学位論文

Search for the Lepton Flavor Violating
Muon Decay µ+ → e+γ with a Sensitivity
below 10−12 in the MEG Experiment
訳

MEG 実験による 10−12 以下の感度でのレプトンフレーバーを破るミューオン崩壊 µ+ → e+γ の探索

平成 25 年 12 月 博士（理学）申請
東京大学大学院理学系研究科
物理学専攻

藤井 祐樹

December 2013

Abstract
The Standard Model (SM) of the particle physics has been established by many experimental tests. However, there are still several puzzles which cannot be explained by the SM
and it is thought that the SM is an approximation of more fundamental physics theory.
The MEG experiment searches for the Lepton Flavor Violating decay: µ+ → e+ γ as anevidence of the new physics, since this decay is strictly forbidden in the SM. We perform
a µ+ → e+ γ search using new data sample and improved analysis methods. Comparedwith the previous MEG result, the total statistics are doubled by adding the data taken
in 2011. In this search, several methods of calibrations and analysis are improved to maximize the experimental sensitivity. In the analysis, 1) new pileup identification method in
the gamma reconstruction, 2) the oﬄine noise reduction for the positron reconstruction
and 3) new track fitting algorithm are newly implemented and performed. The physics
analysis is improved as well. Therefore, the data of 2009–2010 are analyzed again with
the new analysis and the new analysis gives a 20% better sensitivity than that with theprevious analysis. By analyzing the dataset of 2009–2011, we obtain a branching ratio
sensitivity of
S = 7.7 × 10−13 .
This is the first search for the µ+ → e+ γ decay with a sensitivity below 10−12 . The result
is consistent with background-only hypothesis, therefore we set only the upper limit. The
observed 90% C.L. upper limit on the existence on of the µ+ → e+ γ decay is calculated
to beB(µ+ → e+ γ) < 5.7 × 10−13 .
This is a four times more stringent upper limit on the existence of the µ+ → e+ γ decay
than the previous best limit set by MEG.

Contents
1 Physics Motivation
1.1 Muon Decay in the Standard Model . . . . . . . . . . . . . . . . . . . . .
1.1.1 Michel Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 The µ+ → e+ γ Decay in the Standard Model . . . . . . . . . . . .
1.2 The µ+ → e+ γ decay and Physics Beyond the Standard Model . . . . . .
1.2.1 SUSY GUT SU(5) . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 SUSY Seesaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 The µ+ → e+ γ Decay and (g − 2)µ Anomaly . . . . . . . . . . . .
1.3 Searches for the CLFV Processes . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Experimental History of the µ+ → e+ γ Decay Searches . . . . . .
1.3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Comparison with Other CLFV Searches . . . . . . . . . . . . . .

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2 The MEG Experiment
2.1 Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Muon Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Beam Transport Solenoid . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Gamma Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Liquid Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Positron Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 COBRA Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Timing Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Front-end Electronics and Data Acquisition System . . . . . . . . . . . .
2.4.1 DRS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 MIDAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Simulation and Analysis Framework . . . . . . . . . . . . . . . . . . . . .
2.5.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Run Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Run 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Run 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Run 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS
3 Event Reconstruction
3.1 Gamma Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Position Reconstruction . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Energy Reconstruction . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Time Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.5 Pileup Identification . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.6 Cosmic-ray Rejection . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Positron Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Hit Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Cluster Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Track Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Track Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Time Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.6 Impact Point in Timing Counter . . . . . . . . . . . . . . . . . .

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4 Calibrations
4.1 LXe Monitoring and Calibration Methods . . . . . . . . . . . . . . . . .
4.1.1 Charge Exchange Calibration . . . . . . . . . . . . . . . . . . . .
4.1.2 Cockcroft-Walton . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Neutron Generator . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Cosmic Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.5 LED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.6 Alpha Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.7 Light Yield History . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Drift Chamber Calibrations . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Hit Z Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Hit Time Calibration . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Optical Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Cosmic Ray Alignment . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Michel Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Relative Alignment . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Timing Counter Calibrations . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Gain Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Time Oﬀsets Correction . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 CW-Boron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Relative Time Oﬀset . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Performance
5.1 Performance of the LXe Detector . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Gamma Selection . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Gamma Position Resolution . . . . . . . . . . . . . . . . . . . . .
5.1.3 Gamma Energy Response . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Gamma Timing Resolution . . . . . . . . . . . . . . . . . . . . . .
5.1.5 Gamma Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

5.2

5.3

5.4

Performance of the Positron Spectrometer . . . . . . . . . . . . . . . . .
5.2.1 Positron Selection . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Single Hit Resolution . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Positron Energy Response . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Angular and Vertex Resolutions . . . . . . . . . . . . . . . . . . .
5.2.5 Positron Timing Resolution . . . . . . . . . . . . . . . . . . . . .
5.2.6 Positron Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . .
Combined Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Vertex Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Timing Resolution . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Contributions of Reconstruction Improvements . . . . . . . . . . .
Performance Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Physics Analysis
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6.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1 Pre-Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.2 Blind Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.3 Analysis Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.4 Sideband Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1.5 Time Sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1.6 Angle Sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Likelihood Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.1 Category PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.2 Signal PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.3 Accidental Background PDF . . . . . . . . . . . . . . . . . . . . . 117
6.3.4 RMD PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.5 Illustration of PDFs . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4 Confidence Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4.1 Full-MC study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.5 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5.1 Analysis Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5.2 Michel Normalization . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5.3 RMD Normalization . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5.4 Combined normalization factors . . . . . . . . . . . . . . . . . . . 129
7 Results and Discussions
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7.1 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2 Sideband Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2.1 Results with Time Sidebands . . . . . . . . . . . . . . . . . . . . 134
7.2.2 Results with Angle Sidebands . . . . . . . . . . . . . . . . . . . . 136
7.3 Result in Analysis Region . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.4 Eﬀects of Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . 142
7.4.1 Uncertainties of Parameters . . . . . . . . . . . . . . . . . . . . . 142
7.4.2 Comparison of Systematic Uncertainties . . . . . . . . . . . . . . 144
7.5 Further Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

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CONTENTS

7.6

7.5.1 Result from Category PDF . . . . . . . . . . . . . . . . . . . . . .
7.5.2 Analysis Compatibility . . . . . . . . . . . . . . . . . . . . . . . .
7.5.3 Event-by-Event Check and Comparison . . . . . . . . . . . . . . .
Discussions and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.2 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Conclusion

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A Geometrical Model of Correlations
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A.1 Implementation of Correlations . . . . . . . . . . . . . . . . . . . . . . . 158
B Radiative Muon Decay

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List of Figures
1.1
1.2
1.3

1.4
1.5

1.6

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

Feynman diagram for the µ+ → e+ γ decay through the neutrino oscillation. 19
Feynman diagrams for the µ+ → e+ γ decay in SU(5) SUSY GUT. . . . .
20
Correlation between B(µ → eγ) and B(τ → µγ) as a function of mN 3 , forconstraint MSSM assuming universality of the soft-SUSY breaking at thescale of gauge coupling unification, MX so-called minimal supergravity(mSUGRA) framework from [16]. Diﬀerent colored regions correspond to◦◦◦◦θ13 = 1 , 3 , 5 and 10 (red, green, blue and pink, respectively) and recent◦
experiments report the large value of θ13 ∼ 10 [17]. Dashed line labeled“MEG2011” is added to the original plot and it represents the 90% C.L.
upper limit given by the previous result of MEG [4]. . . . . . . . . . . . .
21
Simplified Feynman diagram of the µ+ → e+ γ decay (left) and the muon
magnetic moment (right). . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Correlation between B(µ+ → e+ γ) and ∆aµ from [20]. Horizontal dashedline labeled “MEG2011” is added to the original plot and it represents the
90% C.L. upper limit given by the previous result of the MEG experiment.
Vertical two dashed lines are also added and they represent the 1σ regionof the (g − 2)µ anomaly measured in the E821 experiment [18]. Assumed
12
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parameters are as follows;|δLL| = 10−4 , |δLL
| = 10−2 . Here the relatively
small slepton mass and large tan β are assumed. The inner red area satisfy
23
the B-physics constraints. . . . . . . . . . . . . . . . . . . . . . . . . . .
Left figure shows positron energy spectrum of unpolarized µ+ → e+ νe ν µdecay (Michel spectrum). A radiative decay correction [25] is included.
Right one shows gamma Energy spectrum of unpolarized µ+ → e+ νe ν µ γ
decay. Positron energy and the angle between a positron and a gamma are26
integrated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Top and side view of all the experimental apparatus with coordination. .
3D view of all the experimental apparatus with coordination. . . . . . . .
590 MeV proton ring cyclotron at PSI. . . . . . . . . . . . . . . . . . . .
Schematic view of the πE5 beam channel including the beam transportsystem and detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Muon target (a) and installed view inside the COBRA magnet (b). . . .
Schematic views of the LXe detector. . . . . . . . . . . . . . . . . . . . .
LXe detector before closing a flange of the outer vacuum vessel. . . . . .
LXe detector inside with PMTs installed on all surfaces. . . . . . . . . .

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LIST OF FIGURES

2.9

Concept of the COBRA magnetic field compared with the uniform field.
Positrons emitted around 90◦ are quickly swept away (c) while they staylonger in a uniform field case (a). Positrons which emitted diﬀerent θangles have same curvatures in the COBRA magnetic field (d). On theother hand, they have diﬀerent curvatures in a uniform case (b). . . . . .
2.10 Contour plot of absolute values of the gradient magnetic field generated
by the COBRA magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Positron hit rate per 1 cm2 in each drift chamber module with a 3 × 107 Hz
muon stopping rate as a function of the radius. Green triangle markers
show those with an uniform magnetic field and orange markers show thosewith the COBRA magnetic field. . . . . . . . . . . . . . . . . . . . . . .
2.12 Distribution of the magnetic field around the LXe detector. The PMTs of
the LXe detector are placed along the trapezoidal box shown in this figure.
2.13 Drift chamber modules installed inside the COBRA magnet. . . . . . . .
2.14 Schematic side views of a drift chamber module. One plane has nine anodewires and all anode and potential wires are supported by a open shapeframe as shown in (a). The module is covered by the cathode foil whichhas 5 cm periodical zig-zag shape called “vernier pattern” as shown in (b).
2.15 Configuration of cells inside the chamber module. Two planes are divided
with a gap of 3 mm and the position of the sense wires are staggered.
Green horizontal lines show the cathode pads. . . . . . . . . . . . . . . .
2.16 Field map and drift lines calculated by GARFIELD simulation with the nominal operated condition in MEG. . . . . . . . . . . . . . . . . . . . . . . .
2.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.18 Timing counter bars covered with fibers for z position measurement. . . .
2.19 Schematic of DRS principle. . . . . . . . . . . . . . . . . . . . . . . . . .
2.20 Illustration of the trigger system. . . . . . . . . . . . . . . . . . . . . . .
2.21 Illustration of the full programming chain for the analysis and the MC
simulation in MEG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.22 Number of stopped muons on the target during physics data taking of
2009–2011. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3

3.4
3.5

(a) An example of a PMT raw waveform from the waveform digitizer. (b)An example of a waveform with high-pass filter. . . . . . . . . . . . . . .
An example event with pileup gamma ray which is identified by using the
distribution of the light yield. Magenta circles show the pileup cluster. .
An event with pileup which identified by the peak search. Left plot shows
the normal waveform after moving-average and right plot shows the diﬀerential one. Magenta stars show the found peaks. . . . . . . . . . . . . . .
An waveform as same event as shown in Fig. 3.3 before (left) and after(right) applying the pileup elimination. . . . . . . . . . . . . . . . . . . .
Two dimensional scatter plot of the charge ratio on the inner/outer faces
vs the gamma ray interaction depth. Black points show the data fromcosmic ray runs, and green ones shows the signal Monte Carlo events. The
two blue lines show the selection criteria for the cosmic ray rejection. . .

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LIST OF FIGURES

3.6

Example of waveforms in one cell in an event. Top two waveforms are fromboth ends of the anode wire, and others are from each side of cathodes.
Red horizontal lines show the calculated baseline of each waveform. Red
vertical lines at the left edge of each peak show calculated hit timing. . .
3.7 Correlation between the vernier circle and za . . . . . . . . . . . . . . . .
3.8 Relation between distance from anode wire and hit timing calculated byusing the Garfield simulation. . . . . . . . . . . . . . . . . . . . . . . .
3.9 Example of DCH waveforms in a cell which taken in the 2011 noisy run
period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Example of DCH waveforms taken after hardware investigations. . . . . .
3.11 Power spectrum of accumulated DCH waveforms taken in the 2011 noisy
run period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Raw DCH waveform (black solid line) and after the filtering (red solid line)
in noisy run (a). The charge integration is done for the area between 2
blue solid lines. (b) is the power spectrum of waveform (a) before/after
filtering (black/red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.13 A positron trajectory fitted by using the previous track fitting in an event.
3.14 Diﬀerences of reconstructed hits by using new/old Kalman in X-Y view(left) and X-Z view (right). Red stars show the hits reconstructed byusing new Kalman and blue ones show those reconstructed by using old
Kalman. Magenta circles in the left plot shows measured hits belonging to
the reconstructed track and the radius of each circle shows the measured
drift distance. Black points shown in the left plot represent the positionof wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2

4.3

4.4
4.5
4.6

4.7
4.8

BGO (NaI) mover(a) and two dimensional plot of gamma ray energies
measured by the LXe detector and the BGO detector(b). . . . . . . . . .
Reconstructed positions of 25 α sources inside the LXe detector. In thisparticular run, the half of the detector was filled with liquid xenon and theother part was filled with gaseous xenon. . . . . . . . . . . . . . . . . . .
Upper plot shows the light yield history measured by using CW-Li peak(black) and background scale (red). To calibrate the light yield history,
other possible sources (CEX 55 and 83 MeV gamma, 9 MeV gamma withneutron generator, cosmic-ray, and alpha source peak) are used. Combined
history curve is also shown in this plot with a gray curve. Bottom plotshows same history after the correction by using the combined curve sothat it becomes flat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlation between za and the cathode vernier pad asymmetry (black
markers), fitted by a sinusoidal function (red line). . . . . . . . . . . . . .
Corner cubes mounted on the downstream side of the drift chambers. . .
3D view of 10 cosmic-ray counter bars (shown in cyan). Green line showsa simulated cosmic-ray, which pass through both drift chambers and thecosmic-ray counter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reconstructed positron vertex distribution in Z-Y plane in 2011. . . . . .
An illustration of the software target alignment with Michel positrons incase of the displacement of the target in horizontal axis. . . . . . . . . .
8

61
62
63
64
65
66

67
70

71
75

77

79
80
81

82
83
83

LIST OF FIGURES

5.1

Reconstructed position distribution projected on the v-axis with a leadcollimator in CEX runs. . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
5.2 Fitted energy response of the LXe detector for 55 MeV gamma-ray using
CEX data taken in 2011 for wγ > 2 cm(a). (b) shows measured resolution
map in sigma for wγ > 2 cm. . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.3 Residual distributions of Z and R in 2011. Red lines are double Gaussianwhich are used for the fitting. . . . . . . . . . . . . . . . . . . . . . . . .
91
5.4 A Ee distribution from the data taken in 2011 with a fitting result byusing Eq. (5.6). Dashed blue line shows a resolution function centered at
52.8 MeV. Dashed black line is the theoretical Michel spectrum. In thebottom plot, the acceptance function is shown. . . . . . . . . . . . . . . .
92
5.5 Diﬀerences of φe (a), θe (b), ye (c) and ze (d) between two turns from 2011
data fitted by the double Gaussian function. . . . . . . . . . . . . . . . .
93
5.6 Relative time distribution from 2011 data. . . . . . . . . . . . . . . . . .
96
5.7 Residuals of the single hit z-position with/without the noise reduction. .
97
5.8 Ee spectrum in 2011 noisy run period. Red/black lines are with/without
99
the noise reduction. Right figure shows the log scale of the left figure. . .
5.9 Distributions of ∆x(x = Ee , φe , θe , ye , ze ) from two-turn events in 2011
noisy run period. Red/black lines are with/without the noise reduction. . 100
5.10 Ee spectrum in 2009–2011 combined data. Red/black lines are with/without the noise reduction. Right figure shows the log scale of the left figure. 101
5.11 Distributions of ∆x(x = Ee , φe , θe , ye , ze ) from two-turn events in 2009–2011 combined data. Red/black lines are with/without the noise reduction. 101
5.12 Reconstructed energy spectrum of positrons with new track fitting (red)in comparison with that with the previous one (dashed black line). Bothspectra are normalized by using the area of each histogram. . . . . . . . . 102
5.13 Energy spectra of gamma rays in comparison between that by applying new
pileup elimination algorithm (red), previous algorithm (blue-dot-dashed)
and without the pileup elimination (black-dotted). . . . . . . . . . . . . . 103
6.1

6.2

Two dimensional event distribution in teγ vs Eγ taken in 2009–2011. The
blank box at the center shows the blind box and the inner box with bluelines shows the analysis region. Left and Right boxes with dashed blacklines show the sideband data and two boxes with solid magenta lines show
the time sidebands which are used for the maximum likelihood fit. A bottom center box shows the energy sideband used to evaluate the expected
number of RMD events and the teγ resolution. For the illustration purpose, following loose cuts are applied;40 ≤ Ee ≤ 60 MeV;40 ≤ Eγ ≤60 MeV;|θeγ | ≤ 200 mrad; |φeγ | ≤ 200 mrad;|teγ | ≤ 4 ns. . . . . . . . . . .
Two dimensional event distribution in θeγ vs φeγ taken in 2009–2011 intime sidebands. The center box with blue lines shows the analysis region in relative angles. Neighboring four boxes with dashed lines show
the angle sidebands which we perform the maximum likelihood fitting.
For the illustration purpose, following loose cuts are applied;40 ≤ Ee ≤60 MeV;43 ≤ Eγ ≤ 60 MeV;|θeγ | ≤ 200 mrad; |φeγ | ≤ 200 mrad;|teγ | ≤ 4 ns
and |teγ | > 1 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9

107

108

LIST OF FIGURES

6.3

Calculated covariance matrices, (a) for the accidental background PDF,
and (b) the signal PDF for 2009 data set. . . . . . . . . . . . . . . . . . .
6.4 (a) teγ PDFs for diﬀerent categories (2011). (b) Event fractions of thecategories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 An example of the spectrum of the accidental background gamma rays used
for the PDF in depth is larger than 2 cm in time sideband. Green line isextracted from the MC distribution of RMD and AIF, Blue line show that
convolved with resolution and pedestal, namely acceptance. Solid blackline is contribution from cosmic-ray background and Red one shows sumof the blue one and black one. . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Average PDFs for the 2009–2011 combined dataset. The green, red andmagenta lines show the signal, RMD and BG PDFs, respectively. The
correlations between the errors of positron observables are corrected except
for (f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Examples of per-event PDF in typical two events observed in the timesideband. (a) shows PDFs in which the positron-track shows good fitting
quality. (b) shows ones in the event with worse fitting quality. Greenlines show the signal PDFs, Magenta dashed lines show the accidental
background PDFs and red lines show the RMD PDFs. Blue lines are sumof those three PDFs and black stars indicate measured variables. . . . . .
6.8 (a) Average distributions of Ee (left), θeγ (center) and φeγ (right) of the
full-MC (black markers) and the toy-MC (blue solid line). (b) Averagecorrelations of δEe v.s. δφeγ (left) and δθeγ v.s. δφeγ of the full-MC (black
markers) and the toy-MC (blue markers). . . . . . . . . . . . . . . . . . .
6.9 (a) Distribution of MC signal events on vertex-angle plane. Missing-turn
events make a characteristic pattern on this plane appearing as a bandparallel to the red arrow. (b) Scatter plot of the MC events on teγ v.s. a newvariable defined by the projection along the red arrow in (a). The missing
turn events ranging at the negative teγ are clearly distributed around −28
on the new variable. (c) and (d) show the projected distributions of MC
and data (2011), respectively. Fraction of the peak events is estimated byfitting a Gaussian and a quartic function to the distribution and results
are shown in (c) and (d). . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Distribution of RMD events in 2009–2010 data. Black dots show the measured distribution. Black solid histograms show the expected distribution
calculated from the normalization factor from Michel channel and the best
estimate value for each parameter, and the gray bands show the uncertainty. Red dashed ones show the distribution with the best-fit values ofnormalization and systematic parameters. . . . . . . . . . . . . . . . . .
6.11 Distribution of RMD events in 2011 data. Black dots show the measured
distribution. Black solid histograms show the expected distribution calculated from the normalization factor from Michel channel and the best
estimate value for each parameter, and the gray bands show the uncertainty. Red dashed ones show the distribution with the best-fit values ofnormalization and systematic parameters. . . . . . . . . . . . . . . . . .

10

112
117

119

121

122

124

126

129

130

LIST OF FIGURES

7.1

Distributions of the upper limits of the background-only pseudo experiments. (a) 2009-2010, (b) 2011, (c) 2009-2011. The red arrows show themedians of the distributions. The black arrows show the observed limits.
In (a), two arrows are overlapped by chance. . . . . . . . . . . . . . . . .
7.2 Confidence level curves in time sidebands (a)–(c) before normalization and(d) after normalization. (a) 2009-2010, (b) 2011, (c)(d) 2009-2011. Redlines show the results of the negative time sidebands and blue lines show
the results of the positive ones. . . . . . . . . . . . . . . . . . . . . . . .
7.3 Event distributions in Eγ v.s. Ee and teγ v.s. cos Θeγ . Corresponding
datasets are 2009-2010 combined (a)(b) and 2011 only (c)(d), respectively.
The signal two-dimensional PDFs are superimposed as contours at 1, 1.645,
2 σ as blue dashed, solid, and dotted lines respectively. . . . . . . . . . .
7.4 Event distributions in the (Ee , Eγ ) plane(a) and (cos Θeγ , teγ ) plane(b).
Corresponding dataset is 2009-2011 combined. Definition of blue contours
are same as those in Fig. 7.3. . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Likelihood fit result for 2009–2010 combined dataset in the analysis region.
Black markers show the data. Red and pink lines show the best fit for RMD
and BG, respectively. Green line shows the best fit for the signal (almost
invisible because of the small Nsig ) and blue one shows total best fit. . . .
7.6 Likelihood fit result for 2011 dataset in the analysis region. The definitionsof each components are the same as in Fig. 7.5. . . . . . . . . . . . . . .
7.7 Likelihood fit result for 2009–2011 combined dataset in the analysis region.
The definitions of each components are the same as in Fig. 7.5. . . . . . .
7.8 Profile likelihood curves . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 Confidence level curves (a)–(c) before normalization and (d) after normalization. (a) 2009-2010, (b) 2011, (c)(d) 2009-2011. . . . . . . . . . . . . .
7.10 The average signal and background PDFs for the 2011 dataset. The parameters are randomized for each curve according to their uncertainties.
Dark green (blue) lines are the signal (BG) PDFs without randomizing
parameters and green (cyan) lines are PDFs with randomized parameters.
7.11 Diﬀerence of reconstructed teγ (upper left), Ee (upper center), Eγ (upper
right), θe (bottom left) and φe (bottom center) in the 2009–2010 data withthe old and the new reconstruction algorithms. The definition of ∆x is asfollows:∆x = xnew − xold . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.12 Distribution of ∆Nsig upper limits. The black arrow shows the observed
diﬀerence in the 2009–2010 combined dataset. . . . . . . . . . . . . . . .
7.13 3D view of reconstructed events which have the first(a) and second(b)largest Rsig value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.14 Event-by-event comparison for five high rank events found in new/previous
analysis shown by red/black markers. Numbers represent the event of Rsig
ordering. The signal two-dimensional PDFs are superimposed as contours
at 1, 1.645, 2 σ as blue dashed, solid, and dotted lines respectively. . . .
7.15 (a) shows B(µ → eγ) v.s. B(τ → µγ) correlation (Fig. 1.3 in Sec. 1.2.2[16]) with the new upper limit of B(µ+ → e+ γ). (b) shows B(µ → eγ)
v.s. muon g-2 correlation (Fig. 1.5 in Sec. 1.2.3 [20]) with the new upper
limit of B(µ+ → e+ γ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11

133

135

138

139

140
140
141
141
142

144

147
148
148

149

150

LIST OF FIGURES

7.16 Sensitivity curve as a function of the accumulated DAQ time. Black solid
curve shows the curve of the expected sensitivity and green and yellow
regions express the 1 and 2 σ error bands, respectively. Magenta markers
are observed 90% C.L. upper limits. Red solid curve shows a 3 σ discovery
potential of the experiment for the µ+ → e+ γ decay. . . . . . . . . . . . .

152

A.1 Extraction of the δφe v.s. δEe and δye v.s. δEe correlations. The red verticalline shows the target intersection with the transverse plane passing fromthe muon decay point V = (X, Y ). . . . . . . . . . . . . . . . . . . . . .
A.2 Extraction of the δθe v.s. δze correlation. . . . . . . . . . . . . . . . . . .

157
158

12

List of Tables
1.1
1.2
1.3

Decay modes and branching fractions of muon [7]. . . . . . . . . . . . . .
All particles and sparticles in the MSSM. . . . . . . . . . . . . . . . . . .
History of the µ+ → e+ γ search experiments in the era of meson factories
with 90% C.L. upper limits. The resolutions are given as full width at half
maximum (FWHM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Recent the most stringent experimental limits given by the BABAR andBelle experiments. Only four famous decay modes are shown here. . . . .

18
20

35
44

2.4

Several characteristics of LXe in comparison with other scintillators. . . .
Characteristics and Parameters of DCH. . . . . . . . . . . . . . . . . . .
Various trigger settings used in the MEG experiment and the prescaling
factors in physics run. Approximately QL ∼ 30 MeV,QH ∼ 40 MeV,TN ∼20 ns and TW ∼ 40 ns. DM represents the direction-match algorithm
given by the trigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of some important events which are prepared in MEGMC. . . . . . . .

4.1

Several calibration and monitoring methods prepared for the LXe detector. 74

5.1
5.2
5.3
5.4
5.5
5.6
5.7

Single hit resolutions of core/tail in each year. . . . . . . . . . . . . . . .
Performance comparison in 2009. . . . . . . . . . . . . . . . . . . . . . .
Performance comparison in 2010. . . . . . . . . . . . . . . . . . . . . . .
Performance comparison in 2011 noisy run period. . . . . . . . . . . . . .
Performance comparison in 2011 run after hardware investigations. . . .
Performance comparison for 2009-2011 all combined data. . . . . . . . . .
Performance summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Parameters for positron energy response. Since errors on all variables are
implemented as a covariance matrix in the PDF, they are not shown inthis table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sigmas of pulls for positron angular and vertex responses. . . . . . . . . .
Parameters for the correlations and their uncertainties. . . . . . . . . . .
Breakdown of analysis eﬃciencies . . . . . . . . . . . . . . . . . . . . . .
Michel normalization parameters for 2009 data in comparison with theprevious estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michel normalization parameters for 2010 data in comparison with theprevious estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michel normalization parameters for 2011 data with old/new Kalman . .

1.4
2.1
2.2
2.3

6.2
6.3
6.4
6.5
6.6
6.7

13

24
30

48
50

90
98
98
99
100
100
104

112
113
114
125
127
128
128

6.8
7.1
7.2

Summary of normalization factors calculated for 2009, 2010 and 2011 dataset.131

Results from the previous analysis presented in [4]. . . . . . . . . . . . . 132
Calculated sensitivities in each dataset. Systematic uncertainties are included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3 Time sideband results without including systematic uncertainties. The
results in the negative teγ sidebands are labeled as ‘negative’ and those in
opposite sides are labeled as ‘positive’. . . . . . . . . . . . . . . . . . . . 134
7.4 Angle sideband results with constraining the number of backgrounds. . . 136
7.5 Angle sideband results without constraints on the number of backgrounds(Uncertainties are in 1σ). Parameters in bracket show the expected numberof backgrounds and those with a hat show the best fit results. . . . . . . 137
7.6 Expected number of background events and the observed number of eventsin the analysis region defined in Sec. 6.1.3. . . . . . . . . . . . . . . . . . 137
7.7 The best fit number of events and the corresponding branching ratios. The
uncertainties are MINOS (i.e. profile likelihood ∆NLL method) 1.645σ
errors. The numbers with hats show the best fit values. . . . . . . . . . . 139
7.8 Confidence intervals (upper limits and lower limits if they are set) on the
2009-2010, 2011 and 2009-2011 datasets calculated from confidence level
curves shown in Fig. 7.9(d). The numbers in the first table are written in
Nsig and those in the second are in B × 1013 . A confidence level at 0 isshown only when N̂sig is positive. The results are including the systematic
uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.9 Uncertainties associated to the drift chamber alignment and the magnetic
field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.10 Relative contributions of uncertainties to upper limit of B in RMS of ∆∆NLL.145
7.11 Relative contributions of the uncertainties of the correlations in the positronobservables to the upper limit of B in RMS of ∆∆NLL. . . . . . . . . . . 146
7.12 Results from category PDF and comparison without systematic uncertainties.146
7.13 Detector performance which were written in the proposal and those obtained from data taken in 2011. . . . . . . . . . . . . . . . . . . . . . . . 150

Introduction
Recently, a Higgs boson was discovered by the ATLAS and the CMS collaborations [1]using Large Hadron Collider (LHC) which materialized the collisions of proton-protonwith the world highest center of mass energy, up to 8 TeV. Since the Higgs boson was thelast missing elementary particle predicted by the Standard Model of the particle physics(SM), this discovery proves the accuracy of the SM in a high precision. Even if the
SM well agrees with most of experimental measurements, there are also several issues
that cannot be explained by the SM, for example the tiny masses of neutrinos and theexistence of dark matter which is the most dominant component and yet unknown matterin our universe. It clearly indicates the existence of new physics beyond the SM (BSM).
In the fermion sector, it is found that there are three generations in quark and lepton sectors as well so-called “Flavor” and the flavor mixing was found in the quark
sector which is taken into account by the CKM matrix. In the recent decade, the flavor oscillation in the neutrino sector called “neutrino oscillations” was also confirmed
by various experiments, e.g. Super Kamiokande, Daya Bay and T2K. Therefore, thelepton flavor violation in the charged lepton sector (CLFV) has only been undiscovered
yet. The µ+ → e+ γ decay is one of the simple and famous decay modes of the muon
CLFV processes. In the SM framework, CLFV processes are strictly forbidden and even
if non-zero neutrino mixing is considered, the branching fraction of the µ+ → e+ γ decay is calculated to be ∼10−50 . On the other hand, the branching fraction of CLFV
processes can be enhanced in many popular BSM models, e.g. SUSY-GUT, and they
predict detectable branching ratios of various CLFV processes. For example, the branching fraction of the µ+ → e+ γ decay is expected to be in the range of 10−11 –10−15 order ofmagnitude depending on the values of parameters in SUSY-GUT. From the experimental
point of view, the upper limit on the branching ratio of the µ+ → e+ γ decay was given
as B(µ+ → e+ γ) < 1.2 × 10−11 at 90% C.L. by the MEGA experiment [2] and the value
was already close to the region predicted by the new physics. Accordingly, the search forthe µ+ → e+ γ decay is one of the eﬀective probes to search for the BSM.
Aiming to search for the µ+ → e+ γ decay with higher sensitivity, the MEG (Mu toElectron Gamma) experiment was proposed by a group of Japanese and Russian physicists
and approved by the research committee of the Paul Scherrer Institut (PSI) in 1999 [3] andthen the international collaboration was established. The MEG experiment started thephysics data taking in 2008. By using 2009–2010 combined dataset, the MEG experimenthas already set the upper limit of 2.4 × 10−12 at 90% C.L. [4]. This means that we set a
5 times tighter upper limit than that of set by the MEGA experiment. If the µ+ → e+ γ
decay is discovered, it should be an evidence of new physics. Otherwise, the result will
set constraint to the region of parameters allowed in the BSM and give an important hint
to investigate the unknown mechanism of the BSM. Therefore, it is important to improve
15

LIST OF TABLES

the sensitivity of the µ+ → e+ γ search as much as possible.
In order to improve the sensitivity, the MEG experiment continued the physics datataking until the end of August in 2013. In this thesis, we analyze the data taken during
2009–2011. In 2011, we collected data with 1.85×1014 stopped muons on the target whichis the almost same data statistics as collected in 2009 and 2010. In the new analysis of
the µ+ → e+ γ decay search, we have four improvements as follows,• apply oﬄine noise filtering for the drift chamber waveform analysis,• revise the fitting algorithm for the track reconstruction of positrons,• improve the gamma ray analysis in order to get higher eﬃciency and suppress thepileup events,• take the event-by-event fitting uncertainties on positron observables into accountin the physics analysis.
We analyze 2009–2010 data again with these improvements in order to enhance the sensitivity as much as possible. Owing to these eﬀorts, the sensitivity is improved by a factor
2 by analyzing 2009–2011 combined dataset [5] in comparison with the previous result.
In this thesis, the details of the analysis procedure, analysis and calibration improvements, and results by using 2009–2011 combined dataset is described. In Chapter 1,
the details of searching for the µ+ → e+ γ decay and few related theories are introduced.
The apparatus of the MEG experiment and the details of data taking during 2009–2011
are explained in Chapter 2. In Chapter 3, the method of the event reconstruction is
described. The calibration tools for the MEG experiment are introduced in Chapter 4
in detail. Chapter 5 summarizes the performance and several improvements due to the
new reconstruction algorithms. In Chapter 6, details of the analysis to search for theµ+ → e+ γ decay are described. The results of the analysis are written and discussed in
Chapter 7. In Chapter 8, we conclude this thesis.

16Chapter 1
Physics Motivation
The standard model, which is based on the gauge theory of the strong and electroweak
interactions, has withstood many experimental tests since it was established. However,it cannot explain the existence of tiny masses of neutrinos, the origin of dark matter,and the reason of the large discrepancy between electromagnetic scale (O(100) GeV)
and Planck scale (O(1019 ) GeV) called “hierarchy problem”. Hence the SM is thought
to be an approximated theory of the more fundamental nature principle. In the present
elementary particle physics, the most important subject is to find out the BSM which can
explain above issues. Although many experiments have searched for the evidence of the
BSM up to now, it has not been discovered yet. In many of the BSMs, new symmetries
are introduced and new particles are predicted by those symmetries. Since the achievableenergy range of direct searches for new particles such like LHC are limited by present
collider technologies, indirect searches are also eﬀective methods because undiscovered
heavy particles could exist virtually as a higher order eﬀect. Therefore, searching for
the CLFV process is one of the powerful methods to investigate the BSM. In particular,muon CLFV searches are good probes because a large amount of muon can be produced
more easily than the case of the tau lepton because of its longer life time (2.2 µs). In
addition, the final state of muons are simpler than those of the tau lepton because of thesmaller mass of muon (mµ = 105.6 MeV). The µ+ → e+ γ decay is one of the famous muon
CLFV processes. In this chapter, several decay modes of the muon in the Standard Modeland those in new physics are described. Possible sources of background in searching forµ+ → e+ γ decay are also discussed.

1.1

Muon Decay in the Standard Model

The µ+ → e+ γ decay has been explored for more than 50 years since the muon was
discovered in 1937 [6]. Table 1.1 shows decay modes of muon and its branching fractions
or upper limits. As written in this table, µ± decays into e± and two neutrinos in almost
100% probability. This decay mode is called “Michel ” decay [13] and the total lepton
number is conserved in this process.

17

1.1. MUON DECAY IN THE STANDARD MODEL

Table 1.1: Decay modes and branching fractions of muon [7].
Decay mode
Branching ratio
Reference−−µ → e νµ ν e∼100%(1.4 ± 0.4)% (for Eγ > 10 MeV) [8]µ− → e− νµ ν e γµ− → e− νµ ν e e+ e− (3.4 ± 0.4) × 10−5
[9]−−µ → e νe ν µ< 1.2 %
[10]µ+ → e+ γ< 2.4 × 10−12
[4]
+
+ + −
−12
µ →e e e< 1.0 × 10
[11]µ− → e− γγ< 7.2 × 10−11
[12]

1.1.1

Michel Decay

In the SM, muon decay is determined by the V − A interaction. The diﬀerential decayrate of Michel decay is given by [14]√d2 Γ(µ+ → e+ νe ν µ )mµ4 2x2 − x20=
WG
eµF
dxd(cos θ)
4π 3
×(FIS (x) + Pµ cos θe FAS (x))ˆ×(1 + P~e (x, θe · ξ)),(1.1)

where Weµ = (m2µ + m2e )/(2me ), x = Ee /Weµ and x0 = me /Weµ . Here Ee is the energyof the e+ and me and mµ are the masses of electron(positron) and muon, respectively.
Allowed region of x is from x0 to 1 by its definition. θe is the angle between the muonpolarization (P~µ ) and the positron momentum, and ξˆ is the directional vector of themeasurement of the positron spin polarization. FIS (x) and FAS (x) are the isotropic andanisotropic parts of the positron energy spectrum, respectively and they are given by [15]2
FIS (x) = x(1 − x) + ρ(4x2 − 3x − x20 ) + ηx0 (1 − x),
9(
)
√
√
1
2
2
2
2
FAS (x) =ξ x − x0 1 − x + δ[4x − 3 + ( 1 − x0 − 1)] ,
3
3

(1.2)
(1.3)

where ρ, η, ξ and δ are called Michel parameters. In the SM, Michel parameters arecalculated to be:
3
3
(1.4)
ρ = , η = 0, ξ = 1, δ = .
4
4

1.1.2

The µ+ → e+ γ Decay in the Standard Model

The µ+ → e+ γ decay is strictly forbidden in the SM because of the law of lepton flavor
conservation. If the neutrino oscillation is considered, the µ+ → e+ γ decay can happen
through the Feynman diagram shown in Fig. 1.1. However, the calculated branching
fraction is approximately 10−50 and almost undetectable with any existing experimental
technologies. It means that if the µ+ → e+ γ decay would be discovered, it should be theevidence of the BSM.

18

1.2. THE µ+ → E+ γ DECAY AND PHYSICS BEYOND THE STANDARD MODEL

W+

µ+

νµ

γ

νe

e+

Figure 1.1: Feynman diagram for the µ+ → e+ γ decay through the neutrino oscillation.

1.2

The µ+ → e+γ decay and Physics Beyond the Standard Model

Many BSMs predict that the µ+ → e+ γ decay can happen in a detectable range of the
branching fraction. Here several BSMs that can enhance the branching fraction of theµ+ → e+ γ decay are introduced. The eﬀective Lagrangian for the µ+ → e+ γ decay isgiven by [14]4GF
Lµ→eγ = − √ [mµ AR µR σ µν eL Fµν + mµ AL µL σ µν eR Fµν + H.c.].
2

(1.5)

where the term of GF is the Fermi coupling constant, AR and AL are coupling constants
that correspond to the left-handed process and right-handed one, respectively. From
Eq. (1.5), the diﬀerential angular distribution of µ+ → e+ γ decay is given bydB(µ+ → e+ γ)= 192π 2 [|AR |2 (1 − Pµ cos θe ) + |AL |2 (1 + cos θe ))],d(cos θe )(1.6)

where θe is the angle between the muon polarization and the e+ momentum vectors.
The supersymmetric (SUSY) model is one of the candidates of the BSMs, which cananswer several issues in the SM. In the SUSY, new gauge symmetry called “supersymmetry” is introduced as an extension of the SM. As a result, all of elementary particles in the
SM have their own supersymmetric partners called “sparticles”. Table 1.2 shows all particles and sparticles which are contained in the minimal supersymmetric standard model(MSSM). If the supersymmetry is not broken, each sparticle should have the same massof its own supersymmetric partner. However, such particles have not been discovered yet.
Therefore the supersymmetry is thought to be broken. In the MSSM, the diﬀerent masses
between sparticles and their own partners are generated by the soft SUSY-breaking terms.
As a result, the branching fractions of CLFV processes are enhanced by the flavor mixing
in the slepton sector.

1.2.1

SUSY GUT SU(5)The SUSY grand unified model (SUSY GUT) is one of attractive models of the SUSY
extensions, because three gauge coupling constants of strong, weak, and electromagnetic
19

1.2. THE µ+ → E+ γ DECAY AND PHYSICS BEYOND THE STANDARD MODEL

Table 1.2: All particles and sparticles in the MSSM.
Names
squarks,qquarksu(×3 families)dsleptons,leptons l(×3 families)e
Higgs,higgsinos Hu
Hd
Names
gluino, gluon
winos, W bosons
bino, B bosonµ̃Rspin 0spin 1/2(ũL d˜L )(uL dL )
∗
ũR
u†R
d˜∗R
d†R
(ν̃L ẽL )(ν eL )∗ẽR
e†R
(Hu+ Hu0 ) (H̃u+ H̃u0 )(Hd0 Hd− ) (H̃d0 H̃d− )spin 1/2
spin 1
g̃
g
±
0
±
W̃ W̃
W W0
B̃ 0
B0

SU (3)C , SU (2)L , U (1)Y(3, 2, 1/6)
(3, 1, −2/3)(3, 1, 1/3)
(1, 2, −1/2)(1, 1, 1)(1, 2, 1/2)
(1, 2, −1/2)
SU (3)C , SU (2)L , U (1)Y(8, 1, 0)(1, 3, 0)(1, 1, 0)

τ̃R

ẽR

µ̃R
µ

χ̃0

µ

e

ẽR
χ̃0

e

Figure 1.2: Feynman diagrams for the µ+ → e+ γ decay in SU(5) SUSY GUT.

interactions are unified to a single SU (5) gauge coupling constant at a scale of the orderof 1016 GeV (called GUT scale) in the SUSY GUT. The oﬀ-diagonal elements of theright-handed slepton mass matrix in the SUSY GUT are given by
(
)
3
MP
2
∗
33 2 22(mẽR )ij = − 2 (VR )i3 (VR )j3 |yu | m0 (3 + |A0 |) ln
.
(1.7)
8π
MG
where VR is a matrix element to diagonalize the Yukawa coupling constant for leptons,
MP and MG represent the reduced Planck mass (∼ 2 × 1018 GeV) and the GUT scale(∼ 2 × 1016 GeV), respectively. Here m0 is the universal scalar mass and A0 is theuniversal trilinear coupling. As a result, Eq. (1.7) becomes a source of the µ+ → e+ γ
decay. Figure 1.2 shows possible Feynman diagrams for the µ+ → e+ γ decay in SUSY
GUT. In the SU (5) SUSY GUT, CLFV processes appear only in the right-handed slepton
sector because of moderate values of tan β, which is defined by the ratio of two Higgs
vacuum expectation values (tan β ≡ hH20 i/hH10 i). Therefore only µ+ → e+
R γ decay isallowed in this model.

201.2. THE µ+ → E+ γ DECAY AND PHYSICS BEYOND THE STANDARD MODEL

Figure 1.3: Correlation between B(µ → eγ) and B(τ → µγ) as a function of mN 3 , forconstraint MSSM assuming universality of the soft-SUSY breaking at the scale of gauge
coupling unification, MX so-called minimal supergravity (mSUGRA) framework from◦◦◦◦[16]. Diﬀerent colored regions correspond to θ13 = 1 , 3 , 5 and 10 (red, green, blue and◦pink, respectively) and recent experiments report the large value of θ13 ∼ 10 [17]. Dashedline labeled “MEG2011” is added to the original plot and it represents the 90% C.L. upper
limit given by the previous result of MEG [4].

1.2.2

SUSY Seesaw

In the SUSY seesaw model, right-handed heavy Majorana neutrinos are introduced to
explain the tiny masses of neutrinos by the seesaw mechanism. By including the seesaw
mechanism in the SUSY standard model, the right-handed neutrino supermultiplets,
the Majorana mass matrix and a new Yukawa coupling constant matrix are includedin the part of the lepton sector Lagrangian. Because of the presence of the Yukawa
coupling constant matrix for the neutrino sector as same as the charged lepton sector,
the lepton flavor would no longer be conserved separately for each generation in the
SUSY seesaw. Hence the branching fractions of CLFV processes are expected to beenhanced. If we assume that the neutrino mixing mostly originates from the neutrino
Yukawa coupling constants, (yν )ij , the branching ratios from µ+ → e+ γ and τ → µγ
decays can be evaluated by using the neutrino mixing parameters. Figure 1.3 shows thescatter plot of the correlation between B(µ → eγ) and B(τ → µγ) in an example of the
SUSY model which including the large mass of right-handed Majorana neutrino. In thismodel, the µ+ → e+ γ decay can happen in somewhere experimentally achievable range.

211.3. SEARCHES FOR THE CLFV PROCESSES

1.2.3

The µ+ → e+ γ Decay and (g − 2)µ Anomaly

In 2004, the E821 experiment reported about the muon anomalous magnetic moment(aµ = (g − 2)µ /2) [18]. From their result, the average of the muon magnetic moment isgiven as:aµ = 11 659 208(6) × 10−10 (0.5 ppm),(1.8)
while the theoretical calculation with the SM is given as [19]:aSM = 11 659 184(6) × 10−10 (0.7 ppm).

(1.9)

The discrepancy between aµ and aSM is calculated to be 2.9σ.
Since both the µ+ → e+ γ decay and the muon magnetic moment are generated bydipole operators as shown in Fig. 1.4, the branching ratio of the µ+ → e+ γ decay correlates
to the value of ∆aµ = (gµ − gµSM )/2 if ∆aµ is generated by the contribution from the newphysics [20]. For example, the B(li → lj γ) can be expressed by using ∆aµ as)()2 (f2c (M22 /Ml̃2 , µ2 /Ml̃2 )B(li → lj γ)
48π 3 α ∆aµ
ij 2
|δLL
|,
(1.10)=
×
B(li → lj νli νlj )G2F
m2µg2c (M22 /Ml̃2 , µ2 /Ml̃2 )where µ is the supersymmetric-invariant mass term of the Higgs potential, Ml̃ is a massof left-handed slepton, , f2c (x, y) and g2c (x, y) are defined as−x2 − 4x + 5 + 2(2x + 1) ln x −y 2 − 4y + 5 + 2(2y + 1) ln y−,2(1 − x)42(1 − y)4
(3 − 4x + x2 + 2 log x) (3 − 4y + y 2 + 2 log y)
g2c (x, y) =−,(1.11)(x − 1)3(y − 1)3f2c (x, y) =ij
in the MSSM extension with the seesaw mechanism. Here δLL
is the left-handed slepton
mass matrices read asijδLL= cν (yν† yν )ij ,(1.12)

where cν is a numerical coeﬃcient. Figure 1.5 shows correlation between the B(µ+ → e+ γ)and ∆aµ including the non-linear eﬀect not shown in Eq. (1.10). Therefore if the reported(g − 2)µ anomaly is really exist, the branching ratio of the µ+ → e+ γ decay might beenhanced. Otherwise, there should be some mechanism which can suppress the lepton
flavor violation. Consequently, both the measurement of the muon magnetic moment andthe µ+ → e+ γ decay search play the important roles to find out the unknown mechanismof new physics.

1.3

Searches for the CLFV Processes

1.3.1

Experimental History of the µ+ → e+ γ Decay Searches

Recent results from experimental searches for the µ+ → e+ γ decay are shown in Table 1.3.
Before starting the MEG experiment, the best upper limit on the µ+ → e+ γ decay was set
to be 1.2×10−11 by the MEGA experiment. The MEG experiment already achieved more
stringent upper limit by using 2009–2010 combined dataset in 2011. As already discussed,
the B(µ+ → e+ γ) is expected to be within 10−11 –10−15 in several BSMs. According to
the upper limit given by the previous result in the MEG experiment, we already achieved
the sensitivity which is able to explore the new physics.
22

1.3. SEARCHES FOR THE CLFV PROCESSES

γ

µ

γ

µ

e

µ

Figure 1.4: Simplified Feynman diagram of the µ+ → e+ γ decay (left) and the muon
magnetic moment (right).

Figure 1.5: Correlation between B(µ+ → e+ γ) and ∆aµ from [20]. Horizontal dashed line
labeled “MEG2011” is added to the original plot and it represents the 90% C.L. upper
limit given by the previous result of the MEG experiment. Vertical two dashed lines arealso added and they represent the 1σ region of the (g − 2)µ anomaly measured in the
1223| = 10−4 , |δLL
| = 10−2 .
E821 experiment [18]. Assumed parameters are as follows;|δLL
Here the relatively small slepton mass and large tan β are assumed. The inner red areasatisfy the B-physics constraints.

23

1.3. SEARCHES FOR THE CLFV PROCESSES

Table 1.3: History of the µ+ → e+ γ search experiments in the era of meson factories
with 90% C.L. upper limits. The resolutions are given as full width at half maximum(FWHM).
Year

Site

1977
SIN
1977 TRIUMF
1979 LAMPF
1986 LAMPF
1999 LAMPF
2011
PSI

1.3.2

Beam Rate(Hz)
5 × 105 µ+
2 × 105 π +2.4 × 106 µ+
4 × 105 µ+
1.3 × 107 µ+
3 × 107 µ+

∆Ee
(%)
10
8.7
8.8
8
1.2
1.6

∆Eγ
(%)
8.7
9.3
8
8
4.5
4.7

∆teγ
(ns)
6.7
1.4
1.9
1.8
1.6
0.35

∆Θeγ
(mrad)

37
87
15
30

Upper Limit
90% C.L.
1.0 × 10−9
3.6 × 10−9
1.7 × 10−10
4.9 × 10−11
1.2 × 10−11
2.4 × 10−12

Ref.
[22]
[23]
[24]
[12]
[2]
[4]

BackgroundTwo diﬀerent types of background sources are considered in the search for the µ+ → e+ γ
decay. First one is the radiative muon decay (RMD) which is called “prompt background”.
In RMD, a muon decays into two neutrinos, a positron and a gamma ray. Since there
are two invisible particles in the final state, it can be misidentified as a signal event if
both a positron and a gamma ray have high energy close to 52.8 MeV. The other isthe accidental background (BG) which is the most dominant one in recent µ+ → e+ γ
search experiments. In the accidental background, a gamma from RMD, annihilation inflight (AIF) of a positron, or bremsstrahlung of a positron accidentally overlaps with apositron from Michel decay. In the MEG experiment, RMD and AIF are main sourcesof the background gamma ray and the fractions are found to be nearly equivalent froma simulation study. Here the details of each background source are explained and thebackground situation in the MEG experiment is discussed.
1.3.2.1

Prompt BackgroundThe decay of RMD can be written as µ+ → e+ νe ν µ γ. If two neutrinos carry only a◦small amount of energy, the opening angle between e+ and γ ray becomes close to 180 .
Moreover, the e+ and the γ have approximately the same energy as in signal event in thiscase. The decay width of RMD, dΓ(µ → eννγ) is given bydΓ(µ → eννγ) =G2F m5µ α
8 4
3×
[2π

(
)
]
1 2
× (1 − x) (1 − Pµ cos θe ) + 4(1 − x)(1 − y) − z (1 + Pµ cos θe )2
×dxdyzdzd(cos θe ),(1.13)
2

where θe is the opening angle between the muon spin and the positron direction, x =2Ee /mµ , y = 2Eγ /mµ , and z is determined by the opening angle between e+ and γ ray

24

1.3. SEARCHES FOR THE CLFV PROCESSES(Θeγ ) as z = π − Θeγ , The diﬀerential decay width of RMD is given by∫ 1∫ 1∫ min[δz,2√(1−x)(1−y)]dΓ(µ → eννγ)
1dB(µ → eννγ) =
dx
dy
dz
,
Γ(µ → eνν) 1−δxdxdydz1−δy0α=[J1 · (1 − Pµ cos θe ) + J2 · (1 + Pµ cos θe )]d(cos θe ),(1.14)
16π
( )2
( )4
( )8
8
δz
δz
δz
1 1
3
2
J1 =
(δx) (δy)− 2(δx)+,(1.15)
232
2
3 (δy)
2
( )6
( )2
( )4
8 δz
δz
δz
2
2
J2 = 8(δx) (δy)− 8(δx)(δy)+,(1.16)
22
3 2
where δx, δy and δz are half-widths of the signal region for x, y and z, respectively.
Here Γ(µ → eνν) is the total√ muon decay width and Eq. (1.15) and Eq. (1.16) are determined by assuming δz ≤ 2 δxδy and this assumption fits into the situation of the MEG
experiment. In case fully depolarized muon, Eq. (1.14) read asαdB(µ → eννγ) =[J1 + J2 ].
(1.17)
8π
For example, we can calculate the eﬀective branching ratio with Eq. (1.17) in the situationof the MEGA experiment by using numerical numbers written in Table 1.3 by keeping
signal eﬃciency to be 90% as;δx = 0.0084, δy = 0.032, δz = 0.021, δteγ = 1.12 ns.

(1.18)

By using these numbers, the branching ratio of the prompt background is given as,
B(µ → eννγ) ∼ 4.4 × 10−15 .

(1.19)

Therefore the prompt background is negligible in recent µ+ → e+ γ search experiments.
1.3.2.2

Accidental Background

The accidental background is the most dominant background to search for the µ+ → e+ γ
decay in the recent experiments including MEG. The eﬀective branching fraction (Bacc )is given bydωeγ
Bacc = Rµ · fe0 · fγ0 · () · (2δteγ ),(1.20)
4π
where Rµ is the instantaneous muon intensity, the back-to-back resolution (dωeγ /4π) is
given by (dωeγ /4π) = (dz)2 /4, fe0 and fγ0 are integrated spectra of background gamma
rays and positrons within the signal region, respectively. Therefore the excellent angularand gamma energy measurements are required to suppress the accidental background.
The factor fe0 can be approximately estimated to be ≈ 2(δx) by integrating Michel
spectrum over 1 − δx ≤ x ≤ 1 since it is almost flat at x ' 1 as shown in Fig. 1.6(a).
In order to make a situation simpler, fγ0 is assumed to be dominated by the RMD eventsand muons not to be polarized. In this case, fγ0 is given by∫ 1∫
dB(µ → eννγ)0dy d(cos θγ )fγ =dyd cos θγ
1−δy
(α)
≈
(δy)2 [ln(δy) + 7.33].
(1.21)
2π25Differential Branching Ratio

Differential Branching Ratio

1.3. SEARCHES FOR THE CLFV PROCESSES

1
0.8
0.6
0.4
0.2
0
0

1
10

-1

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

0.2

10-7
0

0.4
0.6
0.8
1
Normalized Positron Energy (x)

(a)

0.2

0.4
0.6
0.8
1
Normalized Gamma Energy (y)(b)

Figure 1.6: Left figure shows positron energy spectrum of unpolarized µ+ → e+ νe ν µ decay(Michel spectrum). A radiative decay correction [25] is included. Right one shows gamma
Energy spectrum of unpolarized µ+ → e+ νe ν µ γ decay. Positron energy and the angle
between a positron and a gamma are integrated.

From Eq. (1.20), the branching fraction of the accidental backgrounds is proportional toRµ .
A gamma ray from annihilation-in-flight (AIF) of a positron is other possible source of
the accidental background. The energy spectrum and the production rate of the gamma
ray from AIF depends on the materials inside the tracking volume. The rate of the AIF
background is also proportional to the instant muon rate (Rµ ) since almost all positrons
originate from Michel decay of muons.
By way of example, the branching ratio of the accidental background is calculated byusing resolution parameters given by the MEGA experiment. In the calculation, it has tobe considered that pulsed muon beam was used in the MEGA experiment. The instantbeam intensity was 2.6 × 108 in MEGA. The eﬀective branching ratio is given as
Bacc ∼ 2.4 × 10−12 .

(1.22)

Therefore the accidental background was dominant in their experiment and it could bea serious problem on searching for the µ+ → e+ γ decay with higher sensitivity of 10−13level.
1.3.2.3

Experimental Requirement for the µ+ → e+ γ Search

As discussed in above sections, it is the most important to reduce the accidental background to achieve the sensitivity below 10−12 . Since the background rate is proportional
to an instant muon rate, direct-current (DC) muon beam is strongly preferred in orderto suppress the accidental background while keeping the total statistics high enough.
Since the incident rate of the AIF background depends on the materials along thepositron trajectories, namely the target and the tracking devices, the total amount ofmaterials inside the tracking volume should be as small as possible and they should be26

1.3. SEARCHES FOR THE CLFV PROCESSES

made of materials which have small Z in order to reduce the AIF background. As shownin Fig. 1.6(a), high background rate is expected around the signal region of positrons.
Therefore the high rate capability is required for the tracking detector.
Figure 1.6(b) shows the background gamma spectrum from the RMD events. As
shown in this figure, background rate is highly suppressed if the gamma energy can bemeasured very precisely. In MEGA case, they used the pairs of a positron and an electron,
which are generated by the gamma ray conversion, in order to measure the energy of the
gamma ray. This method helps to achieve the 4.5% precise energy measurement in
FWHM, however it causes the trade-oﬀ eﬀect of the low detection eﬃciency which wasmeasured to be 2.4% in MEGA. Consequently, the single event sensitivity was limited intheir experiment. In order to achieve the sensitivity less than 10−12 , the same level ofthe energy resolution for gamma rays as MEGA with much higher detection eﬃciency isrequired for the gamma ray detector.

1.3.3

Comparison with Other CLFV Searches

Here some of other CLFV processes and related experimental searches are described
together with the associated models of new physics for comparisons. The µ+ → e+ e+ e−
decay and the µ− − e− coherent conversion are promising channels to search for the BSM
as same as the µ → eγ decay by using muons. As one can easily imagine, the branching
fraction of those two processes are strongly correlated to that of the µ+ → e+ γ decay.
In case of the µ+ → e+ γ decay, contributions for the eﬀective Lagrangian are dominatedby photon-penguin diagrams as shown in Fig. 1.2. On the other hand, those of theµ+ → e+ e+ e− decay and the µ− − e− conversion are enhanced by mediating the directfour-fermion interactions and the contributions from the photonic interactions of theseprocesses are relatively small due to an additional coupling between a gamma line and afermion line in the Feynman diagrams. Here each of muon CLFV processes are briefly
discussed together with the relation to the µ+ → e+ γ decay and the experimental aspects.
1.3.3.1µ+ → e+ e+ e− decay

If only photon-penguin diagrams contribute to the µ+ → e+ e+ e− decay, a model-independent
branching ratio of µ+ → e+ e+ e− can be derived by using the branching ratio of µ+ → e+ γ,
as follows [14]:[]m2µ 11αB(µ+ → e+ e+ e− )'ln 2 −= 0.006.
(1.23)
B(µ+ → e+ γ)
3πme4The signal event can be well identified by using following two requirements:∑∑• The conservation of momentum (| i p~i | = 0) and energy ( i Ei = mµ ),• Timing coincidence between two positrons and one electron.
For the µ+ → e+ e+ e− decay search, two main background sources are considered. One
is a prompt background; µ → e+ e+ e− νe ν µ , which is allowed in the SM and can be a fakeof the signal event, when two neutrinos have tiny energies. The branching ratio of thisdecay is (3.4 ± 0.4) × 10−5 [9]. The other background is an accidental coincidence ofan e+ from normal muon decay with an uncorrelated e+ e− pair, for example, produced27

1.3. SEARCHES FOR THE CLFV PROCESSES

from Bhabha scattering of e+ . The former one can be suppressed by measuring the totalenergy of all three charged particles, and latter can be reduced by measuring the relative
angles and the timing between three charged particles.
The most stringent upper limit on the µ+ → e+ e+ e− decay search is given as 1.0×10−12(90% C.L.) by the SINDRUM experiment [11] as shown in Table 1.1 and no experiments
are running to search for the µ+ → e+ e+ e− up to now.
One experiment, aiming to search for the µ+ → e+ e+ e− decay with 104 times higher
sensitivity than the present upper limit, was proposed in PSI [26]. They proposed astaged approach in the sensitivity to BR∼ 10−15 in phase I and further improvements
to BR∼ 10−16 at later stages if a new high-intensity muon beam (HiMB), which is a
future planned beamline at PSI, is realized. If the photonic contributions are dominant,the current upper limit corresponds to the 10−10 order of magnitude of that for theµ+ → e+ γ search and the goal sensitivity of proposed new experiment corresponds to10−14 level.
1.3.3.2µ− − e− Conversion

In the SM, a negative muon, which is captured by an atom and forms muonic atom, caneither decay in an orbit (µ− → e− νµ ν e ) or is captured by a nucleus of mass number A
and atomic number Z, namely,µ− + (A, Z) → νµ + (A, Z − 1).

(1.24)

However, the µ− − e− conversion in a muonic atom, such asµ− + (A, Z) → e− + (A, Z),(1.25)

is expected to occur in the BSM, which violates the conservation of the lepton flavor numbers as same as in the µ+ → e+ γ decay. From here, the µ− − e− conversion is expressedas µ− N → e− N . Similar to Eq. (1.23), the ratio of B(µ+ → e+ γ)/B(µ− N → e− N ) is
given by [14]G2F m4µB(µ− N → e− N )B(A, Z)
12
=
×
3
×
10
B(A,
Z)
∼
,
B(µ+ → e+ γ)
96π 3 α428(1.26)

where only the photonic contributions are assumed and B(A, Z) represents the factor
depending on the mass number (A) and the atomic number (Z) of the target nucleus.
For example, for the titanium nucleus, the values of B(A, Z) is calculated to be 1.8–
2.2, based on diﬀerent approximations. Therefore, the ratio of B(µ− N → e− N ) andB(µ+ → e+ γ) is calculated to be B(µ− N → e− N )/B(µ+ → e+ γ) ∼ 0.0042–0.0051. In themuonic atom, most muons transit to the ground state before decaying or being capturedby the nucleus. Therefore if µ− − e− would happen, the energy of emitted electron should
be monochromatic as
Eµe = mµ − Bµ ,(1.27)
where Bµ is the binding energy of the 1s muonic atom. Since the value of Bµ is diﬀerent
for various nuclei, the peak energy of the µ− − e− conversion signal also changes, forexample, Eµe = 104.3 MeV for titanium. Event signature is very simple that, only one
28

1.3. SEARCHES FOR THE CLFV PROCESSES

electron emitted with about 105 MeV energy. Thus µ− − e− search does not suﬀeredfrom the accidental combinatrial background, which would be the serious background forsearches of the µ+ → e+ γ and µ+ → e+ e+ e− decays. There is one physics backgroundcomes from the normal muon decay in orbit (DIO) of the muonic atom. The end-point
energy of the DIO corresponds the muon half mass (52.8 MeV) and it is small enoughcompared with the energy of the signal. Very small fraction of them are given the nuclear
recoil energy, but are still negligible for currently planned two experiments which are
introduced below. A radiative pion capture (π − + (A, Z) → (A, Z − 1) + γ) could be oneof the background sources. However, they can be suppressed by reducing the amount ofpions which reach the muon target by using long enough muon transport solenoid. Other
backgrounds could be caused by the decay in flight or interactions of particles (muon,
pion, (anti)proton) in a primary proton beam. Since the muonic atoms have lifetimes ofthe order of 1 µs, the beam origin backgrounds can be suppressed by using high-intensity
pulsed beam when the measurements are only done during the beam intervals. In thiscase, the purity of the pulse, which called “extinction”, is essential since the remaining
particles between two pulses could make the background.
The current upper limit was set by the SINDRUM-II experiment at 7×10−13 (90% C.L.)[21], which used the negative DC muon beam at the πE5 area in PSI.
In near future, two diﬀerent experiments are being prepared to search for the µ− − e−by using high-intensity pulsed beam at the diﬀerent places. One is the Mu2e experiment
at Fermilab [27] and the other is the COMET experiment at J-PARC [28]. The goal
sensitivities of both experiments are similar, at 10−16 level. Since the present upper limit
and the goal sensitivity are similar to those of µ+ → e+ e+ e− search, similar calculation
can be done for the µ− − e− conversion as well.
For both the µ+ → e+ e+ e− decay and the µ− − e− conversion searches, the branching
fractions could be expected to be enhanced if some other contributions are considered.
For example, in the little higgs model with T-parity, the lepton flavor violation can behappen through the additional mirror sector and the lepton flavor is not conserved any
more [29]. In this model, the ratios between diﬀerent three processes become comparable
as namely B(µ → eγ) ∼ B(µ → eee) ∼ B(µ − e). Accordingly, the more information
about the new physics can be obtained by comparing the results of diﬀerent three CLFV
modes.
1.3.3.3

CLFV in τ decay

Here, four CLFV decay modes of tau lepton are briefly described. The experimental
searches for CLFV processes of tau lepton channel are done by using those generated bycolliders since the tau lepton beam cannot be produced by present technologies becauseof its extremely shorter lifetime (0.29 ps) compared to that of muon. Current upperbounds are given by BABAR and Belle as shown in Table 1.4. In the near future, BelleII collaboration aiming to search for the tau LFV with higher intensity electron positron
collider. The LHCb experiment will also search for the tau on CLFV processes by using
the Large Hadron Collider.

29

1.3. SEARCHES FOR THE CLFV PROCESSES

Table 1.4: Recent the most stringent experimental limits given by the BABAR and Belleexperiments. Only four famous decay modes are shown here.
Mode
90% upper limit Experiment
B(τ → µγ)
4.4 × 10−8
BABAR[30]B(τ → eγ)
3.3 × 10−8
BABAR[30]−8B(τ → µµµ) 2.1 × 10
Belle[31]B(τ → eee) 2.7 × 10−8
Belle[31]30Chapter 2
The MEG Experiment
The MEG (Mu to Electron Gamma) experiment started physics data taking in 2008
at Paul Scherrer Institut (PSI) in Switzerland to search for the µ+ → e+ γ decay witha sensitivity goal of 10−13 . The experimental proposal was submitted to the science
committee of PSI in 1999 and the MEG collaboration was established in collaboratingwith 50–60 scientists from the institutes in Japan, Switzerland, Italy, Russia and UnitedStates. Dedicated detectors were constructed in order to achieve the sensitivity goal forthe µ+ → e+ γ search [32]. The MEG detector is placed in the πE5 area in which the world
most intense Direct-Current (DC) muon beam is provided by the proton ring cyclotron
at PSI.
The schematic views of the MEG experiment and the global experimental coordinate
is shown in Fig. 2.1 and Fig. 2.2. The center of the coordinate is set a origin at the centerof the magnet. Other coordinate expressions as follows are used as well:√r = x2 + y 2 , φ = tan−1 (y/x), θ = tan−1 (r/z), z = z,
(2.1)
with a same origin of the (x, y, z) coordinate. For gamma ray measurement, an innovative
detector using large volume liquid xenon (900 litters), which works as a total absorption
calorimeter, was developed and scintillation photons are read by surrounding 846 Photo
Multiplier Tubes (PMTs) to determine the timing, the position and the energy of incident
gamma rays. In order to reconstruct the track of positrons, a set of drift chambers builtwith very low mass materials are used inside a superconducting magnet which has a
specially designed gradient magnetic field. At each end of the magnet warm bore outside
the drift chambers, 15 plastic scintillating bars are placed to measure the impact timeof positrons in several tens ps of precision. For the data taking, we use fast waveform
digitizers together with the trigger system consists of Flash Analog-to-Digital Converters(FADC) and a Field Programmable Gate Array (FPGA) system.
This chapter describes the details of the experimental apparatus, the analysis method,
and the run conditions in 2009, 2010 and 2011 data taking.

2.1

Beamline

In order to get a high intensity DC muon beam, the proton ring cyclotron located at PSI isa unique choice for the MEG experiment, because it can provide the world most powerful312.1. BEAMLINE

Figure 2.1: Top and side view of all the experimental apparatus with coordination.

Figure 2.2: 3D view of all the experimental apparatus with coordination.

32

2.1. BEAMLINE

proton beam up to 2.2 mA (1.3 MW) with a 590 MeV energy [33]. The frequency of the
proton beam is 50.6 MHz and the width of the bunch is 0.3 ns.

Figure 2.3: 590 MeV proton ring cyclotron at PSI.

2.1.1

Muon Beamline

In order to avoid muons to form the muonic atoms with materials inside the target, a
positive muon (µ+ ) is adopted for the experiment. At the πE5 beamline, a positive muon
beam is delivered from the main proton ring cyclotron.
In the upstream of the πE5 beamline, muons only from pions decaying at the surfaceof the pion production target (E-target), are collected (so-called “surface muon”) andthey are delivered to the MEG detector as shown in Fig. 2.4.
Therefore, almost all muons are fully polarized and have almost same momenta of
28 MeV/c with 5–7% of spread in FWHM (Full Width at Half Maximum). Because of
the smaller momentum spread, high purity and high intensity muons are selectable. A
large contamination of positrons are separated by using both electric and magnetic fields
in a Wien filter with a 8.1σ separation. The low momentum of the surface muon allows
to use the thin target to stop muons, which is important to suppress the production of
the background gamma rays inside the target. Furthermore, easier modifications of thebeam size and the transportation are possible because of its small momentum spread.
Since the frequency of the proton beam is high enough (50.6 MHz) compared with the
decay time of pions (τπ± ≈ 26 ns) and that of muons (τµ ≈ 2.2 µs), the muon intensity isalmost continuous. At the background reduction point of view, the continuous beam isquite important as already explained in Sec. 1.3.2.2. The stopping rate at the target istuned to be 3.0 × 107 µ+ /s at a 2.2 mA of proton current.

2.1.2

Beam Transport Solenoid

The superconducting beam transporting solenoid (BTS) is placed to connect between theπE5 beamline and the detector part. Since the BTS is directly couple to the beamline,
the inside of the BTS is evacuated. In order to avoid the distortion of the magnetic field,
33

2.2. GAMMA DETECTOR

Figure 2.4: Schematic view of the πE5 beam channel including the beam transport systemand detectors.

the BTS consists of iron-free cryogenics and magnets. At the central part of the BTS, a
300 µm thick Mylar R degrader is placed to reduce the momentum of muons so that mostof them can be stopped in the thin target at the center of the detector magnet with aminimum multiple scattering contribution to the beam. In order to focus the muon beamat the degrader position and inside the detector, a 0.36 T magnetic field is generated byusing the double-layered iron-free coil.

2.1.3

TargetIn order to get high stopping power of muons and to reduce the materials which cangenerate the background, the target is implemented as a 205 µm thick sheet of a lowdensity layered-structure of a polyethylene and polyester with an elliptical shape andplaced on the center of the superconducting magnet with a 20◦ slant angle along the beamaxis. The length of semi-major and semi-minor axes are 10 cm and 4 cm respectively.
There are six holes on the target in order to perform the software target alignment byanalyzing the data as shown in Fig. 2.5(a). The target is placed at the almost center of
the COBRA magnet (See Fig. 2.5(b)). At the target, the muon polarization is measured
to be 89 ± 4% by using the angle distribution of Michel positrons and it is confirmed byanalyzing RMD events [34].

2.2

Gamma Detector

With a gamma detection part, we need to detect the signal-like high energy gamma rays
with a high detection eﬃciency. On the other hand, the background contamination mustbe highly suppressed. In order to realize such a kind of detector, following three points,
1. to reconstruct Eγ with an excellent resolution,
2. to measure the first conversion time of gamma rays precisely,
3. to reconstruct the first conversion point with a mm precision,34

2.2. GAMMA DETECTOR

(a)

(b)

Figure 2.5: Muon target (a) and installed view inside the COBRA magnet (b).

Table 2.1: Several characteristics of LXe in comparison with other scintillators.

3

Density (g/cm )Radiation length (cm)Moliere radius (cm)Decay time (ns)Wavelength (nm)
Relative light yield

LXe LAr
2.98 1.40
2.77
14
4.2
7.2
45 1620178 127
75
90

NaI(Tl)
3.67
2.59
4.13
230
410
100

CsI(Tl)
4.51
1.86
3.57
1300
560
165

BGO
7.40
1.12
2.23
300
480
21

are needed to be satisfied for the gamma detector. In the MEG experiment, a Liquid
Xenon (LXe) detector is adopted as the gamma detector to fulfill those three requirements. In this section, several properties of the LXe and the principle of the LXe detector
are described.

2.2.1

Liquid Xenon

Since the LXe can realize non-segmented volume, the detector response can be more
uniform than that of the detector based on segmented crystal. The faster decay timeof the LXe helps to reduce the pileup probability eﬀectively and provides good time
resolution. Because of its short radiation length and high density, high detection eﬃciency
is achievable. The characteristics of LXe are summarized in Table 2.1. Even though LXe
has feasible scintillation properties to measure the energy of the gamma ray, there aremainly three diﬃculties to overcome as follows:
1. Low and the narrow operational temperature range between 161.4–165.1 K in orderto keep the xenon in the liquid phase [7]352.2. GAMMA DETECTOR

Figure 2.6: Schematic views of the LXe detector.

2. Short wavelength of the scintillation light (178±14 nm, peak±FWHM) [35]3. High purity of the liquid xenon to avoid absorption of the scintillation light bycontamination
In order to develop the LXe detector satisfying these requirements, R&D works, performance check using prototype detectors, design and construction of final detector weredone [36].

2.2.2

Detector Design

Figure 2.6 shows schematic views of the LXe detector. The C-shaped structure is suitable
for the outer radius of the magnet. The radii of the inner and outer faces of the active
volume are 67.85 cm and 106.35 cm respectively. The 38.5 cm depth of the LXe detector
is corresponding to 14 X0 . The angular ranges for the the gamma rays from the target◦◦◦◦
are −60 < φ < +60 and 60 < θ < 120 .
A cryostat was constructed with a vacuum layer to keep xenon in liquid phase (SeeFig. 2.7). In order to minimize the energy deposition of incident gamma rays before
reaching the inner face of the detector, the inner face of the cryostat consists of analuminum honeycomb panel with inner vessel carbon fiber and the stainless steel partof the window in thickness of only 0.4 mm. The outer vessel of the cryostat is made of
0.7 mm-thick stainless steel. A high pressure tolerance of the cryostat is up to 0.5 MPa
while total thickness of the entrance window is only 0.075 X0 .
The inner vessel of the cryostat is filled with 900 litters of LXe. In order to monitor
the detector condition, temperature, pressure and liquid level sensors are installed insidethe detector. A custom-designed 200 W pulse-tube cryocooler [37] is installed on the
top of the detector to maintain the LXe temperature. A circulation liquid purification
36

2.3. POSITRON SPECTROMETER

Figure 2.8: LXe detector inside with PMTs installed on all surfaces.
Figure 2.7: LXe detector before
closing a flange of the outer vacuum
vessel.

system with the molecular sieves (MS13A) and a cryogenic fluid pump was developed toeﬃciently remove H2 O from LXe.
All sides of the LXe active volumes are covered by 846 2-inch PMTs as shown in
Fig. 2.8. Since the wavelength of LXe is in the region of the Vacuum Ultra Violet (VUV)
and shorter than that of standard crystal scintillators, the VUV-sensitive PMT (R9869)
was developed for the MEG experiment to obtain the high quantum eﬃciency (Q.E.) in
collaboration with HAMAMATSU Photonics [38]. Each PMT is equipped with a quartz
window that transmits VUV photons and a bialkali photo-cathode sensitive to VUVphotons. Aluminum strips are added to the surface of the cathode to reduce the sheet
resistance at low temperature. The average Q.E. is about 15% and the average gain is
about 1.8 × 106 at 850 V.
For a convenience of the analysis, a local coordinate (u, v, w) is defined as:
u = z
v = Rinner × tan−1 (y/x)√
w =x2 + y 2 − Rinner
where
Rinner = 67.85 cm,(2.2)

in the LXe detector.

2.3

Positron Spectrometer

For the MEG experiment, there are several requirements for the positron spectrometeras follows;372.3. POSITRON SPECTROMETER• operational in high rate environment,• to contain as small materials as possible,• to measure momentum, angular, vertex and timing at a very high precision.
In order to satisfy these requirements, the positron spectrometer is constituted by asuperconducting magnet with a graded magnetic field, 16 drift chamber modules made ofultra low mass materials to be used for the positron-track reconstruction, and the timing
counter for the impact time measurement of positrons. In this section, the details of eachpart are described.

2.3.1

COBRA Magnet

Figure 2.9: Concept of the COBRA magnetic field compared with the uniform field.
Positrons emitted around 90◦ are quickly swept away (c) while they stay longer in a
uniform field case (a). Positrons which emitted diﬀerent θ angles have same curvaturesin the COBRA magnetic field (d). On the other hand, they have diﬀerent curvatures ina uniform case (b).

The COBRA magnet is a superconducting magnet developed for the MEG experiment[39]. The field of the COBRA magnet is graded by five coils with three diﬀerent radii;
one central coil, two gradient coils and two end coils. The central part of the magnet wall
is as thin as 0.197 X0 in order to reduce the energy loss of gamma rays which enter the
LXe detector as much as possible.
Owing to the graded field, positrons emitted from the target are swept away quickly
and the trajectory of positrons with the same momenta have equal radii independentlyof their emission angles as shown in Fig. 2.9. This is the reason why the magnet is called38

2.3. POSITRON SPECTROMETER

Figure 2.10: Contour plot of absolute values of the gradient magnetic field generated by
the COBRA magnet

COBRA (COnstant Bending RAdius). The hit rate of positron inside the drift chambers
is significantly reduced because of those two reasons as shown in Fig. 2.11.
Figure 2.10 shows the field strength of the COBRA magnet. Since the responseof PMTs inside the LXe detector can be aﬀected by the magnetic field, the normal
conductive magnets are installed at the both sides of the superconducting magnet as acompensation coil to keep the field strength inside the LXe detector below 5×10−3 T as
shown in Fig. 2.12.
Total length of the COBRA magnet is ∼2.8 m along the beam axis and the minimum
radius at its center is ∼30 cm. The warm bore of the COBRA magnet is filled withalmost pure helium to reduce the background gamma rays and the multiple scattering of
positrons. Small amount of air ( 5%) is added to the helium gas to avoid the discharging
at the front-end part of the drift chambers.
The magnetic field was measured in 2006 with a commercial three-axis Hall probe. The
Hall sensors are mounted on a movable stage, which moves along z, r and φ. The precisionof the sensors themselves are 0.05% and the planar Hall eﬀect is less than 0.2%, the eﬀect
from the temperature coeﬃcient of the sensor is estimated to be less than 0.06%. We
also calculated the field by assuming an axi-symmetric coil model based on the measured
coil dimensions and the alignment with a finite element method. By comparing these two
field, it was found that there is larger discrepancy between the measured and calculated
field than the precision of the Hall sensors. We evaluated the eﬀect from the misalignmentof the Hall probes and then found that the angular displacement of Hall sensors is crucial
since the main strongest component Bz can contaminate the other field component, Br
and Bφ . Also the oﬀset can cause the systematic eﬀect. Therefore we firstly estimated theglobal oﬀset by comparing the measured Bz and the calculated one and by minimizing thediﬀerence between these two, the oﬀset is found to be (x0 , y0 , z0 ) = (+0.5, −3.0, −0.5) mm.
After the global oﬀset correction, Br and Bφ were calculated from the measured Bz using
the Maxwell equations. Subsequently, we checked the diﬀerence between the measured
and the calculated fields again, then it was found that the discrepancy became smaller.
This reconstructed magnetic field is used for the oﬃcial simulation and the analysis in
MEG.

39

2.3. POSITRON SPECTROMETER

Figure 2.11: Positron hit rate per 1 cm2 in each drift chamber module with a 3 × 107 Hz
muon stopping rate as a function of the radius. Green triangle markers show those withan uniform magnetic field and orange markers show those with the COBRA magnetic
field.

Stray field measured here

26.5mT

Photon detector(LXe)

11.5mT
5mT

Compensationcoil

60.8mT
0.14T

0.32T
1.7T

End coil
Gradient coil
Central coilFigure 2.12: Distribution of the magnetic field around the LXe detector. The PMTs of
the LXe detector are placed along the trapezoidal box shown in this figure.

40

2.3. POSITRON SPECTROMETER

Figure 2.13: Drift chamber modules installed inside the COBRA magnet.

2.3.2

Drift Chamber

In order to minimize the production of the background gamma rays by the interactionof the Michel positrons in the tracking volume, gaseous detector is suitable as a tracking
device because it can be constructed from lower Z materials than other devices such as
silicon trackers. We therefore developed a Drift CHamber system (DCH) which consistsof 16 independent drift chamber modules [40][41]. Drift chamber modules are placed at
the bottom center of the COBRA magnet with 10.5◦ intervals in φ direction and aligned
radially as shown in Fig. 2.13.
2.3.2.1

Drift Chamber Module

Each module is constructed with ultra low mass materials to reduce the multiple scattering of positrons and the production of background gamma rays inside the tracking
volume. Figure 2.14 shows schematic views of each module. As shown in Fig. 2.14 andFig. 2.15, each module consists of trapezoidal shaped two planes. Two planes are separated by thin cathode foils with 3.0 mm of gap and operated independently and eachwidth is 7.0 mm. Each plane has 9 anode wires and cells are divided to 9.0 mm by potential wires in between each anode wire. Field potential and drift lines of ionized electrons
are shown in Fig 2.16. Staggering of 2 layers by half of a cell helps to solve the left-rightambiguities as shown in Fig. 2.15.
Each module is filled with helium-ethane mixed gas in the ratio of 50:50 and thehelium based gas mixture helps to suppress the production of background gamma raysfrom annihilation in flight of positrons and reduce the multiple scattering of positrons412.3. POSITRON SPECTROMETER

(a)

(b)

Figure 2.14: Schematic side views of a drift chamber module. One plane has nine anodewires and all anode and potential wires are supported by a open shape frame as shown in(a). The module is covered by the cathode foil which has 5 cm periodical zig-zag shapecalled “vernier pattern” as shown in (b).

because of its small Z value. The open shaped support structure of the module is madeof carbon fiber in order to keep the tightness with the small Z mass material. Resistive
Ni-Cr wires are used for the anodes with a resistance per unit length of 2.2 kΩ/m anda 25 µm diameter. Potential wires are made of Be/Cu with a 2:98 ratio and 50 µm
diameter. Cathode foils have ‘zig-zag’ shaped 5.0 cm periodical pattern called ‘Vernier
pattern’ to improve the z resolution and is constructed by 12.5 µm-thick polyamide foilwith an aluminum deposition of 250 nm with a resistance per unit length of 50 Ω/m.
The details of the vernier method are described in Chap. 3 and in [42]. The total chargeof ionized electrons are amplified by the pre-amplifier boards, which are mounted at eachend of the support structure, with a total gain of ∼50.
In total, the mean of the radiation length for signal positrons inside the tracking regionis 2 × 10−3 X0 , which is small enough to reduce the multiple scattering of positrons for the
better tracking resolution and reduce the gamma ray background generated inside DCH.
Table 2.2 shows the summary of the characteristics and design parameters of DCH.

2.3.3

Timing CounterAs shown in Fig. 2.1, Timing Counters (TC) are placed at each end of the COBRA warm
bore outside the drift chambers in z-axis to measure the impact time of positrons. Eachbarrel-shaped sector consists of a longitudinal part (called TICP) and a transversed part(called TICZ) and covers the region from −150◦ to 10◦ in φ and from 28 cm to 109 cm
in |z|. The longitudinal part is composed of 15 scintillating bars (Saint-Gobain BC404)
to measure the timing with 10.5◦ gaps in the φ direction. Since the TC has to be operated inside the COBRA magnet, scintillation photons are collected by 2-inch fine-mesh42

2.3. POSITRON SPECTROMETER

y−Axis [cm]y−Axis [cm]Figure 2.15: Configuration of cells inside the chamber module. Two planes are divided
with a gap of 3 mm and the position of the sense wires are staggered. Green horizontal
lines show the cathode pads.

x−Axis [cm]x−Axis [cm](a) Contour plot of the potential(b) Drift lines with isochron map

Figure 2.16: Field map and drift lines calculated by GARFIELD simulation with the nominal
operated condition in MEG.

43

2.3. POSITRON SPECTROMETER

Table 2.2: Characteristics and Parameters of DCH.
16 modules aligned with 10.5◦ intervals in φ direction
2 layers staggered by a half of cell, 9 cells per layer
Chamber Dimension Trapezoidal shape
Rmin =193.38 mm, Rmax =278.88 mm (fiducial volume)
Zmin =385.66 mm, Zmax =827.31 mm (fiducial volume)
Cell Dimension
Rectangle cells divided by potential wires
9.0 mm×7.0 mm size, 4.5 mm wire spacing
Sense Wire
Materials : Ni/Cr (80:20)
Diameter : 25 µm
Tension : 50 gf
Resistance : 2200 Ω/m
Potential Wire
Materials : Be/Cu (2:98)
Diameter : 50 µm
Tension : 120 gf
Cathode Foil
12.5 µm-thick polyimide with aluminum deposition of 250 nm
Vernier Pads
4 pads per cell
Pattern period : 50.0 mm
Resistance : 47.5 Ω/mGas
He:C2 H6 = 50:50
Typical drift velocity : 4 cm/µs
Typical electron diﬀusion : 140 µm/cm at 1 kV/cm
Frame
Open shape, carbon fiber
Voltage
Sense Wire : + 1750–1850 V
Potential Wire and Cathode : GroundNumber of Channels 6 channels/cell × 9 cells × 2 layers × 16 modules = 1728 channels
Structure44

2.4. FRONT-END ELECTRONICS AND DATA ACQUISITION SYSTEM

PMTs which are operational in a high magnetic field. Figure 2.17(a) shows assembled
longitudinal sector and Fig. 2.17(b) shows the schematic cross views of scintillator bars.

(a) 15 timing counter bars placed in 10◦ inter- (b) Cross view of a timing counter bar with PMT
val.
on X-Y plane and along the z-axis.

Figure 2.17:
The transversed part consists of 128 curved scintillating fibers to measure the z position of positrons at the surface of the TC as shown in Fig. 2.18. Scintillation photons
are detected by 5×5 mm2 silicon Avalanche Photo-Diodes (APDs) which are attached atboth ends.

2.4

Front-end Electronics and Data Acquisition System

In order to get the precise charge estimation, an event-by-event baseline calculation isrequired to eliminate the influence of the baseline fluctuation due to the low frequency
noises. Furthermore, it is needed to separate the pileup events in time down to 10 ns.
Therefore we decided to use a fast waveform digitization for the Data AcQuisition system(DAQ) in MEG. All of data from each sub-detector are collected as waveforms by using
a fast waveform sampling device called Domino Ring Sampler (DRS) chip which is developed in PSI to satisfy the requirements for the MEG experiment. Events are triggeredby using Field Programmable Gate Array (FPGA) system to select the signal-like events
eﬃciently. The programmable trigger system also allows us to build the many kind oftriggers for the calibration purpose in a flexible way and it enables to modify the trigger
parameters easily.

2.4.1

DRS4

The latest version of DRS which is called DRS4 is the waveform sampler developed in
PSI for the fast waveform sampling. The basic idea of DRS is using analog Switched45

2.4. FRONT-END ELECTRONICS AND DATA ACQUISITION SYSTEM

Figure 2.18: Timing counter bars covered with fibers for z position measurement.

46

2.4. FRONT-END ELECTRONICS AND DATA ACQUISITION SYSTEM

Capacitor Arrays (SCA) and writing the data for each capacitor in high sampling speed.
There are two chains of capacitors which are connected each other as the analog sampling cells to write waveform data as a diﬀerential, which reduces the cross-talk between
diﬀerent channels, in each cell. This structure called domino wave circuit. Figure 2.19
shows the schematic view of the principle of DRS. Each chip has 8 channels and all channels connected to a single external 12-bit ADC (Analog-to-Digital Converter) by using a
multiplexer. This design helps to reduce the cost and space on the board. The sampling
speed is adjustable up to 5 GHz by modifying a control voltage Uspeed , which is fixed byusing a Phase-Locked Loop (PLL) to an external clock with high precision. Since the
LXe and TC detectors require precise time measurements, 1.6 GHz sampling frequency
was chosen. In the DCH, 0.8 GHz frequency was chosen because they need less timing
requirements than those for other detectors.

Inverter "Domino" Chain

Rotating
signalInput

OutputShift Register
Figure 2.19: Schematic of DRS principle.

2.4.2

Trigger

In order to collect the µ+ → e+ γ like events with high eﬃciency in a high backgroundcondition, we combined FADC and FPGA to construct the trigger. For physics data, thefollowing observables are used in the online trigger selection.
• Eγ selection,• teγ selection,• direction match selection (DM ).
Hereafter, this trigger is called “MEG trigger”.
In the Eγ selection, total charge of PMTs in the LXe detector is used with online
collection factor for each PMT to select the high energy gamma rays eﬀectively. The teγ
selection is done by calculating the time diﬀerence between tLXe and tTC . The position of
gamma ray is determined by the PMT which detects the largest amount of scintillation
photons and convert to the direction of the gamma ray emission assuming the centerof the target. Since the signals from DCH are collected in an order of µs time range,
they cannot be used for MEG trigger. Instead, the signals from the timing counter have47

2.4. FRONT-END ELECTRONICS AND DATA ACQUISITION SYSTEM

enough information for MEG trigger, because each sector of TC is segmented in the φdirection and there are correlations between φ position at TC and the emission angle ofpositrons with high momenta.
Many types of trigger channels are prepared for the calibration purposes for each subdetector as well. All of important triggers are summarized in Table 2.3 with their basesof logic.
Table 2.3: Various trigger settings used in the MEG experiment and the prescaling factors
in physics run. Approximately QL ∼ 30 MeV,QH ∼ 40 MeV,TN ∼ 20 ns and TW ∼ 40 ns.
DM represents the direction-match algorithm given by the trigger.
Id
0
1
2
3
4
5
6
7
8
9
10
12
14
16
18
19
22
31

Descriptionµ+ → e+ γµ+ → e+ γ, low Qµ+ → e+ γ, wide angleµ+ → e+ γ, wide timeµ+ → e+ νe ν µ γ /Dalitz decay in π 0µ+ → e+ νe ν µ γ wide /CW boron
π 0 with NaI (BGO)

Prsc.
1
50
500
200
1000

Logic
(QLXe > QH ) ∧ (|∆T | < TN ) ∧ (narrow DM )(QLXe > QL ) ∧ (|∆T | < TN ) ∧ (narrow DM )(QLXe > QL ) ∧ (|∆T | < TN ) ∧ (wider DM )(QLXe > QL ) ∧ (|∆T | < TW ) ∧ (narrow DM )(QLXe > QL ) ∧ (|∆T | < TN )-(QLXe > QL ) ∧ (|∆T | < TW )-(QLXe > QH )∧ NaI(BGO) time coincidence signal

π 0 w/o pre-shower
NaI (BGO) alone
LXe alone High Q
LXe alone Low Q /CW lithium / α
α selection
LED
Michel DC track+ TC hit
DCH track
Cosmic DCH
TC alone
Pedestal-(QLXe > QH )∧ pre-shower time coincidence signal
QN aI > Qthreshold
20000 QLXe > QH
QLXe > QL
22000 (QLXe > Q0 )∧ α selection
10
100 Hz pulse from LED module
DCH hits ∧ TIC hits
107
107
20000

DCH self
Trigger by Cosmic-ray counters
TIC self
Clock for pedestal

The trigger id 22 is used to calculate the normalization factor for the analysis bycollecting the Michel positrons. From now on, the trigger id 22 is called “Michel trigger”.
We estimate the eﬃciency of the positron spectrometer by using “Michel trigger” and thetrigger id 18 since the trigger id 18 is independent of the timing counter. The method of
the eﬃciency calculation is described in Sec. 5.2.6 in detail.
Figure 2.20 shows illustration view of the trigger system. The system is mainly divided
into 3 sectors. The first one is used to digitize the signals from each sub-detector withthe first pre-selection algorithm. The second layer receives data from the first layer andsend the data to the third one. The final board, which has an overall view of the event,48

2.5. SIMULATION AND ANALYSIS FRAMEWORK

Figure 2.20: Illustration of the trigger system.

generates the trigger signal.

2.4.3

MIDAS

The MIDAS (Maximum Integration Data Acquisition System) which is developed in PSI
and TRIUMF [43] is used as a DAQ system in the MEG experiment. The MIDAS is used
not only for controlling the DAQ but also for the recording data in special formatted filefor MIDAS, the detector monitoring and the alarm system. The MIDAS provides web
interfaces in order to enable the real time monitoring as well.

2.5

Simulation and Analysis Framework

Figure 2.21 shows the full chain of the simulation and the analysis framework developed
for the MEG experiment. Here we mention both the simulation and the analysis.

2.5.1

Simulation

In order to simulate the any kind of necessary events inside the MEG detectors precisely,
the simulation software called MEGMC was developed based on GEANT3 [44] which is Monte
Carlo (MC) simulation tool written in FORTRAN 77. The drift lines of electrons insideeach cell of the DCH are simulated by GARFIELD [45]. In the MEGMC, primary events canbe generated as shown in Table 2.4. Generated events are written in file with ZEBRA
format which was developed in CERN.
49

2.5. SIMULATION AND ANALYSIS FRAMEWORK

Figure 2.21: Illustration of the full programming chain for the analysis and the MC
simulation in MEG.

Table 2.4: List of some important events which are prepared in MEGMC.
Event Id
1
2
11
12
21
22
23
30
50
63
67
68
69

Descriptionµ+ → e+ γ signal event
Radiative Muon Decay (RMD) event
e+ at 52.8 MeV
e+ from Michel decay
γ at 52.8 MeV
γ from RMD
γ from e+ annihilation in flightµ+ beam
Cosmic-ray µ
α from 241 Am source on wire
π 0 → γγ decayp+ from CW accelerator + LiF target
p+ from CW accelerator + B target50

2.6. RUN STATUS

Since all of data are accumulated as waveforms in the physics run, waveform digitization part should be reproduced for the realistic simulation. Since the analysis frameworkof the MEG experiment is based on ROOT [46], it is required to generate from ZEBRA format file which contains information about the physics interaction inside each sub-detector
to ROOT format files which contain physics information and data as waveforms. The software called MEGBartender was developed based on C++ programming to simulate thewaveform digitization. The event mixing by assuming any rate can be also done in the
MEGBartender.

2.5.2

Analysis

As already mentioned, raw data from each sub-detector are stored as waveforms in the
MIDAS file. In order to get physical quantities from those raw data files, the analysis
software called MEGAnalyzer based on the Root based Object oriented Midas Extension(ROME) [47] developed at PSI is used. Details of waveform analysis and the event
reconstruction are explained in Chap. 3, and details of the physics analysis are describedin Chap. 6.

2.6

Run Status

This section describes the MEG data taking in 2009, 2010 and 2011, because we analyze
all the datasets with revised reconstruction and analysis methods. Several drift chamber
modules, which had large number of dead channels or completely dead plane due todischarging, were replaced at the beginning of each year’s run with new ones tested ina lab. In the lab test, we check the functionality of each new module by using the test
apparatus which consists of a vessel which contains helium and air with the same mixture
ratio as used in the MEG data taking. Two plastic scintillator bars are mounted at the
top and bottom sides of the test apparatus and they are used to trigger the cosmicray events which pass through the drift chamber. Due to exchanging the modules, thetracking eﬃciency was recovered at the beginning of each run.
In the shutdown period between each year, we did an optical survey in order tomeasure the positions of the target and each drift chamber module.
Muon stopping rates at proton current of 2.2 mA were set to 3 × 107 µ+ /s in all datataking period and eleven diﬀerent type of triggers were mixed with MEG trigger withprescaling factors written in Table 2.3.
We also took the calibration data in third of twice per week. In each year, we took
the charge exchange (CEX) calibration in order to obtain the absolute energy scale of the
LXe detector and to measure the energy resolution. Details of calibrations are written in
Chap. 4.

2.6.1

Run 2009

In 2009, MEG data taking was done for two months with two diﬀerent setups of muon
momentum degraders of 200 µm and 300 µm, which were used for the first and the second
half of the data taking, in order to optimize the muon stopping eﬃciency at the target.

51

2.6. RUN STATUS

After the degrader replacement to 300 µm, the stopping eﬃciency was recovered from
56% to 82%.
In 2009, the APDs of the TICZ fibers were turned-oﬀ because the large noise wereobserved on the waveforms from the drift chambers when they were turned-on.
The amount of collected data corresponds to 6.5 × 1013 stopped muons in the target
in 35 DAQ days.

2.6.2

Run 2010

Before the 2010 run, we investigated the noise sources on the DRS boards and the PLL
exchanged to ones which have more temperature stability. As these results, the time jitter
between diﬀerent DRS channels were improved to 40–50 ps from those of 110–170 ps.
The APD channels of TICZ detector was turned-on during most MEG data taking in
2010, because they did not make the large noise which was observed in 2009. However,they are not used in the analysis because the large noises were observed in waveforms
from APDs themselves.
In 2010 run, physics data taking started in August and ended in November, which
was one month earlier than scheduled due to an unexpected quench of the BTS. However
the total statistics reached 1.1 × 1014 stopping muons in the target with the data takingof 56 DAQ days.

2.6.3

Run 2011

In 2011 run, 3×3 NaI crystals were replaced with 4×4 BGO crystals in order to better
calibrate the energy scale and measure the energy and timing resolutions of the LXedetector in the CEX calibration run. Because of installing the BGO crystals, detection
eﬃciency of CEX events increased and resolution of the opening angle of the CEX events
was improved due to the larger stopping power of each crystal compared with that of the
NaI crystal (See Table 2.1) used for the CEX calibration taken before 2011. Details of
the CEX calibration are described in Chap. 4.
Before starting 2011 run, new method of the optical survey using the laser tracker
system was done for more precise alignment for drift chambers. Few more details aredescribed in Sec. 4.3.1.
In 2011, a multiple buﬀer read-out scheme was implemented in the DAQ system. In
this method, data from waveform digitizers are written in the other buﬀer if the datataken by the last trigger is still stored in the first buﬀer. Owing to this method, DAQ live
time increased from 72% in 2010 to 99%, significantly. This large reduction of the deadtime allowed us to use wider direction match criteria. Consequently, the trigger eﬃciency
is also improved from 92% in 2010 to 97% in 2011.
At the beginning of the physics run, huge noises which we cannot ignore were observedin waveforms of drift chambers. After that, we pinned down that there were the following
two main noise sources:• APD channels used for TICZ generate the noise of mainly 40 MHz frequency,• High voltage modules for DCH cause the noise of manly 14 MHz frequency,52

Nµ on target (× 1012)

2.6. RUN STATUS

350
300
250
200
150
100
50
0

Dec/2009

Dec/2010

Jan/2012

Figure 2.22: Number of stopped muons on the target during physics data taking of
2009–2011.

and we turned oﬀ 25% of APD channels which caused the noise component of 40 MHz
frequency, and high voltage modules for the drift chambers were replaced with low noisecommercial high voltage devices. Because of these eﬀorts, two main noise components
almost disappeared. Furthermore, we developed the oﬄine noise reduction in order torecover the eﬃciency and resolutions during the noisy run period. The details of theoﬄine noise reduction are explained in Sec. 3.2.1.3.
In 2011, there was an unexpected power-cut in the πE5 area and 1.5 weeks DAQ time
were lost. Nevertheless, we could collect the physics data corresponding to 1.85 × 1014
muons stopped on the target as shown in Fig. 2.22 in a six month operation including thecalibration periods. The total statistics is approximately equivalent to the sum of thoseobtained in 2009 and 2010.
The total statistics collected in 2009–2011 corresponds to 3.58 × 1014 muons stopped
on the target.

53

Chapter 3
Event Reconstruction
In this chapter, the details of the event reconstruction algorithms are described. Since we
newly improve the following three reconstruction algorithms, each of them is explained
in detail as well.
• New pileup identification and unfolding method using summed waveform• Oﬄine noise reduction in the waveform analysis of drift chambers• Change the code for the track fitting

3.1

Gamma Reconstruction

In order to reconstruct the position, the energy and the timing of gamma rays which
enter the LXe detector, the 846 PMTs mounted inside the LXe detector are used asalready mentioned in Sec. 2.2. The waveform analysis is performed on the waveformfrom each PMT to calculate the charge and the timing of photons detected by the PMT
in each event. Based on the waveform analysis, position, energy, timing of a gamma
ray are reconstructed [48][49][50]. Pileup events are identified and subtracted by using
the light yield distribution and the PMT time distribution. A pileup unfolding using
the summed waveform analysis is newly implemented in the analysis for 2009–2011 data.
The background from cosmic-rays are reduced by applying the topological cut.

3.1.1

Waveform Analysis

The number of detected scintillation photons and the hit time of photons are calculatedby analyzing the waveforms from all the PMTs. Figure 3.1(a) shows typical waveform
before and after the filtering. The baseline is estimated by averaging the points in theregion before the pulse on an event-by-event basis. The pulse time for each PMT isdetermined by applying the digital constant fraction method with 30% fraction for the
given maximum pulse height. In order to maximize the signal-to-noise ratio for the charge
determination, a digital high-pass filter, based on a moving-average method, is applied
to the raw waveform as shown in Fig. 3.1(b). In this method, the fast component of
the waveform is extracted by subtracting the averaged waveform from the original one.
The number of moving-average points is 90–100, corresponds to about 10 MHz cutoﬀ
54

Amplitude (mV)

Amplitude (mV)

3.1. GAMMA RECONSTRUCTION

0
-5
-10

30%

-15
-20

10

50 ns

0
-10

-25
-20

-30
-35

-30

-40

Mov-Ave(high-pass) 89pnts

-600

-500

-400

-300

-600

Time (nsec)

-500

(a)

-400

-300

Time (nsec)(b)

Figure 3.1: (a) An example of a PMT raw waveform from the waveform digitizer. (b)An example of a waveform with high-pass filter.

frequency. The charge is calculated by integrating the filtered pulse found in each PMTwith a 50 ns time window.
The number of photo electrons detected by i-th PMT is calculated by using theequation below.
Npe,i = Qi /(e · Gi ),(3.1)
where Qi and Gi are measured charge and the gain of i-th PMT, respectively. The gainof each PMT is determined by using the LED calibration data (See Sec. 4.1.5). Then the
total number of scintillation photons detected in i-th PMT is given by
Nnpho,i = Npe,i /(QEi ),(3.2)

where QEi is the quantum eﬃciency of the i-th PMT. The QEi is determined by analyzing
the calibration data with point-like α sources (See Sec. 4.1.6).

3.1.2

Position Reconstruction

The gamma ray conversion point is reconstructed by using the distribution of the scintillation light detected by the PMTs close to the incident position. The three-dimensional
position is determined by fitting the expected light distribution on the PMTs, calculated
from the solid angles, to the observed distribution. The interaction position (u, v, w) is
determined by minimizing χ2position defined as
χ2position =

PMT
∑
i

Npho,i − c × Ωi (u, v, w),
σpho,i (Npho,i )(3.3)

where c is the free parameter for fitting as well as (u, v, w) and Ωi (u, v, w) is the solid
angle subtended by the photo-cathode of the i-th PMT. In order to minimize the eﬀectof the shower fluctuation, the fit is done iteratively.
55

3.1. GAMMA RECONSTRUCTION

3.1.3

Energy Reconstruction

The reconstruction of the gamma ray energy (Eγ ) is based on the sum of the numberof photons detected by PMTs. The weight of each PMT is corrected depending on its
location, for example, the weight of the outer face PMTs is 2.6 times less than that on
the inner face PMTs. Therefore the energy of the gamma ray is calculated as∑
Eγ = F (u, v, w) × S(u, v, w) × T (t) × C ×(Npho,i × Wi ),(3.4)
iwhere F (u, v, w) is a non-uniformity correction factor, S(u, v, w) is a correction factor
which depends on the solid angle, T (t) is a correction factor of the light yield changing intime, C is the conversion factor from the number of photons to energy, Wi is the constantweight and Npho,i is the number of photons defined by Eq. (3.2).

3.1.4

Time Reconstruction

We can calculate the time of the first interaction point from each PMT time (thit,i ) if
the position of gamma ray interaction point is determined. Then the hit time (tγ ) isreconstructed by combining each measurement by using following function,
χ2time =∑ thit,i − tγi

σt,i (Npe )2

,

(3.5)

where σt,i (Npe )2 is the time resolution of each PMT as a function of the number of photoelectrons. The gamma emmision time is reconstructed by minimizing the χ2time in Eq. (3.5)
and subtracting the time-of-flight between the vertex reconstructed by fitting of positron
track and the reconstructed conversion point in the LXe detector.

3.1.5

Pileup Identification

At 3 × 107 µ+ /s beam rate, around 15% of MEG trigger events suﬀer from pileup. Therefore, it is important to recognize and unfold pileup events in such a high rate environment.
In order to identify the pileup events, we apply three diﬀerent methods.
First one is called “pileup elimination”, which searches for the peaks by using thelight yield distribution in each event. If the second peak is found by this method andthe position is far enough from the first peak which has larger reconstructed energy, theexpected shape of the light yield distribution of the first peak is calculated except forpileup region, and then the expectation of outputs around pileup region is calculated
based on the first reconstructed energy. The output photon distribution of the pileup
gamma ray of the second peak is replaced with those expectation and the energy isreconstructed again with replaced outputs of PMTs. Figure 3.2 shows the example of thepileup event which is found by the pileup elimination in an event.
The second one, which is newly implemented in the analysis, is called “pileup unfolding”. In this method, the peak search is done in the summed waveform from all PMTs asshown in Fig. 3.3. If multiple peaks are found in sum of waveforms,and the peak whichhave diﬀerent time from that of the main pulse is identified as a peak due to the pileup
event. Then the pulse, which is identified as a pileup, is subtracted by using template56

3.2. POSITRON RECONSTRUCTION

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

0

100

z

200

300
1

Figure 3.2: An example event with pileup gamma ray which is identified by using the
distribution of the light yield. Magenta circles show the pileup cluster.

waveform unless they are close in time (< 5 ns) as shown in Fig. 3.4. By using thismethod, the pileup events, which are discarded in the previous analysis, are recovered by
7%.
If pileup events cannot be identified by both of two methods, the events, which haveworse χ2time /NDF in the time reconstruction given by Eq. (3.5), are discarded as pileup
events. Similarly, if the pileup event is only found in the light distribution but thesubtracted energy is negative or larger than 10% of the total energy, the event is discarded.

3.1.6

Cosmic-ray Rejection

There is only a few percent contamination of the cosmic-ray events in the physics datataken by the MEG trigger. The cosmic-ray background events are not so crucial for the
analysis, however they are reduced by applying the cosmic-ray rejection algorithm. In
order to reject the cosmic-ray background events in the LXe detector, we use a topologicalcut because most of these events enter the detector from the outer face in contrast to
those of gamma rays from the target. Therefore the ratio of the charges collected on the
inner and the outer faces of the detector is smaller in cosmic-ray events than in gamma
ray events. The reconstructed depth of cosmic ray events are significantly larger thanthose of signal gamma rays for the same reason. Figure 3.5 shows correlation between
the charge ratio and the reconstructed depth. The selection criteria is defined in order tomaximize cosmic-ray rejection eﬃciency while keeping a signal eﬃciency of 99%. The cut
discards 56% of cosmic-ray events. The combined analysis eﬃciency of the cosmic-ray
rejection and the pileup cut is 97%.

3.2

Positron Reconstruction

As we use the spectrometer in order to measure the momentum of positrons, the trajectoryof positrons should be reconstructed inside the magnetic field. The equation of motion
57

Amplitude

3.2. POSITRON RECONSTRUCTION

1000

×109

0
0

-20000-1000-40000-2000-60000-80000

-3000

(a)

-4000

-600 -550 -500 -450 -400
Time (nsec)(b)-600 -550 -500 -450 -400
Time (nsec)

Amplitude

Figure 3.3: An event with pileup which identified by the peak search. Left plot shows thenormal waveform after moving-average and right plot shows the diﬀerential one. Magentastars show the found peaks.

10000

10000

(a)

0

0

-10000-10000-20000-20000-30000-30000-40000-40000(b)-50000-50000
-600

-550

-500
Time (nsec)

-600

-550

-500
Time (nsec)

Figure 3.4: An waveform as same event as shown in Fig. 3.3 before (left) and after (right)
applying the pileup elimination.

58

3.2. POSITRON RECONSTRUCTION

Figure 3.5: Two dimensional scatter plot of the charge ratio on the inner/outer faces vs
the gamma ray interaction depth. Black points show the data from cosmic ray runs, andgreen ones shows the signal Monte Carlo events. The two blue lines show the selection
criteria for the cosmic ray rejection.

593.2. POSITRON RECONSTRUCTION

of charged particles in the magnetic field can be written as
dp= qv × B.
dt

(3.6)

where p is momentum vector, v is velocity vector and B is vector of the magnetic field.
The diﬀerential of this equation can be written as
 


1
z

 x  
x0
 


0




y
d  y   √

=
(3.7)
0
0
0
0
  ρ 1 + x02 + y 02 [(r × B)x − x (r × B)z ]  ,xdz  0   √
 y   ρ 1 + x02 + y 02 [(r 0 × B)y − y 0 (r 0 × B)z ] ρρ0where prime represents the diﬀerential of variables on z and ρ = q/p. By using thisequation, track of charged particles can be reconstructed from discrete points measured
by the DCH. The track reconstruction of positrons is generally divided by four steps asfollows:
1. Hit reconstruction from waveforms in each cell of drift chambers,
2. Cluster finding to reduce the accidental hits from all hits and make the track finding
easier by using clusters.
3. Track finding from clusters.
4. Track fitting from found track candidates.
Details of each step is described below.

3.2.1

Hit Reconstruction

The first important step for the track reconstruction is to measure the position of each
hit in the best precision. As we use the drift chamber for the tracking device, the position
along the wire (z) and the drift distance between the hit and the wire are the measured
variables [51]. In order to do that, we analyze six waveforms from each cell of the drift
chamber as shown in Fig. 3.6. In the z measurement, the vernier method is used asalready mentioned. In this method, charge from an anode wire and cathode foils are
combined to get better resolution than that reconstructed by only using charges from theanode wire. The drift distance is calculated from the drift time of ionized electrons.
3.2.1.1

Z Position

The charge ratio of anode wire a is defined by using charge from upstream wire end (QU )and one from downstream (QD ) as:a ≡QU − QD.
QU + QD
60(3.8)

Amplitude [mV]

Amplitude [mV]

3.2. POSITRON RECONSTRUCTION

200
150
100
50

80
70
60
50
40
30
20
10
0

0

-10
-500

0

Time [ns]Amplitude [mV]

Amplitude [mV]

-1000

25
20
15
10
5

-1000

-500

-1000

-500

-1000

-500

0

Time [ns]

70
60
50
40
30
20
10

0

0
-5

-10
-500

0

Time [ns]Amplitude [mV]

Amplitude [mV]

-1000

100
80
60
40

0

Time [ns]

20
15
10
5

20

0

0

-5
-10

-20
-1000

-500

0

Time [ns]0Time [ns]Figure 3.6: Example of waveforms in one cell in an event. Top two waveforms are fromboth ends of the anode wire, and others are from each side of cathodes. Red horizontal
lines show the calculated baseline of each waveform. Red vertical lines at the left edge ofeach peak show calculated hit timing.

61

3.2. POSITRON RECONSTRUCTION(a) Plot of the vernier circle calculated by (b) Calculated α from the vernier circle vs z position, whichcharge ratio between each vernier pad.
is normalized with the wire length, reconstructed by the anode.

Figure 3.7: Correlation between the vernier circle and za .

Then the reconstructed z position from an anode wire za can be written as:
(
)
L Z
za =
+
· a .
2ρ(3.9)

where L is the length of the anode wire, Z is input impedance and ρ represents the resistivity of the wire. Since the resolution of the z measurement by using anode charge is only
∼1 cm, the vernier method is essential to get the resolution of single hit z measurement
down to O(100) µm. In the vernier method, the z position is given by(αn)
z = l·+i−,(3.10)
2π2where l is the length of one vernier period, which corresponds to 5 cm, n is the numberof vernier patterns, and i is the vernier turn from downstream side. Here α is defined as
( )
2
−1
α = tan,(3.11)
1
where 1 and 2 are charge ratios measured by using inner and outer vernier pads respectively in the same way as Eq. (3.8). As shown in Fig. 3.7(b), i can be determined bycomparing α and za .
3.2.1.2

Drift DistanceThe drift distance of ionized electrons is calculated from the drift time which is determinedfrom the waveform of the anode wire. In the drift chamber waveform analysis, the peaksearch is done in the waveform from each wire end. Then the time where the peak crosses
the threshold which determined by the RMS of pedestal is defined as a hit time as shownin Fig. 3.6. The hit time is translated to the drift time by subtracting the time oﬀset from62

drift time (µ sec)

3.2. POSITRON RECONSTRUCTION

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 3.8: Relation between distance from anode wire and hit timing calculated by using
the Garfield simulation.

cables, electronics and the distance between the hit position and the end of anode wires.
Then the drift distance is calculated by using the drift time. Figure 3.8 shows drift time
as a function of drift distance for each magnetic field strength. Once the positron track
is reconstructed from selected hits with success to propagate to the timing counters, each
hit time is refined by using corresponding impact time measured by using the TC. Thenthe calculation of the drift distance is repeated by using refined drift time.
3.2.1.3

Oﬄine Noise Reduction

Since the z position is reconstructed from the charges calculated by using waveforms ofanode wires and cathode pads, the resolution of the z measurement strongly depends on
the signal-to-noise ratio of the charge measurement. The performance of the positron
measurements can therefore be improved by using the oﬄine noise reduction. At the
beginning of 2011 run, strong noise components are observed (see Fig. 3.9). Althoughalmost all periodical and strong noise components disappeared after hardware investigations (see Fig. 3.10), physics data corresponding to approximately a month was aﬀectedby the large noise.
We therefore developed the oﬄine noise reduction. It was found that the noise reduction improves the performance in other run periods in 2009–2011 as well and we applied
it to all data. Figure 3.11 shows accumulated power spectrum taken in a 2011 noisy
run. Large periodical noise components can be seen around 14 MHz and around 40 MHz.
Noise components around 40 MHz come from the timing counter APD system and were63

3.2. POSITRON RECONSTRUCTION

Chamber 12 plane 1 cell 2

50

Amplitude (mV)

Amplitude (mV)

Chamber 12 plane 1 cell 2

40
30
20
10
0
-10
-1000

-500

50
40
30
20
10
0
-10

0

-1000

-500

Time (ns)Chamber 12 plane 1 cell 2

Amplitude (mV)

Amplitude (mV)

Chamber 12 plane 1 cell 2

50
40
30
20
10
0
-10

50
40
30
20
10
0
-10

-1000

-500

0

-1000

-500

Chamber 12 plane 1 cell 2

50
40
30
20
10
0

0

Time (ns)
Amplitude (mV)Time (ns)
Amplitude (mV)0Time (ns)-10Chamber 12 plane 1 cell 2

50
40
30
20
10
0
-10

-1000

-500

0

-1000

Time (ns)-5000Time (ns)Figure 3.9: Example of DCH waveforms in a cell which taken in the 2011 noisy run
period.

64

3.2. POSITRON RECONSTRUCTION

Chamber 8 plane 1 cell 1

50

Amplitude (mV)

Amplitude (mV)

Chamber 8 plane 1 cell 1

40
30
20
10
0
-10
-1000

-500

50
40
30
20
10
0
-10

0

-1000

-500

Chamber 8 plane 1 cell 1

50
40
30
20
10
0
-10

Chamber 8 plane 1 cell 1

50
40
30
20
10
0
-10

-1000

-500

0

-1000

-500

Time (ns)Chamber 8 plane 1 cell 1

50
40
30
20
10
0

0

Time (ns)
Amplitude (mV)

Amplitude (mV)0Time (ns)
Amplitude (mV)

Amplitude (mV)Time (ns)-10Chamber 8 plane 1 cell 1

50
40
30
20
10
0
-10

-1000

-500

0

-1000

Time (ns)-5000Time (ns)Figure 3.10: Example of DCH waveforms taken after hardware investigations.

65

3.2. POSITRON RECONSTRUCTION

Figure 3.11: Power spectrum of accumulated DCH waveforms taken in the 2011 noisy
run period.

suppressed after initializing the system. The 14 MHz component is related to the HV
distributors for DCH and was reduced by replacing the HV distributors in 2011.
There are 2-dimensions called imaginary part and real part in the frequency domain(fj ) as:fj = <(fj ) + i · =(fj )(3.12)
where i is an imaginary unit and the phase of component j can be calculated by <(fj )and =(fj ). The reduction is done for components above 20 MHz and around 14 MHz
component. The procedure of the noise reduction is as follows:
1. do the FFT for waveforms in the DCH cells in which hits are found in both wireends,
2. suppress both <(fj ) and =(fj ) around a frequency of 14 MHz region by a factor of
1010 ,
3. suppress both <(fj ) and =(fj ) which have larger amplitude than 0.1 in the power
spectrum above 20 MHz region by a factor of 1010 ,4. do the inverse FFT.
This means that the FFT filtering works as an oﬄine low pass filter. The FFT is done forapproximately 120 ns narrower time window than the analysis window to make analysis
faster. As shown in Fig. 3.12 (a), waveform after the noise reduction shows less noise
and the pulse shape becomes smoother than that before filtering. In 2011 noisy run,the RMS of pedestal is reduced from 2.4 mV to 1.2 mV owing to the noise reduction.
66

Amplitude (V)Power (mV · Npoints / MHz)3.2. POSITRON RECONSTRUCTION(a)

Time (sec)(b)

Figure 3.12: Raw DCH waveform (black solid line) and after the filtering (red solid line)
in noisy run (a). The charge integration is done for the area between 2 blue solid lines.
(b) is the power spectrum of waveform (a) before/after filtering (black/red).

The performance improvements are observed in data taken in 2009 and 2010 as well (SeeSec. 5.3.3.1). Therefore the oﬄine noise reduction is applied for all the data taken in
2009, 2010 and 2011. The detailed performance improvements due to the oﬄine noise
reduction are written in Sec. 5.3.3.1.

3.2.2

Cluster Finding

Since reconstructed hits include a lot of accidental hits which come from pileup events or
are not related to positrons, for example from delta rays or gamma rays, a cluster finding
algorithm is applied to choose the hits belonging to the track of positrons eﬀectively. In
the cluster finding, neighboring hits in the same chamber module are combined as the
single cluster by using following criteria:• Hits in same plane : ∆z < 0.2 cm + 0.02 · z1 cm, two hits in neighboring cells.
• Hits in diﬀerent plane : ∆z < 0.6 cm + 0.04 · z1 cm, two hits within one skipped
cells.

3.2.3

Track Finding

After the cluster finding, the seeds of the positron tracks are searched for by connecting
the neighboring clusters starting from the hit found in the largest R position in order
to select the clusters belonging to positron tracks with high enough momentum around
50 MeV. Since the absolute z positions of positrons increase according to the increasingof the DCH module index, track finding algorithm is given by the selection criteria as
follows:|zi+1 − zi | < a · |zi | + b,
|ri+1 − ri | < ∆rmax ,where, a = 0.2, b = 2 cm, ∆rmax = 1 cm.
67(3.13)

3.2. POSITRON RECONSTRUCTION

In the track finding, a circle fitting is done at the same time and the result of the fitting
is used as initial information in the track fitting.

3.2.4

Track Fitting

The track fitting is performed for each track candidate found by the track finding. The
algorithm of the fitting is based on a Kalman filter technique. It is totally revised in the
new analysis and is described in detail in this section.
3.2.4.1

Kalman Filter

In order to reconstruct the trajectories of charged particles precisely, we need to get thebest estimate at any measuring points. Here the state vector and the position vector
at i-th measuring point are defined as si and mi respectively. If there are no random
fluctuation eﬀects such as a Multiple coulomb Scattering (MS), an i-th state vector canbe calculated by a function as:
si = f (si−1 ),mi = h(si ),(3.14)

where the f is the function of track modeling and the h is the function to convert fromglobal coordinates to local coordinates of detectors. On the other hand, fluctuation eﬀects
should be included in the propagation of particles inside materials. For this reason, thefitting function becomes more complex. Since the Kalman filter was developed as a linear
estimator for the state of a dynamic system in presence of noise contributions [52], thistechnique is suitable to reconstruct the track of particles with random fluctuations, suchas the multiple scattering eﬀects, and it is recently used in many particle experiments[53]. Therefore, we adopted this technique for the base of the track fitting of positrons.
From here, a basic procedure of the Kalman filter technique is introduced. In the
Kalman filter method, D × D matrix T (called “transport matrix”) and M × D matrix
which are Jacobians of the f and h functions are introduced. Here D is the number ofparameters of the state vectors and M is the number of measured parameters at eachpoint. Then Eq. (3.14) can be modified to the linear approximation assi = si,0 + Ti−1 ((si−1 − si−1,0 ),hi = hi,0 + Hi (si − si,0 ).

(3.15)(3.16)

where si−1,0 is a reference starting state and si,0 = f (si−1,0 ). In this scheme, the contribution from the noise in the propagation (MS and energy loss in tracking applications)i−1
can be introduced as a covariance matrix Ci−1
. By using the covariance matrix, the
and
the
i-th
covariancematrix are given by“predicted” i-th state vector si−1i
= f (si−1
si−1i−1 ),
i
i−1
i−1 T
Ci
= Ti−1 Ci−1
Ti−1 + Qi ,(3.17)

i−1
where Qi represents the contribution from the noise and si−1
is estimated state vector
extracted from the measurements m0 ...mi−1 (this step is called “Prediction” in Kalman683.2. POSITRON RECONSTRUCTION

filter). Then a parameter written as χ2 which should be minimized in the fitting is givenby(3.18)
χ2 = sTi Cii−1 si + (mi − hi (si ))T Vi hi (si )),where Vi are the M × M correlation matrices of the single measurements. The minimization is done by the following formulas:
Ki = Cii−1 HiT (Hi Cii−1 HiT + Vi )−1 ,si = si−1+ Ki (mi − hi,0 ),iiCi = (1 − Ki Hi )Cii−1 ,(3.19)

where the D × M matrix Ki is called “Kalman gain”. Once the state vector si is updatedby Eq. (3.19), si and Cii are including the information carried by the measurement points
up to mi and this process is called “Filtering” in Kalman filter. After repeating “Prediction” and “Filtering” from 0-th measurement point up to N -th one, same procedurecan be repeated starting from N -th measurement point and going backward. The best
estimate at the i-th measurement point is obtained by averaging the results of the forward
and backward filter and this final step is called “Smoothing”.
In practice, the state vector is defined as si = (x, y, z, px , py , pz , t)i where t is timein the track fitting and it can also be translated to the coordinate which is used in the
MEG experiment as (x, y, z, p, θ, φ) where p is absolute value of the positron momentum.
After the fitting for hits belonging to the found track candidate is done, both ends ofthe positron trajectory are prolongated toward the target and the timing counter. The
propagation is stopped when it reaches the middle plane of the target. Figure 3.13 showsthe example of the positron trajectory reconstructed by the track fitting.
3.2.4.2

New Track Fitting

In order to improve the precision of the track fitting, the algorithm of the fitting is revisedin the new analysis. In the new track fitting, which we call “new Kalman”, uncertainties
of each single hit measurement are regarded as functions of the characteristics of eachhit. The function of the z uncertainty depends on the charge over the noise ratio (Q/N )and the track angle θ with respect to the wire. In contrast, the r uncertainty is defined
as a constant double Gaussian, because no obvious dependence is observed in data. The
definition of the local coordinate and the measurement points are modified as well. In theprevious algorithm (called “old Kalman”), the reconstructed cluster position by combining hits found in both planes in a module is used as a single measurement point. In case of
the new Kalman, each single hit is regarded as an independent measurement point in new
Kalman. Accordingly, a local coordinate is defined at each state vector inside each cell of
the drift chamber as (d, z). Here d represents the distance between an anode wire to thehit position of the positron track with a point of closest approach, namely, d corresponds
to the drift distance of each hit, and z represents the position along the anode wire, as
the same in the global coordinate. In the old Kalman, the position of each cluster, whichis reconstructed in each module, is used with the global coordinate. Figure 3.14 shows
diﬀerence of the single state of a fitted positron-track defined in new/old Kalman.
The calculation method of energy depositions inside the detector is also modified. In
the new Kalman, the fitting algorithm based on GEANE [54] is used to calculate the interactions of particles inside the detectors, which is already defined in MEGMC (See Sec. 2.5.1).
69

Y [cm]3.2. POSITRON RECONSTRUCTION

5
0
-5
-10
-15
-20
-25
-20

-10

0

10

20

X [cm]

Z [cm](a) A sample of fitted track shown as blue line in X-Y view.
50
40
30
20
10
0
-10
-20
-30
-40
-50

-20

-10

0

10

20

X [cm](b) A sample of fitted track shown as blue line in X-Z view.

Figure 3.13: A positron trajectory fitted by using the previous track fitting in an event.

70Z (cm)Y (cm)

3.2. POSITRON RECONSTRUCTION

-10
-12

12
11

-14

10

-16

9

-18

8

-20

7
6

-22

5
-24

4
-15

-10

-5

0

-15

-10

-5

X (cm)

0
X (cm)

Figure 3.14: Diﬀerences of reconstructed hits by using new/old Kalman in X-Y view(left) and X-Z view (right). Red stars show the hits reconstructed by using new Kalman
and blue ones show those reconstructed by using old Kalman. Magenta circles in theleft plot shows measured hits belonging to the reconstructed track and the radius of eachcircle shows the measured drift distance. Black points shown in the left plot represent
the position of wires.

GEANE also provides the error propagation during the track fitting by using Kalman filter
technique.
In the analysis, we also use the previous fitting algorithm (called “old Kalman”) forthe compatibility check.

3.2.5

Time Reconstruction

The positron time is measured by using 15 scintillating bars as already mentioned inSec. 2.3.3. If a positron enters a TC bar at time tTC , the measured time of the inner andouter PMTs of each bar are read as;
L/2 + z,veﬀ
L/2 − z= tTC + bout + T Wout +veﬀ

tin = tTC + bin + T Win +tout(3.20)

where bin,out are time oﬀsets, T Win,out are contributions from Time Walk eﬀect, veﬀ isthe eﬀective velocity of light in the bar, L is the bar length and zbar is the impact point
along the main axis of the bar and its starting point is in the middle of the bar. From
Eq (3.20), the impact time of positrons at TC (tTC ) is calculated as;tTC =tin + tout bin + bout T Win + T Wout
L
−
−
−
,
2
2
2
2veﬀ

(3.21)

which is independent on the positron impact point (zbar ). Then the emission time of
positrons (te ) at the target is given by
te = tTC − tTOF ,71(3.22)

3.2. POSITRON RECONSTRUCTION

where tTOF represents the time of flight of the positron track from the target to the TC
determined by using the reconstructed track length (Ltrack ) astTOF = Ltrack /c,(3.23)

where c is the speed of light.

3.2.6

Impact Point in Timing Counter

The positron impact point can be calculated from two diﬀerent ways. First one called
charge ratio method is using the charge ratio of the charges at the inner and outer PMTs;
Qin = EGin e− L/2+zΛ

Qout = EGout eeﬀ,− L/2−zΛeﬀ,(3.24)

where E is the energy deposit inside the bar, Gin,out takes into account the contributions from the scintillator yield, Q.E. of the PMT and the gain, and Λeﬀ is the eﬀective
attenuation length of the bar. From Eq. (3.24), zbar is given by
(
)
Λeﬀ
Qout
Gout
zbar =ln− ln.
(3.25)
2Qin
Gin
The impact point also can be calculated by using tin,out from Eq. (3.20) as:zbar =

veﬀ
× (tin − tout − (bin − bout ) − (T Win − T Wout )) .
2

(3.26)

The former is used for the direction match algorithm in the trigger since it can be calculated fast, while the latter is used for the oﬄine analysis since the resolution is better
than that by using the former algorithm.
3.2.6.1

DC-TC Matching

The new tracking code enables to estimate the matching quality between the reconstructed track by the drift chambers and the hit position in the timing counter bars
during the fitting. In the new Kalman, the reconstructed track is propagated to thetiming counter volume. Then the DC-TC matching quality is classified into the following
four categories:• If the track is successfully propagated inside the volume where the timing counterbar is defined in GEANT and the diﬀerence between z position of reconstructed
track and zT C (∆ZT C ) is less than 12 cm, zero is assigned to a T CM atchingQuality,• If the matching above fails, a larger bar volume which also includes PMTs is usedwith same threshold for ∆zT C . If this matching succeeds, one is assigned to a
T CM atchingQuality,• If both methods above fail, the track is propagated to the point approaching closest
to the bar axis. If the radial distance between the track and the hit in TC satisfies|∆RT C − Roﬀset | < 5 cm, two is assigned to a T CM atchingQuality.
72

3.2. POSITRON RECONSTRUCTION• If all aboves are not succeeded, the larger value (999) is assigned to a T CM atchingQuality
as it is the track without the matched TC,
If positrons reconstructed by DCH satisfy at least one of the above three conditions, thedrift time of positrons at each hit point is refined by using tTOF at that point.
3.2.6.2

Event-by-Event Uncertainties

Since the new Kalman enables more precise calculation of the error propagation inside
the materials of each detector, the fitting error on each variable can be considered as
event-by-event uncertainty, which are called “per-error”, of the track fitting. Measured
variables at the positron side are as follows:• Ee : momentum on the target.
• φe : φ emission angle on the target.
• θe : θ emission angle on the target.
• ye : y position on the target plane.
• ze : z position on the target plane.
• te : timing at the timing counter.
Except for te , per-errors of all variables associated to the positron-tracks are determinedby the diagonal components of the calculated error matrix at the target. We define theper-errors as:
σ 0 = (σE0 e , σφ0 e , σθ0 e , σy0 e , σz0 e ),(3.27)
where ‘σx0 ’ represents the uncertainty of parameter ‘x’. Even though same parameters
can be extracted in the previous fitting as well, there were inconsistency between fitted
uncertainties and measured resolutions. In the new Kalman, we observed more reliable
uncertainties extracted from the error matrix. We therefore decide to use the per-errors
in the analysis as an additional parameters which can provide the precise tracking quality
on an event-by-event basis.

73

Chapter 4
Calibrations
In order to improve the performance of each sub-detector, many calibration methods aredeveloped in MEG. The detail of each method is described in this chapter.

4.1

LXe Monitoring and Calibration Methods

It is important to monitor and calibrate the LXe detector to check the stability for thelong term data taking. For these purposes, several methods are performed as shown in
Table 4.1.

4.1.1

Charge Exchange Calibration

To define the energy scale of the LXe detector, the data using pion Charge EXchange(CEX:π − p → π 0 n) interaction are taken by using a Liquid-Hydrogen (LH2 ) target insteadof the target which is used for the MEG data taking. The neutral pion after the CEXinteraction has 28 MeV/c momentum and immediately decays into two gamma rays.
These two gamma rays are emitted back-to-back in the π 0 rest frame with an energy of
mπ 0
Eγ∗ =
' 67.5 MeV.
2
In the laboratory frame, the gamma-ray energies are determined by
mπ 0
Eγ1,2 = γ
(1 ± β cos θ∗ )(4.1)
2
Table 4.1: Several calibration and monitoring methods prepared for the LXe detector.
Name
PurposeCharge Exchange Energy Scale Calibration
Cockcroft-WaltonLight Yield Monitoring
Neutron generator Light Yield Monitoring
Cosmic Ray
Light Yield Monitoring
LED
Gain Monitoring
Alpha Source
Q.E. Monitoring

74

Period1/year1/week1/week1/week2/week2/week4.1. LXE MONITORING AND CALIBRATION METHODS

where β is the π 0 velocity and θ∗ is the center-of-mass angle between the gamma ray
and the π 0 direction. Therefore monochromatic 55 MeV and 83 MeV gamma rays aredetectable by selecting back-to-back gamma rays with the BGO (NaI for 2009 and 2010run) detector placed opposite to the LXe detector as shown in Fig 4.1(a). Figure 4.1(b)shows the correlations of gamma ray energies measured by using the BGO (NaI) detector
and the LXe detector. Therefore the absolute energy scale factor of the LXe detector toconvert the Npho to the absolute energy is able to be extracted by using those 55 MeV and
83 MeV gamma ray events. The position-dependent energy resolution is also measured
by the CEX calibration to construct the signal Eγ PDF. In the CEX runs, the time
resolution of the LXe detector is also calculated by using the gamma ray which converted
inside the lead converter placed in front of the plastic scintillator bars. Moreover, the

LXe Energy (MeV)

LXe energy vs BGO energy

100
80
60
40
20
0
0

20

40

60
80
100
BGO Energy (MeV)(b)(a)
Figure 4.1: BGO (NaI) mover(a) and two dimensional plot of gamma ray energies measured by the LXe detector and the BGO detector(b).

CEX run is also used to evaluate the position resolution of the detector (See Sec. 5.1.2).
It is the most important calibration run for the LXe detector since the absolute energy
scale is determined by the CEX run. However, this calibration requires a replacementof the normal muon target with the dedicated LH2 target and the beam setting must bechanged. For this reason, the CEX calibrations are done only once per year. In order tomonitor the detector stability and the variation of the energy scale to be corrected, othercalibration methods which are written below, are performed more frequently.

4.1.2

Cockcroft-Walton

For monitoring the light yield stability and extracting the position dependent correction
factor, calibration runs using a dedicated Cockcroft-Walton (CW) proton accelerator are
performed once or twice per week during the MEG data taking. The CW accelerator
75

4.1. LXE MONITORING AND CALIBRATION METHODS

was installed in the downstream of the detector and the protons are delivered by a CW
beamline. At the CW beamline, protons can be accelerated up to 1 MeV and they aretransported to a lithium tetraborate (Li2 B4 O7 ) nuclear target which is inserted into the
COBRA center by an insertion system. In the target, 17.6 MeV gamma rays are produced
8from the reaction 11
5 Li(p,γ)4 Be. By using the 17.6 MeV gamma rays, we can monitor thelight yield stability, correct the non-uniform detector response together with the CEX
patch scan, and refine the Q.E. estimation by comparing the peak of α source events (SeeSec. 4.1.6).

4.1.3

Neutron Generator

In this calibration, a 9.0-MeV gamma ray line which is generated from nickel 5 8Ni(n, γ9.0 )5 9Ni
by using a Neutron Generator. The CEX calibration and CW calibration cannot be donein the muon beam-on condition. In contrast with above two calibration methods, theneutron generator can be done in various beam conditions because it is unnecessary tomodify the beamline setups. Therefore this calibration method is suitable for frequent
light yield monitoring and was started to be used from 2010 MEG data taking.

4.1.4

Cosmic Ray

In order to check the light yield stability, cosmic-ray data are taken as a high energy
source. Although the energies of cosmic-rays are not monochromatic, the landau peaksof cosmic-rays are used to monitor the light yield stability. The energy of the landau
peak is around 150 MeV.

4.1.5

LED

The blue light LEDs are attached inside the LXe detector to calibrate the PMT gain (G)as stable light sources. Since the intensity of photons from LEDs is stable enough, G isgiven by
q
G=
,
(4.2)
e · Npewhere q is the average of charge, e represents the elementary charge and Npe is the average
number of photo-electrons in each PMT. Then the square deviation of observed charge(σq2 ) is given by an equation,222
σq2 = (G2 + σG
) · e2 · (σpe
+ σLED) + σ02 .

(4.3)

2= Npe when the number of photoelecThe deviation of the photoelectrons is given by σpetrons obeys Poisson distribution. Then the equation above becomes2) × e2 × Np e + σ002
σq2 = (G2 + σG

=

(G

2

2
+ σG
)×e

G

× q + σ002 ,

(4.4)
(4.5)

where σ00 is a constant term and diﬀers from a σ0 because it contains σLED and the gain.
2
Because σG
is small enough compared with G, the relation between σq2 and q can be76

4.1. LXE MONITORING AND CALIBRATION METHODS

100
80
60
40
y (cm)

20
0

-20
-40
-60
-80
-100
10090
8070
40
6050
0 20
x (c
40 -40 -20m)
z (cm)

Figure 4.2: Reconstructed positions of 25 α sources inside the LXe detector. In thisparticular run, the half of the detector was filled with liquid xenon and the other partwas filled with gaseous xenon.

regarded as linear. The light intensity can be changed in nine steps and it enables us tocalculate the gain of each PMT.
The LED trigger is also included in the MEG physics run with the scaling factor inorder to monitor the stability of the PMTs in the muon beam condition.

4.1.6

Alpha Source

The light source with a known intensity is required to evaluated the Q.E. of each PMT byusing the following relation; Npho = Npe /QE. Therefore 25 241 Am sources are attachedand used to calculate the Q.E. of each PMT inside the LXe detector. Each 241 Am source
emits the 5.5 MeV α particle with ∼ 1 kBq decay rate. Five α sources are mounted on
a 100 µm gold-plated tungsten wire as point-like sources of scintillation light and fivewires are positioned in a staggered pattern to optimize the range of angles and distances
so that they are visible from the PMTs Figure 4.2 shows reconstructed positions of theα sources with the detector filled with liquid xenon and gaseous xenon. Since the mean
free path of the α particle in liquid xenon is short, the reconstructed positions of α source
events form the ring images due to the shadow of the wire from PMTs opposite to theemission direction of the α particle as shown in this figure. By applying a topologicalcut on the rings, the position of the corresponding α source can be identified. Q.E. of77

4.2. DRIFT CHAMBER CALIBRATIONS

each PMT is evaluated from the measured charge of α source events compared with thosefrom simulation. In order to reduce the systematic uncertainties related to the α source
positions and angles between sources and PMTs, the gamma ray from CW-Li peak eventsare used for the correction (See Sec. 4.1.2).
The peak of α source events is used for the light yield monitoring as well.

4.1.7

Light Yield History

By combining the results from calibration methods above and the time-dependence of theenergy scale obtained by fitting the theoretical spectrum folded with the detector response
function to the measured background spectrum (See Sec. 6.3.3.2), time-dependence of thelight yield, which is mainly caused by the impurity of the liquid xenon, is determined
and then it is corrected. Figure 4.3 show the light yield history of 2009–2011 data before
and after the correction. The uncertainty of this correction is less than 0.2% for all thedatasets.

4.2

Drift Chamber Calibrations

Since the drift chamber is a position sensitive detector, it is important to measure thepositions of each module and wire to improve the position resolutions. In order to measure
the position of the DCH, a laser tracker measurement, a Michel positron alignment, anda cosmic-ray alignment are performed. Relative gains and time oﬀsets of all channels arealso calibrated by using tracks of Michel positrons and cosmic-rays. An accuracy of theposition measurement for positrons is achieved down to 1 mm for z measurement anddown to a few hundred µm for r position reconstruction by those calibration methods.
Here each calibration is described in detail.

4.2.1

Hit Z Calibration

The z reconstruction of the DCH is determined by charges from two anode ends and four
cathode pads. Therefore the reconstructed results depend on the relative gain diﬀerence
between both sides of pre-amplifiers. This is calibrated by reconstructed tracks of Michel
positrons and cosmic-rays. The relative gain of anode signals is corrected by comparing za
given by Eq. (3.9) and the charge ratios of the cathodes ends (1 and 2 used in Eq. 3.11).
Figure 4.4 shows the correlation between za and 1 with a fitting result using a sinusoidal
function.
Gain corrections for cathode pads are done in the similar way by using the z positionsof reconstructed tracks (ztrack ) instead of za . In this case, a vertical oﬀset of the sinusoidal
function indicates a relative gain diﬀerence between upstream and downstream cathodepads and the gain correction is done so that the oﬀset becomes zero.

4.2.2

Hit Time Calibration

The time oﬀset of each channel is calibrated by using the Michel positrons and cosmicrays. In this calibration, we calculate the position of the leading edge of the pulse fromtracks of positrons or those of cosmic-rays. After subtracting the time diﬀerence due to78

4.2. DRIFT CHAMBER CALIBRATIONSEnergy scale (a.u.)

1.04

1.04

1.04

2009

2010

2011

1.03

1.03

1.03

1.02

1.02

1.02

1.01

1.01

1.01

1

1

1

0.99

0.99

0.99

0.98

0.98

0.98

0.97

0.97

0.97

0.96

02/Nov

02/Dec0.96

29/Aug28/Sep0.96

28/Oct25/Jun

25/Jul24/Aug23/Sep23/Oct(a)Energy scale (a.u.)

1.04

1.04

1.04

2009

2010

2011

1.03

1.03

1.03

1.02

1.02

1.02

1.01

1.01

1.01

1

1

1

0.99

0.99

0.99

0.98

0.98

0.98

0.97

0.97

0.97

0.96

02/Nov

02/Dec0.96

29/Aug28/Sep0.96

28/Oct25/Jun

25/Jul24/Aug23/Sep23/Oct(b)
Figure 4.3: Upper plot shows the light yield history measured by using CW-Li peak(black) and background scale (red). To calibrate the light yield history, other possible
sources (CEX 55 and 83 MeV gamma, 9 MeV gamma with neutron generator, cosmic-ray,and alpha source peak) are used. Combined history curve is also shown in this plot witha gray curve. Bottom plot shows same history after the correction by using the combined
curve so that it becomes flat.

79

AH

4.3. ALIGNMENT

0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-40

-20

0

20

40
zanode [cm]Figure 4.4: Correlation between za and the cathode vernier pad asymmetry (black markers), fitted by a sinusoidal function (red line).

the z position of each end, time oﬀsets of both ends are corrected by using average time
diﬀerence. Time oﬀsets between diﬀerent channels are also corrected by comparing the
average time diﬀerence with same tracks which go through the diﬀerent cells in an event.

4.3

Alignment

As already mentioned, the alignment for the position of each wire is essential to achieve theexcellent precision of the track reconstruction. For the drift chamber alignment, we use
three diﬀerent methods. First one is an optical survey measurement and it is used as an
initial input for other two software alignments. The alignment by using Michel positronsand cosmic-rays is performed based on the positions of drift chambers measured by the
optical survey measurement.

4.3.1

Optical Survey

As an initial alignment for each DCH module, the optical survey is done before starting
the MEG run in each year [55]. In 2009 and 2010, the survey measurements using
cross-hairs and the theodolite were performed. The cross-hair markers are mounted on
the all drift chamber modules both upstream and downstream side and the position ofeach marker is measured by the theodolite which is placed three diﬀerent positions inorder to measure the three dimensional position (x, y, z). At the same time, the position
measurements for the target and the support structure are performed by using cross prints
80

4.3. ALIGNMENT

Figure 4.5: Corner cubes mounted on the downstream side of the drift chambers.

as shown in Fig. 2.5(a) and the small pins, respectively. The accuracy of the position
measurement using the theodolite is calculated to be 0.3 mm for x and y, and 0.5–2 mm
for z. In 2011, a laser tracker system was newly introduced by using corner cube reflectors
mounted at both ends of the top-side of each DCH module as shown in Fig. 4.5. Since
the corner cubes are attached to both sides of each module, we can measure not only
the position of the module, but also the angular displacement based on the shape of thedrift chamber which is measured in the lab. The measurement uncertainty of the laser
tracker itself can reach a several tens of µm and the total accuracy including the cornercube displacement is calculated to be 200 µm.

4.3.2

Cosmic Ray Alignment

The cosmic-ray tracks without the magnetic field can be regarding as the simple alignment
tool because of their almost straight trajectories while the Michel positrons are bended
by the magnetic field. Therefore the cosmic-ray data are taken for the purpose of thedrift chamber alignment. In order to trigger the cosmic-ray events, which pass throughthe drift chambers, dedicated 10 cosmic-ray counters (CRC) are mounted on the outerwall of the COBRA magnet at the opposite side of the LXe detector as shown in Fig. 4.6.
These counters are used for the time oﬀset correction and the relative alignment as well(See Sec. 4.2.2 and Sec. 4.3.4.1, respectively).
The alignment procedure is to minimize the residuals of hit position of DCH modules
with respect to straight muon tracks, according to the Millipede algorithm [56]. The
result is checked by using a sample of Michel positrons which enter twice in the drift
chamber volume so-called “two-turn event” in the magnetic field and the edge of Michel
spectrum. After the cosmic-ray alignment, the accuracy for each coordinate becomes
better than 150 µm.

4.3.3

Michel Alignment

The alignment result can be cross-checked by using the track of Michel positrons, whichis called Michel alignment. In the Michel alignment, the measured position of each hit

81

4.3. ALIGNMENT(a) X-Y view(b) Arbitrary view

Figure 4.6: 3D view of 10 cosmic-ray counter bars (shown in cyan). Green line shows asimulated cosmic-ray, which pass through both drift chambers and the cosmic-ray counter.

and the position reconstructed by the track fitting are compared. If the oﬀsets between
the hit positions and the fitted positions are found , then they are corrected.
The Michel alignment is also used to check the alignment of the muon target since
the track of positrons are projected to the target plane as shown in Fig. 4.7. If there
is a misalignment for example, the horizontal displacement, the reconstructed positionof target holes should be aﬀected as shown in Fig. 4.8. Therefore if the reconstructed
hole position found to be depend on the emission angle of positrons, we can collected
this displacement. By using all holes, either vertical displacement or rotations can becorrected with a few hundred µm accuracy.

4.3.4

Relative Alignment

If there is a relative misalignment between the LXe detector and the positron spectrometer, it causes the systematic shift of the opening angle between a gamma ray and apositron. In order to correct that kind of possible misalignment, we use three diﬀerent
methods, a cosmic-ray relative alignment, a CW alignment and an AmBe source alignment. In order to reduce the possible measurement errors, we combined the results fromthe three approaches above and the result is incorporated into the analysis.
4.3.4.1

Cosmic Ray Relative Alignment

In order to evaluate the relative position diﬀerence between the positron spectrometerand the LXe detector, cosmic rays, which pass through both the LXe detector and thedrift chambers, are used. The diﬀerence of the position is defined as∆z = ztrack − zrec ,∆v = vtrack − vrec ,

(4.6)
(4.7)

where xtrack (x = z or v) is the measured position of the cosmic ray on the inner surface of
the LXe detector by extrapolating the cosmic ray track reconstructed by the drift cham82

Y VERTEX [cm]4.3. ALIGNMENT

Entries
Mean xMean yRMS x
RMS yIntegral

333482200.3285± 0.001404
0.1576± 0.0005382
2.569 ± 0.0009929
0.9848± 0.0003806
3.348e+06

700

2

600

1

500
400

0

300
-1
200
-2

100

-3
-8

-6

-4

-2

0

2

4

6
8
Z VERTEX [cm]0

Figure 4.7: Reconstructed positron vertex distribution in Z-Y plane in 2011.

Yassumed target
positione+apparent holeposition (φe>0)
true holeposition
apparent holeposition (φe<0)

true target
positione+X'

Figure 4.8: An illustration of the software target alignment with Michel positrons in caseof the displacement of the target in horizontal axis.

834.3. ALIGNMENT

bers, and xrec (x = z or v) represents the position of the same cosmic ray reconstructedby using the LXe detector. Since the drift chambers are aligned with respect to the MEG
global coordinate system, the position diﬀerence between the DCH and LXe indicatesthe misalignment of the calorimeter. As a result, the position diﬀerence is measured tobe ∆z = −6.1 ± 0.7 mm, here the position accuracy of the drift chambers ∼ 0.5 mm isincluded in the uncertainty, e.g. in 2010 data. We also measure the position displacementin v direction as ∆v = −1.9 ± 0.7 mm, however we observe that cosmic rays pass throughboth LXe and DCH in only limited area in v direction. Therefore the measurement alongv axis is not sensitive to the displacement of other v region.
4.3.4.2

CW Alignment

A dedicated CW data were taken in 2010 to directly measure the positions of the PMTs
mounted inside the LXe detector. In the CW alignment run, lead cubes are put in front of
the entrance window of the LXe detector. Then the positions reconstructed by using the
LXe detector are compared with the theoretical positions which lead cubes are placed.
The results are as follows:• ∆z = −7.3 ± 1.4 mm. The uncertainty includes that from the the LiB target
position (1 mm), lead cube position (1 mm), statistical uncertainty (0.2 mm).
• ∆φ = 0.3±1.1 mrad. Th uncertainty consists of lead positions (1 mm) and statistics(0.5 mm) and translated to the angular uncertainty.
4.3.4.3

AmBe Alignment

A position measurement was performed by using an AmBe source as well in 2009. In thisalignment, an AmBe source was placed in front of the LXe detector and the reconstructed
position of 4.4 MeV gamma ray events is compared with the source position which wasmeasured by using a laser tracker system. The gamma ray from the source is collimatedwith 20 mm-thick lead-collimator with a 10 mm-φ hole. The measurements were performed in front of two PMTs at z = 0 and diﬀerent v positions, −9.3 and −49.6 cm.
Since the alignment of the collimator along the v-axis is not easy due to its roundedshape, only z positions are measured in the AmBe alignment. The diﬀerence between
the reconstructed and the source z position is calculated to be −5.5 ± 1.8 mm, where
error is a quadrature sum of the error of the source position (1.7 mm) and statisticalerror (0.4 mm). Here the data taken at diﬀerent positions are averaged. The result is
considered to be displacement of the calorimeter with respect to the center of the MEG
coordinate system.
4.3.4.4

Combined Result

The relative alignment performed by the three measurements above are combined bytaking a weighted mean. Since we need to know the relative alignment between thecalorimeter and the spectrometer, the uncertainty of the drift chamber position is added
to that of CW and AmBe result. A combined result is obtained as ∆z = −6.2 ± 0.6 mm.
In order to take into account that the measurement √is diﬀerent from the position expectedfrom the design value, the diﬀerence divided by 1/ 12 is added to the uncertainty. The
84

4.4. TIMING COUNTER CALIBRATIONS

final number to be used in the physics analysis is ∆z = −6.2±2.3 mm, which corresponds
to ∼ 9.1 ± 3.4 mrad. Although there were no significant shift observed in φ, the sameamount of uncertainty is assigned to φ conservatively. By adding the uncertainty of thepositron analysis (0.75 mrad), ∆φ is 0 ± 3.4 mrad.

4.4

Timing Counter Calibrations

4.4.1

Gain Equalization

Since the signal from TC is used to trigger positrons in the MEG trigger, the gain of each
PMT needs be equalized to avoid the possible trigger biases. Therefore calibration datafrom cosmic rays is taken because the distribution of cosmic rays is almost uniform on
the longitudinal axis of each bar to equalize the gain of inner and outer PMT.

4.4.2

Time Oﬀsets Correction

Inter-bar time oﬀsets between inner and outer PMTs on each scintillator bar are evaluatedby using cosmic rays. Since the distribution of cosmic rays is uniform and isotropic, thetime diﬀerence between inner and outer PMTs should be flat and centered at zero. The
time oﬀsets between diﬀerent scintillator bars can be calibrated by using two or threehits events in consecutive scintillator bars from both cosmic rays and the positron fromMichel decays.

4.4.3

CW-BoronIn the CW calibration runs, a LXe-TC time coincidence trigger to collect events of apair of two gamma rays from 11 B(p, γ4.4 , γ11.6 )12 C nuclear reaction (Eγ =4.4 MeV and11.6 MeV). By using this data, possible drifts in the relative timing between the TC andLXe detector can be monitored.

4.5

Relative Time Oﬀset

RMD events make a peak in the distribution of the relative timing between gamma ray
and positron since both a positron and a gamma are coincident in this decay. Therefore
the alignment of the relative timing can be done by using the timing peak observed in the
RMD events. The resolutions of the relative time can be measured by the RMD events
as well.

85

Chapter 5
Performance
We evaluate the performance of each sub-detector by using the data taken in 2009–2011.
The performance improvements due to changing the reconstruction algorithm, which are
explained in Chapter 3, are presented here as well.

5.1

Performance of the LXe Detector

In this section, the performance of the LXe detector is described in detail.

5.1.1

Gamma Selection

In order to select the gamma ray events coming from muon decays in the target, theselection criteria called GammaSelection are defined as follows:• Without pileup or pileup unfolded with success,• χ2time /N DF of the time fitting less than three,• Gamma ray converted inside the fiducial volume,• Cosmic-ray rejection cut.
The fiducial volume of the LXe detector is defined as,• |u| < 25.0 cm, |v| < 71.0 cm, 0 < w < 38.5 cm.

5.1.2

Gamma Position Resolution

The position resolution of the LXe detector is evaluated by a Monte Carlo simulationand is validated in dedicated CEX runs by placing a 1.8 cm thick lead collimator atseveral positions in front of the LXe detector. Figure 5.1 shows the result of position
reconstruction projected onto the v coordinate in CEX runs with the collimator. The
lead collimator has two slits with 1 cm width along the u coordinate at v = 0 and 6 cm.
The average position resolutions are estimated to be σu,v ≈ 5 mm and σw ≈ 6 mm,
respectively, and they are in good agreement with the values estimated by using the MC
simulation.
86

Entries / 0.12cm

5.1. PERFORMANCE OF THE LXE DETECTOR

220
200
180
160
140
120

χ2 / ndf

117.6 / 121

Scale 0mean0

119.9 ± 4.9

σ0-5.706 ± 0.027
0.6592 ± 0.0271

Scale 1130.1 ± 5.0

mean1

0.05354 ± 0.02542

σ1
Floor

0.6748 ± 0.0274
60.4 ± 0.9

100
80
60
40
20
0

-15

-10

-5

0

5

v (cm)

Figure 5.1: Reconstructed position distribution projected on the v-axis with a lead collimator in CEX runs.

5.1.3

Gamma Energy Response

The Eγ resolution is evaluated by using 55 MeV peak in π 0 run at each scanning positionof the LXe detector. The fitting function is given as
f (x) = f0 (x) ⊗ hπ0 (x),and f0 (x) is the response function of the LXe detector parameterized as( [])A · exp 2t t − (x − x0 )x ≤ x0 + t,
( σup 22 )f0 (x) =A · exp (x−x20 )x > x0 + t
−2σ

(5.1)

(5.2)

up

where A is a scale parameter, x0 is a peak-position parameter, t is a transition parameter,and σup is a resolution parameter that indicates the spread of the distribution in higher
energy side. Here hπ0 is extracted from the pedestal to take the pileup eﬀect into account.
The lower tail is due to interactions with materials in front of the fiducial volume andshower escapes mainly from the inner face. Since there is a position dependence ofthe energy response, for example the energy resolutions become worse in the narrow
region or edge part of the fiducial volume of the detector, the fitting is performed witheach subdivided position. Figure 5.2(a) shows fitted result using 2011 CEX data at one
subdivided position and Figure 5.2(b) shows resolution map in sigma in u-v plane forw > 2 cm. The systematic uncertainties of σup is less than 10% and those of x0 is
less than 0.3% for all positions. For the physics analysis, hπ0 is deconvoluted and thepedestal function which is extracted from MEG trigger is convoluted to take into accountthe diﬀerence of the beam settings.

5.1.4

Gamma Timing Resolution

In order to calculate the tγ resolution, the data taken at the CEX calibration is used. In
this calculation, time diﬀerence between two gamma rays detected by the LXe detector
87

3
60

σup (%)V (cm)

5.1. PERFORMANCE OF THE LXE DETECTOR

Number of events /(0.50 MeV)

2.8
40
2.6

2500
20

2000
1500

2.4

σE γ = 1.56 ± 0.03 %0

FWHME γ = 4.54 ± 0.11 %

2.2
-20
2

1000
-40

1.8

500
-60

0

1.6

20

40

60
E γ (MeV)

(a)

-20

-10

0

10

20
U (cm)(b)

Figure 5.2: Fitted energy response of the LXe detector for 55 MeV gamma-ray using
CEX data taken in 2011 for wγ > 2 cm(a). (b) shows measured resolution map in sigma
for wγ > 2 cm.

88

5.2. PERFORMANCE OF THE POSITRON SPECTROMETER

and a plastic scintillation counter with the lead converter mounted in front of the BGO(NaI in 2009 and 2010) detector is investigated. After subtracting the contribution fromthe resolution of plastic scintillation counter and the contribution from the spread of thepion CEX interaction point inside the LH2 target, the timing resolution is evaluated tobe σtγ = 67 ps for the 52.8 MeV gamma ray events.

5.1.5

Gamma Eﬃciency

Due to a high stopping power of the LXe for the gamma ray with the energy around
50 MeV, the LXe detector has good detection eﬃciency. However, gamma rays also be
converted before entering the inner surface of the detector and that kind of events makethe ineﬃciency because of the energy leakage. The eﬃciency of the LXe detector is foundto be γ = 65% in the Monte Carlo simulation for the 48 MeV energy threshold. The
diﬀerence between the simulation and data is checked by using the π 0 calibration data andthe detection eﬃciency for 55 MeV gamma rays is found to be γ =64–67%, consistent
with the MC. Taking into account the analysis eﬃciency of 97% (See Sec. 3.1.6), thecombined analysis and detection eﬃciency for the gamma rays is 63 ± 3%. This detection
eﬃciency is used to calculate the normalization factor (See Sec. 6.5.2).

5.2

Performance of the Positron Spectrometer

The performance of the positron spectrometer are described here in detail.

5.2.1

Positron Selection

Since there are sometimes several reconstructed positrons in an event due to pileup eventsand duplicates of a positron, we apply the selection in order to select the positron which
has large enough momentum and good tracking quality from all of positron-tracks byusing measured parameters. The selection criteria called PositronSelection are defined
as follows;• Nturn ≤ 2,
• reconstructed tracks which have suﬃcient matching quality with the timing counterby using following criteria;
1. |zTC − ztrack | < 15 cm,
2. |rTC − rtrack | < 15 cm,
3. |tTC − ttrack | < 85 nsec,• good fitting quality as follows;
1. Nhits ≥ 7,
2. χ2 < 12,
3. σE0 e < 1.1 MeV,
4. σθ0 e < 15 mrad,89

5.2. PERFORMANCE OF THE POSITRON SPECTROMETER

Table 5.1: Single hit resolutions of core/tail in each year.
Year∆Z (µm)∆R (µm)2009 664(core,68%) / 1683(tail) 159(core,67%) / 423(tail)2010 730(core,64%) / 1869(tail) 138(core,64%) / 374(tail)2011 795(core,61%) / 2022(tail) 132(core,68%) / 361(tail)5. σφ0 e < 55 mrad,• Target constraint,• Best ranked track.
In the track finder, sometimes same cluster is shared with diﬀerent track candidates
because of the accidental hits. Therefore the track fitting frequently reconstructs morethan one candidate associated to a single real track and this causes the replicated tracks,
which we call ghosts. Here the parameter to select the best ranked track, called Ghost
Rank (GRank ) is defined as
GRank = 50.716σφ0 e + 184.934σθ0 e + 4192.5σE0 e + 0.0327215χ2track − 0.224026Nhits ,(5.3)

to identify the fitting quality of each positron. The positron, which has the smallest
GRank value, is selected as the best ranked positron.

5.2.2

Single Hit Resolution

Since the true hit position cannot be known for the reconstructed positron-track, thesingle hit resolutions of DCH are calculated from the residuals of the reconstructed tracks.
Residuals are defined as follows:∆Z = Zhit − Ztrack ,∆R = Rhit − Rtrack .

(5.4)
(5.5)

Figure 5.3 shows residual distributions of data taken in 2011. The single hit resolutions
are evaluated by fitting double-Gaussian functions to the residual distributions. In Table
5.1, numerical results are summarized for 2009–2011 data. The diﬀerence of resolutions
is mainly due to the diﬀerent noise situation, but it is also aﬀected by the gain of anode,
misalignment or deformation.

5.2.3

Positron Energy Response

Although the per-error of momentum is used as the resolution for the analysis, average
momentum resolution is required to check the reliability of the value of σE0 e . In orderto evaluate the momentum resolution, Michel spectrum in the time sideband data (thedetail of the sideband data is written in Sec. 6.1.4) is fitted by the theoretical Michel

90

×103

Entries / (20 µm)

Entries / (100 µm)

5.2. PERFORMANCE OF THE POSITRON SPECTROMETER

3500
3000
2500

×103
4500
4000
3500
3000

2000

2500

1500

2000
1500

1000

1000

500

500

0
-1.5

-1

-0.5

0

0.5

0
-0.3

1
1.5
∆Z (cm)

-0.2

-0.1

(a)

0

0.1

0.2
0.3
∆R (cm)(b)

Figure 5.3: Residual distributions of Z and R in 2011. Red lines are double Gaussianwhich are used for the fitting.

spectrum multiplied by an acceptance function convolved with a double Gaussian as theresolution of the energy measurement:
P (Ee measured ) = ((Michel) ∗ (Acceptance))(Ee true ) ⊗ (Resolution),(5.6)

where the acceptance function is defined as:
1 + erf( Ee√2σ−µAcc )true

Acceptance(Ee true ) =

Acc

2

,

(5.7)

Figure 5.4 shows the fit result on 2011 data. As a fit result, the resolutions of σEe =313 ± 2 keV in core (84%) and 1.13 ± 0.12 MeV in tail are obtained. Results from thefitting with 2009 and 2010 data are similar to that for 2011 data and shown in Table 5.7
in Sec. 5.4. For the analysis, only the acceptance function is used for and the fit results
are shown in Table 6.1 in Sec. 6.3.2.1.

5.2.4

Angular and Vertex Resolutions

The per-errors are used for angular and vertex resolutions in the physics analysis assame for the momentum resolution. The average resolutions are calculated from theanalysis using two-turn events. In the two-turn method, a virtual target plane is assumed
between the first turn and the second turn. Then we compare the diﬀerences between
reconstructed variables from the first turn at the plane and reconstructed variables fromthe second turn at the same plane.
Figure 5.5 shows diﬀerences of φe , θe , ye and ze between the two√ turns. The resolutions
are extracted from the fitted sigmas of those plots divided by 2 by assuming an equal
weight for each turn. We correct the diﬀerence between two-turn method and the true
resolution by using the scaling factor s = σ MC, true /σ MC, two turn which is calculated fromthe signal monte carlo simulation. The correlations, the details of which are described inSec. 6.3.2.3, are also taken into account.
91

Relative efficiency

Number of events /(0.25 MeV)

5.2. PERFORMANCE OF THE POSITRON SPECTROMETER

6000

4000

2000

0
1
0.5
0
40

45

50

55
E e (MeV)

Figure 5.4: A Ee distribution from the data taken in 2011 with a fitting result by using
Eq. (5.6). Dashed blue line shows a resolution function centered at 52.8 MeV. Dashedblack line is the theoretical Michel spectrum. In the bottom plot, the acceptance function
is shown.

92Events / (3 mrad)

Events / (3 mrad)5.2. PERFORMANCE OF THE POSITRON SPECTROMETER

4000
3500
3000
2500

5000
4000
3000

2000
2000

1500
1000

1000

500
0

-0.1

-0.05

0

0

0.05
0.1
φ - φ (rad)
1

-0.1

-0.05

0.05
0.1
θ1 - θ2 (rad)(b)
Events / (600 µm)(a)
Events / (600 µm)

0

2

9000
8000
7000
6000
5000
4000

3500
3000
2500
2000
1500

3000

1000

2000
500

1000
0
-3

-2

-1

0

1

0
-3

2
3
Y1 - Y2 (cm)

(c)

-2

-1

0

1

2
3
Z1 - Z2 (cm)(d)

Figure 5.5: Diﬀerences of φe (a), θe (b), ye (c) and ze (d) between two turns from 2011 data
fitted by the double Gaussian function.

935.2. PERFORMANCE OF THE POSITRON SPECTROMETER

5.2.5

Positron Timing Resolution

The time resolution of the positron spectrometer is calculated by combining the contribution from the fluctuation of the fitted track length and that from the resolution of thetime measurement of the timing counter. It is calculated to be 107 ps after subtracting
the eﬀect from front-end electronics.

5.2.6

Positron Eﬃciency

The spectrometer eﬃciency for the signal positron is determined by using the data takenby the Michel trigger which is included in the MEG physics run. The eﬃciency is determined bye = p(DCH|acc) × p(TC|DCH, acc) × p(e trg.|DCH, TC, acc),(5.8)

where p(DCH|acc) is the probability to detect the positrons if they are inside the acceptance, p(TC|DCH, acc) is the fraction of positrons detected by the timing counter in all
positrons detected by DCH. p(e trg.|DCH, TC, acc) represents the trigger eﬃciency forMichel trigger and it is calculated to be ∼ 99%. By using the data taken with trigger#18 defined in Table 2.3, p(TC|DCH, acc) is given byp(TC|DCH, acc) =Nmatched
,
Ngood

(5.9)

where Ngood is the number of reconstructed positrons and Nmatched is the number ofreconstructed positrons with success to find the related hit in TC.
The numerical value of p(DCH|acc) is evaluated from the data taken with the Michel
trigger. Here the number of selected positrons can be calculated by using the total numberof stopped muons at the target (Nµstop ), which is estimated from the proton current andnumeric values are written in Sec. 2.6.1, 2.6.2 and 2.6.3, as,
Obs
NMichel× Prescaling × Pcorr = Nµstop ×
fMichel ×p(DCH|acc) ×p(TC|DCH, acc) ×p(e trg.|DCH, TC, acc),(5.10)

Obs
where NMichel, Prescaling, Pcorr and fMichel are defined as follows:Obs• NMichel:
The number of observed positrons passing through the oﬃcial PositronSelection.

• fMichel :The fraction of Michel positrons in the momentum range from 50 to 56 MeV. It is
extracted from MC.
• Prescaling :The pre-scaling factor used for the Michel trigger.
945.3. COMBINED PERFORMANCE• Pcorr :The correction factor for Michel trigger to take into account the probability ofmulti-pileup events. This factor is calculated from the Eq.(5.11) using the live time
and the number of fired trigger (Ntrigger]22 ).
Rmeas = Ntrigger]22 /(LiveTime)= Rtrue × exp(−Rtrue · ∆T ),Pcorr = Rtrue /Rmeas .

(5.11)

They are also used for the normalization calculation (See Sec. 6.5.2). Once we obtain thevalue of p(TC|DCH, acc), p(DCH|acc) is derived from Eq. (5.10). For example, in 2011,
the positron eﬃciency is calculated to be e = (31 ± 3)%.

5.3

Combined Performance

In the physics analysis, the timing resolutions, angular resolutions, vertex resolutions andgamma position resolutions are merged into three observables; teγ , φeγ and θeγ . Here we
present the combined resolutions of these three parameters.

5.3.1

Vertex Resolutions

For the analysis, resolutions of relative angles between a gamma-ray and a positron in φand θ should be given by θe , φe , positron vertex resolutions, the position resolutions of a
gamma ray position. The relative angles are defined as
φeγ = (π + φe ) − φγ ,
θeγ = (π − θe ) − θγ ,(5.12)(5.13)

where θγ and φγ are defined by assuming that a gamma ray decays at the same position
as a positron decay vertex (xe , ye , ze ) as
(
)
z
−
z
γ
e
θγ = cos−1 √(5.14)(xγ − xe )2 + (yγ − ye )2 + (zγ − ze )2 )()
yγ − ye−1φγ = tan.
(5.15)
xγ − xe
Although the resolution of each variable is defined on an event-by-event basis, averagevalues can be evaluated by using sum of the PDF which is defined later in Sec. 6.3.2.2.
They are calculated to be ∼9 mrad and ∼15 mrad for φeγ and θeγ , respectively.

5.3.2

Timing Resolution

Overall timing resolution is evaluated from the RMD timing peak observed in the datacollected by the MEG trigger. In order to extract the teγ response from the data, adouble Gaussian for RMD peak and a flat distribution for accidental background areused. The fitting result from 2011 data is shown in Fig. 5.6. The uncertainty on themean is calculated to be 15 ps.
95

Number of events /(0.080 nsec)

5.3. COMBINED PERFORMANCE

12000
10000
8000
6000
4000

N RMD = 16430 ± 374
σt eγ = 130 ± 4 ps

2000
0
-2

-1

0

1
2
t eγ (nsec)

Figure 5.6: Relative time distribution from 2011 data.

5.3.3

Contributions of Reconstruction Improvements

We discuss the performance improvements due to the newly implemented algorithms in
the following.
5.3.3.1

Oﬄine Noise Reduction

As already described in Sec. 3.2.1.3, the oﬄine noise reduction is newly implemented in
the reconstruction of the positron track. In order to check the performance improvement
due to the noise reduction, portions of 2009–2011 data were produced with the oﬃcial
configurations except for one condition that the noise reduction is turned oﬀ.
Single Hit z Resolution
The single hit z resolution is evaluated by comparing the z hit positions with the z
positions estimated by tracks reconstructed by old Kalman. As shown in Fig. 5.7, thesingle hit z resolution is improved by applying the oﬄine noise reduction. The total
number of hits increased as well.
Single Hit r Resolution
Since the r is determined by reading the edge of the rise time, the r-resolution also
can depend on the noise. The noise reduction before reading the edge is therefore tested.
In the test, it is found that the r resolution doesn’t change so much and it is decided toapply the noise reduction only to measure the z position.
Performance Comparisons
In order to evaluate the eﬃciency improvement, Michel positrons in sideband data96

Events/(200 µm)

Events/(200 µm)

5.3. COMBINED PERFORMANCE

w/ FFT
RMS = 2.18 mm

105

w/o FFTRMS = 2.41 mm

105

RMS = 2.13 mm

104

103

103

102

-1

0

1

2
∆Z (cm)-2

w/ FFT
RMS = 2.18 mm
w/o FFTRMS = 2.41 mm

10

-1

0

1

2
∆Z (cm)(c) 2011 noisy data
Events/(200 µm)

Events/(200 µm)(a) 2009 data

105

w/o FFTRMS = 2.58 mm

104

-2

w/ FFT

w/ FFT

10

5

RMS = 2.43 mm
w/o FFTRMS = 2.59 mm

104

4

103

103

-2

-1

0

1

102
-2

2
∆Z (cm)(b) 2010 data

-1

0

1

2
∆Z (cm)(d) 2011 normal data

Figure 5.7: Residuals of the single hit z-position with/without the noise reduction.

97

5.3. COMBINED PERFORMANCE

are used. Almost all selection criteria are the same as the selection used as the selectionused for the physics analysis, but some parameters are loose as written below. For the
resolution checks, only tracks which have two turns are selected.
• −5 < teγ < 5 ns and |teγ | > 1 ns,• −300 < φeγ < 300 mrad,• −300 < θeγ < 300 mrad,• Eγ : no thresholds.
The relative improvements of the resolutions are calculated by comparing two-turn tracks
reconstructed by new Kalman with/without the noise reduction. For the eﬃciency, thenumbers of Michel positrons reconstructed with/without the noise reduction are compared. The average number of hits per track is also calculated from Michel positrons
because it is related to the performance.
2009 Data
The improvements are summarized in Table 5.2.
Table 5.2: Performance comparison in 2009.
Parameters
Number of Michel e+
Average number of hits
RMS of Ee
RMS of φe
RMS of θe
RMS of ye
RMS of ze

w/o FFT
25370713.0
756 keV27.5 mrad
16.9 mrad4.65 mm
4.72 mm

w/ FFT Relative improvement (%)
268403613.2
2751 keV<1
27.1 mrad2
16.6 mrad
2
4.57 mm24.52 mm42010 Data
Table 5.3 shows the improvements due to the noise reduction in 2010 data.
Table 5.3: Performance comparison in 2010.
Parameters
Number of Michel e+
Average number of hits
RMS of Ee
RMS of φe
RMS of θe
RMS of ye
RMS of ze

w/o FFT
26877812.7
800 keV30.7 mrad19.6 mrad
4.85 mm
6.16 mm

w/ FFT Relative improvement (%)
290548812.9
2795 keV<128.7 mrad
7
17.8 mrad
94.75 mm2
5.48 mm1198

5.3. COMBINED PERFORMANCE

2011 Noisy Data
In 2011 noisy data, both eﬃciency and resolutions are significantly improved as shownin Table 5.4. The eﬃciency improvement is shown in Fig. 5.8. Figure 5.9 shows thediﬀerence of ∆x (x = Ee , φe , θe , ye , ze ) from two-turn method with/without the noise
reduction. The θe and the ze resolutions are directly improved by the noise reduction andthe φe and the ye resolutions improved as well because more hits are used for the fitting.

Table 5.4: Performance comparison in 2011 noisy run period.
Parameters
Number of Michel e+
Average number of hits
RMS of Ee
RMS of φe
RMS of θe
RMS of ye
RMS of ze

w/o FFT
10085012.8
813 keV31.6 mrad19.6 mrad5.01 mm
6.15 mm

w/ FFT Relative improvement (%)
110153913.7
7784 keV428.3 mrad
11
17.5 mrad
11
4.77 mm55.31 mm14

Ee in 2011 noisy
Events/(50 keV)

Events/(50 keV)

Ee in 2011 noisy
2000
1800
1600
1400

3

10

1200
102

1000
800
600
400

10

200
0
50

51

52

53

54

55
56
Ee (MeV)

50

51

52

53

54

55
56
Ee (MeV)Figure 5.8: Ee spectrum in 2011 noisy run period. Red/black lines are with/without thenoise reduction. Right figure shows the log scale of the left figure.

2011 Data
After several hardware investigations, most of strong noise components were suppressed in 2011 run. Even in the lower noise condition, the noise reduction slightly
improves performance and is applied also to the data. The performance comparison
between with/without the noise reduction is shown in Table 5.5.
2009–2011 All Combined Data
The eﬀect for the 2009-2011 all combined data is also evaluated. The improvements
are summarized in Table 5.6 and shown in Fig. 5.10 and Fig. 5.11. As shown in Fig. 5.10,
total reconstructed positrons increase while the tail component does not.
99

1200
1000
800

Events / (4 mrad)

Events / (4 mrad)

Events / (0.2 MeV)

5.3. COMBINED PERFORMANCE

600
500
400

900
800
700
600
500

600

300

400

400

200

300

200

100

200

-4

-2

0

2

0

4
∆Ee (MeV)

Events / (1 mm)

Events / (1 mm)

0

1000
800

100
-0.1

-0.05

0

0.05

0.1
∆φe (rad)

-1

0

1

2
∆Ze (cm)

0

-0.1

-0.05

0

0.05

0.1
∆θe (rad)

700
600
500

600

400

400

300
200

200

100

0
-2

-1

0

1

2
∆Y e (cm)0-2Figure 5.9: Distributions of ∆x(x = Ee , φe , θe , ye , ze ) from two-turn events in 2011 noisy
run period. Red/black lines are with/without the noise reduction.

Table 5.5: Performance comparison in 2011 run after hardware investigations.
Parameters
Number of Michel e+
Average number of hits
RMS of Ee
RMS of φe
RMS of θe
RMS of ye
RMS of ze

w/o FFT
10711212.0
796 keV29.6 mrad18.5 mrad4.82 mm5.86 mm

w/ FFT Relative improvement (%)
110272
3
12.2
2764 keV429.5 mrad<1
18.5 mrad<14.75 mm15.86 mm<1

Table 5.6: Performance comparison for 2009-2011 all combined data.
Parameters
Number of Michel e+
Average number of hits
RMS of Ee
RMS of φe
RMS of θe
RMS of ye
RMS of ze

w/o FFT
52178812.5
796 keV29.8 mrad18.8 mrad
4.84 mm5.84 mm

w/ FFT Relative improvement (%)
5529226
12.8
2773 keV329.0 mrad3
18.1 mrad
44.75 mm25.60 mm4100

5.3. COMBINED PERFORMANCE

Ee in 2009-2011
Events/(50 keV)

Events/(50 keV)

Ee in 2009-2011
10000
8000

104

3

6000

10

4000
102

2000
0
50

51

52

53

54

55
56
Ee (MeV)

50

51

52

53

54

55
56
Ee (MeV)

6000
5000
4000

Events / (4 mrad)

Events / (4 mrad)

Events / (0.2 MeV)

Figure 5.10: Ee spectrum in 2009–2011 combined data. Red/black lines are with/without
the noise reduction. Right figure shows the log scale of the left figure.

2500
2000
1500

3000
2000

1000

1000

500

4000
3500
3000
2500
2000
1500
1000

-4

-2

0

2

0

4
∆Ee (MeV)

5000

Events / (1 mm)

Events / (1 mm)

0

4000
3000

500
-0.1

-0.05

0

0.05

0.1
∆φe (rad)

-1

0

1

2
∆Ze (cm)

0

-0.1

-0.05

0

0.05

0.1
∆θe (rad)

3000
2500
2000
1500

2000
1000
1000
0
-2

500
-1

0

1

2
∆Ye (cm)

0-2

Figure 5.11: Distributions of ∆x(x = Ee , φe , θe , ye , ze ) from two-turn events in 2009–2011
combined data. Red/black lines are with/without the noise reduction.

101

dN/dEe (MeV-1)

5.3. COMBINED PERFORMANCE

10-1

10-2

10-3
50

52

54

56
Ee (MeV)Figure 5.12: Reconstructed energy spectrum of positrons with new track fitting (red)in comparison with that with the previous one (dashed black line). Both spectra arenormalized by using the area of each histogram.

5.3.3.2

New Track Fitting

Figure 5.12 shows the positron energy spectrum with new track fitting including the oﬄine
noise reduction. Owing to the better precision of the new track fitting, the tail componentof the positron energy is reduced. We also obtained a 6% eﬃciency improvement byusing new Kalman in comparison with that by using old Kalman by looking the numberof reconstructed positrons in the time sideband and it is confirmed by comparing the
calculated normalization factors with new/old Kalman (See Sec. 6.5.2, Table 6.7).
5.3.3.3

Sum Waveform Analysis

As described in Sec. 3.1.5, a new algorithm to reduce the pileup events is applied in the
new analysis. Due to the newly implemented pileup unfolding technique, the events inhigher energy part of Eγ spectrum are reduced as shown in Fig. 5.13, while the gamma
ray detection eﬃciency is increased by 7%.

102Number of events / (0.5 MeV)

5.3. COMBINED PERFORMANCE

105

104
103

102

10
45

50

55

60

65
70
Eγ (MeV)Figure 5.13: Energy spectra of gamma rays in comparison between that by applying new
pileup elimination algorithm (red), previous algorithm (blue-dot-dashed) and without thepileup elimination (black-dotted).

103

5.4. PERFORMANCE SUMMARY

Table 5.7: Performance summary.
Variable
Gamma Resolutions
Eγ (%)
uγ ,vγ (mm)wγ (mm)tγ (ps)Positron Resolutions
Ee (MeV)φe (mrad)θe (mrad)ye (mm)ze (mm)te (ps)Combined Resolutions
φeγ (mrad)
θeγ (mrad)teγ (ps)Eﬃciencyγ (%)
e (%)
trg (%)

5.4

2009

2010

2011

1.9 (w > 2 cm),
2.4 (w < 2 cm)
5696

1.9 (w > 2 cm),
2.4 (w < 2 cm)
5667

1.7 (w > 2 cm),
2.4 (w < 2 cm)
5
6
67

0.31
6.6
9.4
1.1 (core)
1.1
107

0.32
7.2
11.0
1.1 (core)
1.7
107

0.31
7.5
10.6
1.2 (core)
1.9
107

8.9
15.0
156

9.0
16.1
123

8.9
16.2
127

63
28
91

63
35
92

63
31
97

Performance Summary

The performance of the MEG detector in 2009, 2010 and 2011 data is summarized inTable 5.7.

104

Chapter 6
Physics Analysis
This chapter shows the details of the analysis used for the estimation of the branchingratio of µ+ → e+ γ decay from data.
A maximum likelihood analysis is performed to estimate the number of signal events(Nsig ) since only the small number of events are expected to be observed in the signal
region. A maximum likelihood function is defined with probability density functions(PDFs). In order to avoid possible human biases, we use a blind analysis procedure. The
events close to the µ+ → e+ γ signal region were hidden until calibrations and analysis
have been fixed for 2011 data. The datasets of 2009 and 2010 are re-analyzed in thesame analysis procedure which is used for the 2011 dataset. Although the signal region isclosed, all necessary parameters for the analysis can be extracted from events outside of
the blind box, the data from the calibration runs, the data taken by the diﬀerent trigger
and Monte Carlo simulation.
The sensitivity is calculated in the same way as our data analysis by using many
pseudo experiments assuming null-signal-hypothesis.
In order to translate the number of events to the branching ratio, the normalization
factor is calculated by combining results from two diﬀerent ways, which are called Michel
normalization and RMD normalization. The normalization factors for 2009 and 2010
datasets are also calculated again, because of the eﬃciency improvements due to theoﬄine noise filtering, new Kalman and the pileup unfolding.
In order to improve the sensitivity, the per-errors of observables associated to thepositron measurement, which is introduced in Sec. 3.2.6.2, are incorporated into the
PDFs. The detailed analysis using the per-errors is described in Sec. 6.3.

6.1

Datasets

We analyzed combined dataset taken in 2009–2011 with the MEG trigger. The datasets
taken in 2009 and 2010 are re-analyzed with the new reconstruction methods in orderto gain the performance and the experimental sensitivity as already mentioned. The
analysis using several sidebands are also performed.

105

6.1. DATASETS

6.1.1

Pre-Selection

We took 10,343, 12,168 and 30,852 physics runs in 2009, 2010 and 2011, respectively.
One run contains ∼ 5,000 MEG trigger events. To make analysis faster, we apply the
pre-selection by using preliminary calibration factors. The selection criteria for the preselection are as follows:• −6.875 < tγ − tTC < 4.375 ns,• |tTrackCandidate − tTC | < 50 ns,
where tTrackCandidate is the best estimated time of the hit calculated by the track candidate.
Since the time diﬀerence between tTrackCandidate is set to wide enough, this cut corresponds
to the requirement of which at least one track candidate is reconstructed in an event. The
events which are not selected by the pre-selection are used only for the calibrations.

6.1.2

Blind Box

The blind box is defined by using the gamma ray energy and the time diﬀerence between
a gamma ray and a positron as• 48 ≤ Eγ ≤ 58 MeV,
• |teγ | ≤ 1 ns,
Before opening the blind box, we fix the calibrations and the analysis. The analysis on
the datasets in time sidebands, angle sidebands and the reliability check for the newly
implemented PDFs by using full Monte Carlo simulation (full-MC) are also done before
the un-blinding.

6.1.3

Analysis Region

In order to perform the maximum likelihood fitting, the analysis region, which includes
the µ+ → e+ γ like gamma-positron pair event, is defined as follows;• 48 ≤ Eγ ≤ 58 MeV,
• 50 ≤ Ee ≤ 56 MeV,• |φeγ | ≤ 50 mrad and |θeγ | ≤ 50 mrad,• |teγ | ≤ 0.7 ns,• a pair of a positron and a gamma is selected by using the PositronSelection and
the GammaSelection.

106

Eγ (MeV)

6.1. DATASETS

60
58

160

56

140

54

120

52

100

50

80

48

60

46

40

44

20

42
-4

-2

0

2

4

0

Teγ (nsec)

Figure 6.1: Two dimensional event distribution in teγ vs Eγ taken in 2009–2011. The
blank box at the center shows the blind box and the inner box with blue lines shows theanalysis region. Left and Right boxes with dashed black lines show the sideband data
and two boxes with solid magenta lines show the time sidebands which are used for the
maximum likelihood fit. A bottom center box shows the energy sideband used to evaluate
the expected number of RMD events and the teγ resolution. For the illustration purpose,
following loose cuts are applied;40 ≤ Ee ≤ 60 MeV;40 ≤ Eγ ≤ 60 MeV;|θeγ | ≤ 200 mrad;|φeγ | ≤ 200 mrad;|teγ | ≤ 4 ns.

6.1.4

Sideband Data

We define several sideband datasets in order to perform the calibration, evaluate theparameters required to construct the PDFs and perform the maximum likelihood analysis
in the same way as used for the analysis region. Figure 6.1 shows distribution of the datataken with MEG trigger in 2009–2011. As shown in this figure, accidental backgroundis distributed flat in the relative time. Therefore the parameters which are required toconstruct the accidental background PDFs can be extracted from the data in oﬀ-time
region called time sidebands. In the same plot, there is the event concentration at thecenter of the relative time below the blind box (called energy sideband). These eventsshould be the radiative muon decay (RMD) and we can use these events to extract thenormalization factor and to calculate the number of RMD events in the analysis region
by using the energy sideband. Figure 6.2 shows the two dimensional distribution of
the events in relative angles in time sidebands. The asymmetric shape is caused by thetrigger direction match algorithm with lower momentum positrons than that of the signal
positron.

1076.1. DATASETS

φeγ (mrad)

200
140

150

120

100

100

50
0

80

-50

60

-100

40

-150

20

-200
-200

-100

0

100

200

0

θeγ (mrad)

Figure 6.2: Two dimensional event distribution in θeγ vs φeγ taken in 2009–2011 in timesidebands. The center box with blue lines shows the analysis region in relative angles.
Neighboring four boxes with dashed lines show the angle sidebands which we performthe maximum likelihood fitting. For the illustration purpose, following loose cuts areapplied;40 ≤ Ee ≤ 60 MeV;43 ≤ Eγ ≤ 60 MeV;|θeγ | ≤ 200 mrad; |φeγ | ≤ 200 mrad;|teγ | ≤4 ns and |teγ | > 1 ns.

6.1.5

Time Sidebands

The regions of the time sidebands are defined as similar to that of signal, but with anegative or a positive side of the time diﬀerence between a gamma ray and a positron.
In order to perform the maximum likelihood fitting in the time sidebands, two regions
are defined as follows:
Negative time:48 ≤ Eγ ≤58 MeV,50 ≤ Ee ≤56 MeV,−2.7 ≤ teγ ≤ −1.3 nsec,−50 ≤ θeγ ≤50 mrad,−50 ≤ φeγ ≤50 mradPositive time:48 ≤ Eγ ≤58 MeV,50 ≤ Ee ≤56 MeV,1.3 ≤ teγ ≤2.7 nsec,−50 ≤ θeγ ≤50 mrad,−50 ≤ φeγ ≤50 mradA pair of a positron and a gamma is selected by PositronSelection and GammaSelection.
Time sidebands are used to extract the parameters for the accidental backgroundas well since the accidental background events distribute uniformly in the relative time
distribution. For this purpose, loose selection criteria are applied according to the observable, for example, for the background Ee PDF, criteria associated to the gamma ray
measurement can be loosen.

1086.2. LIKELIHOOD ANALYSIS

6.1.6

Angle Sidebands

Since the RMD events are not included in the time sidebands, the datasets of angle sidebands are helpful to check the maximum likelihood fitting with non-zero RMD conditions.
Before opening the blind-box, only four time coincident parts without back-to-back region, which are shown in Fig. 6.2, are partially opened and analyzed. The maximum
likelihood fitting is performed in four angle sidebands as follows:
Negative φ:48 ≤ Eγ ≤58 MeV,50 ≤ Ee ≤56 MeV,−0.7 ≤ teγ ≤0.7 nsec,−50 ≤ θeγ ≤50 mrad,−150 ≤ φeγ ≤ −50 mrad
Positive φ:48 ≤ Eγ ≤58 MeV,50 ≤ Ee ≤56 MeV,−0.7 ≤ teγ ≤0.7 nsec,−50 ≤ θeγ ≤50 mrad,50 ≤ φeγ ≤ 150 mrad
Negative θ:
48 ≤ Eγ ≤58 MeV,50 ≤ Ee ≤56 MeV,−0.7 ≤ teγ ≤0.7 nsec,−150 ≤ θeγ ≤ −50 mrad,−50 ≤ φeγ ≤50 mradPositive θ:
48 ≤ Eγ ≤58 MeV,50 ≤ Ee ≤56 MeV,−0.7 ≤ teγ ≤0.7 nsec,
50 ≤ θeγ ≤ 150 mrad,−50 ≤ φeγ ≤50 mradA pair of a positron and a gamma is selected by PositronSelection and GammaSelection.

6.2

Likelihood Analysis

In order to take into account the event-by-event diﬀerence of the detector responses dueto the time-dependence, position-dependence and the per-errors, the likelihood function
is defined as
(N
−hN
i)
RMD i)
e−Nobs − (NRMD2σ−hN− BG 2 BG
2
2σBG
RMD
L(Nsig , NRMD , NBG ) =ee×Nobs !
N
obs
∏
→
→
→
(Nsig S(−xi ) + NRMD R(−
xi ) + NBG B(−
xi ))
2

2

(6.1)

i=1

where Nsig , NRMD and NBG are best fit values, which maximize the likelihood, of the number of signal events, radiative muon decay events and the accidental background events
respectively, and hNRMD i and hNBG i are expected number of RMD and that of accidental
background calculated from Eγ and time sideband data respectively. Here hNRMD i and
hNBG i are evaluated from the time and the angle sidebands data and constrained by→−→
σRMD and σBG , which are also evaluated from the sidebands. S(−xi ), R(→xi ) and B(−
xi )
109

6.3. PDF−
represent PDFs, which depend on the vector of the event-by-event observables(→xi ), forsignal events, RMD events and accidental backgrounds, respectively. Here we define the→observable as −xi = (Eγ , Ee , teγ , φeγ , θeγ )i .

6.3

PDF

Since the detector responses depend on the gamma ray positions inside the LXe detector, the fitting quality of the positron track, the correlated measurement values and soon, the event-by-event PDFs of the observables are defined for signal and backgrounds.
The background PDFs are determined for the radiative muon decay background (RMD)
and the accidental background (BG) separately. Most of parameters of PDFs are extracted from the data outside of the blind box and some of them are calculated from
the MC simulation. In the new analysis, per-errors are introduced as already mentioned
in Sec. 3.2.6.2. Since we adopt the blind analysis procedure, all PDF parameters are
determined and fixed before opening the blind-box. In this section, the details of each
PDF, the methods to extract the all parameters, and the systematic uncertainty on eachvariable are described.

6.3.1

Category PDF

In the previous analysis, two categories called P ositronCategory of Ee PDFs were defined
depending on the quality of the positron track reconstruction (high or low quality tracks),
instead of using the per-errors. The positrons are categorized as high quality tracks if
they satisfy the following conditions:• the number of hits is larger than nine,• less fitting uncertainties of the momentum, the angle and the chi squared of fit,
• diﬀerences between the positions reconstructed by the track fitting and the positions
which are measured by the timing counter are less than 4 cm.
Then the high quality Ee PDF and the low quality Ee were constructed separately. In each
PDF, the energy resolution is defined as a constant parameter. In order to distinguish the
PDFs used in previous/present analysis easily, they are called a category/per-event PDF,
respectively, hereafter. For the new datasets, we also tested the maximum likelihood
fitting by using the category PDF. The P ositronCategory is also used in the per-event
PDF as a parameter to separate the precision of the positron time measurement.

6.3.2

Signal PDF

In the signal Ee , φeγ and θeγ PDFs, per-errors from the positron-track fitting are newly
introduced, while with the previous analysis the averaged PDFs are used with two categories divided by the quality of the position-track fitting. It means that a resolution(σ) is replaced by a product of a sigma of pull (s) and a per-event error (σ 0 ) which is
calculated by the new Kalman. The correlations between pulls (p0 ) are used instead of

110

6.3. PDF

those between raw observables (p) because (p0 ) is found to be constant in the MC while
p depends on the per-errors.
The notations used in this section are following.
δ An error; a diﬀerence between the estimated and the true valueµ A center of the PDF
σ A resolution of the variable
σ 0 An uncertainty of the estimated values A σ of the pullf A fraction of the component when a PDF consists of several components
p A slope parameter between two variables
p0 A slope parameter between pulls of two variables
ρ A correlation parameter between two variables
ρ0 A correlation parameter between pulls of two variables
The signal PDF is written as:
S(Eγ , Ee , teγ , φeγ , θeγ |uγ , vγ , wγ , xe , ye , ze , φe ) = S(teγ |Eγ , Ee ) ×S(Eγ |uγ , vγ , wγ ) ×
S(Ee |φe ) ×S(φeγ |uγ , vγ , wγ , xe , ye , ze , θeγ , Ee , φe ) ×S(θeγ |uγ , vγ , wγ , xe , ye , ze , Ee ).
(6.2)
6.3.2.1

Ee PDF

The positron-energy response is evaluated by fitting the kinematic edge of the Michel
spectrum in the time sideband. By using the same data, parameters for the accidental
Ee PDF are measured as well as for the signal Ee PDF. In order to extract the PDF
parameters, the theoretical Michel spectrum multiplied by an energy-dependent detector
acceptance approximated by an error function is convolved with a response function
modeled by a sum of two Gaussian functions with σE0 e scaled by sigmas of pulls (score , stail ).
The fit results are shown in Table 6.1. The energy scale is also calibrated with the fitting.
The fitting function is similar to Eq. (5.6), but it is modified to adapt the per-event
scheme asN
data∑((Michel) ∗ (Acceptance)(Ee true ) ⊗ (Resolution)i ),(6.3)
iwhere (Resolution)i is defined as(Resolution)i = fcore · G(µcore , score · σE0 e ) + (1 − fcore ) · G(µtail , stail · σE0 e ),(6.4)

where G is a Gaussian function and fcore represents the fraction of core Gaussian. We
measure the dependence of energy-bias on φe and θe and correct them by shifting data to
make the Michel edge independent of the angles. We do not correct the remaining global
biases after correcting the angle dependence, but the mean parameters of the PDFs are
modified to fit to the biases. In order to extract the score,tail and µcore,tail , we fit thefunction defined by Eq. (6.3) to the data, which is outside of the teγ center to avoid thecontamination of the positrons from RMD, while fixing the parameters of the acceptance111

6.3. PDF

function. Acceptance parameters are extracted from the result of the fitting to the samedata with Eq. (5.6).
The systematic uncertainties of PDF parameters are given as covariance matrices fromthe fitting since there are correlations between diﬀerent parameters as shown in Fig. 6.3.
For the signal PDF, µcore , score and acceptance parameters (µAcc and σAcc , the mean andthe width parameters of the error function for representing the acceptance) are fixed.
For the accidental background PDF, the correlations in the acceptance parameters andresolution parameters are taken into account.

1

µcore

1

0.8

µ

0.6

score

0.8
tail

0.6

0.4
stail

0.4

0.2

0.2
stail

0

0

f core

-0.2

-0.2

µAcc

-0.4

-0.4

σAcc

-0.8

-0.6

µcore

score

stail

f core

µ

Acc

σAcc

-0.6

f core

-0.8

-1

µ

tail

(a)

stail

f core-1(b)Figure 6.3: Calculated covariance matrices, (a) for the accidental background PDF, and(b) the signal PDF for 2009 data set.

The global energy scale uncertainty is estimated to be 25 keV by the study with MC;
the accuracy of the determination of the energy scale with the Michel fit on the MC
samples.
Table 6.1: Parameters for positron energy response. Since errors on all variables are
implemented as a covariance matrix in the PDF, they are not shown in this table.
Parameter
fcoreµcore (keV)scoreµtail (keV)
stail (keV)µAcc (MeV)
σAcc (MeV)2009 2010
0.85 0.85−5−12
0.98 0.925
−5
−12
5.1
4.5
48.64 49.49
2.74 2.73

112

2011
0.855
14
0.85
14
4.12
49.2
2.62

6.3. PDF

6.3.2.2

Angular PDF

The positron angular and vertex resolutions are formed from σ 0 and s and then thecorrelations are taken into account; they are therefore event-by-event. Correlations arealso implemented as per-event. The general definitions of the per-event slope and thecorrelation parameter (p0xy and ρ0xy ) between two observables (x and y) are:dµy/σy0dx/σx0= p0xy= ρ0xy

(6.5)
sy
.
sx

(6.6)

In the likelihood analysis, we can identify:
σx = sx σx0
σy0
pxy = p0xy 0 .
σx

(6.7)
(6.8)

When a correlation is corrected in the PDF, a resolution is reduced by using sinner definedas:√sxy,inner = sxy 1 − ρ02
(6.9)
xy .
The angle and the positron-energy PDF for signal is written as:
P (φeγ , θeγ , Ee ) = P (φeγ | θeγ , Ee ) × P (θeγ | Ee ) × P (Ee ),(6.10)

where P (Ee ) is defined in Sec. 6.3.2.1. In Sec. 6.3.2.3, we describe each correlation. The
constant parameters used in the PDFs as well as the systematic uncertainties are summarized in Table 6.2 for sigmas of pulls and in Table 6.3 for correlations. In Sec. 6.3.2.4,
the incorporation of the per-event errors and the correlations to the PDF is described.
Table 6.2: Sigmas of pulls for positron angular and vertex responses.
Parametersφe ,minsθesye ,inners ze

6.3.2.3

2009
0.73 ± 0.07
1.17 ± 0.05
0.74 ± 0.03
1.16 ± 0.06

2010
0.74 ± 0.07
1.22 ± 0.05
0.73 ± 0.03
1.28 ± 0.06

2011
0.74 ± 0.07
1.25 ± 0.05
0.72 ± 0.03
1.31 ± 0.07

Correlations

There are several correlations among the positron observables which are known to be dueto the constraint from the fixed target plane, for example, if a positron momentum (Ee )is mis-reconstructed, the φ angle of the positron (φe ) and y position of the positron at
the target (ye ) are moved from their true values according to the δEe = Ee reconstructed −113

6.3. PDF

Table 6.3: Parameters for the correlations and their uncertainties.
Parameter
2009
2010
2011
σφe (0) [mrad]8.7 ± 1.4
9.5 ± 1.6
9.8 ± 1.6
kφ [mrad]7.6 ± 1.1
9.2 ± 1.3
8.4 ± 1.2
cφ [mrad]−1.7 ± 0.4 −1.7 ± 0.4 −1.7 ± 0.4
pEe ye [mm/GeV]
537 ± 5513 ± 5516 ± 5p0θe ze
0.9 ± 0.2
0.9 ± 0.2
0.9 ± 0.2

Ee true . In the previous analysis, the correlation was already included based on the simple
geometrical model (See Appendix A). The source of the correlations is almost sameas it was included in the category PDF, however, the treatment of them in the new
analysis diﬀers from those in the previous analysis. In the category PDF, correlation
parameters px and ρx are implemented as constant parameters while they are no longer
constants in the per-event PDF, since these parameters depend on the energy, angularand vertex resolutions and values of resolutions are diﬀerent in the per-event PDF. We
therefore define new correlation parameters p0x and ρ0x instead of using px and ρx as alreadymentioned and they are independent on the values of the per-event resolutions.
Most of correlation parameters are extracted by analyzing two-turn events and others
are calculated from the monte carlo simulation.
σφe depending on φe The φe resolution σφe (φe ) is a function of φe . This eﬀect isembedded in σφ0 e . In the PDF, sφe ,inner , where the δφe -δEe correlation is corrected for, isused. Because the correlation depends on φe , sφe ,inner also depends on φe as:v
uu 1 − (cφ 2 − 2cφ kφ tan φe )/σφ2 e (0)sφe ,inner = sφe ,min u,(6.11)()2tkφ
1 + σφ (0) tan φeewhere sφe ,min is sφe ,inner (φe ) at φe where it is minimized. An oﬀset parameter cφ is added.
The parameters σφe (0) and kφ are extracted from data by using the two-turn method,and cφ is extracted from the signal MC with a 25% uncertainty conservatively.
δφe v.s. δEe correlation The center of φeγ PDF is shifted event-by-event by using
p0Ee φe , which is φe dependent as:cφ − kφ tan φe.
p0Ee φe = √σφ2 e (0) + (kφ tan φe )2(6.12)The eﬀect on the resolution is included in sφe ,inner . The constant term in the δφe -δEe correlation was used for a systematic uncertainty in the previous analysis, but not includedin this time because this uncertainty is included in the systematic uncertainties of µφe
and µθe ; the numbers are written in Sec. 7.4.

114

6.3. PDF

δye v.s. δEe correlation The same correlation used in the previous analysis is used forthe per-event PDF since pEe ye = p0Ee ye · (σy0 e /σE0 e ) is found to be almost constant. The
value of pEe ye is the same as in the average PDF. It is used to calculate the center of φeγ
and θeγ PDFs.
(6.13)µδye = pEe ye δEe
The value of pEe ye is extracted from the data by using the two-turn method. The eﬀect
on the resolution is included in sye ,inner .
δze v.s. δθe correlation Unlike the δye -δEe correlation, the δze -δθe correlation can not
be used for shifting the center of the PDF because the true value of θe of the signal event
can not be known. Due to the correlation, δθe and δθγ are not independent; thereforeσθeγ is not a quadratic sum of σθe and σθγ . The correlation is taken into account whenσθeγ is calculated by adding a correlation term using ρθe θγ defined as,ρθe θγ = pθe θγ

σθe
σθγ

(6.14)

sθe σθ0 e√= Cz pθe ze
Cz2 s2ze σz02e + σθ2XEC(6.15)

sθe σz0 e= Cz p0θe ze √,Cz2 s2ze σz02e + σθ2XEC(6.16)

where σθXEC is a component from the gamma position resolutions in σθeγ . It is assumed
that δze can be linearly scaled to δφγ by using a conversion factor Cz .
The correlation term of the θeγ resolution is√02ρθe θγ σθe σθγ = 2ρθe θγ sθe σθe Cz2 s2ze σz02e + σθ2XEC(6.17)= 2sθe σθ0 e Cz p0θe ze sθe σz0 e .

(6.18)In the PDF, Cz p0θe ze sθe σz0 e is replaced by a general function to translate σye and σze to theresolution of θeγ as in Eq. (6.23). The value of p0θe ze is extracted from the signal MC andthe 25% uncertainty is conservatively assigned.
δφeγ v.s. δθeγ correlation A slope parameter p0φeγ θeγ is used to shift the center of the
φeγ PDF. To modify the event-by-event φeγ resolution, ρ0θe φe and ρ0ze φe are used as thetwo correlations are independent. Since these correlations are evaluated from MC, 25%
uncertainties are conservatively assigned.
p0θeγ φeγ = −0.17 + 0.114 φe − 0.294 φe 2 − 0.195 φe 3p0θe φep0ze φe= 0.60 − 0.13 φe − 0.21 φe2(6.20)= 0.24 − 0.036 φe + 0.14 φe .
2115

(6.19)(6.21)

6.3. PDF

6.3.2.4

Incorporation of correlations in signal PDF

The center of P (θeγ | Ee ) is shifted for the correlation by:µθeγ = Fθ (pEe ye δEe ),(6.22)

where Fθ (δye ) is a function to translate δye to δθγ with taking into account the positionof the muon vertex, the gamma-ray conversion position and the target geometry. The θeγresolution is modified to take into account the correlations as:√(6.23)σθeγ = s2θe σθ02e + Gθ2 (sye ,inner σy0 e , sze σz0 e ) + 2Gθ (0, p0θe ze sθe σz0 e )sθe σθ0 e + σθ2XEC ,where Gθ (σye , σze ) is a function to translate σye and σze to the resolution of θeγ with taking
into account the position of the muon vertex, the gamma-ray conversion position and thetarget geometry.
Similarly, the center of the φeγ PDF is shifted by:µφeγ = p0Ee φe

σφ0 eγσφ0 e
0δEe + pθeγ φeγ 0 θeγ + Fφ (pEe ye δEe ),
σE0 e
σθeγ

(6.24)

where Fφ (δye ) translates δye to δφγ similarly to Fθ (δye ). The φeγ resolution is modified
as:
√
(
)(
)
2
2
0
0
σφeγ = s2φe ,inner σφ02e 1 − ρ02
1 − ρ02(6.25)
θe φe
ze φe + Gφ (sye ,inner σye , sze σze ) + σφXEC ,where Gφ (σye , σze ) is a function to translate σye and σze to the resolution of φeγ similarly
to Gθ (σye , σze ), and σφXEC is σφ from the gamma-ray position resolutions.
6.3.2.5

Eγ PDF

Since we measure the energy responses at diﬀerent regions of the LXe detector from the
CEX data as already described in Sec. 5.1.3, the position dependent Eγ PDF is used on an
event-by-event basis according to the reconstructed first conversion point of the gamma
ray. In the muon beam condition, the pileup eﬀect diﬀers from that in the pion beam
condition. Therefore the convoluted pedestal function (hπ0 ) in Eq. (5.1) is disentangled
and the pedestal function in the muon beam (hµ ) is convoluted and the peak position of
the PDF is shifted from 55 MeV to 52.8 MeV.
6.3.2.6

teγ PDFThe new tracking algorithm provides new information about the TC-DC matching,
T CM atchingQuality. The definition of the variable is given in Sec. 3.2.6.1. The eventsare classified by this parameter as well as the positron category (P ositronCategory) representing the quality of positron time measurements. Figure 6.4(a) shows the measured
teγ PDFs for the diﬀerent categories. It is found that the variable T CM atchingQuality
well separates the events having diﬀerent time centers while P ositronCategory separates
events with diﬀerent precision. Therefore, in this analysis we separately implement teγ
PDFs of signal and RMD for categories defined by the combination of the two variables(in total six categories). The event fractions of the six categories are shown in Fig. 6.4(b).
In each category, the PDF is defined as a double Gaussian and parameters are determinedby fitting the RMD peak.
116

3.5
MQ0, LQ3

MQ0, HQ
2.5

Event fraction

Probability density

6.3. PDF

MQ1, LQ2

MQ1, HQ
MQ2, LQ

1.5

MQ2, HQ

0.4
0.3
0.2
0.1

1

0

0.5
0
-1

-0.5

0

0.5

Pos1i (HQ)
tron
Cate 0 (LQ)
g o ry1t eγ (nsec)

(a)

0
1
lity
a
u
2
ngQ
i
h
c
t
a
TCM

(b)

Figure 6.4: (a) teγ PDFs for diﬀerent categories (2011). (b) Event fractions of the categories.

δteγ v.s. δEe correlation The correlation between teγ and δEe , which is observed in
the signal MC, is considered in the analysis. The correlation between teγ and δEe is givenbyµδteγ = pEe teγ δEe ,(6.26)
where the slope (pEe teγ ) is extracted to be 52.81±1.6 psec/MeV from the signal MC.

6.3.3

Accidental Background PDF

The accidental background (BG) PDF is evaluated from the sideband data except for theteγ , which is assumed to be a constant function.
B(Eγ , Ee , teγ , φeγ , θeγ |uγ , vγ , wγ , φe ) = B(Eγ |uγ , vγ , wγ ) ×
B(Ee |φe ) ×B(teγ ) ×B(φeγ |vγ ) ×B(θeγ |uγ ).
6.3.3.1(6.27)

Background Ee PDF

As already mentioned in Sec 6.3.2.1, background Ee PDF is extracted from the data in thetime sidebands. These data are fitted by the combined function of the theoretical Michel
spectrum and the acceptance error function convolved with the double Gaussian response
function according to the event-by-event positron momentum resolution calculated by the
per-errors and the correlations.
6.3.3.2

Background Eγ PDF

The background Eγ PDF is determined position-dependently in the same way as thesignal Eγ PDF because the fractions of RMD, AIF and cosmic ray (CR) events, which117

6.3. PDF

are all components considered as accidental backgrounds, depend on the position inside
the LXe detector. The Eγ spectrum obtained in each position is used as the positiondependent PDF. In order to measure the energy scale stability as mentioned in Sec. 4.1.7,
following background modeling is applied. The accidental background Eγ PDF consistsof
1. spectrum of gamma ray from RMD in MC,
2. spectrum of gamma ray from AIF in MC,
3. pileup eﬀect from pedestal,
4. detector response,
5. cosmic-ray spectrum in real data,
6. eﬃciency curves due to the lower and higher trigger threshold.
In order to make the background Eγ PDF including eﬀects above, we use the following
methods with the data in both time sidebands. First, the MC distribution of energy
deposition in LXe by a single event with gamma ray from RMD or AIF is smeared byusing the measured pedestal distribution to take into account the pileup eﬀect. Here thepedestal distribution is shrunk by taking into account the eﬀect of the pileup identification algorithm The shrink factor is determined before fitting and is fixed in the fitting.
Then the distribution is convolved with the position-dependent detector response function which is given as a Gaussian; sigma corresponds to the detector resolution and themean to the energy scale. After adding the CR spectrum measured in the CR runs, theeﬃciency curves due to the trigger threshold are applied. Here, the trigger eﬃciency curveis fixed in the fitting. Therefore the fraction of RMD, the fraction of AIF, the resolution,
the energy scale, the fraction of CR background and the vertical scaling factor are freeparameters when we fit the data with this function. Figure 6.5 shows the background Eγ
PDF.
6.3.3.3

Background Angular PDFs

We extracted relative angle (φeγ and θeγ ) PDFs from the time sideband data by using
the selection which used for the analysis region apart from looser angular cuts as |φeγ | <200 mrad, |θeγ | < 200 mrad and Eγ > 46 MeV in order to gain the statistics. The
background φeγ (θeγ ) distribution is divided in five slices of vγ (uγ ) region and each ofthem is fitted with a 5th degree polynomial function. The uncertainties on the backgroundangular PDFs are taken into account in the likelihood analysis using the error covariancematrices of the fitting.

6.3.4

RMD PDFIn the RMD events, Eγ , Ee , φeγ and θeγ are kinematically correlated. In order to construct
the RMD PDFs, the correlations between diﬀerent observables are taken into account by118

6.3. PDF

Figure 6.5: An example of the spectrum of the accidental background gamma rays used
for the PDF in depth is larger than 2 cm in time sideband. Green line is extracted from
the MC distribution of RMD and AIF, Blue line show that convolved with resolution andpedestal, namely acceptance. Solid black line is contribution from cosmic-ray backgroundand Red one shows sum of the blue one and black one.

119

6.4. CONFIDENCE REGION

using the theoretical formula of the RMD given as [14]dB(µ+ → e+ νe ν µ γ) =αdyβdx dΩe dΩγ ×
3
64π
y
[
]
F (x, y, d) − β P~µ · p̂e G(x, y, d) − P~µ · p̂γ H(x, y, d) .(6.28)

Here P~µ is the muon polarization vector;~pe and p~γ are momenta of the positron and the
gamma in the muon rest frame, respectively;p̂e and p̂γ are their unit vectors defined asp̂x ≡ p~x /|~px | (x = e or γ);β is defined as β ≡ |~pe |/Ee ;d is given by d ≡ 1 − β · p̂γ ; andx and y are normalized positron and gamma energies. The allowed ranges of x and y
are constrained by the four-body kinematics, electron mass and muon mass. F (x, y, d),G(x, y, d) and H(x, y, d) in the SM are given in Appendix B. As already mentioned inSec. 2.1.3, we measure the muon polarization at the stopping target. Therefore the muonpolarization is set to be P~µ = 0.89 in Eq. (6.28). The RMD PDF is defined as follow:
R(Eγ , Ee , teγ , φeγ , θeγ |uγ , vγ , wγ , φe , θe ) = R(Eγ |uγ , vγ , wγ , θe , Ee , φeγ , θeγ ) ×R(Ee |φe , θe , φeγ , θeγ ) ×R(teγ |Eγ , Ee ) ×R(φeγ |uγ , vγ , wγ , φe , θe , θeγ ) ×R(θeγ |uγ , vγ , wγ , θe ),(6.29)
where R(Eγ |uγ , vγ , wγ , θe , Ee , φeγ , θeγ ), R(Ee |φe , θe , φeγ , θeγ ), R(φeγ |uγ , vγ , wγ , φe , θe , θeγ ) andR(θeγ |uγ , vγ , wγ , θe ) are determined by using Eq. (6.28) considering the detector performance (e.g. resolutions, acceptance of each sub-detector).
6.3.4.1

RMD teγ PDF

The same teγ PDF, as for the signal, but without correcting the δteγ v.s. δEe correlation,
is used as the teγ PDF of RMD event. because the mean value of Ee cannot be definedin the RMD event.

6.3.5

Illustration of PDFs

Figure 6.6 shows the average PDFs for the 2009–2011 combined dataset in the analysis
region. They are shown only for illustrative-purposes, and not used in the analysis.
In comparison with the average PDFs, two examples of the event-by-event PDFs areshown in Fig. 6.7. Figure 6.7(a) shows the PDFs in the event, in which the positron-trackreconstructed with the good quality, observed in the time sideband. On the other hand,
Figure 6.7(b) shows the PDFs with worse tracking quality,

6.4

Confidence Region

In order to estimate the upper limit on the number of signal events, we adopt the statistical approach, which is known as the Feldman-Cousins approach [58], with the profilelikelihood ordering as already done in the past analysis [4] because that approach is120

6.4. CONFIDENCE REGION(a) Eγ PDF(b) Ee PDF(c) θeγ PDF(d) φeγ PDF(e) teγ PDF(f) φeγ PDF without correcting correlations

Figure 6.6: Average PDFs for the 2009–2011 combined dataset. The green, red and magenta lines show the signal, RMD and BG PDFs, respectively. The correlations between
the errors of positron observables are corrected except for (f).

121

6.4. CONFIDENCE REGION

0.8

0.35

0.7

teγ

0.3
0.25
0.2

0.1
0.05
-9

1.5

2

2.5

0.18

× 10

θeγ

0.16
0.14

0.35

0.4

0.2

0.3

0.15

0.2

0.1

0.1

0.05

0
0.05

Eγ

0.3
0.25

0.5

0.15

0

Ee

0.6

0.051 0.052

0.053 0.054 0.055

0
0.048

0.056

0.05

0.052

0.054

0.056

0.058

φeγ

0.25
0.2

0.12
0.15

0.1
0.08

55592-1433

0.1

0.06
0.04

0.05

0.02
0

-0.04

-0.02

0

0.02

0

0.04

-0.04

-0.02

0

0.02

0.04

(a) PDFs with better tracking quality
0.25

0.25

teγ

0.2

Ee

0.2

Eγ

0.3
0.25

0.15

0.15

0.35

0.2

0.1

0.1

0.05

0.05

0.15
0.1

0

0.05
-9

1.5

2

2.5

0.18

θeγ

0.16
0.14
0.12

× 10

0
0.05

0.051 0.052

0.053 0.054 0.055

0.3

0.056

0.05

0.052

0.054

0.056

0.058

φeγ

0.25
0.2

0.1

55549-625

0.15

0.08
0.06

0.1

0.04

0.05

0.02
0

0
0.048

-0.04

-0.02

0

0.02

0.04

0

-0.04

-0.02

0

0.02

0.04

(b) PDFs with worse tracking quality

Figure 6.7: Examples of per-event PDF in typical two events observed in the time sideband. (a) shows PDFs in which the positron-track shows good fitting quality. (b) shows
ones in the event with worse fitting quality. Green lines show the signal PDFs, Magenta
dashed lines show the accidental background PDFs and red lines show the RMD PDFs.
Blue lines are sum of those three PDFs and black stars indicate measured variables.

1226.4. CONFIDENCE REGION

known to be more accurate method to treat a small number than others. In this method,
we generate many toy Monte Carlo experiments (called “toy-MC”) by using all PDFs
defined in Sec. 6.3 to calculate the confidence level for a given fixed number of signals.
The procedure for the calculation of the confidence level is as follows;
1. Calculate the profile-likelihood at j-th toy-MC defined as
q j (Nsig ) = −2 ln λjp (Nsig ),ˆ j (N ), N̂ˆ j (N ))L(Nsig , N̂
sig
sig
RMD
BG
λjp (Nsig ) =,jjj
L(N̂sig , N̂RMD , N̂BG )(6.30)(6.31)

where N̂k (k = sig, RMD, BG) is the best fit value of the maximum likelihood fit
ˆ j (N ) and N̂ˆ j (N ) are the best fit values for a fixed N .
and N̂
sig
sig
sig
RMD
BG
2. Calculate qiObs on the data in the same way as 1.
3. Find the probability to observe q(Nsig ) which is larger than that observed on thedata (q Obs (Nsig )) as follows;
P (Nsig ) =

MC
N
∑

j

{
H(q j (Nsig ), q Obs (Nsig )) =H(q j (Nsig ), q Obs (Nsig )),N MC(6.32)

1 (q j (Nsig ) > q Obs (Nsig )),0 (otherwise).

(6.33)

4. Repeat 1–3. and find Nsig where the probability become less than 10% to find 90%
C.L. intervals.
obs
In the frequentist approach, the probabilities of what Nsig
becomes larger than fixed Nsig
at each value of Nsig are calculated. We therefore set the 90% C.L. lower or upper limit
with this method if P (Nsig ) becomes less than 10% at certain lower/upper points of fixed
Nsig and the bands inside the lower and upper limits are called 90% C.L. intervals. In
order to include the eﬀect from the systematic uncertainties, the parameters are randomly
fluctuated with their evaluated uncertainties in each toy-MC experiment.

6.4.1

Full-MC study

A reliability check for the newly implemented PDFs is done by a full-MC study. In
this study, 100 k signal events are generated using the toy-MC in a wide angular range(−0.19 ≤ φeγ ≤ 0.19 rad and −0.19 ≤ θeγ ≤ 0.19 rad) with PDF parameters extracted
from the signal full-MC. Then, the agreements of average distributions and correlations
between the toy-MC and the full-MC are checked. Figure 6.8(a) shows the comparison of
the average true resolutions and those from the toy-MC and they agree well each other.
Figure 6.8(b) shows the diﬀerence of the correlations observed in the full-MC and thosein the toy-MC. θeγ vs φeγ correlation shows a small discrepancy between the full-MC andthe toy-MC in the large θeγ region. However, the event fraction of this region is small andthe diﬀerence is within the level of the systematic uncertainties, namely, the diﬀerence istaken into account in the analysis.
123

6.4. CONFIDENCE REGION

φeγ

θe γ

Ee
1800

900

900

800

800

1400

700

700

1200

600

600

1000

500

500

800

400

400

600

300

300

400

200

200

200

100

1600

0
0.05 0.051 0.052 0.053 0.054 0.055 0.056

0

100
-0.05

0

0

0.05

-0.05

0

0.05

φeγ (rad)φeγ (rad)

(a)
0.06
0.04
0.02

0.06
0.04
0.02

0

0

-0.02

-0.02

-0.04

-0.04

-0.06
50

-0.06
51

52

53

54

55
56
Ee (MeV)

-0.05

0

0.05
θe (rad)(b)
Figure 6.8: (a) Average distributions of Ee (left), θeγ (center) and φeγ (right) of the fullMC (black markers) and the toy-MC (blue solid line). (b) Average correlations of δEev.s. δφeγ (left) and δθeγ v.s. δφeγ of the full-MC (black markers) and the toy-MC (bluemarkers).

124

6.5. NORMALIZATION

6.5

Normalization

The normalization factor (k) is defined by
Nsig(6.34)kto normalize the number of observed events to the branching fraction. In order to calculate
the normalization factor, we combine the results from Michel normalization and RMD
normalization to reduce the uncertainty. Here we describe the details of the method foreach normalization channel and the combined results.
B(µ+ → e+ γ) =6.5.1

Analysis Eﬃciency

The analysis ineﬃciency comes mainly from the tail events outside the analysis window.
The escape from angle windows is due to the reconstruction tail, while that from the time
window is mainly due to the mis-reconstruction of positron trajectory where one of turns
is missed. The ineﬃciency is evaluated from the MC simulation taking into account thepileup and the detector conditions. The breakdown of the ineﬃciencies is summarizedin Table 6.4. The probability of the missing turn events is also evaluated by comparing
the fraction of events showing a characteristic distribution of the reconstructed angle andvertex (Fig. 6.9) between MC and data, and a correction to the MC evaluation is applied
from the diﬀerence.
Table 6.4: Breakdown of analysis eﬃciencies
Selection
Michel normalization RMD normalization|teγ | < 0.7 ns
0.933
1
0.920
0.986
|teγ | < 0.7 ns, |θeγ | < 50 mrad|teγ | < 0.7 ns, |θeγ |,|φeγ | < 50 mrad
0.863
0.925

6.5.2

Michel Normalization

The normalization factor from the Michel decay channel (µ+ → e+ νe ν µ ) is written as
Obs
NMichelsignalkMichel =× (Prescaling) × (Pcorr ) × pDM ×× γ × analysis .
fMichelMichel(6.35)

And the definitions of parameters in Eq.(6.35) are written in Sec. 5.2.6 and below.
• pDM :The eﬃciency of the trigger direction match for signals since it is not used in theMichel trigger.
• signal /Michel :The eﬃciency ratio of signal/Michel positrons in the analysis region (50–56 MeV).
It is calculated by fitting the function, which consists of the theoretical Michel spectrum multiplied with the acceptance function and convolved with the momentum
resolution, to data.
1256.5. NORMALIZATION(b)
Number of events

Number of events

(a)
600
500
400
300

2500

2000

1500

1000

4.0 ± 0.4 %200

3.4 ± 0.2 %

500

100
0
-40

-35

-30

0
-40

-25
-20
-15
Axis of the projection

(c)

-35

-30

-25
-20
-15
Axis of the projection(d)Figure 6.9: (a) Distribution of MC signal events on vertex-angle plane. Missing-turn
events make a characteristic pattern on this plane appearing as a band parallel to thered arrow. (b) Scatter plot of the MC events on teγ v.s. a new variable defined by the
projection along the red arrow in (a). The missing turn events ranging at the negative teγ
are clearly distributed around −28 on the new variable. (c) and (d) show the projecteddistributions of MC and data (2011), respectively. Fraction of the peak events is estimatedby fitting a Gaussian and a quartic function to the distribution and results are shown in(c) and (d).

126

6.5. NORMALIZATION• γ :The detection and the analysis eﬃciency due to the pileup identification and thecosmic-ray rejection for signal γ. See Table 5.7 and Sec. 5.1.5 for more detail.
• analysis :See Sec.6.5.1.
Because of using the ratio of eﬃciencies for signal and Michel (signal /Michel ) to calculate
the normalization factor, the large uncertainty due to the instability of the beam intensity
and the time-dependence of the eﬃciency due to the dead channels of DCH are cancelled
out. Since the algorithm of the positron-track fitting in the new analysis diﬀers from thatin the previous one, the eﬃciency ratios are diﬀerent between previous/new estimates.
The results and variables for parameters are shown in Table 6.5 and Table 6.6 for2009 and 2010, respectively, together with the previous estimates used for the previous
publication [4]. In 2011 data, the new waveform analysis for the gamma ray reconstruction has already been included. This means there are only the diﬀerences between old
Kalman and new Kalman shown in Table 6.7. By comparing the normalization factor of
the previous track fitting with that of the new one, the improvement in the eﬃciency due
to the new track fitting is calculated to be 6%. The number of selected positrons is also
increased by 6% from the FFT filtering newly applied to the positron track reconstruction. Since a 7% relative eﬃciency improvement is obtained by new pileup elimination
algorithm for gamma rays, we obtained ∼20% total gain of the statistics in comparison
with the previous reconstruction algorithms.
Table 6.5: Michel normalization parameters for 2009 data in comparison with the previous
estimation
Parameter names previous estimate new estimate
Obs
NMichel
15600
19385
Prescaling
1 × 107
1 × 107
1.14
1.14
Pcorr
pDM
0.915
0.915
signal /Michel
1.17
1.09
γ
0.99×0.58
0.984×0.645
analysis
0.94
0.86
fMichel0.09910.10062009−12kMichel (×10 )1.03±0.09
1.21±0.07

6.5.3

RMD Normalization

The normalization factor from RMD channel is written as
kRMD =Obs
NRMD
signal×× analysis .
BRMD
RMD

(6.36)

Since the same data sample as µ+ → e+ γ search is used and both positron and gamma-ray
are measured, the formula is simpler than Eq. (6.35). Each term is defined as below.
1276.5. NORMALIZATION

Table 6.6: Michel normalization parameters for 2010 data in comparison with the previous
estimation
Parameter names previous estimate new estimate
Obs
NMichel
30670
40597
7
Prescaling
1 × 101 × 107
Pcorr
1.13
1.13
pDM
0.922
0.922
signal /Michel
1.23
1.15
γ
0.99×0.59
0.631
analysis
0.94
0.86
fMichel
0.09730.10062010
kMichel(×10−12 )2.21±0.20
2.66±0.13

Table 6.7: Michel normalization parameters for 2011 data with old/new Kalman
Parameter names old Kalman new Kalman
Obs
NMichel
51574
59999
7
Prescaling
1 × 101 × 107
1.107
1.107
Pcorr
pDM
0.97
0.97
signal /Michel
1.24
1.18
0.99×0.628 0.99×0.628
γanalysis
0.91
0.86
0.10060.1006
fMichel2011
kMichel(×10−12 )3.85±0.21
4.10±0.20

Obs• NRMD:The number of RMD events in a pre-defined window is measured by a binnedlikelihood fit to the teγ distribution.

• BRMD :The partial branching ratio of RMD in the relevant window is calculated from thetheoretical formula (at tree level) convolved with the detector responses.
• signal /RMD :The eﬃciency ratio of signal/RMD can be decomposed into that of positron detection, that of the gamma-ray detection and that of direction match. The positron
relative eﬃciency is evaluated using the acceptance curve extracted from the Michelspectrum fit. In addition, the momentum-dependent probability of the missing turnis incorporated. The gamma ray relative eﬃciency is evaluated from the comparisonof spectra obtained by the MEG trigger and the trigger]1 (lower Eγ threshold). The
relative eﬃciency of direction match is evaluated from the distribution of accidental
background. The eﬀect of muon polarization, which makes the background distribution non-flat (asymmetric) even in case of a full eﬃcient detector and trigger, is128

6.5. NORMALIZATION

newly taken into account for this analysis from the distribution of each particle inthe self-trigger data.
• analysis :The same definition as the Michel normalization except for the condition on time
window, which is already taken into account in the RMD detection. See Table 6.4.

2000

Number of events

Number of events

We evaluate kRMD for 24 statistically independent windows, and the best estimate iscomputed by a χ2 fit. The systematic uncertainty on each item correlating among thediﬀerent windows is inserted in the χ2 as a pull term.
45<E e <48 MeV
45<E γ <48 MeV

(a)

1500
1000
500

0

1500
1000

45<E e <48 MeV
40<E γ <45 MeV(c)

0-0.3 -0.2 -0.1

0.1 0.2 0.3
θ eγ (rad)Number of events

Number of events

48<E e <53 MeV
45<E γ <48 MeV

(b)

500

0
-0.3 -0.2 -0.1
2000

2000

1500
1000
500
0
-0.3 -0.2 -0.1

2000

0

0.1 0.2 0.3
θ eγ (rad)48<E e <53 MeV
40<E γ <45 MeV

(d)

1500
1000
500

0

0.1 0.2 0.3
θ eγ (rad)0-0.3 -0.2 -0.1

0

0.1 0.2 0.3
θ eγ (rad)Figure 6.10: Distribution of RMD events in 2009–2010 data. Black dots show the measured distribution. Black solid histograms show the expected distribution calculated fromthe normalization factor from Michel channel and the best estimate value for each parameter, and the gray bands show the uncertainty. Red dashed ones show the distribution
with the best-fit values of normalization and systematic parameters.
The distribution of RMD events is shown in Fig. 6.10 and Fig. 6.11 in 2009–2010 data
and 2011 data, respectively.

6.5.4

Combined normalization factors

Since the normalization factors from Michel and RMD are calculated by using diﬀerent
data samples (Michel trigger and MEG trigger), the two methods are almost independent.
129

Number of events

Number of events

6.5. NORMALIZATION

45<E e <48 MeV
45<E γ <48 MeV

(a)
2000
1500
1000
500

2000
1500
1000
500

0

45<E e <48 MeV
40<E γ <45 MeV

(c)
2000

0
-0.3 -0.2 -0.1

0.1 0.2 0.3
θ eγ (rad)Number of events

0-0.3 -0.2 -0.1
Number of events

48<E e <53 MeV
45<E γ <48 MeV

(b)

1500
1000
500
0
-0.3 -0.2 -0.1

0

0.1 0.2 0.3
θ eγ (rad)48<E e <53 MeV
40<E γ <45 MeV

(d)
2000
1500
1000
500

0

0.1 0.2 0.3
θ eγ (rad)0-0.3 -0.2 -0.1

0

0.1 0.2 0.3
θ eγ (rad)Figure 6.11: Distribution of RMD events in 2011 data. Black dots show the measured
distribution. Black solid histograms show the expected distribution calculated from thenormalization factor from Michel channel and the best estimate value for each parameter,
and the gray bands show the uncertainty. Red dashed ones show the distribution withthe best-fit values of normalization and systematic parameters.

130

6.5. NORMALIZATION

Therefore the uncertainties on both factors can be reduced by combining them. The
combined normalization factors are shown in Table 6.8. The uncertainty become only
4–5% depending on the dataset.
Table 6.8: Summary of normalization factors calculated for 2009, 2010 and 2011 dataset.
kMichel (×1012 ) kRMD (×1012 ) kcombined (×1012 )2009
1.21 ± 0.07
1.19 ± 0.06
1.20 ± 0.06
2010
2.66 ± 0.13
2.41 ± 0.13
2.52 ± 0.09
2011
4.10 ± 0.20
3.96 ± 0.24
4.05 ± 0.16

Then the total normalization factor for the combined datasets are calculated to be2009+2010
kcombined= (3.72 ± 0.15) × 1012 ,2009+2010+2011
kcombined= (7.77 ± 0.31) × 1012 .

131(6.37)(6.38)

Chapter 7
Results and Discussions
In this chapter, we describe the likelihood analysis results on the 2009–2010, 2011 and
2009–2011 datasets. Here the limits are calculated at 90% C.L. if not mentioned explicitly.
The results and those of reliability and compatibility between the results from the previous
analysis are discussed as well. For the comparison, the results from the previous results
are shown in Table 7.1.

7.1

Sensitivity

We define an experiment sensitivity as a median of the upper limits for an ensemble ofpseudo experiments with the background-only hypothesis. The 90% C.L. upper limit foreach pseudo experiment is calculated by the same way as already described in Sec. 6.4.
The median is generally used as an expected outcome of an experiment rather than themean because it is less aﬀected by anomalous outliers and it is metric independent. Figure
7.1 shows the distribution of the upper limits on the number of signals in 3000 pseudo
experiments. The medians of the ensembles are 4.7, 4.6 and 6.0 in number of signals
for 2009–2010 combined, 2011 and the 2009–2011 combined datasets, respectively. Afterthe normalization, they correspond to B of 1.3 × 10−12 , 1.1 × 10−12 and 7.7 × 10−13 ,respectively. These results are summarized in Table 7.2. The systematic uncertainties(described later) are included. Since the sensitivity definition is not changed from theprevious analysis, they can directly compared and the sensitivity is improved by factor two
as shown in Table 7.1 and Table 7.2. Owing to the several reconstruction improvements
and the newly implemented per-event PDF, the sensitivity is found to be improved by
20% in the 2009–2010 dataset, which was also analyzed by using the previous analysis
method. By adding the 2011 dataset to the 2009–2010 combined dataset, the sensitivity
reaches unprecedented region, the order of the 10−13 .

Table 7.1: Results from the previous analysis presented in [4].
Data set
Sensitivity
2009–2010 combined 1.6 × 10−12

Observed best fit−1.5 × 10−12132

Lower limit–

Observed upper limit
2.4 × 10−12

7.1. SENSITIVITY

Table 7.2: Calculated sensitivities in each dataset. Systematic uncertainties are included.
Data set
Sensitivity in Nsig Sensitivity in B
2009–2010 combined
4.7
1.3 × 10−12
2011 only
4.6
1.1 × 10−12
2009–2011 combined
6.0
7.7 × 10−13

250

250

200

200

150

150

100

100

50

50

0
0

5

0
0

10
15
20
Nsig upper limit at 90% C.L.

(a)

5

10
15
20
Nsig upper limit at 90% C.L.

(b)

200
180
160
140
120
100
80
60
40
20
0
0

5

10
15
20
Nsig upper limit at 90% C.L.

(c)
Figure 7.1: Distributions of the upper limits of the background-only pseudo experiments.
(a) 2009-2010, (b) 2011, (c) 2009-2011. The red arrows show the medians of the distributions. The black arrows show the observed limits. In (a), two arrows are overlappedby chance.

1337.2. SIDEBAND RESULTS

Table 7.3: Time sideband results without including systematic uncertainties. The resultsin the negative teγ sidebands are labeled as ‘negative’ and those in opposite sides arelabeled as ‘positive’.
Dataset
B best fit
2009-2010 negative
7.7 × 10−13
2009-2010 positive −2.8 × 10−13
2011 negative
3.7 × 10−15
2011 positive
2.1 × 10−13
2009-2011 negative
3.5 × 10−13
2009-2011 positive
7.8 × 10−14

7.2

B upper limit
3.1 × 10−12
1.1 × 10−12
1.7 × 10−12
1.4 × 10−12
1.6 × 10−12
8.1 × 10−13

Sideband Results

In order to check the reliability of the implemented likelihood function, two kinds of sidebands (time and angle) are analyzed. In time sidebands, the shape of the background
events should be consistent with those of implemented accidental background PDFs since
signal events and RMD events are not included in the these region. Therefore calculated
upper limits should be consistent with those of expected sensitivities calculated in Table
7.2. In angle sidebands, best fit values of NRMD should be consistent with those expected from the RMD analysis. In the sideband analysis, the systematic uncertainties
except for those of NRMD and NBG are not incorporated since the eﬀect of the systematic
uncertainties are found to be not so large as described in Sec. 7.4.
When the time sidebands are analyzed, the center of the time PDF of signal is shifted
so that it is at the center of the sideband time window (±2 ns). Similarly, when the angle
sidebands are analyzed, the center of the θeγ or φeγ signal PDF is shifted ±50 mrad.

7.2.1

Results with Time Sidebands

For both time sidebands, the same expected numbers of accidental background eventsused for the analysis region are used. The analysis results of B are summarized in Table 7.3. Figure 7.2 shows the C.L. curves calculated in the Feldman-Cousins method intime sidebands. When positron tracks which have real multi-turns with one or a few
missing turns in the reconstruction, the reconstructed emission time on the target mustbe earlier than the true value. Since single missing turn events of RMD events with
50–56 MeV positron momenta correspond to 2–2.5 nsec in the time-of-flight, only thenegative sidebands are aﬀected by the missing turn events of RMD as shown in Fig. 7.2.
It is therefore expected that the B upper limits in the negative sidebands become larger
than those in the positive sideband as seen in Table 7.3. On the other hand, the calculated
B upper limits in the positive sidebands look more likely consistent with the evaluatedsensitivity in the analysis region.

134

1

Confidence Level

Confidence Level

7.2. SIDEBAND RESULTS

0.9

1

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

5

10

0
0

15
20
Number of signals

5

15
20
Number of signals(b)

1

Confidence Level

Confidence Level

(a)

10

0.9

1

0.9

0.8

0.8

0.7

0.7

0.6

0.6
2009-2010 negative

0.5

0.5

0.4

0.4

2011 negative

0.3

0.3

2011 positive

0.2

0.2

2009-2011 negative

0.1

0.1

2009-2010 positive

2009-2011 positive

0
0

5

10

0
0

15
20
Number of signals

(c)

1

2

×10-12
3
4
Branching ratio(d)Figure 7.2: Confidence level curves in time sidebands (a)–(c) before normalization and(d) after normalization. (a) 2009-2010, (b) 2011, (c)(d) 2009-2011. Red lines show theresults of the negative time sidebands and blue lines show the results of the positive ones.

1357.3. RESULT IN ANALYSIS REGION

Table 7.4: Angle sideband results with constraining the number of backgrounds.
Dataset2009-2010 negative φ2009-2010 positive φ2009-2010 negative θ
2009-2010 positive θ2011 negative φ2011 positive φ2011 negative θ2011 positive θ
2009-2011 negative φ2009-2011 positive φ2009-2011 negative θ
2009-2011 positive θ

7.2.2

B best fit
7.6 × 10−13−2.9 × 10−12
5.3 × 10−13−3.5 × 10−13
1.2 × 10−12−1.4 × 10−13−6.1 × 10−13
1.2 × 10−13
9.7 × 10−13−1.1 × 10−12
1.9 × 10−14−1.1 × 10−13

Results with Angle Sidebands

The accidental background events in 1 ≤ |teγ | ≤ 3.7 nsec in the same angles regions arecounted and scaled to 1.4 nsec to be used for the constraint. The number of RMD eventsare calculated from the RMD analysis using the Eγ sideband data. The analysis results in
B are summarized in Table 7.4. Hereafter, N̂sig , N̂RMD and N̂BG are used to represen thebest fit values of the number of signal, RMD and the accidental background respectively.
In the angle sidebands, we can study RMD as well as the accidental backgrounds. To
check the RMD PDF implementation and background evaluation, we repeated the fitsomitting the constraint terms of NBG and NRMD in the likelihood function. The results
are summarized in Table 7.5. The observed numbers are in good agreement with the
expectations.

7.3

Result in Analysis Region

The definitions of the analysis region are described in Sec. 6.1.3. The positron is selectedby using the selection criteria explained in Sec. 5.2.1 and the gamma selection is doneby using the criteria written in Sec. 5.1.1. The blind region of 2011 data is opened after
all PDF parameters are determined. From the sideband data, the expected numbersof RMDs and the accidental background events in the analysis region are calculated
and constrained. The numeric numbers are summarized in Table 7.6 with the observed
number of events in the analysis region. The event distributions around the µ+ → e+ γ
signal region are shown in Fig. 7.3 and Fig. 7.4. For the illustration purpose, only theevents in cos Θeγ < −0.99963 and |teγ | < 2.443 ps regions are shown in plots of Eγ vs.
Ee , and 51 < Eγ < 55.5 MeV and 52.385 < Ee < 55 MeV regions are shown in plots ofteγ v.s. cos Θeγ . Thresholds for Ee , cos Θeγ and teγ correspond to 90% signal region andthose for Eγ correspond to 75% signal region, which is arbitrary defined.
The best fits of NBG , NRMD and Nsig , maximizing the profile likelihood shown in
136

7.3. RESULT IN ANALYSIS REGION

Table 7.5: Angle sideband results without constraints on the number of backgrounds (Uncertainties are in 1σ). Parameters in bracket show the expected number of backgrounds
and those with a hat show the best fit results.
Dataset2009-2010 negative φ2009-2010 positive φ2009-2010 negative θ
2009-2010 positive θ2011 negative φ2011 positive φ2011 negative θ2011 positive θ
2009-2011 negative φ2009-2011 positive φ2009-2011 negative θ
2009-2011 positive θ

hNBG ihNRMD i
1120 ± 17
34 ± 31169 ± 17
36 ± 31123 ± 17
39 ± 4877 ± 1524 ± 21130 ± 18
35 ± 41189 ± 18
37 ± 41131 ± 18
41 ± 4976 ± 1725 ± 32228 ± 24
69 ± 72365 ± 25
73 ± 72251 ± 24
80 ± 81855 ± 22
49 ± 5

Nobs
1120
1247
1120
962
1163
1241
1233
1016
2283
2488
2353
1978

N̂BG
1077 ± 35
1212 ± 39
1083 ± 36
963 ± 34
1132 ± 37
1195 ± 38
1208 ± 38
979 ± 34
2210 ± 51
2404 ± 54
2292 ± 52
1939 ± 48

N̂RMD
40 ± 1546 ± 1835 ± 161 ± 1227 ± 1547 ± 1627 ± 1537 ± 1466 ± 2193 ± 24
61 ± 2241 ± 18

Table 7.6: Expected number of background events and the observed number of events inthe analysis region defined in Sec. 6.1.3.
Year
hNBG i2009-2010 1195.7±17.6
1219.3±17.8
2011
2009-2011 2415.0±25.0

137

hNRMD iNobs
83.8±8.4 1291
85.5±8.9 1282
169.3±17.0 2574

58

t eγ (nsec)

E γ (MeV)

7.3. RESULT IN ANALYSIS REGION

2

1.5

56

1

0.5

54

0
52

-0.5

50

-1
-1.5

48
50

51

52

53

-2
-1

54 55 56
E e (MeV)(b)teγ v.s. cos Θeγ ,51 < Eγ < 55.5 MeV,
52.385 < Ee < 55 MeV

58

t eγ (nsec)

E γ (MeV)(a)Eγ v.s. Ee ,cos Θeγ < −0.99963,|teγ | < 2.443 ps-0.9995 -0.999 -0.9985
cosΘeγ

2

1.5

56

1

0.5

54

0
52

-0.5
-1

50

-1.5
48
50

51

52

53

54 55 56
E e (MeV)(c)Eγ v.s. Ee ,cos Θeγ < −0.99963,|teγ | < 2.443 ps-2-1-0.9995 -0.999 -0.9985
cosΘeγ(d)teγ v.s. cos Θeγ ,51 < Eγ < 55.5 MeV,
52.385 < Ee < 55 MeV

Figure 7.3: Event distributions in Eγ v.s. Ee and teγ v.s. cos Θeγ . Corresponding datasetsare 2009-2010 combined (a)(b) and 2011 only (c)(d), respectively. The signal twodimensional PDFs are superimposed as contours at 1, 1.645, 2 σ as blue dashed, solid,
and dotted lines respectively.

138

58

2

t eγ (nsec)

E γ (MeV)

7.3. RESULT IN ANALYSIS REGION

1.5

56

1

0.5

54

0
52

-0.5
-1

50

-1.5
48
50

51

52

53

54 55 56
E e (MeV)-2-1(a)Eγ v.s. Ee ,cos Θeγ < −0.99963,|teγ | < 2.443 ps-0.9995 -0.999 -0.9985
cosΘeγ(b)teγ v.s. cos Θeγ ,51 < Eγ < 55.5 MeV,
52.385 < Ee < 55 MeV

Figure 7.4: Event distributions in the (Ee , Eγ ) plane(a) and (cos Θeγ , teγ ) plane(b). Corresponding dataset is 2009-2011 combined. Definition of blue contours are same as thosein Fig. 7.3.

Table 7.7: The best fit number of events and the corresponding branching ratios. The uncertainties are MINOS (i.e. profile likelihood ∆NLL method) 1.645σ errors. The numberswith hats show the best fit values.
Year
N̂BG
2009-2010 1198.4+26.1−26.0
1215.2+26.4
2011
−26.3
2009-2011 2413.6+37.1
−37.0

N̂RMD
83.4+12.9−12.8
84.8+13.4−13.3
167.5+24.2
−24.0

N̂sig
0.3+4.1−1.5−1.4+3.8−1.3−0.4+4.8−1.9B̂
8.9 × 10−14−3.5 × 10−13−5.8 × 10−14

Fig. 7.8, are summarized in Table 7.7. The corresponding best estimate of Bµ+ →e+ γ fits
are also shown in the table. The corresponding fit results are shown in Fig. 7.5, 7.6 and7.7.
Figure 7.9 shows the C.L. curves calculated in the Feldman-Cousins method. On the
all dataset, only UL are obtained, namely Nsig = 0 is included in the intervals. Table
7.8 summarizes the confidence intervals. In 2011 dataset, the result shows a smaller Nsig
upper limit than the sensitivity. The diﬀerence between the observed Nsig upper limit
and the sensitivity is shown by arrows in Fig. 7.1(b); the probability of observing, in 2011
dataset, a Nsig upper limit smaller or equal to the observed value is 24%, which is within
the statistical fluctuation.
By analyzing the 2009–2011 combined dataset, the upper limit on the branching ratioof the µ+ → e+ γ decay is calculated as:
B(µ+ → e+ γ) < 5.7 × 10−13 ,
139

(7.1)

80
70
60
50

Events / (0.4 MeV)

Events / (0.24 MeV)

Events / (56 psec)

7.3. RESULT IN ANALYSIS REGION

140
120
100
80

40
60

30

40

20

20

10
-0.5

0

Time (nsec)

0
50

0.5

Events / (4 mrad)

Events / (4 mrad)

0

80
70
60
50

20

10

10
20

0

40

56

50

52

54

56

Gamma Energy (MeV)

58

50

30

0

55

60

20

θ (mrad)

54

70

40

-20

53

80

30

-40

52

90

40

0

51

Positron Energy (MeV)

240
220
200
180
160
140
120
100
80
60
40
20
0
48

-40

-20

0

φ (mrad)

20

40

80
70
60
50
40

Events / (0.4 MeV)

Events / (0.24 MeV)

Events / (56 psec)

Figure 7.5: Likelihood fit result for 2009–2010 combined dataset in the analysis region.
Black markers show the data. Red and pink lines show the best fit for RMD and BG,
respectively. Green line shows the best fit for the signal (almost invisible because of thesmall Nsig ) and blue one shows total best fit.

140
120
100
80
60

150

40

20

50

20

10
-0.5

0

Time (nsec)

0
50

0.5

Events / (4 mrad)

Events / (4 mrad)

200

100

30

0

250

80
70
60
50

51

52

53

54

55

Positron Energy (MeV)

56

0
48

50

52

54

56

Gamma Energy (MeV)

58

70
60
50
40

40
30

30

20

20

10

10
0

-40

-20

0

θ (mrad)

20

40

0

-40

-20

0

φ (mrad)20

40

Figure 7.6: Likelihood fit result for 2011 dataset in the analysis region. The definitionsof each components are the same as in Fig. 7.5.

140

140
120
100
80

Events / (0.4 MeV)

Events / (0.24 MeV)

Events / (56 psec)

7.3. RESULT IN ANALYSIS REGION

250
200

500
400
300

150
200

60

100

40

-0.5

0

Time (nsec)

0
50

0.5

Events / (4 mrad)

Events / (4 mrad)

0

160
140
120
100

51

52

53

54

55

Positron Energy (MeV)

56

0
48

50

52

54

56

Gamma Energy (MeV)

58

140
120
100
80

80
60

60

40

40
20

20
0

100

50

20

-40

-20

0

θ (mrad)

20

40

0

-40

-20

0

φ (mrad)20

40

Figure 7.7: Likelihood fit result for 2009–2011 combined dataset in the analysis region.
The definitions of each components are the same as in Fig. 7.5.

(a)

(b)
Figure 7.8: Profile likelihood curves1417.4. EFFECTS OF SYSTEMATIC UNCERTAINTIESat 90% C.L. Since no significant excesses of the signal events are observed in all datasets,lower limits (LL) are not given in the results.
1

Confidence Level

Confidence Level

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

5

0
0

10
15
Number of signals

5

10
15
Number of signals

(a)

(b)
1

Confidence Level

Confidence Level

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

5

0
0

10
15
Number of signals

(c)

2011
2009-20102009-2011

0.5

1

1.5

×10-122
2.5
Branching ratio(d)Figure 7.9: Confidence level curves (a)–(c) before normalization and (d) after normalization. (a) 2009-2010, (b) 2011, (c)(d) 2009-2011.

7.4

Eﬀects of Systematic Uncertainties

7.4.1

Uncertainties of Parameters

In this section, systematic uncertainties which were not described in Sec. 6.3 are explained.
The position of the calorimeter was measured from the several kinds of data (SeeSec. 4.3.4.1) and the uncertainty is estimated to be 2.4 mrad in θγ . We assign the samevalue to φγ as described in Sec. 4.3.4.1. The uncertainty became smaller than the previous
estimate because the average position including the new measurement using 2011 cosmicray data is closer to the hardware-alignment than the previous average.
The eﬀect of correlations due to the global bias of Ee is included by fluctuating thebias in the flat distribution between 0 and the observed global bias. The systematic142

7.4. EFFECTS OF SYSTEMATIC UNCERTAINTIES

Table 7.8: Confidence intervals (upper limits and lower limits if they are set) on the
2009-2010, 2011 and 2009-2011 datasets calculated from confidence level curves shown in
Fig. 7.9(d). The numbers in the first table are written in Nsig and those in the second
are in B × 1013 . A confidence level at 0 is shown only when N̂sig is positive. The results
are including the systematic uncertainties.
Year
LL (90% C.L.)
2009-2010–2011–
2009-2011–

UL (90% C.L.)
4.7
2.7
4.4

Dataset
LL (90% C.L.)
2009-2010–2011–
2009-2011–

UL (95% C.L.)
5.8
3.7
5.6

UL (90% C.L.)
13
6.7
5.7

C.L. at 00.71––

UL (95% C.L.)
16
9.0
7.2

Table 7.9: Uncertainties associated to the drift chamber alignment and the magnetic
field.
Variable2009µθe [mrad]
1.4µφe [mrad] 1.0µze [cm]0.05µye [cm]
0.013

Year
2010
2
2
0.03
0.008

2011
1.8
3
0.10
0.03

uncertainties associated to the drift-chamber alignment and the magnetic field appear as
non-zero oﬀsets parameters in the two-turn correlation as:µδy = c0 + pxy · δx,(7.2)

where c0 is ideally zero while they are measured to be non-zero in the data for parameterx (x = θe , φe , ze , ye ). The systematic uncertainties shown in Table 7.9 correspond to theoverall biases of the two-turn diﬀerences after corrections.
The uncertainties of Eγ scale, which are calculated by combining the eﬀect from lightyield correction and the fitting of the 55 MeV gamma ray peak in the CEX run, are 0.33,
0.31 and 0.32% in 2009, 2010 and 2011, respectively. The uncertainties of γ position
resolutions are 0.3 mm in (u, v) and 0.7 mm in w. The uncertainty of the center of teγ is
15 psec which comes from the fitting uncertainty. The normalization uncertainty is 4%
as written in Table 6.8. For all PDF shapes extracted from the sideband or calibration
data, namely teγ signal, Eγ signal, Ee signal, Eγ BG, Ee BG and Angle BG, the fitting
errors including the correlations between parameters are used.
Figure 7.10 shows the average PDFs for 2011 data with parameters randomized according to their uncertainties. As shown in this figure, the accidental background PDFs
143

7.4. EFFECTS OF SYSTEMATIC UNCERTAINTIES

are well controlled and the uncertainties are smaller than those in the signal PDFs.

0.045

0.09

0.05
0.04
0.03
0.02
0.01

0.08

0.04

0.07

0.035

0.06

0.03

0.05

0.025

0.04

0.02

0.03

0.015

0.02

0.01

0.01

0.005

-9

× 10

0

-0.5

0

0.5
Te γ (sec)00.05 0.051 0.052 0.053 0.054 0.055 0.056
Ee (GeV)

0.03

0.06

0.025

0.05

0.02

0.04

0.015

0.03

0.01

0.02

0.005

0.01

0

-0.04

-0.02

0

0.02

0.04
θe γ (rad)

0

-0.04

-0.02

0

0.02

0
0.048

0.05

0.052

0.054

0.056 0.058
Eγ (GeV)0.04
φe γ (rad)(a)
Figure 7.10: The average signal and background PDFs for the 2011 dataset. The
parameters are randomized for each curve according to their uncertainties. Dark green(blue) lines are the signal (BG) PDFs without randomizing parameters and green (cyan)lines are PDFs with randomized parameters.

7.4.2

Comparison of Systematic Uncertainties

The total eﬀect of the systematic uncertainties is a 1% change in the upper limit of the
2009-2011 combined dataset. The eﬀects of uncertainties of NRMD and NBG are confirmed
to be much smaller than other uncertainties from the calculation of the upper limit by
fixing NRMD and NBG in the fitting. The eﬀects of other uncertainties are comparedin this section in the following procedure. This study is done only to disentangle thecontribution of each uncertainty. The results are not used for calculating confidence
intervals.
1. Generate 100 pseudo experiments with the nominal PDF, Nsig = Poisson(4.4). The
number of signals of 4.4 is the upper limit obtained from the data.
2. Fit the pseudo experiments with the nominal PDF. And calculate the likelihood
ratio at the best fit and Nsig = 4.4.

144

7.5. FURTHER CHECKS

Table 7.10: Relative contributions of uncertainties to upper limit of B in RMS of ∆∆NLL.
Center of θeγ and φeγ
Positron correlations
Eγ scaleEe biasteγ signal shapeteγ center
Normalization
Eγ signal shapeEγ BG shape
Positron angle resolutions (θe , φe , ze , ye )γ angle resolution (uγ , vγ , wγ )Ee BG shapeEe signal shapeAngle BG shape
Total

0.18
0.11
0.07
0.06
0.06
0.05
0.04
0.03
0.03
0.03
0.03
0.01
0.01
0.00
0.25

3. Do the same thing on the same 100 pseudo experiments using the alternative PDF.
In each experiment, the parameters in the fitting PDF are randomized according totheir uncertainties. For the estimation of the eﬀect of the normalization uncertainty,
PDF parameters are not changed, but the negative-log-likelihood at randomized
Nsig according to the normalization uncertainty is used for calculating ∆NLL.
4. Make a distribution of the diﬀerence of negative-log-likelihood-ratio at 2. and 3. (∆∆NLL).
The third step is repeated three times. The RMS of the distribution of ∆∆NLL is used forcomparing the eﬀects. In this procedure, the eﬀects of the uncertainties on the probability
density function of the likelihood ratio around the upper limit are measured. Table 7.10
shows the relative contributions (RMS of ∆∆NLL) of the uncertainties and Table 7.11
shows those of the correlations in positron observables. The center of θeγ and φeγ are
determined by the relative alignment (See Sec. 4.3.4.4). From this test, it is found thatthe largest contribution to the result is the relative displacement between the positron
spectrometer and the LXe detector.

7.5

Further Checks

We describe a compatibility check between the present and the previous published results[5][4] due to the diﬀerence from the improved analysis method as written in Sec. 6.3, andthe change in the observables by the diﬀerences of the reconstruction algorithms. An
event-by-event check for the highest ranked events are also done.

1457.5. FURTHER CHECKS

Table 7.11: Relative contributions of the uncertainties of the correlations in the positronobservables to the upper limit of B in RMS of ∆∆NLL.
δφeγ :δθeγ correlation
σφe v.s. φe
δφe :δEe correlation
Correlation due to Ee bias
δze :δθe correlation
δye :δEe correlation
δze :δφe correlation

0.09
0.04
0.02
0.02
0.01
0.00
0.00

Table 7.12: Results from category PDF and comparison without systematic uncertainties.
Dataset2009-2010 category PDF
2009-2010 per-event PDF
Dataset2009-2010 category PDF
2009-2010 per-event PDF

7.5.1

Nsig sensitivity
5.4
4.7

Nsig best fit
0.4
0.3

B best fit
1.1 × 10−13
8.5 × 10−14

B sensitivity
1.45 × 10−12
1.3 × 10−12
Nsig UL
B UL
5.7
1.5 × 10−12
4.7
1.3 × 10−12

Result from Category PDF

The likelihood fitting and the full frequentist analysis are performed also with the category
PDFs which were used in the previous analysis. The results in the analysis region are
shown in Table 7.12. In particular for 2009–2010 combined dataset, it is important toanalyze this dataset for a direct comparison. The calculated sensitivity is 5.4 (4.7) in
number of signals for the category (per-event) PDF on the 2009-2010 dataset and it
corresponds to B of 1.45 × 10−12 (1.3 × 10−12 ) after the normalization. By comparing
these two sensitivities, the improvement due to the per-event PDF is directly calculated
to be ∼10%.

7.5.2

Analysis Compatibility

To evaluate the compatibility, we generate pseudo experiments with taking into accountthe diﬀerences of the reconstruction and the eﬃciency based on the 2009–2010 combined
dataset. A pair of toy-MC experiments, corresponding to the previous and the new
analyses, are generated from a common “true” events where the detector responses are
not included. The two experiments are generated with taking into account the correlations between the old and the new reconstruction algorithms extracted from data (seeFig. 7.11). The diﬀerence of the eﬃciency are also included in the generation.
Each toy is fitted with the constant PDFs, then the 90% C.L. upper limits are calculated. Figure 7.12 shows the the distribution of the diﬀerence in upper limit in Nsig
146

7.5. FURTHER CHECKS

2000
2000

1800

3500

1800

1600

1600

3000

1400

2500

1200

2000

1000

800

1500

800

600

1000

1400

1200
1000

600
400

400
500

200
0

-0.4

-0.2

0

0.2

0.4
∆ T (nsec)

1800

1800

1600

1600

1400

1400

1200

1200

1000

1000

800

800

600

600

400

400

200
0

200

0
-3

-2

-1

0

1

-0.02

0

0.02

2
3
∆ E (MeV)

0
-3

-2

-1

0

1

2
3
∆ Eγ (MeV)

200
-0.04

-0.02

0

0.02

0.04
∆ θ (rad)

0

-0.04

0.04
∆ φ (rad)Figure 7.11: Diﬀerence of reconstructed teγ (upper left), Ee (upper center), Eγ (upper
right), θe (bottom left) and φe (bottom center) in the 2009–2010 data with the old andthe new reconstruction algorithms. The definition of ∆x is as follows:∆x = xnew − xold .

from the analysis on the pair of pseudo experiments which generated from a common true
events. In the previous analysis, the observed Nsig upper limit is 7.7 and in the present
analysis it is 5.7 with the category PDF. The probability for observing a diﬀerence equal
to or larger than the observed value is 31%. Therefore the diﬀerence of the calculated Nsig
upper limit between the present and previous analysis can be explained by the diﬀerenceof the reconstruction methods.

7.5.3

Event-by-Event Check and Comparison

Since we use the unbinned maximum likelihood analysis with event-by-event PDF, events
which have high signal-likelihood can be identified and compared. The signal like ordering
is determined by using Rsig , which is defined as
Rsig = log10

Lsig.
0.1 · LRD + 0.9 · LBG(7.3)

An event which has the largest Rsig value and one has the second largest Rsig in 2011
dataset are shown in Fig. 7.13(a) and (b), respectively. In the highest ranked event, one
pileup gamma ray is observed, but it is correctly subtracted. Other event are also checkedby looking light yield distribution, waveforms of each detector and the shape of PDF ineach event and we found no strange behavior.
Figure 7.14 shows five events, which show the highest Rsig value in the new/previous
analysis. As shown in this figure, observables of the most of these events are changed
within the level which we can predicted from Fig. 7.11. The event with third highest
Rsig disappeared in the new analysis while the diﬀerent third highest ranked event labeled with red colored number newly appeared in the new analysis. They are found to147

7.5. FURTHER CHECKS

600
500
400
300
200
100
0
-15

-10

-5

0
5
10
15
NsigUL90 -NsigUL90
New

Old

Figure 7.12: Distribution of ∆Nsig upper limits. The black arrow shows the observed
diﬀerence in the 2009–2010 combined dataset.

(a)

(b)

Figure 7.13: 3D view of reconstructed events which have the first(a) and second(b) largest
Rsig value.

148

teγ (nsec)

Eγ (MeV)7.6. DISCUSSIONS AND PROSPECTS

55
54

1

1
3

2

51

5

1
2

0

4

4

16

50

1

14
5

3

23

52

3
0.2
0.1

2

53

0.3

4
14

16

5

5

23
-0.2

10

3

10

2

-0.1

4

49
52

52.5

53

-0.3
-1

53.5

Ee (MeV)(a)-0.9995-0.999-0.9985cosΘeγ(b)

Figure 7.14: Event-by-event comparison for five high rank events found in new/previous
analysis shown by red/black markers. Numbers represent the event of Rsig ordering.
The signal two-dimensional PDFs are superimposed as contours at 1, 1.645, 2 σ as blue
dashed, solid, and dotted lines respectively.

be appeared/disappeared due to the diﬀerent reconstructed values therefore one cannot
satisfy the selection criteria and the other pass the selection criteria.

7.6

Discussions and Prospects

7.6.1

Discussions

From the analysis on 2009–2011 dataset, we set the most stringent upper limit for theµ+ → e+ γ decay with increased statistics by adding the 2011 dataset and several analysis
improvements. Because of the oﬄine noise reduction, a 6% higher positron eﬃciency anda few % better angular resolutions are obtained. New track fitting also improves theeﬃciency by ∼6% and enables us to use the event-by-event positron PDFs. We obtained
a 10% sensitivity improvement by adding the event-by-event uncertainties in the analysis
instead of using the two category PDFs. The new pileup elimination algorithm gains
the eﬃciency by 7% for gamma rays and reduces the energy tail due to pileup events.
For these reasons, we obtained a 20% sensitivity improvement in 2009–2010 dataset with
the new analysis than that calculated by the previous analysis. From the compatibility
check, the diﬀerence in the observed upper limits is consistent with the expectation witha 31% probability.
As a result, we obtain 20 times more stringent upper limit than that of MEGA [2].
Here we summarize the current experimental bounds (90% C.L.) on other muon CLFV
processes as already introduced in Sec. 1.3.3 together with a new result which we obtained.
B(µ+ → e+ γ) < 5.7 × 10−13
B(µ+ → e+ e+ e− ) < 1.0 × 10−12
B(µ− N → e− N , Au) < 7.0 × 10−13
149

[5]
[11]
[21]

7.6. DISCUSSIONS AND PROSPECTSMSSM with large tanβ
heavy squarks
not heavy sleptons
B-physics constraint

New Result

New Result

(a)

(b)

Figure 7.15: (a) shows B(µ → eγ) v.s. B(τ → µγ) correlation (Fig. 1.3 in Sec. 1.2.2 [16])with the new upper limit of B(µ+ → e+ γ). (b) shows B(µ → eγ) v.s. muon g-2 correlation(Fig. 1.5 in Sec. 1.2.3 [20]) with the new upper limit of B(µ+ → e+ γ).

Table 7.13: Detector performance which were written in the proposal and those obtained
from data taken in 2011.
Parametersin proposal [3]in 2011 data
8
Rµ (Hz)1 × 103 × 107
0.7
1.3
Ee (%, FWHM)
Eγ (%, FWHM)
1.4
4.0(w > 2 cm)/5.6(w < 2 cm)
opening angle (mrad, FWHM)
921(φ)/38(θ)
teγ (ns, FWHM)
0.150.3e (%)9531
γ (%)
70
63

We set the most stringent upper limit in comparison with other CLFV decay searches.
If the photonic contributions are dominant in the new physics (See Sec. 1.3.3), the branching ratio of 10−13 corresponds to 10−15 branching ratio of the µ+ → e+ e+ e− decay andthe µ− N → e− N conversion. Figure 7.15 shows the same plots shown in Fig. 1.3 andFig. 1.5 with the 90% C.L. upper limit obtained in the new result. By using this result,
we set more stringent constraint on many new physics models than that by using theprevious upper limit.
Table 7.13 shows the detector performance which are written in the proposal and thoseobtained in 2011 data. The muon beam is operated at 3 times lower rate in experiment,
it is because the rate capability of the drift chamber is worse than the design value.
Besides, worse resolutions cause the smaller signal-to-background ratio and the higher
rate does not promise the better sensitivity in this case. Since the scattering eﬀect due
to cables and pre-amplifier boards between drift chambers and the timing counter is notconsidered in the proposal, positron eﬃciency shows large discrepancy between the design
value and the measured one. The single event sensitivities (SES) can be calculated by150

7.6. DISCUSSIONS AND PROSPECTS

using numbers which are shown in Table 7.13 by using an equation given as
SES(µ+ → e+ γ) =1,
Rµ · T · (Ωeγ /4π) · e · γ · eγ(7.4)

where Rµ is the muon stopping rate, T is the total eﬀective run time, Ωeγ /4π is ratio ofthe detector acceptance ratio and is calculated to be Ωeγ /4π ' 0.1. The results are as
follows:
SESproposal (T = 200 days LV ) = 1.1 × 10−14 ,SESMEG2011 (T = 200 days LV ) = 1.1 × 10−13 .
Here 200 days of DAQ live time is assumed. The eﬀective branching ratio of the accidental
background can be calculated from Eq. (1.20) and the results are as follows:
Bacc,proposal = 4.1 × 10−15 ,Bacc,MEG2011 = 3.6 × 10−13 .

(7.5)
(7.6)

This means that the SES of the current MEG experiment is already close to the rate of
background and thus the evolution of the sensitivity is not proportional to the statistics.
Therefore, more background reduction is necessary to search for the µ+ → e+ γ decay
below 10−13 .

7.6.2

Prospects

The MEG experiment finished the physics data taking at the end of the summer in 2013.
The final statistics will be doubled by adding the data taken in 2012–2013 to the data
which we already analyzed. There are few additional studies which can possibly improvethe experimental sensitivity as follows.
• The new measurement of the COBRA magnetic field is under preparation in orderto improve the precision down to 0.1%, which is 2–3 times better than that in thelast time measurement. This may help to reduce the systematic uncertainties on
the positron observables.
• We are now studying the AIF background by searching for the candidate of the
AIF gamma ray associated with a positron track which disappears before exiting
the drift chamber volume. By using the AIF analysis, the relative alignment canbe measured by the similar way as used in the cosmic-ray relative alignment (SeeSec. 4.3.4.1). This can help to reduce the uncertainties on the relative angles (θeγand φeγ ) and they are the most largest uncertainties in the present analysis as shownin Table 7.10. Moreover, it would be a potential tool to identify the backgroundgamma rays from the AIF events and this can help to reduce the background ratein the physics analysis.
Figure 7.16 shows the curve of the sensitivity evolution as a function of the accumulated
DAQ time. The sensitivity of 2009–2013 all combined dataset is expected to be reachthe left edge of this sensitivity curve, which corresponds to 5 × 10−13 at 90% C.L. The
151

Branching ratio

7.6. DISCUSSIONS AND PROSPECTS

Obtained 90% UL
90% UL sensitivity

MEGA
-11

±1σ

10

±2σ
3σ discovery

MEG 2011

10-12
MEG 2013

10-13
0

100

200
300
Accumulated DAQ days

Figure 7.16: Sensitivity curve as a function of the accumulated DAQ time. Black solid
curve shows the curve of the expected sensitivity and green and yellow regions express the
1 and 2 σ error bands, respectively. Magenta markers are observed 90% C.L. upper limits.
Red solid curve shows a 3 σ discovery potential of the experiment for the µ+ → e+ γ decay.

1527.6. DISCUSSIONS AND PROSPECTS

analysis using this dataset will be presented in near future with further improvements as
introduced above.
As already mentioned in Sec. 1.3.3, three experiments which will search for the+µ → e+ e+ e− and the µ− N → e− N have been proposed and they are aiming to startphysics data taking in the next decade. The proposed physics sensitivity of the µ+ → e+ e+ e−
experiment, the COMET experiment and the Mu2e experiment are expected to be exceed
the final expected sensitivity of the current MEG experiment (5 × 10−13 ). Therefore, it
is very important to search for the µ+ → e+ γ decay with a better sensitivity in orderto realize the complementary CLFV searches together with those experiments. However,the sensitivity is already close to the background dominated region by using present MEG
detector and is not expected to be improved if we continue the experiment with current
detectors, according to the Eq. (7.6). Therefore, the significant improvements of resolutions with the higher detection eﬃciency are required in order to search for the µ+ → e+ γ
decay with a sensitivity better than the 10−13 level.
For this purpose, we already proposed the upgrade experiment of MEG called MEG II[60], in order to search for the µ+ → e+ γ decay with a 10 times higher sensitivity than thatof MEG. As already discussed, we lose much eﬃciency between the drift chamber and thetiming counter. This will be fixed by replacing 16 moduled drift chambers to the longer
size of the stereo-wire drift chamber with a single gas volume. Rate capability shouldbe also a matter for the MEG II experiment and now the test is ongoing. The photosensors mounted on the inner surface of the LXe detector will be replaced with smaller
size silicon-photomultipliers (SiPM). Because of their small size and high granularity, theposition and the energy resolutions will be improved, in particular, the events in which
gamma rays are converted in narrow region of the detector. Timing counter will be
replaced with the subdivided plastic scintillators with small counter with SiPM read-out.
Since positrons pass through the several counters, the timing resolution will be improvedby using average timing. In MEG II, some optional detectors are also proposed to identify
the background, or to improve the resolutions. The upgrade proposal was approved by
the science research committee of PSI and many detector R&Ds are ongoing now.

153

Chapter 8
Conclusion
The µ+ → e+ γ decay search is one of the powerful probes of the BSM since many new
physics models predict the large branching ratio of the µ+ → e+ γ decay although it isforbidden in the SM. In order to search for the µ+ → e+ γ decay with a sensitivity which
has never been reached before, we developed
1. An innovative gamma ray detector using 900 l LXe and 846 VUV-sensitive PMTs,
2. A positron spectrometer, which is operational in the high rate environment, consistsof(a) A superconducting magnet with graded field specialized to the 52.8 MeV signal
positron,(b) Drift chambers, made of ultra low mass materials, which can measure theposition in O(100) µm precision,(c) Timing counters which can measure the positron timing less than 100 ps precision,and the world most intense DC muon beam at PSI allows us to collect the high statistics
while suppressing the accidental background rate.
In this thesis, we analyze the data taken during 2009–2011 with several improvements
as follows:• by adding the 2011 data, statistics are doubled in comparison with the previous
result of the MEG experiment [4],• gamma ray pileup unfolding helps to reduce the pileup events and increases theeﬃciency by 7%,• apply the oﬄine noise reduction, which gives 6% eﬃciency improvement and resolutions are also improved by a few %,• new Kalman filter gains the statistics by 6% and it enables us to use the positron
per-event PDFs,• per-event PDFs give a 10% sensitivity improvement.
154

As a result, we obtained a sensitivity of 7.7 × 10−13 . This is the first µ+ → e+ γ searchwith a sensitivity below 10−12 . No signal excess is observed in the analysis region andthe maximum likelihood fitting shows the result consistent with a null-signal hypothesis.
Therefore we set 90% C.L. upper limit, which calculated to beB(µ+ → e+ γ) < 5.7 × 10−13 ,on the existence of the µ+ → e+ γ decay. This is a four times more stringent upper limit
than the previous best limit set by MEG.

155

Appendix A
Geometrical Model of Correlations
In order to understand the source of the correlation, the simple geometrical model gives
good enough solutions. Here we assume the helicoidal approximation of the positron
track.
For the extraction of the δφe vs. δEe and δye vs. δEe correlations, a circle shape of apositron track is assumed to be projected on the transverese plane as shown in Fig. A.1.
When the position of the center of the circle defined as (X0 , Y0 ) = (xe + dx , ye + dy ), they
are given as(π)
dx = R cos− φe = R sin φe ,
(A.1)
( π2
)
dx = R sin− φe = −R cos φe ,(A.2)
2
where R is the radius of the circle and φe is the emission angle of the positron. Since thetransverse momentum of a positron is proportional to the radius of the circle, an error of
the positron energy can be approximated to δEe /Ee = δR/R. From Eq. (A.1), the δφevs. δEe correlation is read asδφe = −2 tan φe

δEe
.
Ee

(A.3)

The δye vs. δEe correlation can be calculated from the vertical shift of the circle centerasδye = δY0 − δdy ,= cos φe δR + cos φe δR − R sin φe δφe ,2R δEe=.
cos φe Ee(A.4)

The ze vs. θe correlation is extracted by using the same way as two correlations above.
Figure A.2 shows the track projected on a plane that is parallel to the Z-axis and tangent
to the track helix at the muon vertex. Here dz is the distance between the vertex V andthe position of the track after one turn and the Z0 is absolute Z position at that point.
Therefore the vertex position ze and dz are given as
Z = Z0 − dz ,dz = 2R cot θe .
156

(A.5)
(A.6)

Figure A.1: Extraction of the δφe v.s. δEe and δye v.s. δEe correlations. The red verticalline shows the target intersection with the transverse plane passing from the muon decay
point V = (X, Y ).

157

A.1. IMPLEMENTATION OF CORRELATIONS

Figure A.2: Extraction of the δθe v.s. δze correlation.

Assuming that in this case Z0 does not change, being constrained by the hits, δze is given
as
(
)
1
δEe
δze = −δdz = 2Rδθe − cot θe
.
(A.7)
Ee
sin2 θe

A.1

Implementation of Correlations

In the case of the signal positron, the value of Ee is identical to Ee true = 52.8 MeV.
Therefore the shifts of the angular and vertex variables due to the geometrical constraint
can be corrected by using δEe = Ee Measured − Ee true . Besides, the contributions of the correlations to the angular or positrion resolutions can be disentangled since the momentum
shift from the true value is identical for the signal case. The practical parameterizationsof correlations are described in Sec. 6.3.2.3 in detail.

158

Appendix B
Radiative Muon Decay
The diﬀerential branching ratio of RMD is given in Eq. 6.28. The functions in Eq. 6.28,
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 (1) ,G = G(0) + rG(1) + r2 G(1) ,H = H (0) + rH (1) + r2 H (1) ,(B.1)

where r = (me /mµ )2 .
]8[ 2y (3 − 2y) + 6xy(1 − y) + 2x2 (3 − 4y) − 4x3d []+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),[]
32y(3 − 2y)= 2 −− (3 − 4x) + 2xdx]
8[
+ y(6 − 5y) − 2x(4 + y) + 6x2
d[]+8 x(4 − 3y + y 2 ) − 3x2 (1 + y) + 6dx2 y(2 + y),[]
32 (4 − 3y)
48y
= 2
−3 +
d
x
d

F (0) =

F (1)

F (2)

(B.2)

(B.3)
(B.4)
(B.5)

]
8[
xy(1 − 2y) + 2x2 (1 − 3y) − 4x3d []+4 −x2 (2 − 3y − 4y 2 ) + 2x3 (2 + 3y) − 4dx2 y(2 + y),32
8= 2 (−1 + 2y + 2x) + (−xy + 6x2 ) − 12x2 (2 + y),dd96= − 2,d

G(0) =

G(1)
G(2)

(B.6)
(B.7)
(B.8)
(B.9)

159

]
8[ 2
y (1 − 2y) + xy(1 − 4y) − 2x2 yd []+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)
= 2 −
+ 2y
d
x
8
+ [y(2 − 5y) − xy] + 6dx2 y 2 ,d
96y 48y= − 2 +.
dxdH (0) =

H (1)

H (2)

160

(B.10)(B.11)(B.12)

Acknowledgement
Here I would like to mention many thanks to people who gave me innumerable but manyadvices and supports up to this day. This thesis must not be able to be finished without
their huge amount of contributions.
Prof. Toshinori Mori gave me the chance to study in the MEG experiment and many
precious advices during my research period until now and for writing this thesis.
I also express my great appreciation to Prof. Wataru Ootani, Dr. Toshiyuki Iwamoto,
Dr. Ryu Sawada, Dr. Yusuke Uchiyama, Prof. Satoshi Mihara, Dr. Hajime Nishiguchi for
their technically detailed advices and a large number of supports for me to live in the
Switzerland. I thanks to Dr. Hiroaki Natori, Dr. Yasuhiro Nishimura, Dr. Xue Bai andother people in MEG Japanese group for their many advices and helps for my researchand life. I have learned plenty of things from above people since I first joined to the
MEG-J group.
I greatly thanks to Mr. Daisuke Kaneko and Ms. Miki Nishimura for me to have an
exiting research life in Switzerland together with them.
I would like to express my special thanks to my colleagues who are working on PSI,
especially for Dr. Stefan Ritt, Dr. Malte Hildebrandt and Dr. Peter-Raymond Kettle for
their huge amount of supports and advices.
I am very grateful that I could learn a lot of things about the drift chamber analysis
from Dr. Francesco Renga, Dr. Elisabetta Baracchini and Dr. Fedor Ignatov.
There are more people who are working in the MEG collaboration and I express many
gratitude to all of them. Contributions from all collaborators are tremendous to complete
this thesis.
I also appreciate to many students, staﬀs and the secretaries of the International
Center for Elementary Particle Physics (ICEPP) for their much supports. Especially, I
learned many things from students who enter the ICEPP together with me. We studied
together about the particle physics and often discussed each other.
I am thankful to my family and friends for their helps to my life. Finally, I would
like to express my great appreciation to my wife, Satomi Fujii for her deep love andremarkable supports to me.
December 2013,
Tokyo
Yuki FUJII

162

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