline shows Michel spectrum folded withacceptance and eﬃciency function and detector response ﬁtted to the data.

1400

2500

2000
1800

1200

2000

1600
1400

1000

1500

1200

800

1000

600

1000

800
600

400
500

400

200

200
0
-1

-0.98

-0.96

-0.94

-0.92
cosΘeγ

0
-0.4

-0.2

0

0.2

0.4
θeγ (rad)

0
-0.4

-0.2

Figure 8.8: Angular distributions of background in the sideband.

0

0.2

0.4
φeγ (rad)8.3. Maximum Likelihood Fit

8.3

Maximum Likelihood Fit

8.3.1

Formalism

133

Here, we deﬁne the likelihood function which is used to estimate the best values of thenumber of µ+ → e+ γ events (Nsig ), that of RD (µ+ → e+ νe ν µ γ ) events (NRD ), and thatof accidental background events (NBG ).
Let us deﬁne the following variables:• ~x : a set of observed variables of a observation (observation vector);~ : a data set of No independent observations {~x1 , ~x2 , · · · , ~xNo };• X• θ~ : a set of unknown parameters to be estimated.
Then a likelihood function is normally deﬁned as~ =~ | θ)
L(XNo∏~
p(~xi | θ),

(8.1)

i=1

~ is a conditional probability density function (PDF). It gives the probability
where p(~xi | θ)
~ It is a function of ~xdensity of the observation ~xi when the parameter set is equal to θ.
~~and normalized to 1. The left-hand side, L(X | θ), is a likelihood if we regard it as a
~ Then it does not have a meaning of probability, and thus it is not
function only of θ.
normalized. The maximum likelihood ﬁt is a method of estimating unknown parameter set~ has its maximum, given a particular~ | θ)
θ~ by searching for the value θ~best for which L(X~set of observations X.
In high-energy physics, experiments are usually conducted with limitations of timeor beam, and we do not ﬁx the number of measurements in advance. In such case, theobserved number of events statistically ﬂuctuates. When we regard the observed numberof events as one of parameters to be estimated, the likelihood function is extended to~ =N~ | θ)
Lext (XNo
No −N ∏eNo !

~p(~xi | θ),

(8.2)

i=1

where N is the estimated or ﬁtted number of events. This is called the extended likelihood
function. By the extended maximum likelihood ﬁt, we can estimate the number of events
to be observed in the experiment using the ‘shape’ of event distribution, and this estimatednumber can be thought a better estimator of the ‘true’ number of events than the actualobserved number.
We use the extended maximum likelihood ﬁt to estimate the number of events of each
event type. In our case, the parameters to be estimated are
θ~ = (Nsig , NRD , NBG ) and N,(8.3)

and they are connected by a relation,
N = Nsig + NRD + NBG .

(8.4)

Thus the total number of free parameters of the ﬁtting is three. The number of each
event type obeys the Poisson distribution and the errors are independent one anther inCHAPTER 8. µ+ → e+ γ Search Analysis

134

the extended maximum likelihood ﬁt, whereas the numbers of events between event types
are fully anti-correlated and the errors become smaller in the normal maximum likelihood~ as L(Nsig , NRD , NBG ) or simply L.
~ | θ)
ﬁt. After this, we describe Lext (XAs discussed in Sec.2.3, the four kinematic variables, (Ee , Eγ , Θeγ , teγ ), can be used todiscriminate backgrounds. We saw, however, diﬀerent detector responses for the angularmeasurement in θ and φ directions. Therefore, we separately treat these two directions
for the back-to-back condition. Thus, the set of kinematic variables becomes(Ee , Eγ , θeγ , φeγ , teγ ).

(8.5)

We observed that the performance of the detector is position dependent. This means
the PDF is diﬀerent for each event. One way to cope with this situation is using an average
PDF. If we do not care where and when the event was detected, we can form a single
PDF. Measurements would obey this distribution. This is not biased. However, we could
lose some sensitivity because we abandon some useful information. A better way is using
event-by-event PDFs. Then, we can use a good-measured event with a large weight orlarge signiﬁcance while we can also use worse-measured events without discarding them.
To take into account the event-by-event diﬀerence of PDFs, we incorporate also theuncertainties or resolutions of measurements into the observation vector. Thus, our observation vector is given as~xi = (Ee , Eγ , θeγ , φeγ , teγ , δEe , δEγ , δθeγ , δφeγ , δteγ )i = (~xi , δ~xi ),(8.6)

Using Bayes theorem, the probability that an event is observed at ~x = ~xi is given byp(~xi ) = P (sig) · p(~xi | sig) + P (RD) · p(~xi | RD) + P (BG) · p(~xi | BG),

(8.7)

because P (sig|~xi ) + P (RD|~xi ) + P (BG|~xi ) = 1. Thus our PDF can be writtenp(~xi |Nsig , NRD , NBG ) =

Nsig
NRD
NBG
· p(~xi | sig) +· p(xi | RD) +· p(~xi | BG). (8.8)
NNN

Let us write the PDFs likep(~xi | sig) = S(~xi ) = s(~xi |δ~xi )p(δ~xi ),p(~xi | RD) = R(~xi ) = r(~xi |δ~xi )p(δ~xi ),p(~xi | BG) = B(~xi ) = b(~xi |δ~xi )p(δ~xi ),then our likelihood can be formed as
L(Nsig , NRD , NBG )(NoN No e−N ∏
NRD
NBG
Nsig
=
· S(~xi ) +· R(xi ) +· B(~xi )No ! i=1 N
N
N

)

(

=

No
N No e−N ∏
NRD
NBG
Nsig
s(~xi |δ~xi )p(δ~xi ) +r(~xi |δ~xi )p(δ~xi ) +b(~xi |δ~xi )p(δ~xi )No ! i=1 N
N
N

=

No
No
∏
NRD
NBG
N No e−N ∏Nsigs(~xi |δ~xi ) +r(~xi |δ~xi ) +b(~xi |δ~xi ) ,p(δ~xi )No ! i=1
N
N
N
i=1

(

)

)

(8.9)

8.3. Maximum Likelihood Fit

135

where from the second line to the third, we used an assumption that p(δ~xi ) is common∏ oto all the event types2 . The term Nxi ) is then omitted in the likelihood ﬁt because
i=1 p(δ~it is independent of the ﬁt parameters, while p(δ~xi ) is deﬁned by the measured event
distribution for the toy-MC simulation, described later in Sec.8.4.1.
The maximization of the likelihood, or more precisely the minimization of the negative
log likelihood (N LL) is done using the MINUIT package [93].

8.3.2

Probability Density Functions

PDFs for µ+ → e+ γ Decay
The µ+ → e+ γ signal PDF, S (or s), can be written as a product of statistically independent PDFs for the ﬁve kinematic variables.
S(Ee , Eγ , θeγ , φeγ , teγ ) = S1 (Ee )S2 (Eγ )S3 (θeγ )S4 (φeγ )S5 (teγ ),(8.10)

each of the components is deﬁned by the detector response function with measured resolutions. In Chapter 7, we investigated the detector performance and obtained all of the
response functions. S1 (Ee ) is deﬁned by the positron energy response evaluated by theﬁtting of Michel-spectrum edge. It is given as a triple Gaussian. We use a common PDF
over all events in the analysis region. S2 (Eγ ) is deﬁned by the gamma energy response
evaluated by the 55 MeV gamma in π 0 run. It is asymmetry with lower tail. Since
the resolution is position dependent, S2 is diﬀerent for the reconstructed gamma position. S3 (θeγ ) and S4 (φeγ ) are deﬁned by the angular responses formed by combining theevaluated resolutions of gamma-ray position, positron emission angles and muon decay
vertex. Since the gamma position resolution is evaluated position dependently, they arealso dependent on the reconstructed gamma position. S5 (teγ ) is deﬁned by the timing
response evaluated by the RD peak in normal data taking. The resolution depends on thequantity |∆zDCH−T IC |, which is the diﬀerence between the z-coordinate of positron hit
position on the timing φ-counter bar surface reconstructed by the timing counter and the
projected track. It also depends on the reconstructed gamma energy to take into accountthe dependence on deposit energy and collected amount of light. The average PDFs areshown in Figure 8.9.
PDFs for Radiative Decay
The RD (µ+ → e+ νe ν µ γ ) PDF, R (or r), is more complicated because the kinematic
variables are correlated one another except for teγ . It is deﬁned as
R(Ee , Eγ , θeγ , φeγ , teγ ) = R1 (Ee , Eγ , θeγ , φeγ )R2 (teγ ).

(8.11)R2 (teγ ) is the same function as for the signal. R1 (Ee , Eγ , θeγ , φeγ ) is formed with thetheoretical correlated spectrum of RD folded with the detector response functions and2

We may have diﬀerent event distribution between the signal and the background if the polarizationof the muon beam is partly preserved in the target since upstream-downstream asymmetry would be
expected for the background in this case. The asymmetry of the event distribution due to the polarizationis, however, expected to be almost anti-symmetric, while the detector performance is symmetric for both
the gamma-ray detector and the positron spectrometer. Therefore, the distribution of the resolution forthe background is still expected to be the same as for the signal

0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002

51

52

53

54

55

56
Ee [MeV]50

Probability / (0.0010 rad)

Probability / (0.0300 MeV)

0.024
0.022
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
50

Probability / (0.0010 rad)

Probability / (0.0700 MeV)

CHAPTER 8. µ+ → e+ γ Search Analysis

136

0.02
0.018
0.016
0.014
0.012
0.01
0.008

55

60
Eγ [MeV]

0.03
0.025
0.02
0.015
0.01

0.006
0.004

0.005

0.002
-0.05

0

0.05

Probability / (0.0100 nsec)

-0.1

0.1
θe γ [rad]

-0.1

-0.05

0

0.05

0.1
φe γ [rad]

0.025
0.02
0.015
0.01
0.005

-1

-0.5

0

0.5

1
te γ [nsec]

Figure 8.9: PDFs for µ+ → e+ γ signal event. The average PDFs over events in analysis
window are shown here. In ﬁtting, event-by-event PDFs are used.

acceptance functions. The theoretical spectrum given by Eq.2.18 is a three-dimensionalcorrelated function of Ee , Eγ , and cos Θeγ . A three-dimensional convolution is calculated
event-by-event. The projections of the RD PDF on each kinematic variable are shown if
Figure 8.10.
PDFs for Accidental Background
The accidental background PDF, B (or b), can be written as a product of statisticallyindependent PDFs for the ﬁve kinematic variables
B(Ee , Eγ , θeγ , φeγ , teγ ) = B1 (Ee )B2 (Eγ )B3 (θeγ )B4 (φeγ )B5 (teγ ),(8.12)

similar to the signal PDF while each component is deﬁned by the measured background
spectrum. B1 (Ee ) is obtained by a ﬁtting of the measured Michel spectrum. B2 (Eeγ ) is

Probability / (0.0700 MeV)

Probability / (0.0300 MeV)

8.3. Maximum Likelihood Fit

0.018
0.016
0.014
0.012
0.01
0.008
0.006

137

10-2

10-3

10-4

0.004
0.002
51

52

53

54

55

56
Ee [MeV]50

Probability / (0.0010 rad)

Probability / (0.0010 rad)

50

0.006
0.005
0.004
0.003

0.005
0.004
0.003
0.002

0.001

0.001
0

0.05

Probability / (0.0100 nsec)

-0.05

0.1
θe γ [rad]60Eγ [MeV]

0.006

0.002

-0.1

55

-0.1

-0.05

0

0.05

0.1
φe γ [rad]

0.025
0.02
0.015
0.01
0.005

-1

-0.5

0

0.5

1
te γ [nsec]

Figure 8.10: PDFs for µ+ → e+ νe ν µ γ RD event. Here the projections on each variable
are shown, but (a)-(d) are correlated each other.

deﬁned by the background gamma energy spectrum measured in the teγ -sideband. The
spectrum is separately measured in 38 sections in (uγ , vγ , wγ ). Each spectrum is ﬁtted
with the MC one smeared by the detector response. B3 (θeγ ) and B4 (φeγ ) are obtained byﬁtting third-order polynomials to the spectrum measured in eight slices along uγ and vγ
directions, respectively. The asymmetries comes from the trigger bias. B5 (teγ ) is deﬁned
by a ﬂat distribution because the positron and gamma are uncorrelated. The average
PDFs are shown in Figure 8.11.

Probability / (0.0300 MeV)

Probability / (0.0700 MeV)

CHAPTER 8. µ+ → e+ γ Search Analysis

138

0.01
0.008
0.006
0.004

10-2

10-3

10-4
0.002

51

52

53

54

55

56
Ee [MeV]50

Probability / (0.0010 rad)

Probability / (0.0010 rad)

50

0.005
0.004
0.003
0.002
0.001

60
Eγ [MeV]

0.005
0.004
0.003
0.002
0.001

-0.05

0

0.05

Probability / (0.0100 nsec)

-0.1

55

0.1
θe γ [rad]

-0.1

-0.05

0

0.05

0.1
φe γ [rad]

0.005
0.004
0.003
0.002
0.001

-1

-0.5

0

0.5

1
te γ [nsec]

Figure 8.11: PDFs for accidental background event. The averaged PDFs are shownhere.

8.4. Confidence Region

8.4

139

Conﬁdence Region

In past experiments performing likelihood analyses, the shape of likelihood functions
were used to estimate the upper limits on the number of signal events. Namely, the 90%conﬁdence contour is constructed as a contour of N LL where the value of N LL becomes
1.35 larger than its minimum value. This is known as the MINOS (proﬁle likelihood)method [94]. It is, however, now known that likelihood-base conﬁdence intervals are in
general only approximations (asymptotic approximations in the large number of observation), and they are not good approximations for small number. Therefore, we calculate a
conﬁdence region with more statistically accurate method.

8.4.1

Toy MCWe can simulate the experiment using the measured PDFs because they completely describe the statistical behavior of the measurements. We call this simulation ‘toy MC’
and the simulated experiment ‘toy experiment’ to distinguish it from the general full MC
simulation. This toy MC is a powerful tool to study statistical behavior of our experimentand analysis even though the word ‘toy’ is named.
∏
In the toy MC, the distribution of resolutions in the likelihood function, p(δ~xi ),is also necessary, whereas it is omitted in the ﬁtting process. We use the resolution
distribution based on the measured event distribution. The simulation scheme is thefollowing:i) Generate a list of events observed in the analysis window and the teγ -sidebands.
This list contains values of the parameters to describe the PDFs of the likelihood
function associated with each event.
ii) Select an event randomly from the list for every generation of the toy-MC event.
iii) Form PDFs with the parameters (resolutions) associated with the selected event.
iv) Generate the kinematic variables in accordance with the PDFs.
Using the measured event distribution allows to take into account the correlation of theresolution and the eﬃciency between the gamma-ray detector and the positron spectrometer which is expected for the events selected in the analysis window.

8.4.2

Construction of Conﬁdence Region

We construct a conﬁdence region on the (Nsig , NRD )-plane by the Feldman-Cousins approach [95]. The conﬁdence region is constructed in the following procedure:i) Perform a likelihood ﬁt on the data with a constraint on physically allowed region
best0
best0
best0
(Nsig > 0, NRD > 0, NBG > 0) and obtain the best-ﬁt point (Nsig
, NRD
, NBG
).
Here, the subscript 0 means the measurement of the real experiment.
ii) as a sample point in the plane. The picked sample point, NRDii) Pick a point (Nsig
is denoted by a superscript i.

140

CHAPTER 8. µ+ → e+ γ Search Analysisiiii) Calculate a likelihood ratio Rdata
at the sample point as
best0~ 0 | N i , N i , NBG
L(X
)
sig
RD
i
.
Rdata
=
best0
best0
best0
~ 0 | Nsig
L(X
, NRD
, NBG
)

(8.13)

best0iiiv) Simulate many experiments assuming Nsig
, NRD
, and NBG
as the true expectations. The number of simulated events for each event type in each experiment is
randomly ﬂuctuated in accordance with the Poisson distribution. Each toy experiment is denoted by a subscript j.

v) Perform a likelihood ﬁt for each toy experiment with a constraint on physically
best
best
best
allowed region, obtain the best-ﬁt point (Nsig j , NRD j , NBG j ), and calculate a likei
lihood ratio RM
Cj as
i
RM
Cj =

~ j | N i , N i , N best0 )
L(X
BG
sig
RD
bestj
bestj
best .
~
L(Xj | Nsig , N, N j)
RD(8.14)BGvi) Calculate the conﬁdence level at the sample point from the probabilityii
P (Rdata< RM
C)(8.15)

over the simulated experiments. If the probability is 0.9, for example, the sampleiipoint (Nsig, NRD) should be on the contour of 90 % conﬁdence level (C.L.).
iivii) Repeat the procedure ii)-vi) for many choices of (Nsig, NRD) and scan the contour
on the plane. All points with the probability less than or equal to 0.9 form theacceptance region of 90% C.L.

This procedure is illustrated in Figure 8.12. In this way, we can construct a twodimensional conﬁdence region at any conﬁdence level. The point is the ordering principleof the acceptance region by the likelihood ratio. The one-parameter conﬁdence interval
is given by the edges of the contour in the direction.

Figure 8.12: Illustration of conﬁdence region construction. The inverse of the likelihood
ratio corresponds to the distance between the estimated point and the assumed true point.

8.5. Normalization

8.5

141

Normalization

A branching ratio is deﬁned as a ratio of the decay rate for a particular process, forexample Γ(µ → eγ), to the total one Γtot : B(µ → eγ) ≡ Γ(µ → eγ)/Γtot . For the muon
decay, the total decay rate is nearly saturated by that for Michel decay Γ(µ → eνν) with asmall contribution from RD. Thus, we use the Michel decay for the normalization channel.
The number of Michel positrons is counted simultaneously with the signal with the same
analysis cuts. In this way, we can calculate the branching ratio independently of theinstant beam intensity and nearly insensitively to the positron acceptance and eﬃciency
factors which diﬀer only for small momentum-dependent eﬀects between the two modes.
Thus it is an eﬀective method especially for run 2008 where the conditions of the drift
chamber were drastically changing.
We use the TIC-self trigger data mixed in the physics data taking as the Michel data
sample. The number of detected events in the sample after the selection can be written
as the following product of factorsµ
E
NM D = Neνν× Beνν × feνν× Teνν ×

1
Peνν

T IC
DCH
DCH
× tirgeνν × Aeνν × eνν × Aeνν ,(8.16)

where Nµ is the number of stopped muons during the data taking; Beνν is the branchingEratio of Michel decay, and feννis the fraction of Michel spectrum above 50 MeV; T is the
livetime; P is the pre-scale factor of the TIC-self trigger; tirgeνν is the conditional trigger
T IC
eﬃciency; Aeνν is the conditional acceptance of timing counter including the probabilityof DCH-TIC matching; DCH
is the conditional tracking eﬃciency including the selectioneνν
criteria; and ADCH
isthegeometricalacceptance of the drift chamber.
eνν
A similar equation can be written down for the number of detected µ+ → e+ γ eventsfrom the MEG trigger data:
µ
T IC
DCH
Nsig = Nsig
× Beγ × Teγ × ×tirg
× ADCH× γeγ × Aγeγ ,eγ × Aeγ × eγ
eγ(8.17)

where the factors are deﬁned in the same way as for Michel decay. Additional factors are
γeγ for gamma-ray detection and reconstruction eﬃciency including selection criteria, andAγeγ for the conditional acceptance of the gamma ray from µ+ → e+ γ decay.
Here, we explain what we mean by the word ‘conditional’ for each factor. First, wedeﬁne the positron geometrical acceptance ADCH so that the inverted direction of positron
goes to the ﬁducial volume of the gamma-ray detector deﬁned in Sec. 7.1. Therefore, it iscommon to the two modes by deﬁnition. DCH
is the tracking eﬃciency for the positroneνν
T IC
inside the acceptance. A
is the probability of getting at least one timing-counter hit
when the positron is emitted inside the acceptance and the track is reconstructed. tirg is
the probability that the trigger conditions are satisﬁed if the event satisﬁes all the analysis
selection criteria above. Aγeγ is the probability that the gamma-ray from µ+ → e+ γ decay
detected in the ﬁducial volume of the gamma-ray detector when the accompanied positronis detected in the acceptance. It is close to 1 but the ﬁnite detector resolution aﬀects it.
γeγ is the eﬃciency for the gamma inside the acceptance.
The number of stopped muon and livetime is common to the two modes since wemeasure those two modes simultaneously. Then the branching ratio on µ+ → e+ γ decay
can be written as
E
IC
Beγ
Nsig
feνν
trig
DCH
1
ATeνν
1
eνν
=
×
× eνν
×
× γ × γ .
×
trig
T
IC
DCH
Beνν
NM D Peνν
Aeγ
eγ
eγ
Aeγ
eγ

(8.18)

142

CHAPTER 8. µ+ → e+ γ Search Analysis

Number of Michel positron The number of detected Michel positron is NM D = 11414
in the analysis range 50 < Ee < 56 MeV.
Pre-scale factor The pre-scale factor of the TIC-self trigger is Peνν = 107 , while thatof the MEG trigger is 1.

Fraction of Michel spectrum The fraction of Michel spectrum included in the analE
ysis range is calculated to be feνν
= 0.101 ± 0.006 by integrating the theoretical Michel
spectrum over 50 MeV. The systematic uncertainty stems from that of positron energy
scale of 200 keV in the Michel spectrum ﬁtting.

Trigger-eﬃciency ratio The trigger eﬃciency consists of three components corresponding to the three trigger criteria described in Sec.3.3.3: gamma energy, time coincidence and direction match. The trigger eﬃciency by the timing-counter hit is canceledout in the ratio. The eﬃciency of gamma energy is estimated by the online energy resolution and evaluated to > 99 % for Eγ > 45 MeV. It is conﬁrmed by looking at the ratioof the trigger rate in the LXe-self trigger with normal and lower thresholds. The timing
eﬃciency is also estimated by checking the online time resolution and conﬁrmed the fulleﬃciency (> 99 %). The direction-match eﬃciency is evaluated to be 0.66 by the MC
simulation for the signal. This number is conﬁrmed by analyzing unbiased data of thededicated RD run which were taken without the direction-match conditions in the trigger.
trig
In total, the trigger-eﬃciency ratio is evaluated to be trig
eγ /eνν = 0.66 ± 0.03.
DCH-TIC matching-probability ratio It is calculated using the DCH-self trigger
data. The probability is evaluated as a function of positron momentum (Figure 7.27).
IC
The ratio is estimated to ATeγIC /ATeνν= 1.11 ± 0.02 by comparing the end point of the plot
and the average above 50 MeV, where the uncertainty is estimated from diﬀerent ﬁttings
to the plot.

Tracking-eﬃciency ratio It is calculated using random trigger data samples. The
ratio of detected events to the full Michel spectrum is calculated as a function of positron/DCH
momentum. The eﬃciency ratio is evaluated to DCH
= 1.02 ± 0.005 by compareννeγ
ing the end-point of the plot and the integral above 50 MeV, where the uncertainty is
estimated from diﬀerent ﬁttings to the plot.
Gamma-ray eﬃciency It is evaluated in Sec.7.7 to be γeγ = 0.63 ± 0.04.
Gamma-ray acceptance The eﬀect that the gamma from µ+ → e+ γ decay slips fromthe ﬁducial volume when the accompanied positron is detected in the acceptance consistsof two opposite eﬀects; missing and gaining. These eﬀects are simulated using measured
angle and position resolutions and evaluated to be Aγeγ = 0.98 ± 0.005.

8.5. Normalization

143

Normalization Factor
Altogether from the above, the normalization factor3 k , which is the eﬀective number ofobserved Michel decay, is given as
k = (5.2 ± 0.5) × 1011 .

(8.19)

With this number, the branching ratio is given by B(µ+ → e+ γ) = Nsig / k. The summaryof the normalization is given in Table 8.1.
This result was conﬁrmed by alternative methods for the normalization. The normalization with radiative decay is described in Appendix A.2, and that with accidental
background is in Appendix B. Table 8.2 summarizes those results.
Table 8.1: List of factors in normalization calculation. Here ratio is given as that ofsignal to Michel.
factorNumber of Michel events
Pre-scale factor
Michel spectrum fraction
Trigger eﬃciency ratio
DCH-TIC matching probability ratio
Tracking eﬃciency ratio
Gamma-ray eﬃciency
Gamma-ray acceptance
Normalization factor

estimated value
11414
107
0.101 ± 0.006
0.66 ± 0.03
1.11 ± 0.02
1.02 ± 0.005
0.63 ± 0.04
0.98 ± 0.005(5.2 ± 0.5) × 1011

relative uncertainty
–
–
5.9 %
4.5 %
1.8 %
0.5 %
5.0 %
0.5 %
10.0 %

Table 8.2: Summary of normalization.

11k (10 )

Michel5.2 ± 0.5RD
4.4 ± 1.1BG
5.2 ± 0.8

The inverse of this number, (1.9 ± 0.2) × 10−12 , can be regarded as the single-event sensitivity (SES)of this measurement. However, we do not deﬁne it as SES here because the meaning of this number isslightly diﬀerent from SES in conventional cut-counting analysis. In the likelihood analysis, the analysis
region is set to wider and including background events.
3

144

CHAPTER 8. µ+ → e+ γ Search Analysis

8.6

Results and Discussion

8.6.1

Analysis on Sideband Data

Before analyzing the analysis window, we applied the maximum likelihood ﬁt on the sideband ﬁctitious windows. We deﬁned two windows in positive and negative teγ -sideband:
1.5 < teγ < 3.5 and −3.5 < teγ < −1.5 ns. Ranges of the other variables are identical tothose of the analysis window. We believe there are no signal (and no RD events) in thesewindows. Thus, these analyses can be a kind of estimation of our sensitivity as well as acheck of the maximum likelihood ﬁt. Table 8.3 gives the results.
Table 8.3: Results of likelihood analysis in the sidebands.

Fit region (ns)
1.5 < teγ < 3.5−3.5 < teγ < −1.5

8.6.2

No
1197
1209

Best-ﬁt values(N
sig , NRD , NBG )
(
)
+38
+3.4
1.6 , 26+17
,
1169
−16
−37)
(
+38
+1.2
+16
0 , 9.6 , 1199−38

Nsig upper limit(90% C.L.)
10.4
4.6

Nsig lower limit
zero-consistentzero-consistent

The Maximum Likelihood Fit on the Data

We opened the hidden box and applied the maximum likelihood ﬁt on the data in theanalysis window. In the analysis window, No = 1189 events are observed. In Figure 8.13,
the event distributions in the analysis window and the ﬁtted spectra are shown as the
projection on each kinematic variable. The best-ﬁt values of θ~ = (Nsig , NRD , NBG ) are at
(

)

+17
+38
θ~best = 4.3+3.9−2.9 , 25−16 , 1159−37 ,(8.20)

where the asymmetric errors are those from MINUIT (so-called MINOS errors at one
sigma). The likelihood function is shown in Figure 8.13(f) in two-dimensional (Nsig , NRD )plane.
The expected number of RD events in the analysis window is calculated to be 41 ± 8.
best
See Appendix A for the detail calculation. The best-ﬁt value of NRD= 25+17−16 is consistent
with the expectation. We will discuss the result of Nsig best-ﬁt value later on.

8.6.3

Conﬁdence Region from the Data

We applied the procedure of constructing conﬁdence region described in Sec.8.4.2 to thedata. In particular, we calculated the conﬁdence region at 90 % level. First, we scanned
the conﬁdence levels at points on a rough grid of (Nsig ,NRD )-plane because it is verytime consuming to cover all the plane. By this rough sampling as well as by the shapeof the likelihood function, we found that there is no correlation between Nsig and NRD .
As described in Sec.8.4.2, the one-parameter interval of Nsig is calculated by taking the
projection of the contour to the Nsig -axis. Thus under the no-correlated conﬁdence region,
the limits on Nsig are on the line of NRD best-ﬁt value. We scanned the conﬁdence levels
best
best
. The distribution of the conﬁdence levels on the NRD
more ﬁnely on the line of NRD

Events / (0.4667 MeV)

Events / (0.2000 MeV)

8.6. Results and Discussion

100
80
60
40

145

102

10

1

20

50

51

52

53

54

10-1

55
56
Ee (MeV)

50

70
60
50
40

70
60
50
40

30

30

20

20

10

10

-0.1

-0.05

0

0.05

0.1
θeγ (rad)

-0.1

-0.05

(c)

0

0.05

0.1
φeγ (rad)

10

15
20
Number of signal events

(d)

70

Number of RD events

Events / (0.0667 nsec)

60
Eγ (MeV)(b)
Events / (0.0067 rad)Events / (0.0067 rad)

(a)

55

60
50
40

80
70
60
50
40

30
30
20

20

10
-1

10
-0.5

0

(e)

0.5

1
t eγ (nsec)

0
0

5

(f)

Figure 8.13: Result of the maximum likelihood ﬁt. Projected event distributions in theanalysis window are shown in (a)-(e). Green solid line, red dot-dashed one, and magenta
dashed one show the ﬁtted signal, RD, and BG PDFs, respectively. Blue solid line is thetotal spectrum. (f) Contour plot of the two-dimensional likelihood function as a functionof the number of signal and that of RD events. The black dot mark shows the best-ﬁtvalue. The dashed-, solid-, dotted-lines show the 1, 1.645, 2 sigma or more precisely 0.5,
1.353, 2 ∆N LL contours, respectively.

C.L.

C.L.

CHAPTER 8. µ+ → e+ γ Search Analysis

146

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

0.5

1

1.5

0

2
Nsignal

10(a) Lower side

12

14

16
Nsignal(b) Upper side

C.L.

C.L.

best
Figure 8.14: Distribution of Nsig conﬁdence level on the NRD-line.

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

2

4

6

8

0

10

40

50

60

70

NRD

(a) Lower side

80

90
NRD

(b) Upper side

best
Figure 8.15: Distribution of NRD conﬁdence level on the Nsig-line.

is shown Figure 8.14. The lower side of the interval is zero-consistent, that is, the 90%conﬁdence interval contains Nsig = 0. The result of this measurement is included in the~ | Nsig = 0) integrated becomes 90 %. In this case,region over which the probability p(Xnot a two-sided interval but an upper limit is set in the Feldman-Cousins uniﬁed approach.
The 90%-conﬁdence upper limit on Nsig is calculated to be at 14.5 by an interpolation of
the plot. The same calculation is done for NRD , yielding a 90%-conﬁdence upper limit of
68 (Figure 8.15).

8.6.4

Systematic Uncertainties

Summary of Systematic Uncertainties
The systematic uncertainties are summarized in Table 8.4.
Incorporating Systematic Uncertainties
Both the numerator and the denominator could have systematic uncertainties in thelikelihood analysis because we actively measure the number of signal using PDFs. The8.6. Results and Discussion

147

Table 8.4: Summary of systematic uncertainties.
factorestimated value impact on Nsig
Gamma-ray energy scale
0.4%
0.4
Gamma-ray energy resolution
10-15%
negligible
1
Positron spectrum–
1.14
Angular resolution
1 mrad0.35
Time resolution
17 psnegligible
Time center
16 ps
negligible
Normalization factor
10%–
given as alternative parameter sets.

Nsig

Nsig

1

14

16
14

12

12

10

10

8

8

6

6

4

4

2

2

0

0

-2

-1.5

-1

-0.5

0

0.5

1
1.5
2
Deviation of energy scale (%)(a) Gamma-ray energy scale

0.4

0.6

0.8

1

1.2
1.4
1.6
1.8
2
Angular resolution relative to nominal value(b) Angular resolution

Figure 8.16: Scan of the deviations of the Nsig best-ﬁt value. The error bars shows
90%-level upper limit by the MINOS method.

impacts of the systematic uncertainties on the number of signals were determined bylooking at the deviation of the best-ﬁt value θ~best when we changed each parameter for its
uncertainty value. Figure 8.16 shows examples of scans of Nsig deviations by changing of
the parameters. The uncertainty of positron-spectrum shape is found to have the largest
impact (1.14 events); the next contribution comes from that of gamma-ray energy scale(0.4 events) and then that of angular resolution (0.35 events). The others are found tohave negligible eﬀects. The systematic uncertainty of Nsig is evaluated to be 1.26 eventsby adding the individual contribution in quadrature.
In our case, the statistical uncertainty is large, and we accept an approximation to incorporate systematic uncertainties in the calculation of the upper limit. Here, we decided
to use a method which takes the statistics-only contour and scales it uniformly along raysfrom the best-ﬁt value. The scaling factor is given by adding systematic uncertainty ofNsig and that of the normalization in quadrature to the statistical uncertainty. When we
incorporate the systematic uncertainties, the upper limit on Nsig becomes 14.7. Generally,
this approximation over-estimates the impact, but the eﬀect is small. The impact of thesystematic uncertainties is less than 2 %.

148

8.6.5

CHAPTER 8. µ+ → e+ γ Search Analysis

Upper Limit on the Branching Ratio µ+ → e+ γ

Since the ﬁt result is zero-consistent, an upper limit on µ+ → e+ γ branching ratio is set.
Together with the normalization factor, the upper limit is calculated to beB(µ+ → e+ γ) < 2.8 × 10−11(8.21)

at 90 % C.L.

8.6.6

Sensitivity of Run 2008

The sensitivity of an experiment is a measure of the result that the experiment is expected
to give for the limit on a parameter in the absence of signal. It describes accuracy of theexperiment, and is independent of the actual data that could ﬂuctuate.
Here, we deﬁne the 90%-conﬁdence sensitivity, S, by the average upper limit at 90 %C.L. over the ensemble of the simulated toy experiments. In each experiment, the expectedor true numbers (not actual-generated ones) of RD and BG are given by the best-ﬁt valuesof the real experiment, (NRD , NBG ) = (25, 1159), while that of signal is zero, Nsig = 0.
We simulated 800 experiments, and calculated a 90%-conﬁdence upper limit on Nsig foreach experiment. The distribution of Nsig upper limits is shown in Figure 8.17. The
mean value of the distribution is at 6.5. The branching-ratio sensitivity of run 2008 is
thus calculated to be
S2008 = 1.3 × 10−11 .

(8.22)

60
50
40
30
20
10
0
0

5

10

15
Nsignal upper limit

Figure 8.17: Distribution of upper limits of toy experiments.

8.6. Results and Discussion

8.6.7

149

Cut Analysis

For a cross check and better understanding of the result, here we perform conventional
cut-counting analyses. They were also done as blind analyses.
We already deﬁned signal boxes before opening the hidden box to see the backgroundlevel. They are a box based on 1.64 σ (A) and an optimized one (B). The event selection
was however not changed, thus not optimized for the cut analyses, but the same one asthat for the likelihood was applied.
The corresponding signal eﬃciencies for the signal boxes are evaluated using the signal
PDFs as ratios to those for likelihood analysis window. We calculated the single-event
sensitivities (SES) by applying the relative eﬃciencies to the normalization factor k. Those
numbers are summarized in Table 8.5.
We observed one event in signal box A and two events in signal box B. The event
distributions in signal box B projected on each variable are shown in Figure 8.18, where
the signal PDFs normalized to 10 events are superimposed. Two-dimensional plots ofEe v.s. Eγ are also shown in Figure 8.19. Then we calculate the conﬁdence intervals inaccordance with the Feldman-Cousins method based on the Poisson statistics(µ + b)n exp(−(µ + b))
P (n | µ) =,n!

(8.23)

where n is observed number, µ is expected mean number of signal events, and b is the
known number of background events. The results are zero-consistent for both signal boxes,
and upper limits are given by 3.39 and 3.81 for signal box A and B, respectively. Together
with the SESs, upper limits on the µ+ → e+ γ branching ratio,
BboxA < 1.7 × 10−11 ,BboxB < 1.3 × 10−11
are obtained.
Table 8.5: Summary of cut analyses.
1.64σ box (A)optimized box (B)range(rel )
range
(rel )Ee (MeV)[52.2, 53.8] (0.85) [52.0, 56.0] (0.92)[51.0, 54.6] (0.64) [51.0, 56.0] (0.70)
Eγ (MeV)|teγ | (ps)< 242
(0.90)< 273(0.94)
φ − Θeγ (mrad)< 33(0.80)< 42(0.94)SES (10−12 )5.0(0.39)
3.5(0.56)exp
exp
NBG + NRD
0.95+0.02
2.08+0.03
obs
N
12
Upper limit (10−11 )
1.68
1.32

(8.24)(8.25)

Events / ( 0.13 MeV )Events / ( 0.28 MeV )CHAPTER 8. µ+ → e+ γ Search Analysis

150

4.5
4
3.5
3
2.5
2

7
6
5
4
3

1.5

2

1
1

0.5
52

54

0

56
Ee (MeV)
Events / ( 0.0001 )Events / ( 0.04 nsec )

0
50

4.5
4
3.5
3

4

3

2

2

1.5

1.5

1

1

0.5

0.5
0

0.5

1
teγ (nsec)

60
Eγ (MeV)

3.5

2.5

-0.5

55

4.5

2.5

0
-1

50

0
-1

-0.999-0.998-0.997-0.996-0.995cosΘeγFigure 8.18: Event distributions in signal box B projected on each variable. Blue
hatched plots show the signal PDF normalized to 10 events.

Eγ (MeV)

8.6. Results and Discussion

151

60
58
56
54
52

3
2

4
5

50
48
46
50

52

54

56
Ee (MeV)Eγ (MeV)(a) Signal box A

60
58
56
1

54
52

3
2

4

50

5

48
46
50

52

54

56
Ee (MeV)(b) Signal box B
Figure 8.19: Ee vs Eγ two-dimensional plots of the signal boxes. Two-dimensional signal
PDF is superimposed. The color axis is arbitrary. Numbers in the plots show the rank of
S/B listed in the Table 8.6.

152

8.6.8

CHAPTER 8. µ+ → e+ γ Search Analysis

Discussion

Figure 8.17 shows that the result of upper limit Nsig < 14.5 is quite rare. The probabilityof having an upper limit greater than 14.5 is less than 1 % in the toy experiments. Even
though the result is zero-consistent at 90 % level, let us check that the rare result is justdue to a statistical ﬂuctuation or indicating something.
Candidate Events
We list up events that are likely to be µ+ → e+ γ events by the ratio of event-type
likelihoods, S/B. The distribution of S/B is shown in Figure 8.20 together with thatfor signal-MC events and background-MC ones. Table 8.6 lists the best ﬁve signal-likeevents.
One event is observed at a large value of S/B signiﬁcantly separated from the distribution. An event display of the event is shown in Figure 8.21. It is found to be a doublepileup event on which the pileup elimination was failed. We successfully eliminated theﬁrst pileup, while the second pileup was not detected with the nominal threshold because
the conversion of the second-pileup gamma occurred at deep position of the detector andthus the light distribution was smeared. If we set lower threshold for the pileup search,then we can eliminate also the second pileup. Then, Eγ of this event becomes 47.7 MeV
and S/B becomes 3.6. The event gets buried in the background distribution. Therefore,
most likely this event is not a signal but a background accidentally getting a large signal
likelihood because of the reconstructed energy of Eγ = 54.1 MeV.
This is a very special event. First it is a double-pileup event. Moreover, one of themwhich has relatively large energy (> 5 MeV) is missed because of the deep conversion
point. Such gamma event is overlapped by a positron collinearly, while the timing is notin coincidence so much. The probability that an event is identiﬁed to a double pileup in
51 < Eγ < 54.5 MeV is 2 % in the sideband data while that of single pileup is 18 %.
The probability of the deep conversion point is about 2 % if the gamma ray came fromthe target. It is possible that the second pileup is a cosmic-ray event with a probabilityof ∼ 3 %. Thus, the probability of having this kind of events is roughly 0.1 % of normal
accidental background in signal box A, that is, 0.001 events are expected.
Impact of the Best Rank Events
To see the eﬀect of such kind of events, we repeated the analysis with a lower thresholdfor the pileup search so that the second pileup is also eliminated. The PDFs are modiﬁedin accordance with the new threshold.
Table 8.6: List of events with large likelihood ratio S/B.
Run/Event Ee (MeV) Eγ (MeV) teγ (ps) θeγ (mrad) φeγ (mrad)
35909/1908
52.7
54.1
-262
9.5
6.3
34221/2058
52.6
50.8
60.2
3.3
1.8
52.5
51.6
-50.8
-11.9
11.3
30109/1371
40330/ 853
53.2
50.7
82.0
-28.6
-2.3
53.8
50.1
143.6
3.0
14.4
40077/1210

S/B
1206.6
303.2
271.9
250.4
128.6

Number of events

8.6. Results and Discussion

153

103
102
10
1

10-1
10-2
0

500

1000

1500

2000
S/B

Figure 8.20: Distribution of the ratio of event-type likelihoods, S/B for data (black
ﬁlled histogram), MC signal (green dashed one), and MC background (magenta blankone). The MC distributions are normalized to the event number of data.

100

21999

2nd pileup
80

100

60

80

40

60

20

40

0

20

1st pileup
6304

1806

517

0

148

-20
42
-40
-60

12

-80

Run -100
35909
Event 1908
-100
z

3
0

100

200

300
1

Figure 8.21: Event display of the ﬁrst-rank event. It found to be a double-pileup event.
The color axis shows the amount of observed light on each PMT.

+37
The best-ﬁt values of the maximum likelihood ﬁt becomes θ~best = (2.0+3.8 , 24+17
16 , 1119−37 ).
The 90%-conﬁdence upper limit on Nsig is then calculated to be at 11.4. Change of thelikelihood function is shown in Figure 8.22. This event is found to have an impact larger
than one event for signal. Now the probability of having an upper limit greater tha 11.4 is≈ 5 %. The upper limit on the branching ratio becomes 2.2×10−11 , where the ineﬃciency
for the signal with the new threshold that amounts to ∼5 % is taken into account.

CHAPTER 8. µ+ → e+ γ Search Analysis
Number of RD events

154

80
70
60
50
40
30
20
10
0

5

10

15
Number of signal events

Figure 8.22: Change of the likelihood function by the change of the pileup search threshold. Red contours show new ones while blue ones are ones of original likelihood function.

Relationship with Cut Analysis
The sensitivity of cut analysis B is comparable with that of the likelihood analysis; thediﬀerence is less than 10 %.
The cut analyses show well consistent results with their background expectations, andno indication of µ+ → e+ γ decay is found. When looking at the event distribution in
Figure 8.19(b), we see some events around the edge of the box. Those events are almost
overlapped with the signal-like candidates in the likelihood analysis. Potentially, some ofthem could be observed inside the box by ﬂuctuations in the measurement or the decisionof the box size. If the closest two or three events were additionally observed in the box,
the upper limit on the branching ratio of cut analysis B would be 2.3×10−11 or 2.8×10−11 ,respectively. They would be comparable with the likelihood-analysis result. Conversely,
if we removed the double-pileup event, it would be 8.8 × 10−12 .
The upper limit obtained by cut analysis B is 2.2 times better than that by the
likelihood analysis. However, we adopt the result of the likelihood analysis as our result,because we decided to carry on with the likelihood analysis before opening the hidden
box. If we decide the analysis after looking at their results, it just introduces bias in the
result.

In conclusion, we consider the result of large Nsig that is about to exclude null signal at
90 % conﬁdence level is due to a statistical ﬂuctuation by a very rare accidental background
event. The result is statistically true, but we cannot say there is some indication of signal
from those data. Since the background PDF is formed by the actual measured spectra, it
takes into account those rare events in principle. We analyzed fully in blinded. Therefore,
the obtained upper limit is valid without any bias from the statistical point of view.
We can also consider that this ﬂuctuation is one of the systematic uncertainty from

8.6. Results and Discussion

155

Table 8.7: Summary of the upper limit and sensitivity.
B upper limit1 (10−11 )Analysis window
2.8
Analysis window (tighter pileup cut)
2.2
Cut analysis (optimized box)
1.3
Sideband (teγ > 0)
2.0
Sideband (teγ < 0)0.89
Sensitivity
1.3
1

before incorporating systematic uncertainties.

the pileup cut criteria. If we take in the systematic uncertainty, the conﬁdence region isbroadened and the zero point is naturally included in the 90%-level conﬁdence region as
we saw in the change of likelihood function.
The obtained limit is not as stringent as the current best limit of 1.2 × 10−11 set by
MEGA. However, when we think of the relation or comparison with other experimental
results, it is more important to consider sensitivities than actual-obtained upper limits
because the latter are subject to statistical ﬂuctuation as see in our result. Our sensitivity
is 1.3 × 10−11 . Therefore, through this analysis we searched for the µ+ → e+ γ decay witha comparable sensitivity to the current limit, and we set an independent upper limit.
The sensitivity is mainly limited by the data statistics. By continuing data taking,we can progressively improve the sensitivity. From here on, new data start to search forthe decay in unexplored region. We discuss prospects of the experiment in the following
section.

156

CHAPTER 8. µ+ → e+ γ Search Analysis

8.7

Future Prospects

MEG resumed data taking at the beginning of November 2009, and will accumulate data
at least for two years. Since PSI has long accelerator-shutdown period (from end of
December to April) every year, the run is divided into sub data sets for every calenderyear. First, let us think of the prospect of the run 2009.
The largest improvement achieved in 2009 is the full operation of the drift chamber.
We found the causes of the problem and solved them during the long shutdown period
after run 2008. All the chambers were modiﬁed. Almost all chambers are now operated
at the nominal voltage.
First, consider the expected data statistics of run 2009. The positron eﬃciency will
improve to e = 0.4 (DCH × AT IC = 0.8 × 0.5), factor 3.0 improvement. The trigger
eﬃciency is also expected to be improved by a better look-up table of direction matchand looser conditions, from 0.66 to 0.88, factor 1.3. We optimized calibration time, which
recover dead time by factor 1.3. The beam rate was adjusted to a slightly higher intensity,
factor 1.1. On the other hand, the beam time of run 2009 is factor 0.56 of that of run
2008. Taking all the factors into account, we expect 3.2 times larger statistics than thatof run 2008.
Owing to the full operation of the drift chamber, the resolutions are also expected
to improve. In [41], a realistic evaluation of the positron spectrometer performance wasconducted. Here, we assume the following resolutions in reference to the study:• Ee resolution with a single Gaussian whose sigma is the same value as that of core
Gaussian in 2008 (σEe = 374 keV).
• Angular resolutions of σθe = 5.2, σφe = 9.0 mrad.
• Time resolution of teγ = 122 ps improved in track length measurement.
PDFs for signal and backgrounds are re-formed using these values.
Now, we calculate the expected sensitivity of run 2009. We use a similar signal box tothat of B in 2008. The window is narrowed in accordance with the resolution improvementwith keeping the same signal eﬃciency. Thus, the expected single-event sensitivity is
SES = (3.6 × 10−12 )/3.2 = 1.1 × 10−12 . The expected number of background events inthe signal box is estimated using the toy MC. In this calculation, we take the higher
beam intensity into account assuming NBG ∝ Rµ2 . The number of background events is
exp
estimated to be NBG= 0.63. The expected 90%-conﬁdence sensitivity is estimated to be
exp
S2009
= 3.2 × 10−12 .

(8.26)We are still limited by the statistics. Note we here only assumed improvements in thedrift chamber.
Finally, let us estimate the expected sensitivity of MEG with current knowledge. MEG
will run at least additional two years toward the sensitivity of O(10−13 ). Let assume beamtime of 100 weeks (8.7 times longer than run 2008). We expect further improvement ofeﬃciencies: trigger eﬃciency (to 95 %), DAQ livetime (to 90 %), and positron eﬃciency

8.7. Future Prospects

157(to 50 %). In total, 7.8 times higher eﬃciency can be expected. Together with the beamtime, 68 times statistics will be accumulated.
Under the conditions assumed in the estimation of run 2009, we expect to observebackground events as large as 13 events. Thus, we have to suppress background further.
We think there is large possibility of improvement in the gamma-ray energy measurement. We are now considering three directions of improvements. First, we will be able
to improve the resolution at least at the center of the detector to 1.3 % level, because we
veriﬁed such resolution using the prototype detector. The diﬀerence is size of the detector
and geometrical arrangement. We think more precise calibration of PMT QE enables to
go to the level. Next, we can improve the worse resolution at the edge part of the detector.
For example, by optimizing the weighting factor of PMT summation for each position, we
can eﬀectively reconstruct the total number of scintillation light using the anti-correlation
between faces of the detector. Preliminary study shows this kind of analysis can improve
resolutions at large part of the detector acceptance to the same level as that at the center.
In the analysis of run 2008, we could not conﬁrm the improvement because of unstableconditions of the calibration run. However, further study with more stable data in 2009
will achieve a resolution of 1.7 % for all acceptance. Last point is improvement of thepileup elimination. It was, for the ﬁrst time, introduced in the analysis of run 2008. We
think we can improve it with further study. Therefore, here we assume the following:• the energy resolution will become 1.7 % for wγ ≥ 2, and 2.5 % for 0 ≤ wγ < 2 cm,
uniformly over the acceptance.
• the pileup level after elimination will go down to the level where we achieved when
we rejected all events identiﬁed as pileup in the 2008 analysis.
In addition, we expect the time resolution of the gamma-ray detector will be improved
to σtγ = 65 ps by the increase in the light yield. According to [41], the positron energy
resolution will improve to σEe = 335 keV.
Using these assumptions, corresponding PDFs are formed. Then we optimize thesignal box. We can calculate the expected number of background and eﬃciency for thesignal using the PDFs. The two-dimensional PDFs and the optimized box are shown in
Figure 8.23. The corresponding signal eﬃciency is 0.45 relative to that of the analysis
window. The SES would be 6.6 × 10−14 . The expected number of background events inthe box is estimated to be 4.0. Then the expected 90%-conﬁdence sensitivity would begiven asS exp = 3.0 × 10−13 .

(8.27)

If we pose ourselves that the background level should be ≤ 1, we can tune the beam
intensity to suppress background. From the relation NBG ∝ Rµ2 , the expected sensitivity
as a function of Rµ can be calculated as shown in Figure 8.24 together with the expected
number of backgrounds. At a beam intensity of Rµ = 1.5 × 107 , half of current nominal
value, the background would be 1. In this case, the sensitivity becomes S exp = 4.4×10−13 .
The 90%-conﬁdence belts in Feldman-Cousins method are shown in Figure 8.25. With
this sensitivity, we can set a two-sided interval at this C.L. when we observed at least
three events. In this case, the conﬁdence interval would be set to
1.3 × 10−14 < B(µ+ → e+ γ) < 8.4 × 10−13 .
at 90 % C.L.

(8.28)

CHAPTER 8. µ+ → e+ γ Search Analysis

E γ (MeV)

158

60
58
56
54
52
50
48
4650

51

52

53

54

55
56
E e (MeV)

0.6 ×10

-12

12

0.5

10

0.4

8

0.3

6

0.2

4

0.1

2

0
1

2

3

NBG

90% C.L. Sensitivity

Figure 8.23: Two-dimensional PDFs with expected performance. Blue contour shows
signal PDF and black one shows BG PDF. The normalization of the two are arbitrary.
Red box shows the optimized signal box.

04

5Rµ ( × 107/sec)µFigure 8.24: Expected 90%-conﬁdence sensitivity as a function of muon stopping rate.

20
18
16
14
12
10
8
6
4
2
0
0

2

4

6

8

10

12
Nobs

Figure 8.25: 90%-conﬁdence belt in Feldman-Cousins method with the Poisson statistics
in presence of background of 1 (colored belt) and 4 events (interval between cross markers).

Chapter 9
Conclusion
The MEG experiment started physics data taking in September 2008 to search for the
lepton-ﬂavor-violating muon decay, µ+ → e+ γ . We searched for the decay on the basis of
the initial three months data of MEG. The accumulated number of 9.5 × 1013 muons were
stopped on the target. Positrons from the muon decays were measured by a spectrometer
which forms a specially graded magnetic ﬁeld. Gamma rays were detected by a 900 literliquid xenon scintillation detector.
To avoid introducing any artiﬁcial bias in the search, we adopted a blind analysis
with a hidden box. We established precise reconstruction and calibration methods, andevaluated detector performance using calibration data samples. The backgrounds were
evaluated using the sideband data.
The 90%-conﬁdence sensitivity of this search is
S2008 = 1.3 × 10−11(9.1)

with the number of observed muon decays of (5.2 ± 0.5) × 1012 . It is comparable with thecurrent best upper limit of 1.2 × 10−11 set by MEGA.
We performed a maximum likelihood ﬁt to detect µ+ → e+ γ signal. The bestestimated value of the number of signal events is 4.3. The 90%-conﬁdence interval includes
Nsig = 0, and the upper limit is at 14.7 including systematic uncertainties. Therefore, we
set an upper limit on the branching ratio
B(µ+ → e+ γ) < 2.8 × 10−11(90 %C.L.).

(9.2)

We also performed an independent analysis by cutting background events and counting
remaining events inside a signal box for a cross check of the analysis. We expect 2.1
background events, and we observed 2 events in the box. This result yields an upper limitof B < 1.3 × 10−11 at 90 % C.L.
We conclude the reason of the large number of Nsig and the large value of upper limit
compared to the sensitivity in the likelihood analysis is a statistical ﬂuctuation. We do notgive a new record of the experimental limit but an independent result from a measurementwith a comparable sensitivity.
The sensitivity is limited by the statistics. The observed number of muon decays
was less than expected mainly because of a problem on the drift chamber system. In
2009, MEG runs with a ﬁxed drift chamber system, and accumulates further data. The
sensitivity is going to improve as the data statistics. It is promising that MEG explores
unexplored region of the branching ratio. The sensitivity will go down to ∼ 3 × 10−12
with the data 2009, and ﬁnally to ∼ 3 × 10−13 .
159

Appendix A
Radiative Decay Analysis
In chapter 2, we introduced the radiative muon decays, µ+ → e+ νe ν µ γ , as one of background sources. On the other hand, we used them as as a powerful timing-calibration
source in chapter 6 and 7. In addition, we can use them to check our analysis. Detecting
RD events as expected demonstrates the capability of our experiment to really observeµ+ → e+ γ events if there are. For example in MEGA, they could not see the RD peakin time distribution because of the worse signal-to-noise ratio as a consequence of theirlower time resolution and higher instant beam intensity1 . We see a clear peak as shownin Figure 7.26.

A.1

Eγ -Sideband Data

We analyzed the RD events in the Eγ -sideband data. Here we deﬁne a sideband wide box• 40 < Eγ < 48 MeV,
• 48 < Ee < 56 MeV,• |θeγ | < 100 and |φeγ | < 100 mrad.
The partial branching ratio of this box is calculated to be 25.5 × 10−10 by numericallyintegrating the formula of the diﬀerential branching ratio.
To calculate the expected number of RD events in the box, detector resolutions andeﬃciencies have to be considered. We put the following inputs for the calculation:• Eγ eﬃciency: An eﬃciency curve as a function of reconstructed energy obtained
from the ratio between the gamma-ray background rate in normal-threshold MEG
trigger (trigger0) and that in lower-threshold MEG trigger (trigger1). It is given as
an error function.
• Ee acceptance: An acceptance curve as a function of the real positron energy estimated by the Michel spectrum ﬁt. It is given as an error function.
1

We do not say they had no capability of detecting RD events. By using all kinematic variables, theydetected RD events in the likelihood ﬁt as expected.

161

162

CHAPTER A. Radiative Decay Analysis• Direction-match eﬃciency: Probability that an event passes the direction-match
trigger condition is estimated by two methods relatively to the signal (back-toback) eﬃciencies. One is using unbiased RD trigger data in the dedicated RD run
with ultra-low beam intensity. The other is by looking at the angle distribution of
accidental background events.
• Eγ response: A representative response function obtained by averaging the positiondependent response function over actual event distribution.
• Ee response: The same response function as for µ+ → e+ γ analysis (a triple Gaussian) obtained by the Michel-spectrum ﬁtting.
• Angular response: The same response function as for µ+ → e+ γ analysis. Treat θeγ
and φeγ separately.

1
0.8
0.6

Relative efficiency

Relative efficiency

Relative efficiency

Those eﬃciency curves are shown in Figure A.1. We applied these eﬀects to the theoretical three-dimensional distribution, that is, multiplied eﬃciency curves and convolved
response functions. Then we integrated the formed distribution over the box. The eﬀecwide
tive branching ratio is calculated to be BRD= (5.6 ± 0.8) × 10−10 , where the uncertainty
comes from those of eﬃciencies and resolutions, mainly from the Ee -acceptance curve.
We performed a maximum likelihood ﬁt on the data in the box. The best-ﬁt value is
at NRD = 244 ± 47, as shown in Figure A.2.
1
0.8
0.6

1
0.8
0.6

0.4

0.4

0.4

0.2

0.2

0.2

0
40

45

50
Ee (MeV)

0

30

40

50

60
Eγ (MeV)

0

170

172

174
176
178
180
Opening angle Θe γ (deg)Figure A.1: Eﬃciency curves applied to the calculation of RD branching ratio. They
are relative to those for the signal. The Ee eﬃciency is mainly due to the design of thespectrometer, and the other two are due to trigger conditions.

A.2

Normalization by Radiative Decay

From the relationk · BRD = NRD ,(A.1)

we calculate the normalization factor k from the radiative decay,kRD = 244/(5.6 × 10−10 ) = (4.4 ± 1.1) × 1011 .

(A.2)

The relative uncertainty is 24 %. The number is consistent with that by Michel decays
in Sec.8.5.

A.2. Normalization by Radiative Decay

163

300

Events / ( 0.32 MeV )Events / ( 0.32 MeV )

500

400

250

300

200
150

200
100
100

52

0
40

54
56
E e (MeV)

42

44

46
48
E γ (MeV)

-0.05

0

0.05

250

250

200

200

150

150

100

100

50

50

0
-0.1

Events / (0.8 nsec )50

Events / ( 0.008 )Events / ( 0.008 )

0
48

50

-0.05

0

0.05

0.1
θeγ (rad)

0
-0.1

0.1
φ eγ(rad)250
Num Entries = 5166
200

NBG = 4921.88 -82,44 +83.87
NRD = 244.12 -46.48 +47.51
NSig = 0.00 (fixed)

150
100
50
0
-1

-0.5

0

0.5

1
t eγ (nsec)

Figure A.2: Result of a maximum likelihood ﬁt on the Eγ -sideband data. Projected
event distributions are shown. Red dot-dashed line and magenta dash line show the ﬁtted
RD and BG PDFs, respectively. The number of signal was ﬁxed to zero in this ﬁt. Bluesolid line is the total ﬁtted spectrum.

164

CHAPTER A. Radiative Decay Analysis

A.3

Calculation of Expected Number of RadiativeDecay Events

We calculate the expected number of RD events in several boxes used in the analyses by
the ratio of the calculated eﬀective branching ratios. The numbers are summarized inTable A.1. We expect 41 ± 8 RD events in the analysis window.
Table A.1: Eﬀective branching ratio of radiative decay in several boxes.
Window
Eγ (MeV) Ee (MeV) θeγ ,φeγ (mrad)
BRD
Analysis window[46, 60][50, 56]100
0.95 × 10−10
Sideband wide window[40, 48][48, 56]100
5.6 × 10−10Signal box A (1.64σ cuts) [51, 54.6] [52.2, 53.8]
331
4.4 × 10−141Signal box B (optimized)[51, 56][52, 56]42
7.6 × 10−14
1

NRD
41
244
0.019
0.033

for π − Θeγ

A.4

Detail Description of Radiative-Decay Diﬀerential Branching Ratio

The diﬀerential branching ratio of RD is given by Eq.2.17. The functions appearing in
Eq.2.17, F (x, y, d), G(x, y, d), and H(x, y, d) in the SM, are given as follows:
F = F (0) + rF (1) + r2 F (2) ,G = G(0) + rG(1) + r2 G(2) ,H = H (0) + rH (1) + r2 H (2) ,(A.3)

where r = (me /mµ )2 .
8 2
{y (3 − 2y) + 6xy(1 − y) + 2x2 (3 − 4y) − 4x3 }d+8{−xy(3 − y − y 2 ) − x2 (3 − y − 4y 2 ) + 2x3 (1 + 2y)}+2d{x2 y(6 − 5y − 2y 2 ) − 2x3 y(4 + 3y)}+2d2 x3 y 2 (2 + y),{}y(3 − 2y)32
− (3 − 4y) + 2x
F (1) (x, y, d) = 2 −dx8{y(6 − 5y) − 2x(4 + y) + 6x2 }d
8{x(4 − 3y + y 2 ) − 3x2 (1 + y)}
6dx2 y(2 + y),{}48y32 (4 − 3y)−3 +,F (2) (x, y, d) = 2dxd8G(0) (x, y, d) ={xy(1 − 2y) + 2x2 (1 − 3y) − 4x3 }d+4{−x2 (2 − 3y − 4y 2 ) + 23 (2 + 3y)}F (0) (x, y, d) =

(A.4)

(A.5)
(A.6)

A.4. Detail Description of Radiative-Decay Differential Branching Ratio

G(1) (x, y, d) =G(2) (x, y, d) =H (0) (x, y, d) =H (1) (x, y, d) =H (2) (x, y, d) =−4dx3 y(2 + y),
32
8(−1 + 2y + 2x) + (−xy + 6x2 ) − 12x2 (2 + y),
2
d
d
96
− 2,
d
8 2
{y (1 − 2y) + xy(1 − 4y) − 2x2 y}d+4{2xy 2 (1 + y) − x2 y(1 − 4y) + 2x3 y}2d{x2 y 2 (1 − 2y) − 4x3 y 2 }2d2 x3 y 3 ,{}32
y(1 − 2y)
−
+ 2y
d2
x
8
+ {y(2 − 5y) − xy}d+4xy(2y − 3x) + 6dx2 y 2 ,96y 48y
− 2 +
.
dx
d

(A.7)
(A.8)
(A.9)

(A.10)(A.11)(A.12)

165

Appendix B
Normalization with Background
B.1

Overall Eﬃciency

We estimated the overall eﬃciency of our measurement with µ+ → e+ γ like background
events. By counting the number of accidental background events in the te γ sideband, theoverall eﬃciency can be calculated.
The number of accidental background events is given byΩΩ∆ω· e ) × (Rµ · fγ0 ·· γ ) ×× ∆t × T
4π
4π
Ω
∆t ∆ω
= Nµ2 · fe0 · fγ0 ·
·
· ,
T 4π

NBG = (Rµ · fe0 ·(B.1)

where Rµ is a muon intensity and Nµ (= Rµ T ) is the total number of muon; T is the total
livetime; fe0 and fγ0 are the fractions within the signal window of Michel spectrum andbackground-gamma spectrum, respectively; ∆ω and ∆t is full width of a signal window forsolid angle and time; Ω is the geometrical acceptance; e and γ are detection (including
selection and analysis) eﬃciencies of positron and gamma; and  is the overall eﬃciency
including the acceptance.
Window We set a window in which we count the number of background events as• 50 < Ee < 56 MeV,
• 46 < Eγ < 60 MeV,
• 1 < |teγ | < 4 ns,• |θeγ | < 100 and |φeγ | < 100 mrad.
The solid angle of the window is calculated to be ∆ω/4π = 3.19 × 10−3 .
Number of background events NBG = 3577 events are observed in the window.
167

168

CHAPTER B. Normalization with BackgroundNumber of muons The number of muons contributing to the background is slightly
diﬀerent from that of muons stopping on the target. For the gamma-ray background,
we assume all muons reaching the target contributes; the average rate is 3.54 × 107 /sec.
Positrons from muon decay oﬀ target less contributes to the background owing to thetracking, but it has large uncertainty; (3.2 ± 0.32) × 107 . The livetime is T = 3.3 × 106
sec. The eﬀective number of muons is calculated to Nµef f = 11.1 × 1013 .
Michel spectrum fraction This is the same as that calculated in Sec 8.5, fe0 = 0.101±
0.006 for Ee > 50 MeV. Correction factors of eﬃciency ratio between signal and Michel
are 1.11 × 1.02 = 1.13 ± 0.023.
Gamma-ray background spectrum fraction The fraction of RD-gamma spectrum
over 46 MeV is calculated to be 1.20 × 10−4 . The eﬀect of detector resolution is evaluatedto be a factor 1.09. Thus, fγ0 = (1.31 ± 0.085) × 10−4 , where the error comes fromthe uncertainty of energy scale. The fraction of gamma from AIF is estimated to be(24.6 ± 1.0) % for 46 < Eγ < 60 MeV. Thus, fγef f = (1.63 ± 0.13) × 10−4 .
Eﬀect of low-energy tail in Eγ response The diﬀerence of survival rate of gamma
rays between signal and background needs to be considered. The fraction of the RD
events over 46 MeV surviving in the window is 50 %, while that of the signal events is
66 %. Thus, we correct the eﬃciency by a factor 66/50 = 1.32.
Direction match eﬃciency ratio In Eq.B.1, a uniform eﬃciency over the window isassumed, which is not the case because of the dependence of the trigger eﬃciency on the
opening angle. The average eﬃciency relative to that at Θeγ = 180◦ is evaluated to be
83 ± 2 %.
Putting numbers above in Eq.B.1, the overall eﬃciency is calculated to be = (5.4 ± 0.8) × 10−3 .

(B.2)

This number is consistent with the multiplication of individual components estimatedseparately, = Ω/4π × e × γ × trigger= 0.1 × 0.14 × 0.63 × 0.66
= 5.8 × 10−3 .

B.2

(B.3)

Normalization Factor

Using the total number of stopping muon, Nµ = 9.5 × 1013 , the normalization factor is
calculated to be
kBG = (5.2 ± 0.8) × 1011 ,which is consistent with Eq.8.19.

(B.4)

Acknowledgment
First of all, I would like to express my great appreciation to my supervisor Prof. ToshinoriMori.
I would like to express my gratitude to Dr. Wataru Ootani, Dr. Toshiyuki Iwamoto,
Dr. Ryu Sawada, Prof. Satoshi Mihara, Dr. Hajime Nishiguchi and Dr. Shuei Yamada for
their helpful advice. I would like to thank Mr. Hiroaki Natori, Mr. Yasuhiro Nishimura,
Dr. Kenji Ozone, Ms. Yasuko Hisamatsu, Ms. Bai Xue, Mr. Daisuke Kaneko, and Mr. Yuki
Fuji for their kind cooperation and joyful research time with them.
I appreciate all the members of the MEG collaboration. I learned a lot from Prof. Akihiro Maki, Prof. Tom Haruyama, and Mr. Katsuyu Kasami; Dr. Stefan Ritt, Dr. Peter R. Kettle, Dr. Malte Hildebrandt, Dr. Oleg Kiselev, Dr. Matthias Schneebeli, and
Ms. Jeanine Adam; Prof. William Molzon, Mr. Feng Xiao, Mr. Ben Golden, Ms. Christina
Topchyan, and Dr. Elisabetta Baracchini; Dr. Dmitry N. Grigoriev, Dr. Boris I. Khazin,
Dr. Fedor Ignatov, and Dr. Yuri V. Yudin; Prof. Flavio Gatti, Dr. Simeone Dussoni,
Dr. Paolo W. Cattaneo, Prof. Gianluigi Boca, Prof. Giancarlo Piredda, Dr. Gianluca
Cavoto, Dr. Cecilia Voena, Dr. Francesco Renga, Dr. Giovanni Signorelli, Dr. Fabrizio
Cei, Mr. Giovanni Gallucci, Prof. Carlo Bemporad, Dr. Angela Papa, Dr. Marco Panareo,
Dr. Marco Grassi, Dr. Donato Nicolò, Mr. Luca Galli, and Dr. Alessandro Baldini, who,
together with Prof. Mori, shares the spokes-person of the MEG experiment. I enjoyeddiscussions with them. I really appreciate an exciting experience with them.
I would like to thank PSI staﬀ for their support of the experiment and also my Swisslife.
I am deeply grateful to all of ICEPP members and secretaries.
I got ﬁnancial support from Japan Society for the Promotion of Science. Part of mytravel cost was also supported by PSI.
Finally, I gratefully acknowledge my family for a lot of encouragement and support.

December 2009,
Tokyo,
Yusuke Uchiyama

169

Bibliography[1] M. Ahmed et al., “Search for the lepton-family-number nonconserving decay µ+ →e+ γ ”, Phys. Rev. D 65, 112002 (2002); M. L. Brooks et al., “New Limit for the
Lepton-Family-Number Nonconserving Decay µ+ → e+ γ ”, Phys. Rev. Lett. 83,
1521-1524 (1999).
[2] T. Mori et al., “Search for µ+ → e+ γ down to 10−14 branching ratio”, Research
Proposal to Paul Scherrer Institut, R-99-05 (1999), http://meg.web.psi.ch/docs/
prop psi/index.html/[3] L. Michel, “Interaction between Four Half-Spin Particles and the Decay of the µMeson”, Proc. Phys. Soc. A 63, 514-531 (1950).
[4] Particle Data Group, Review of Particle Physics, http://pdg.lbl.gov/.
[5] R. R. Crittenden et al., “Radiative Decay Modes of the Muon”, Phys. Rev. 121,
1823-1832 (1961).
[6] W. Bertl et al., “Search for the decay µ− → e− e+ e− ”, Nucl. Phys. B 260, 1.31 (1985).
[7] S. Freedman et al., “Limits on neutrino oscillations from ν̄e appearance”,
Phys. Rev. D 47, 811-829 (1993).
[8] U. Bellgardt et al., “Search for the decay µ− → e− e+ e− ”, Nucl. Phys. B 299, 1-6(1988).
[9] R. D. Bolton et al., “Search for rare muon decays with the Crystal Box detector”,
Phys. Rev. D 38, 2077-2101 (1988); R. D. Bolton et al., “Search for the Decayµ+ → e+ γ ”, Phys. Rev. Lett. 56, 2461-2464 (1986).
[10] M. Gell-Mann, P. Ramond, and R. Slansky, in “Supergravity”, edited by D. Freedman
and P. van Nieuwenhuizen (North-Holland, Amsterdam), 315, (1979).
[11] T. Yanagida, in Proceedings of Workshop on Uniﬁed Theory and Baryon Number in
the Universe, New York, KEK Report No. 79-18, 95.
[12] J. Hisano and D. Nomura, “Solar and atmospheric neutrino oscillations and lepton ﬂavor violation in supersymmetric models with right-handed neutrinos”, Phys. Rev. D
59, 116005, (1999).
[13] R. Barbieri and L. J. Hall, “Signals for supersymmetric uniﬁcation”, Phys. Lett. B
338, 212-218 (1994); R. Barbieri, L. J. Hall, and A. Strumia, “Violations of lepton
ﬂavour and CP in supersymmetric uniﬁed theories”, Nucl. Phys. B 445, 219-251(1995).
171

172

BIBLIOGRAPHY[14] J. Hisano et al., “Exact event rates of lepton ﬂavor violating processes in supersymmetric SU (5) model”, Phys. Lett. B 391, 341-350 (1997).
[15] J. Hisano et al., “Waiting for µ → eγ from the MEG experiment”, arXiv:0904.2080[hep-ph] (2009).
[16] L. Calibbi et al., “Lepton ﬂavour violation from supersymmetric grand uniﬁed theories: Where do we stand for MEG, PRISM/PRIME and a super ﬂavour factory”,
Phys. Rev. D 74, 116002 (2006).
[17] K. S. Hirata et al., “Experimental limits on nucleon lifetime for lepton+meson decay
modes”, Phys. Lett. B 220, 308-316 (1989).
[18] Y. Kuno and Y. Okada, “Muon decay and physics beyond the standard model”,
Rev. Mod. Phys. 73, 151-202 (2001).
[19] M. Raidal et al., ”Flavour physics of leptons and dipole moments”, report of the
CERN workshop “Flavour in the era of the LHC”, arXiv:0801.1826 [hep-ph] (2008).
[20] S. H. Neddermeyer and C. D. Anderson, “Note On The Nature Of Cosmic Ray
Particles”, Pyhs. Rev. 51, 884-886 (1937).
[21] H. Yukawa, “On the Interaction of Elementary Particles”, Proc. Phys. Math. Soc. Jap, 17 48-57 (1935).
[22] M. Conversi, E. Pancini, and O. Piccioni, “On the Disintegration of Negative
Mesons”, Phys. Rev. 71 209-210 (1947).
[23] E. P. Hincks and B. Pontecorve, “Search for Gamma-Radiation in the 2.2Microsecond Meson Decay Process”, Phys. Rev. 73, 257-258 (1947).
[24] J. Steinberger, “On the Range of the Electrons in Meson Decay”, Phys. Rev 74,
500-501 (1948); ibid 75, 1136-1143 (1949).
[25] Lokonathan, S. and J. Steingerger, “Search for Improbable Muon Decay”,
Phys. Rev. 98, 240 (1955).
[26] G. Feinberg, “Decays of the µ Meson in the Intermediate-Meson Theory”,
Phys. Rev. 110, 1482-1483 (1958).
[27] G. Danby et al., “Observation of High-Energy Neutrino Reactions and the Existenceof Two Kinds of Neutrinos”, Phys. Rev. Lett. 9, 36-44 (1962).
[28] A. van der Schaaf et al., “A SEARCH FOR THE DECAY µ+ → e+ γ ”,
Nucl. Phys. A340, 249-270 (1980); H. P. Povel et al., “A NEW UPPER LIMITFOR THE DECAY µ+ → e+ γ ”, Pyhs. Lett. B 72, 183-186 (1977).
[29] P. Depommier et al., “New Limit on the Decay µ+ → e+ γ ”, Phys. Rev. Lett. 39,
1113-1116 (1977); G. Azuelos et al., “New Upper Limit on the Decay µ → eγγ”,
ibid. 51, 164-167 (1983).

BIBLIOGRAPHY

173[30] W. W. Kinnison et al., “Search for µ+ → e+ γ ”, Phys. Rev. D 25, 2846-2868 (1982);
J. D. Bowman et al., “Upper Limit for the Decay µ+ → e+ γ ”, Phys. Rev. Lett. 42,
556-560 (1979).
[31] A. E. Pifer, T. Bowen, and K. R. Kendall, “A HIGH STOPPING DENSITY µ+
BEAM”, Nucl. Instr. Meth. 135, 39-46 (1976); H. -W. Reist et al., “DEVELOPMENT OF A LOW-MOMENTUM “SURFACE” MUON BEAM FOR LAMPF”,
ibid 153, 61-64 (1978).
[32] T. Kinoshita and A. Sirlin, “RADIATIVE DECAY OF THE MUON”,
Phys. Rev. Lett. 2, 177-178 (1959).
[33] C. Fronsdal and H. Überall, “µ-Meson Decay with Inner Bremsstrahlung”,
Phys. Rev. 113, 654-657 (1959).
[34] Y. Kuno and Y. Okada, “Proposed µ → eγ Search with Polarized Muons”,
Phys. Rev. Lett. 77, 434-437 (1996).
[35] Y. Kuno, A. Maki, and Y. Okada, “Background suppression for µ → eγ with polarized muons”, Phys. Rev. D 55, R2517-R2520 (1997).
[36] T. Kinoshita and A. Sirlin, “Radiative Corrections to Fermi Interactions”,
Phys. Rev. 113, 1652-1660 (1959).
[37] W. Wagner et al., “PSI status 2008 - Developments at the 590 MeV proton accelerator
facility”, Nucl. Instr. Meth. A 600, 5-7 (2009).
[38] ROHACELL R , Evonik Röhm GmbH, http://www.rohacell.com/.
[39] W. Ootani et al., “Development of a Thin-wall Superconducting Magnet for the
Positron Spectrometer in the MEG Experiment”, IEEE Trans. Appl. Superconduct.
14, 568-571 (2005).
[40] A. Yamamoto et al., “Development towards Ultra-thin Superconducting Solenoid
Magnets for High Energy Particle Detectors”, Nucl. Phys. B (Proc. Suppl.) 78, 565570 (1999); A. Yamamoto et al., “A Thin Superconducting Solenoid Magnet for
Particle Astrophysics”, IEEE Trans. Appl. Superconduct. 12, 438-442 (2002).
[41] H. Nishiguchi, “An Innovative Positron Spectrometer to Search for the Lepton
Flavour Violating Muon Decay with a Sensitivity of 10−13 ”, Ph.D. Thesis, the University of Tokyo (2008), http://meg.web.psi.ch/docs/theses/nishiguchi phd.
pdf/.
[42] Saint-Gobain Ceramics & Plastics, http://www.detectors.saint-gobain.com/.
[43] Hamamatsu Photonics, http://www.hamamatsu.com/.
[44] S. Dussoni, “Search for Lepton Flavour Violation:
Ph.D. Thesis, Università degli Studi di Genova (2006).

The MEG experiment”,[45] R. Valle, “Development of the Timing Counter”, Ph.D. Thesis, Università degli Studidi Genova (2006), http://meg.web.psi.ch/docs/theses/valle phd.pdf/.

174

BIBLIOGRAPHY[46] M. Rossela et al., “Behaviour in high magnetic ﬁelds of ﬁne-mesh photodetectors forfast time-of-ﬂight detectors”, Nucl. Instr. Meth. A, 567, 200-204 (2006).
[47] K. Ozone, “Liquid Xenon Scintillation Detector for the New µ → eγ Search Experiment”, Ph.D. Thesis, the University of Tokyo (2005), http://meg.web.psi.ch/
docs/theses/ozoned.pdf/.
[48] R. Sawada, “A Liquid Xenon Scintillation Detector to Search for the Lepton Flavour
Violating Muon Decay with a Sensitivity of 10−13 ”, Ph.D. Thesis, the University ofTokyo (2008), http://meg.web.psi.ch/docs/theses/sawada phd.pdf/.
[49] E. Aprile and T. Doke, “Liquid Xenon Detectors for Particle Physics and Astrophysics”, arXiv:0910.4956v1 [hep-ex] (2009).
[50] Photon Cross Sections Data Base, http://physics.nist.gov/PhysRefData/.
[51] S. Kubota, M. Hishida, and J. Raun, “Evidence for a triplet state of the self-trapped
exciton states in liquid argon, krypton and xenon”, J. Phys. C 11, 2645-2651 (1978);S. Kubota et al., “Dynamical behavior of free electrons in the recombination process
in liquid argon, krypton, and xenon”, Phys. Rev. B 20, 3486-3496 (1979).
[52] T. Doke, “FUNDAMENTAL PROPERTIES OF LIQUID ARGON, KRYPTONAND XENON AS RADIATION DETECTOR MEDIA”, Portgal. Phys. 12, 9-48
(1981).
[53] D. Lide ed., “Handbook of Chemistry & Physics 2004-2005”, Crc Pr. I Llc. (2004).
[54] A.C. H. Hallett, “ARGON, HELIUM AND THE RARE GASES” Vol I, Chap IX,
Interscience Publishers, Inc., Yew Nork/London (1961).
[55] C. W. Fabjan and F. Gianotti, “Calorimetry for particle physics”, Rev. Mod. Phys.
75, 1243-1286 (2003).
[56] J. Jortner, L. Mever, S. A. Rice, and E. G. Wilson, “Localized Excitations in Condensed Ne, Ar, Kr, and Xe”, J. Chem. Phys. 42, 4250-4253 (1965); N. Schwenter, E.-E. Koch, and J. Jortner, “Electronic Excitations in Condensed Rare Gases”,
Springer-Verlag, Berlin (1985).
[57] L. M. Barkov, A. A. Grebenuk, N. M. Ryskulov, P. Yu. Stepanov, and S. G. Zverev,“Measurement of the refractive index of liquid xenon for intrinsic scintillation light”,
Nucl. Instr. Meth. A, 379, 482-483 (1996).
[58] V. N. Solovov et al.,”Measurement of the refractive index and attenuation length ofliquid xenon for its scintillation light”, Nucl. Instr. Meth. A, 516, 462-474 (2004).
[59] G. M. Seidel, R. E. Lanou, and W. Yao, “Rayleigh scattering in rare-gas liquids”,
Nucl. Instr. Meth. A, 489, 189-194 (2002).
[60] T. Doke et al.,”Absolute Scintillation Yields in Liquid Argon and Xenon for Various
Particles”, Jpn. J. Appl. Phys. 41, 1538-1545 (2002).

BIBLIOGRAPHY

175[61] A. Hitachi et al., “Eﬀect of ionization density on the time dependence of luminescencefrom liquid argon and xenon”, Phys. Rev. B 27, 5279-5285 (1983).
[62] A. Braem et al.,”Observation of the UV scintillation light from high energy electron
showers in liquid xenon”, Nucl. Instr. Meth. A, 320, 228-237 (1992).
[63] V. Y. Chapel, M. I. Lopes, R. Ferreira Marques, and A.J.P.L. Policarpo, “Puriﬁcationof liquid xenon and impurity monitoring for a PET detector”, Nucl. Instr. Meth. A,
349, 500-505 (1994).
[64] N. Ishida et al., “Attenuation length measurements of scintillation light in liquidrare gases and their mixtures using an improved reﬂection suppresser”, Nucl. Instr. Meth. A, 384, 380-386 (1997).
[65] T. Haruyama et al., “Development of a high-power coaxial pulse tube refrigerator fora liquid xenon calorimeter”, Adv. Cryo. Eng. 49, 1459-1466 (2004).
[66] T. Iwamoto et al., “Development of a large volume zero boil-oﬀ liquid xenon storage
system for muon rare decay experiment (MEG)”, Cryogenics 49, 254-258 (2009).
[67] A. Baldini et al., “A radioactive point-source lattice for calibrating and monitoring
the liquid xenon calorimeter for the MEG experiment”, Nucl. Instr. Meth. A, 565,
589-598 (2006).
[68] Y. Yoshino et al., “Absorption cross section measurement of water vapor in the
wavelength region 120 to 188mm”, Chem. Phys. 211 387 (1996).
[69] S. Himi et al., “Liquid and Solid Argon, and Nitrogen-doped liquid and solid Argon
Scintillators”, Nucl. Inst. Meth. 203, 153 (1982).
[70] S. Mihara et al., “Development of a method for liquid xenon puriﬁcation using acryogenic centrifugal pump”, Cryogenics 46 688-693 (2006).
[71] S. Mihara et al., “Development of a liquid-xenon photon detector - toward the search
for a muon rare decay mode at Paul Scherrer Institut”, Cryogenics 44 223-228 (2004).
[72] MIDAS, Maximum Integration Data Acquisition System, https://midas.psi.ch/.
[73] Analog Devices, AD9218, 10 bit 3V Dual A/D converter, http://www.analog.com/.
[74] Xilinx, Virtex II Pro, http://www.xilinx.com/.
[75] SaRonix, Crystal Clock Oscillator, http://www.pericom.com/saronix/.
[76] S. Ritt, “The DRS chip: cheap waveform digitizing in the GHz range”, Nucl. Instr. Meth. A, 518, 470-471 (2004).
[77] S. Ritt, “Design and Performance of the 5 GHz Waveform Digitizing Chip DRS3”,
Nucli. Sci. Symmp. Conf. Rec. IEEE 4, 2485-2488 (2007).
[78] NACRE - European Compilation of Reaction Rates for Astrophysics, http://pntpm.
ulb.ac.be/Nacre/nacre.htm/.

176

BIBLIOGRAPHY[79] High Voltage Engineering Europe, http://www.highvolteng.com/.
[80] Jinan Jingyi Photonics Technology Co., http://www.jnjydkdp.com/.
[81] A. Papa, “Search for the lepton ﬂavour violation in µ+ → e+ γ . The calibration
methods for the MEG experiment”, Ph.D. Thesis, the University of Pisa (2009),
http://meg.web.psi.ch/docs/theses/Angela.pdf/.
[82] GEANT, Detector Description and Simulation Tool version 3.21, http://wwwasd.
web.cern.ch/wwwasd/geant/.
[83] Garﬁeld, Simulation of gaseous detectors,
garfield/.

http://garfield.web.cern.ch/[84] HEED, Interactions of particles with gases, http://consult.cern.ch/writeup/
heed/.
[85] Magboltz, Transport of electrons in gas mixtures, http://consult.cern.ch/writeup/magboltz/.
[86] ROME, Root based Object oriented Midas Extension, https://midas.psi.ch/rome/.
[87] ROOT, An Object-Oriented Data Analysis Framework, http://root.cern.ch/.
[88] R. E. Kalman, “A New Approach to Linear Filtering and Pbrediction Problems”,
Transaction of the AMSE-J. Basic Engineering D82, 35-45, (1985).
[89] F. Rfühwirth, “APPLICATION OF KALMAN FILTERING TO TRACK AND
VERTEX FITTING”, Nucl. Instr. Meth. A, 262, 444-450 (1987).
[90] J. R. Cash and A. H. Karp, “A variable order Runge-Kutta method for initial value
problems with rapidly varying right-hand sides”, ACM Trans. Math. Software 16,
201-222 (1990).
[91] M. Yamashita, “Time dependence of rate-dependent photomultiplier gain and its
implications”, Rev. Sci. Instrum. 51(6), 768-775 (1980).
[92] J. R. Klein and A. Roodman, “Blind Analysis in Nuclear and Particle Physics”,
Annu. Rev. Nucl. Part. Sci. 55, 141-163 (2005).
[93] MINUIT, Function Minimization and Error Analysis, http://lcgapp.cern.ch/project/cls/work-packages/mathlibs/minuit/home.html/.
[94] F. James, “Statistical Methods in Experimental Physics (2nd Edition)”, World Scientiﬁc Publishing Co. Pte. Ltd. (2006).
[95] G. J. Feldman and R. D. Cousins, “Uniﬁed approach to the classical statistical analysis of small signals”, Phys. Rev. D 57, 3873-3889 (1998).