Doctoral T H E S I S

Search for the Lepton Flavor
Violating Decay µ+ ! e+ WithA Sensitivity of 10 12by

Xue Bai
DEPARTMENT OF PHYSICS, SCHOOL OF SCIENCETHE UNIVERSITY OF TOKYO

December 2012

ii

Abstract

The MEG experiment is a precise rare decay search designed to observeµ+ ! e+ or lack thereof as a sensitive low energy probe of new physics.
In this thesis we present an updated result using the data taken in 2010,
which correspond to 1.1 ⇥ 1014 muon decays in the stopping target.
The previous preliminary result based on the 2009 data gave a higher than
expected upper limit of 1.5 ⇥ 10 11 at 90% C.L. with a few possible eventsin the signal region[1][2][3]. To examine this result, we analyzed the 2010
data which has twice higher statistical sensitivity than the 2009 data. In
addition, we improved several aspects of calibration and analysis such asdetector alignment, implementation of correlations in position observables,
improved magnetic field map and improved likelihood analysis. We appliedthese improvements to the 2009 data and confirmed that that the excess
signals still remain but the sensitivity is improved.

We adopted a ”blind analysis” and a maximum likelihood fit. After unblinding the signal region, the number of of µ+ ! e+ decay events in thedata sample is extracted by a maximum likelihood fit. A 90% confidence
interval is then constructed using the Feldman-Cousins technique.

Weevaluated an expected sensitivity of the 2010 data to be 2.2 ⇥ 10 12 by
toy Monte Carlo experiment, which was also confirmed by analysis of theside band data. All the analysis is done by hiding the signal region until
probability density functions for a likelihood fit are settled upon.

The obtained result is consistent with a null hypothesis and we set an
upper limit on the branching ratio

B(µ+ ! e+ ) < 1.7 ⇥ 10 12 at 90% C.L.

(1)

for the 2010 data andB(µ+ ! e+ ) < 2.4 ⇥ 10 12 at 90% C.L.

(2)

for the combined data of 2009 and 2010.
This result exceeds the previous world’s best limit of 1.2 ⇥ 10 11 set by the
MEGA experiment[4].

Contents
List of Figuresiii

List of Tablesv

1 Introduction

1

2 µ ! e Decay

5

2.1

2.2

Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.1

Standard Model and Beyond . . . . . . . . . . . . . . . . . . . .

5

2.1.2

Muon Decay in the Standard Model . . . . . . . . . . . . . . . .

6

2.1.3

SUSY and Lepton Flavor Violation . . . . . . . . . . . . . . . . .

8Experimental Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

History of µ ! e Search . . . . . . . . . . . . . . . . . . . . . .

13

2.2.1
2.2.2
2.2.3

Signal and backgrounds of µ ! e. . . . . . . . . . . . . . . . .

15

Requirements of µ ! e Search . . . . . . . . . . . . . . . . . . .

18

3 MEG Experiment

21

3.1

Detectors Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2

Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.2.1

Proton Accelerators . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.2.2

⇡E5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.2.3

Beam Transport System . . . . . . . . . . . . . . . . . . . . . . .

23

3.3

Stopping Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.4

Liquid Xenon Gamma-ray Detector . . . . . . . . . . . . . . . . . . . . .

28

3.4.1

Liquid Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.4.2

Photomultiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

i

CONTENTS

3.5

3.6

3.7

3.8

3.4.3

Design of LXe Detector . . . . . . . . . . . . . . . . . . . . . . .

32

3.4.4

Cryogenic System . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.4.5

Purification System . . . . . . . . . . . . . . . . . . . . . . . . .

36

Positron Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.5.1

COBRA Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.5.2

Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.5.3

Timing Counter . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Calibration Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.6.1

Setup For ⇡ 0 Run . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.6.2

Cockcroft-Walton Proton Accelerator. . . . . . . . . . . . . . .

53

Electronics and Data Acquisition . . . . . . . . . . . . . . . . . . . . . .

54

3.7.1

Electronics Chain . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.7.2

Data Acquisition System with MIDAS . . . . . . . . . . . . . . .

55

3.7.3

Waveform Digitization with DRS . . . . . . . . . . . . . . . . . .

56

3.7.4

Trigger System . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Simulation and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4 Run 2010

61

4.1

Physics Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.2

⇡ 0 Run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.3

Calibration Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.4

Run Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.5

BTS problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5 Event Reconstruction
5.1

5.2

5.3

67

Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.1.1

Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.1.2

Hit Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.1.3

Track Reconstruction . . . . . . . . . . . . . . . . . . . . . . . .

69

5.1.4

Track Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70Timing Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.2.1

Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.2.2

Hit Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .

72

DCH-TIC Interconnection . . . . . . . . . . . . . . . . . . . . . . . . . .

72

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CONTENTS

5.4

LXe Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.4.1

Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.4.2

Photon Reconstruction . . . . . . . . . . . . . . . . . . . . . . . .

75

5.4.3

Pileup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

6 Calibration
6.1

6.2

6.3

6.4

83

DRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.1.1

Voltage Calibration. . . . . . . . . . . . . . . . . . . . . . . . .

83

6.1.2

Time Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

6.2.1

z-coordinate Calibration . . . . . . . . . . . . . . . . . . . . . . .

84

6.2.2

Time O↵set Calibration . . . . . . . . . . . . . . . . . . . . . . .

84

6.2.3

Time-to-Distance Calibration . . . . . . . . . . . . . . . . . . . .

84

Timing Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

6.3.1

z-coordinate Calibration . . . . . . . . . . . . . . . . . . . . . . .

85

6.3.2

Inter-Bar Time O↵set Calibration . . . . . . . . . . . . . . . . .

85

LXe Gamma-ray Detector . . . . . . . . . . . . . . . . . . . . . . . . . .

85

6.4.1

PMT calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

6.4.2

Non-Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

6.4.3

Energy Scale and Light yield . . . . . . . . . . . . . . . . . . . .

94

7 Performance
7.1

99

Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

7.1.1

Photon Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

7.1.1.1

Fiducial Volume and Acceptance . . . . . . . . . . . . .

99

7.1.1.2

Pileup Rejection . . . . . . . . . . . . . . . . . . . . . .

99

7.1.1.3

Cosmic Ray Veto

7.1.2

. . . . . . . . . . . . . . . . . . . . . 100

Positron Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.1.2.1

Fiducial Volume . . . . . . . . . . . . . . . . . . . . . . 101

7.1.2.2

Drift Chamber Hit Pattern . . . . . . . . . . . . . . . . 101

7.1.2.3

Quality of Track Fit . . . . . . . . . . . . . . . . . . . . 102

7.1.2.4

Vertex Cut . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.1.2.5

DCH-TIC Matching Cut . . . . . . . . . . . . . . . . . 102

7.1.2.6

Ghost Track Selection . . . . . . . . . . . . . . . . . . . 103

iii

CONTENTS

7.2

Gamma Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3

Positron Energy

7.4

Relative Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.5

7.6

7.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.4.1

Positron Emission Angle . . . . . . . . . . . . . . . . . . . . . . . 107

7.4.2

Muon Decay Vertex . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4.3

Gamma ray Position . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.4.4

Combined Resolution . . . . . . . . . . . . . . . . . . . . . . . . 110

Relative Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5.1

Positron Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.5.2

Gamma-ray Time

7.5.3

Gamma-Positron Relative Time . . . . . . . . . . . . . . . . . . . 111. . . . . . . . . . . . . . . . . . . . . . . . . . 110

Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.6.1

Positron Detection Efficiency . . . . . . . . . . . . . . . . . . . . 113

7.6.2

Gamma ray Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 113

Summary of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8 Improvements for 2010 Data
8.1

115

Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.1.1

DC Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.1.2

Relative Alignment Between DC and LXe . . . . . . . . . . . . . 116

8.1.3

Target Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.2

Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.3

Magnetic Field Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.4

Likelihood Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.5

Update on 2009 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9 Physics Analysis on Data 2010
9.1

9.2

129

Overview of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.1.1

Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9.1.2

Pre-Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.1.3

Blinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.1.4

Definition of Analysis Window . . . . . . . . . . . . . . . . . . . 131

Maximum Likelihood Analysis
9.2.1. . . . . . . . . . . . . . . . . . . . . . . 132

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

iv

CONTENTS

9.3

Probability Density Functions . . . . . . . . . . . . . . . . . . . . . . . . 135
9.3.1

Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.3.2

Radiative Muon Decay . . . . . . . . . . . . . . . . . . . . . . . . 142

9.3.3

Accidental Background . . . . . . . . . . . . . . . . . . . . . . . 143

9.3.4

Confidence Interval . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.4

Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9.5

Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.6

Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.7

9.8

9.6.1

Background Evaluation By Sideband Analysis . . . . . . . . . . . 155

9.6.2

Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Maximum Likelihood Fit on the Data . . . . . . . . . . . . . . . . . . . 163
9.7.1

Upper Limit on Branching Ratio . . . . . . . . . . . . . . . . . . 164

9.7.2

Event Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

10 Prospects

171

Bibliography

173v

CONTENTS

vi

List of Figures
1.1

Event distributions in the analysis region obtained by preliminary analysis of the 2009 data[1][2][3]. The contours show 1, 1.64 and 2 regions of
the signal PDF, which cover 39, 74 and 87% of probability, respectively.
In the left plot, selections in te and ✓e , each of which is 90% efficienton the signal, are applied. In the right plot, a selection in Ee (90% on
the signal) and a selection in E (73% on the signal) are applied. . . . .

2.1

Feynman diagram describing the µ ! e

2decay in the SM with aneutrino-mass extension. The internal fermion line is a neutrino mass
eigenstate. By diagonalizing the neutrino propagator in flavor space,o↵-diagonal flavor couplings, Uµi and Uei , are induced. . . . . . . . . . .

2.2

Predictions for Br(µ ! e ) at various values of tan
the left-handed selectron mass for MR

2.3= 1013 GeV.[5]8as a function of. . . . . . . . . .

10Predictions for Br(µ ! e ) are shown as a function of the universal

gaugino mass for two cases of tan , scanning an LHC relevant space inthe parameters describing the Planck scale masses. Both the PMNS case

2.4
2.5

(green) and the CKM case (red) are explored. . . . . . . . . . . . . . . .

12

Historical progress of µ+ ! e+14

Schematics of µ+ ! e+search.[6] . . . . . . . . . . . . . . . . .

event signature (a), and two types of back-grounds (b)(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6Positron energy spectrum of unpolarized µ+ ! e+ ⌫e ⌫µ decay (Michel

15

spectrum). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.7

Photon energy spectrum of unpolarized mu+ ! e+ ⌫e ⌫¯µ ) decay. . . . .

18

3.1

Overview of the MEG experiment. . . . . . . . . . . . . . . . . . . . . .

22

vii

LIST OF FIGURES

3.2

Layout of detectors in the MEG experiment.

. . . . . . . . . . . . . . .

22

3.3

Proton Accelerators at PSI. . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.4

590MeV ring cyclotron at PSI. . . . . . . . . . . . . . . . . . . . . . . .

25

3.5

⇡E5 beamline components. . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.6

Pion and muon beam intensity at ⇡E5. . . . . . . . . . . . . . . . . . . .

26

3.7

Schematic view of MEG beam transport system. . . . . . . . . . . . . .

26

3.8

Muon stopping target. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.9

Mounting position of the target. . . . . . . . . . . . . . . . . . . . . . .

27

3.10 Photon reaction in liquid xenon. . . . . . . . . . . . . . . . . . . . . . .

29

3.11 Diagram of xenon phase[7]. . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.12 PMT for the LXe detector. . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.13 Schematic view of the liquid xenon detector. . . . . . . . . . . . . . . . .

33

3.14 Development view of the faces of liquid xenon detector. Each face isequipped with 216, 234, 144 ⇥ 2 and 54 ⇥ 2 PMTs for inner, outer,

lateral and top or bottom. . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.15 Liquid xenon detector . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.16 PMTs mounted inside the LXe detector. . . . . . . . . . . . . . . . . . .

35

3.17 Cryogenic system of the LXe detector. . . . . . . . . . . . . . . . . . . .

36

3.18 Liquid xenon tank and gaseous purification system. . . . . . . . . . . . .

37

3.19 Gas xenon storage tanks. . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.20 A photograph of COBRA magnet. . . . . . . . . . . . . . . . . . . . . .

38

3.21 Comparison between COBRA spectrometer and uniform magnetic field.
(a) and (c) show trajectories of positrons emitted transverse to the field.
The uniform field makes many turns inside the detector, whereas thegradient field sweep the positron out of the detector much more quickly.
(b) and (d) show trajectories of mono-energetic positrons emitted atvarious angles. In the uniform field, the bending radius depends on theemission angle, whereas it is independent in the gradient field. . . . . . .

39

3.22 Rate of Michel positrons per cm2 per second as a function of radius
assuming muon decay rate of 3 ⇥ 107 /sec. . . . . . . . . . . . . . . . . .

40

3.23 Magnetic field (Bz ) along beam axis (x = y = 0). . . . . . . . . . . . . .

40

3.24 Map of COBRA magnetic field. . . . . . . . . . . . . . . . . . . . . . . .

41

viii

LIST OF FIGURES

3.25 A photograph of the 16 drift chamber modules mounted on the support
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.26 Drift chamber module. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.27 Section of wires in two layers. . . . . . . . . . . . . . . . . . . . . . . . .

43

3.28 Schematic of vernier pattern on the cathode pads. . . . . . . . . . . . .

44

3.29 Field map and drift lines of drift chamber calculated by the GARFIELD
simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.30 Picture of timing -counter. . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.31 Design of timing

counter. . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.32 Timing z-counter and APDs. . . . . . . . . . . . . . . . . . . . . . . . .

49

3.33 Kinematics of the ⇡ 0 !

50

3.34

decay. . . . . . . . . . . . . . . . . . . . . . .

Overview of the setup in ⇡ 0 run.

. . . . . . . . . . . . . . . . . . . . . .

51

3.35 Liquid hydrogen target. . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.36 NaI detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.37 CW accelerator.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.38 Energy peaks in the LXe detector with CW accelerator. . . . . . . . . .

54

3.39 Schematic of data flow and electronics. . . . . . . . . . . . . . . . . . . .

55

3.40 Schematic of DAQ system. . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.41 Schematic of DRS principle. . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.42 Structure of MEG software. . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.1

Integrated muons during the MEG data taking in 2010. . . . . . . . . .

62

4.2

Degrader system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.3

DRS4 board. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.1

The six waveforms associated with a hit. The horizontal red lines are
the calculated baselines, the vertical red lines are the leading edge time,
and the region between the two dashed blue lines is the expected region
where triggered signal appear. . . . . . . . . . . . . . . . . . . . . . . . .

5.2

68

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

5.3

70

Event display of positron tracking. Fitting is performed with Kalmanfilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

71

LIST OF FIGURES

5.4

Figure 5.10: An example of the waveforms used in the timing counterreconstruction. The black line is a DRS output, which is fitted to atemplate (red). The blue line is the NIM pulse output, which is fitted
to another template (green). The time delay of the NIM pulse withrespect to the DRS waveform is due to an electronic delay in the double
threshold discriminator. . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.5

Typical waveform of a PMT in LXe detector. . . . . . . . . . . . . . . .

74

5.6

A high-pass filtered waveform. . . . . . . . . . . . . . . . . . . . . . . . .

74

5.7

A saturated waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

5.8

Charge estimation with ToT. Pulse shape of template waveform is used

5.9

to get the conversion factor from ToT to charge. . . . . . . . . . . . . .

76

Correlation between in, max and sum of inner PMT Npho for w < 3 cm.

79

5.10 Linearity plot of E vs Nsum . The five points are given from 4.4 and11.7 MeV from CW-B run, 17.6 MeV from CW-Li run, and 54.9 and82.9 MeV from ⇡ 0 run. Red line is the best-fit linear function whose
intercept is fixed to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Distribution of normalized

80

2 of the time fitting on data (dotted line)

and signal MC without pileup (solid line). . . . . . . . . . . . . . . . . .

81

5.12 Photon distribution before and after pileup elimination. . . . . . . . . .

82

6.1

Inside view of LXe detector shows 241 Am source on wire and LED on
the wall. Three LEDs with di↵erent attenuations are mounted on lateralfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Closeup of 241 Am source on wire.

86

Diameter is 100 µm and length isabout 2 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

6.3

Gain calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.4

Gain shift of a worst PMT after beam blocker opened in upper figure
and closed in lower. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

6.5

Gain - HV curve and fit. . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

6.6

History of gain during MEG and ⇡ 0 run. . . . . . . . . . . . . . . . . . .

91

6.7

Reconstructed positions of alpha wires. . . . . . . . . . . . . . . . . . . .

92

6.8

Comparison of alpha peaks between MC and data. . . . . . . . . . . . .

92

6.9

Stability of averaged QE. . . . . . . . . . . . . . . . . . . . . . . . . . .

93

x

LIST OF FIGURES

6.10 Non-uniformity correction on uv plane. . . . . . . . . . . . . . . . . .

94

6.11 Non-uniformity correction on uv plane. . . . . . . . . . . . . . . . . .

94

6.12 Cosmic ray peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

6.13 Gamma ray peaks from AmBe sources. . . . . . . . . . . . . . . . . . . .

96

6.14 Alpha peak of all sources. . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6.15 Ni peak during beam time. . . . . . . . . . . . . . . . . . . . . . . . . .

97

7.1

The accepted regions of the LXe detector are shown in red in the u-v
plane on the left and the u-w plane on the right. . . . . . . . . . . . . . 100

7.2

Distribution of the subtracted energy in the pileup elimination. Red
filled entries are rejected by the selection criteria. . . . . . . . . . . . . . 101

7.3

Cosmic ray cut. Black plot shows the events in the acceptance taken
without beam. Red one shows MC signal events. Blue lines show theselection criteria.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.4

Energy peak of 54.9MeV in ⇡ 0 run. . . . . . . . . . . . . . . . . . . . . . 104

7.5

Energy resolution at 54.9 MeV on inner face (w

7.6Energy Scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.7

Fitting of a Michel spectrum.The dashed black line is the theoretical

2 cm). . . . . . . . . 105

spectrum and the dashed blue line is the resolution function. The acceptance curve is shown in the bottom plot. . . . . . . . . . . . . . . . . 108
7.8

A fit to the distribution of energy di↵erence between two turns. . . . . . 109

7.9

Lead cubes and the support. . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.10 PMT grouping. Blue circles belong to the ’odd’ group while orange
circles belong to the ’even’ group. . . . . . . . . . . . . . . . . . . . . . . 111
7.11 timing di↵erence between LXe datector and reference counter . . . . . . 112
7.12 Time di↵erence between positron and gamma ray in 2010.
8.1. . . . . . . 112

Definition of variables used in CR analysis. Blue lines are the reconstructed trajectory by DCH. . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.2

Trigger patches used in CW runs. . . . . . . . . . . . . . . . . . . . . . . 117

8.3

Target markers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.4

Position of the holes when misalignment happens. Dotted line shows theassumed target and solid line is the actual target.

8.5

Extraction of thee vs. . . . . . . . . . . . 119

E and Y vs E correlations. . . . . . . . . . 120

xi

LIST OF FIGURES

8.6

Extraction of the Z vs

e and

Z vs E correlations.

. . . . . . . . . 120

8.7

Illustration of the double turn track method. . . . . . . . . . . . . . . . 121

8.8

Preliminary result of 2009 data before applying the improvements. (a)(e) show the summed PDFs weighted with best fit values of each types.
The blue lines are total sums of all PDFs. The green, red, and magentalines show the signal, RD, and background PDF respectively. The black
dots are the best fit values. (f) shows the contour plot on Nsig

NRD .

The dashed, solid, and dotted lines show contours of the likelihood function at 1, 1.645, 2 sigma respectively[1][2][3]. . . . . . . . . . . . . . . . . 124
8.9

Event distributions in the analysis region.[2][1][3] The contours show 1,
1.64 and 2 regions of the signal PDF, which cover 39, 74 and 87% ofprobability, respectively. In the left plot, selections in te and ✓e , eachof which is 90% efficient on the signal, are applied. In the right plot, aselection in Ee (90% on the signal) and a selection in E (73% on thesignal) are applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.10 Updated result of sensitivity in 2009.

. . . . . . . . . . . . . . . . . . . 125

8.11 Updated result of 2009 data. The green, red, magenta and blue lines
show the signal, RD, background and total sums of PDF respectively. . 126
8.12 Updated event distributions in the analysis region for 2009 data. In theleft plot, selections in te and ✓e , each of which is 90% efficient on thesignal, are applied (|te | < 0.278 nsec, |✓e | > 178.34 ). In the right plot,a selection in Ee (90% on the signal) and a selection in E (73% on thesignal) are applied (51 < E < 55 MeV, 52.34 < Ee < 55 MeV). . . . . . 126
8.13 Confidence level curve in 2009 data. . . . . . . . . . . . . . . . . . . . . 127
9.1

The blind box, the analysis window, both timing sidebands, and the low
gamma energy sideband are shown on the (E , te ) plane. . . . . . . . . 131

9.2

Probability(E , Ee ,densitye , ✓e , te )functions(PDFs)

ofeachobservableaveraged over all events within the analysis

region. The green, red and magenta lines show PDFs of signal, radiative
muon decay and accidental background (S, R, B), respectively. . . . . . 135

xii

LIST OF FIGURES

9.3

Residual energy distribution in ⇡ 0 due to opening angle resolution. The
true energy is corrected by the energy-opening angle correlation to thekinematic edge using the reconstructed opening angle. (a) is for the 54.9
MeV side and (b) is for the 83.9 MeV side. . . . . . . . . . . . . . . . . 139

9.4

Divisions in (u, v) for signal PDF extraction. Separate but same color
regions are combined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9.5

Examples of ⇡ 0 54.9 MeV peak fit. Red lines are the best fit functions.
Green and yellow bands show the confidence intervals of the function at
68% and 95% C.L. calculated with the covariance matrix evaluated in
fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9.6

BG distribution on (✓e , phie ) measured in time sidebands. The redbox is the analysis region. . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.7

Position dependence of angular distribution for 2010 data. . . . . . . . . 144

9.8

BG ✓e distributions in u slices for 2010 data. The black line is thebest fitted function. The green and yellow band show the 68% and
95% confidence intervals respectively. The red arrow shows the analysis
region in the likelihood fit.. . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.9

BG

e

distributions in v slices for 2010 data. The black line is thebest fitted function. The green and yellow band show the 68% and
95% confidence intervals respectively. The red arrow shows the analysis
region in the likelihood fit. . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9.10 Divisions in (u, v) for background PDF extraction. . . . . . . . . . . . . 148
9.11 Examples of background fit for 2010 data. Black lines are the best fit
functions and blue bands show the confidence intervals at 95% C. L.
calculated with the covariance matrix. . . . . . . . . . . . . . . . . . . . 149
9.12 Components of background. Data points show the-ray background

spectrum taken from the sideband physics data. Green line is the fitted
spectrum from RMD and AIF. Blue line is the green line plus e↵ectsof pileup. Black line is the cosmic ray spectrum and the red line is thefitted result of all background.

. . . . . . . . . . . . . . . . . . . . . . . 150

xiii

LIST OF FIGURES

9.13 Likelihood fit in positive time sideband. Summed PDFs weighted withbest fit numbers of each types are superimposed on each plot. The green,
red, magenta and blue lines show the signal, RD, background and totalsums of PDF respectively.

. . . . . . . . . . . . . . . . . . . . . . . . . 158

9.14 Likelihood fit in negative time sideband. Summed PDFs weighted withbest fit numbers of each types are superimposed on each plot. The green,
red, magenta and blue lines show the signal, RD, background and totalsums of PDF respectively.

. . . . . . . . . . . . . . . . . . . . . . . . . 158

9.15 Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red,magenta and blue lines show the signal, RD, background and total sumsof PDF respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.16 Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red,magenta and blue lines show the signal, RD, background and total sumsof PDF respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.17 Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red,magenta and blue lines show the signal, RD, background and total sumsof PDF respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.18 Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red,magenta and blue lines show the signal, RD, background and total sumsof PDF respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.19 Updated result of sensitivity in 2009.

. . . . . . . . . . . . . . . . . . . 161

9.20 Distribution of Nsig upper limit at 90% confidence level in toy experiments for 2010 data. A median of this distribution (4.8) is defined as a
90% confidence sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.21 Distribution of Nsig upper limit at 90% confidence level in toy experiments for combined data of 2009 and 2010. A median of this distribution(5.2) is defined as a 90% confidence sensitivity. . . . . . . . . . . . . . . 162
9.22 Updated result of 2009 data. The green, red, magenta and blue lines
show the signal, RD, background and total sums of PDF respectively. . 163

xiv

LIST OF FIGURES

9.23 Result of maximum likelihood fit for 2010 data. Summed PDFs weighted
with best fit numbers of each types are superimposed on each plot. The
green, red, magenta and blue lines show the signal, RD, background andtotal sums of PDF respectively. . . . . . . . . . . . . . . . . . . . . . . . 164
9.24 Result of maximum likelihood fit for combined data of 2009 and 2010.
Summed PDFs weighted with best fit numbers of each types are superimposed on each plot. The green, red, magenta and blue lines show thesignal, RD, background and total sums of PDF respectively. . . . . . . . 165
9.25 Confidence level curve for 2009, 2010 and combined data. . . . . . . . . 166
9.26 Updated event distributions in the analysis region for 2009 data. In theleft plot, selections in te and ✓e , each of which is 90% efficient on thesignal, are applied (|te | < 0.278 nsec, |✓e | > 178.34 ). In the right plot,a selection in Ee (90% on the signal) and a selection in E (73% on thesignal) are applied (51 < E < 55 MeV, 52.34 < Ee < 55 MeV). . . . . . 167
9.27 Event distribution in the analysis region for 2010 data. The signal 2D
PDFs are superimposed as contours at 1, 1.64, 2 sigma as blue solid,
dot-dashed, and dashed lines respectively. The number shows the rank
by S/(S + R + B). In plot (a), selections in te and ✓e , each of which
is 90% efficient on the signal, are applied (|te | < 0.278 nsec, cos|✓e | <0.9996). In plot (b), a selection in Ee (90% on the signal) and aselection in E (73% on the signal) are applied (51 < E < 55 MeV,
52.34 < Ee < 55 MeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.28 Event distribution in the analysis region for combined data of 2009 and2010. The signal 2D PDFs are superimposed as contours at 1, 1.64,
2 sigma as blue solid, dot-dashed, and dashed lines respectively. The
number shows the rank by S/(S + R + B). Events with rank (2,3,4)
are from 2009, events with rank (1,5) are from 2010 data.. In plot (a),selections in te and ✓e , each of which is 90% efficient on the signal, areapplied (|te | < 0.278 nsec, cos|✓e | <0.9996). In plot (b), a selectionin Ee (90% on the signal) and a selection in E (73% on the signal) areapplied (51 < E < 55 MeV, 52.34 < Ee < 55 MeV). . . . . . . . . . . . 168
10.1 Future prospects of sensitivity. . . . . . . . . . . . . . . . . . . . . . . . 171

xv

LIST OF FIGURES

xvi

List of Tables
2.1

Decay modes and branching ratios of muon. . . . . . . . . . . . . . . . .

6

3.1

Properties of LXe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.2

Properties of the PMT. . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

7.1

A table of inefficiencies for each cut. The inefficiencies are calculated as
the fraction of events rejected by making only the cut with respect tomaking no cuts. The

acceptance cut inefficiency is calculated when

positron cuts, pileup cut, and cosmic ray cut are already applied. . . . . 104
7.2

Performance in 2010. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.1

Resolution in 2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2

Confidence intervals on 2009 data. . . . . . . . . . . . . . . . . . . . . . 127

9.1

Ee parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.2

Fit results of time response extraction from E -sideband. . . . . . . . . 138

9.3

Fractions of background components in the signal region. All the rest
come from pileup or reconstruction tail. . . . . . . . . . . . . . . . . . . 150

9.4

Systematic uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.5Best fit results of number of signal, radiative muon decay and background
events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.6

Upper limits at 90% C.L. in number of signal events and branching ratio
B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.7

Fit results in angle sidebands. . . . . . . . . . . . . . . . . . . . . . . . . 157

9.8

Confidence intervals on 2009 data. . . . . . . . . . . . . . . . . . . . . . 165

9.9

Confidence intervals on 2010 data. . . . . . . . . . . . . . . . . . . . . . 165

xvii

LIST OF TABLES

9.10 Confidence intervals on combined data of 2009 and 2010. . . . . . . . . . 166

xviii

1

Introduction
Although the Standard Model has been supported by essentially all the experimental
results, it is thought to be a low energy approximation of a more fundamental theoryof particle physics. New physics is thought to come in at a mass scale that has notbeen experimentally explored yet.
One way to explore new physics scenarios beyond the SM is to look for a lepton
flavor violating decay. While lepton flavor symmetry is broken among neutral leptons
by observations of neutrino oscillations, the process among the charged leptons, i.e.
charged-lepton-flavor violation (cLFV) is considered to have too small rates to detectat present. Some models of new physics incorporating supersymmetry, however predictlarge branching ratios that are near the current experimental limits.
The MEG experiment is a µ+ ! e+search aimed at either a discovery or

improving the limit set by the MEGA experiment of Br(µ+ ! e+ ) < 1.2 ⇥ 10 11(90% CL) [4] down to O(10 13 ). The MEG experiment achieves a higher sensitivity
with relatively low accidental background using the world’s most intense continuous
muon beam at Paul Sherrer Institut (PSI), a positron spectrometer with a specially
graded magnetic field, and an innovative 900 liter liquid xenon gamma-ray detector.
In 2009 we took data for two months and had a preliminary result of higher than
expected upper limit with a few possible events in the signal region[1][2][3].

11. INTRODUCTION

The preliminary result of the maximum likelihood fit is+38(Nsig , NRD , NBG ) = (3.0+6.9 , 35+24
22 , 332 36 ), Nobs = 370(1.1)

where the asymmetric errors are 1.645 sigma of MINOS errors obtained fromMINUIT[8]. The negative error of Nsig is not written as it is outside of the defined
fitting region.
The single event sensitivity in 2009 is estimated to be
S2009 = 6.1 ⇥ 10
Analysis of the MEG experiment to search for µ + ! e+ γ decays

12

Ryu Sawada(1.2)

The upper limit of branching ratio ist eγ (nsec)

58
57
56

2
1.5
1

55
54

0.5 6

53

0

81
5
23

52
51

-0.5

7

-1

50

-1.5

49
48
50

5 1
3 10
2
8 9

51

52

53

54

55
56
Ee (MeV)

-2
-1

(a)

-0.9995-0.999-0.9985cosΘeγ(b)

Figure 1: Event distribution in the analysis region. The contours show 1, 1.64 and 2 σ regions of the signal

Figure1.1: coverEvent
the analysis
region(a)obtained
bypositron
preliminary
analysisof
PDF, which
39,distributions
74 and 87 % ofinprobability,
respectively.
Photon andenergy. Cuts
to90 % of timeTheand angle
signal show
PDF arefor the(b) Cosineof opening
the select2009aboutdata[1][2][3].
contours
1,applied
1.64 and2 plot.
regionsof thesignal angle
PDF,andwhichtime difference. Cuts to select about 90 % of photon and positron energy signal PDF are applied for the plot.
cover
39, 74 and 87% of probability, respectively. In the left plot, selections in te and ✓e ,Highly ranked events in terms of the relative signal likelihood are numbered correspondingly.
each of which is 90% efficient on the signal, are applied. In the right plot, a selection inEe (90% on the signal) and a selection in E (73% on the signal) are applied.
Figure 2 shows projections of the fitting result to each variable. The fitting was done on 370
observed events, and the best estimates in the analysis window are Nsig = 3.0 and NRMD = 35+24
22 .
The best estimate of NRMD is consistent with the expectation estimated from the Eγ sideband to
be 32±2. The confidence interval was evaluated using the unified classical frequentist method[4].
Systematic uncertainties are included by fluctuating parameters of PDFs in fittings of Monte Carlosimulations to calculate confidence levels. The2upper limit of the number of signals is 14.51 ,and Nsig = 0 is included in the interval. Using the norminalization factor, the upper limit of the
branching ratio is set as
B(µ + ! e+ γ )11

PoS(ICHEP 2010)263

Eγ (MeV)The sensitivity is estimated as an average upper limit at 90 % confidence level (C.L.) from anensemble of a large number of simulation experiments to be 6.1 ⇥ 10 12 . The upper limits in Teγ+
1112 , andBr(µ!
e+ ) <with
1.5 the⇥ 10
at 90% C.L.
(1.3)
sidebands is 4 ⇠ 6 ⇥ 10
is consistentsensitivity.
Events around the analysis region were unmasked after the calibration, the optimization of the
The
event
distributions
are shownin inFigure
1.1. The
events arelabeledtheanalysisalgorithms
and the backgroundstudy
the sidebands
are completed.
Figure1 showswithadistribution of events after unmasking.
order of the likelihood ratio.
A physics analysis was performed on events in 48  Eγ  58 MeV, 50  Ee  56 MeV, |Teγ | 
0.7 nsec, |φeγ |  50 mrad and |θeγ |  50 mrad.

In 2010, we took data for 67 days and observed 1.1 ⇥ 1014 muon decays.

It

correspond to twice more statistics than the 2009 data. In addition, we improved
several aspects of calibration and analysis such as detector alignment, implementationof correlations in position observables, improved magnetic field map and improved
likelihood analysis.

In this thesis we analyze the 2009 and 2010 data with these

improvements to examine the possible excess seen in the preliminary 2009 data
analysis.
The theme of this thesis is a search for µ+ ! e+using the MEG data taken in2010. The analysis is structured as (a) standard analysis (b) improvements in 2010(c) updated result of 2009 data (d) result of 2010 data (e) combined result of 2009 and2010.
In Chapter 2, the standard model and SUSY-GUT theories are briefly introduced as wellas the experimental searches for µ+ ! e+ . In Chapter 3, the experimental apparatus
of MEG is described in detail. Methods for event reconstruction are described in
Chapter 4. Calibration is discussed in Chapter 5. The performance of the detector isevaluated in Chapter 6. Aspects and conditions of runs in 2010 are given in Chapter 7
and the improvements made in 2010 are discussed in Chapter 8. The analysis to search
for the µ+ ! e+ decay is described in Chapter 9 and the result is discussed. Future
perspectives are discussed in Chapter 10.

3

1. INTRODUCTION4

2µ ! e Decay
We discuss theoretical and experimental features of µ ! e

decay in this chapter.

Firstly, the limits of Standard Model and various scenarios for new physics along with
their predictions for Br(µ ! e ) are discussed. Then the history of µ ! e searchand its experimental features are summarized in the next section.

2.1

Theoretical Background

2.1.1

Standard Model and Beyond

The behavior of fermions and gauge bosons are well described in the Standard Model(SM). Various features have been experimentally verified under TeV-scale energy.
Nonetheless, it contains some fundamental and theoretical problems and is considered
to be a low-energy approximation of a more fundamental theory.
The SM does not account for the existence of three generations of quarks andleptons. This problem is known as a flavor puzzle of the SM. The Standard Model also
lacks a dark matter candidate.
Another problem of the SM is the hierarchy problem. It is related to the hugedi↵erence of energy scales between the weak scale of O(100 GeV) and the reduced
Planck scale of O(1018 GeV), where quantum gravitational e↵ects become important.

5

2. µ ! E

DECAY

Many extended models from SM have been proposed, such as supersymmetry andextra dimensions. One way to verify these models is through higher energy particle
collision, while a search for rare mixings of lepton flavor, which is sensitive to many
new physics models, provides another approach.

2.1.2

Muon Decay in the Standard Model

Muon is the second generation charged lepton that interacts through electromagnetic
and weak interactions. It also couples to the Higgs boson. Muons decay through the
charged-weak current interaction mediated by Wµ± gauge bosons. The decay modes
and their branching ratios are summarized in Table 2.1. The dominant mode is Michel
decay, µ ! e⌫µ ⌫¯e .
Decay mode

Branching ratio

Referenceµ ! e ⌫µ ⌫¯eµ ! e ⌫µ ⌫¯eµ ! e ⌫µ ⌫¯e e+ e

⇠ 100%(1.4 ± 0.4)%(3.4 ± 0.4) ⇥ 10 5

[9]
[10]

µ ! e ⌫e ⌫¯µµ !eµ ! e e+ e
µ !e

< 1.2%< 1.2 ⇥ 10 11< 1.0 ⇥ 10 12< 7.2 ⇥ 10 11

[11]
[12]
[13]
[14]

Table 2.1: Decay modes and branching ratios of muon.

Neutrino Mass and Mixing
In the minimum standard model where neutrinos are massless, lepton flavor isconserved and occurrence of processes such as µ ! e is prohibited. The observationsof neutrino oscillations however indicate neutrino mass terms in the Lagrangian andphysics beyond the SM.
If neutrinos are not massless, their mass matrix will be non-diagonal and complex
just as the case for quark sector. The mass eigenstates are di↵erent from the flavor

62.1 Theoretical Backgroundeigenstates:⌫↵ = ⌃i U↵i ⌫i ,(2.1)

where ⌫↵ = ⌫e , ⌫µ , ⌫⌧ are flavor eigenstates and ⌫i = ⌫1 , ⌫2 , ⌫3 are mass eigenstates with
masses m1 , m2 , m3 . U is a unitary matrix known as Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix.
A neutrino of flavor ↵ at t = 0 evolves after a time interval of t as|⌫↵ (t) >= ⌃i U↵i e iEi t |⌫i (0) > .

(2.2)

The probability of finding flavor ⌫ in ⌫↵ beam at a distance x from the source is givenby2

2P⌫↵ !⌫ = ⌃i |U↵i | |U i | + ⌃i6=j U↵i U ⇤ i U ⇤↵j U j cos
where Lij = 2⇡/(Ei

Ej ) ' 4⇡p/|m2i

✓

2⇡x
Lij

◆

,

(2.3)

m2j | is the oscillation length. For neutrino

oscillation, non-zero neutrino masses and mixing angles are needed.

Neutrino mixing contributes to transition between charged leptons at the one-loop
level as shown in Figure 2.1. However the LFV processes in charged lepton (cLFV)
are severely suppressed because of the GIM mechanism. The branching ratio in theminimum extension of SM is given as
Br(µ ! e ) =3↵m2⌃i U ⇤µi Uei 2 1i
32⇡
mW

2

< 10 54(2.4)

using the current measurements of the di↵erences in the squared neutrino masses[15]:| m232 | ⇡ 2.5 ⇥ 10 3 eV 2(2.5)

m221 ⇡ 7 ⇥ 10 5 eV 2

(2.6)

Thus an observation of µ ! e would demonstrate the existence of new physics

beyond the minimal extension of the Standard Model to include neutrino masses.

The branching ratio in Eq 2.4 needs to be enhanced in order to be detected. In the7

2. µ ! E

DECAY

Figure 2.1: Feynman diagram describing the µ ! e decay in the SM with a neutrinomass extension. The internal fermion line is a neutrino mass eigenstate. By diagonalizing
the neutrino propagator in flavor space, o↵-diagonal flavor couplings, Uµi and Uei , areinduced.

following section we review some examples of such a scenario.

2.1.3

SUSY and Lepton Flavor Violation

The mechanisms for generating neutrino masses have important consequences for lepton
flavor violation.
The sector describing neutrino masses can be written asc⇥~⌫ M⌫ ~⌫ + h.c. = ⌫¯L

c
⌫¯R



⇤ mL mTD ⌫¯Lc
+ h.c.
mD MR ⌫¯R

(2.7)

If no new physics occurs below the Planck scale, neutrino masses mij ⇠ 10 5 eV
are too small. With right-handed neutrinos, a Dirac mass term can be accommodated,⌫yielding neutrino masses mD
ij ⇠ Yij <>. If this is the sole source of neutrino mass,

unnaturally small values of Yukawa couplings are required to reproduce experimental
results.

Another way to explain the small neutrino masses is through seesaw mechanism[16].
If the left-handed Majorana mass is set to zero (mL = 0) and the Dirac mass is muchsmaller than the right-handed Majorana mass (mD ⌧ MR ), then Eq 2.7 gives neutrino
masses as8

2.1 Theoretical BackgroundmN ' MR , m⌫ '

m2D
MR

(2.8)

mD is expected to be of the same order as the lepton masses, since they both arise
though Yukawa terms. This requires right-handed neutrino masses in the range of106

1015 GeV[17] to get the right scale for the light neutrinos.

SUSY
In the minimum SUSY extension of the SM (MSSM), LFV would originate fromthe misalignment between particle and superparticle mass eigenstates. O↵-diagonal
slepton mass terms would contribute to LFV. With no inherent guiding principles tolimit the form of the soft breaking terms, it’s unclear why µ ! e does not take placeat large rates. However, constraints from LFV and FCNC in quark sector suggest asmall fermion-sfermion misalignment. This is known as the SUSY flavor problem[18].
A number of solutions have been proposed. In gauge mediation scenarios, for example, supersymmmetry breaking is delivered to the visible sector by messenger fields
charged under Standard Model gauge groups. The degree of flavor violation is tied
to the scale of supersymmetry breaking; Other proposals involve postulating various
flavor symmetries or kinematic suppression through heavy superpartner masses[19].
The following models focus on the gravity mediation scenario, in which the slepton
mass matrix is taken to be diagonal and proportional to the unit matrix at the Planckscale, MP l ⇠ 1018 GeV.

MSSM with Seesaw
Right-handed neutrinos can be incorporated into the MSSM through a superpotential
for the leptons. If neutrino mixing is assumed to come entirely from Yukawa couplings,
the information from neutrino oscillations can be related to the slepton mixing. Solarand atmospheric neutrino observations may be used to estimate typical values forBr(µ ! e ), displayed in Figure 2.2[5].

92. µ ! E

DECAY

Figure 2.2: Predictions for Br(µ ! e ) at various values of tan
left-handed selectron mass for MR = 1013 GeV.[5]as a function of the

SUSY-GUTS
Grand-unified theories (GUT) try to unify SU (2) ⇥ U (1) electroweak interaction andSU(3) strong interaction in a single simple gauge group.

The simplest GUT model is the minimal SU(5) model. However it cannot unify
the gauge couplings of the SM at the single scale and minimal SU (5) is ruled out bystrong limits on proton decay from Super Kamiokande[20].
With right-handed neutrinos of singlet (SU(5)RN), SU(5) model gets a naturalenhancement of the branching ratio due to the left-handed sleptons mass matrix.
SO(10) gauge group on the other hand has a 16-dimensional representation,
which can naturally accommodate all 15 fermions of a single generation (two quarks
and two anti-quarks per family with three possible colors, a charged lepton andanti-lepton, and a left-handed Majorana neutrino that is its own anti-particle) as well10

2.1 Theoretical Backgroundas a corresponding right-handed neutrino into a single multiplet. It can be extended
in various ways to include a seesaw mechanism.
Most models work in either the CKM (Cabibbo-Kobayashi-Maskawa) case with small
mixing in the neutrino Yukawa couplings or the PMNS (Pontecorvo-Maki-NakagawaSakata) case with large mixing in the neutrino Yukawa couplings. The universal scalarmass m0 , trilinear couplings A0 , and universal gaugino mass M1/2 are free parametersthat tie down the theory at the Planck scale. Other parameters include superpotential,
an undetermined sign in the Higgs potential parameters, and tan . For a fixed tan ,Br(µ ! e ) can be predicted by scanning a range in the other parameters, as shownin Figure 2.3.[21]Summary
In general, cLFV could be induced in many extensions to the SM. While the predicted
rate is highly model dependent, a discovery or stricter limit on µ ! e would provide
a guidance in narrowing down the allowed parameter space in each class of models,and valuable information for the new physics.

112. µ ! E

DECAY

FigurePredictions
Br(µ!!e e ) )areare shownshown as aa function
gauginoFigure2.5: 2.3:
Predictions
for forBR(µfunctionofofthe
theuniversal
universal
gauginomass
forfortwocases
scanningananLHC
LHC
relevant
space
inparameters
the parametersdescribing
mass
twocasesofoftan
tan , scanning
relevant
space
in thedescribing
thethe Planck
Planckscale
scalemasses.
masses.Both
BoththethePMNSPMNScasecase(green)
and
the
CKM
case
(red)
are
explored.
(green) and the CKM case (red) are explored.

31

12

2.2 Experimental Search

2.2

Experimental Search

2.2.1

History of µ ! e Search

The muon was initially discovered by Anderson and Nedermeyer[22] in an experimentthat measured the fractional energy loss of cosmic ray particles.

At first it wasthought to be the mediator of the force between protons and neutrons as predictedby Yukawa[23] on the basis of its mass. In 1947, however, it was shown throughexperiment[24] that the muon does not interact via the strong interaction, and thus
the muon cannot be the ⇡ meson of Yukawa. The muon was thought to decay into aelectron and a gamma ray if it is simply a heavy electron.
The first search for µ+ ! e+was made in 1947 using cosmic-ray muons[25] and it

gave an upper limit on the branching ratio of less than 10%. In 1955, an upper limitof B < 2 ⇥ 10 5 was set using the Nevis cyclotron at the Columbia University[26].
In the late 1950s, it was pointed out that if the form of the 4-fermion interaction
was, as a number of experiments supported, indeed a universal V-A one in whichcurrents interacted with themselves through the exchange of a heavy charged boson,the predicted branching ratio for µ+ ! e+

was about 10 4 [27], at odds with thecurrent experimental limit. It was suggested that the apparent absence of µ+ ! e+

could be explained by associating separate lepton numbers for muons and electrons,
and requiring them to both be conserved. In order to explain the abundantly observedµ ! e⌫⌫, the outgoing neutrinos had to also carry muon (µ⌫ ) and electron number (⌫e ).

Both the two-neutrino hypothesis and the separate conservation of lepton number werefirst confirmed in 1962 at Brookhaven[28].
With this discovery, interest in the search for neutrino-less decay modes of the muon
waned and experiments essentially ended for about 15 years, until a new era of searches
began in 1977 with an experiment making use of the intense muon beam at the Swiss
Institute for Nuclear Research (SIN)[29], which is the site of present day PSI. Moreimprovements in the limit followed. The best pre-MEG upper limit of 1.2 ⇥ 10 11 (90%CL) comes from MEGA experiment at the Los Alamos National Laboratory Meson

Physics Facility (LAMPF)[30]. In an era where beyond the Standard Model physics ispervaded by numerous free parameters weakly constrained by experiments, the prospectfor discovery or even a more stringent limit provides an alluring motivation to look for13

2. µ ! E

DECAY

µ ! e . A summary of the measurements of BR(µ ! e ) leading up to MEG isprovided in Figure 2.4.

Figure 2.4: Historical progress of µ+ ! e+ search.[6]142.2 Experimental Search

2.2.2

Signal and backgrounds of µ ! e

Figure 2.5: Schematics of µ+ ! e+ event signature (a), and two types of backgrounds(b)(c) .

Signal Event Signature
In the muon rest frame, a µ+ ! e+event is identified by a back-to-back positronand photon consistent with simultaneous emission from a common vertex, and withenergies approximately equal to half the muon mass. Further, conservation of theenergy-momentum 4-vector requires Ee+ = 52.83 MeV and E = 52.82 MeV. Signal
events are identified by measured positron energy (Ee ), photon energy (E ), time(te ),and opening angle (⇥e ).

Prompt Background
One source of background comes from radiative muon decay (µ+ ! e+ ⌫e ⌫¯µ ) with anearly back to back positron and photon and two neutrinos that carry o↵ little energy.
The branching ratio for this process is suppressed in that region of phase space. For
Ee > 46 MeV, E > 30 MeV, and no constraint on the opening angle, the radiative
decay branching fraction is of order 10 7 [31].
The expected background can be calculated by integrating tree level di↵erential
decay width over a finite signal box. Let us take x ⌘ 2Ee /mµ , y ⌘ 2E /mµ , andz⌘⇡⇥e , and x, y, z as the respective half-widths of a signal box centered at

x = y = 1 and z = 0, the expected branching ratio for unpolarized muons at the tree15

2. µ ! E

DECAY

level is[32]dBRD (µ+ ! e+ ⌫e ⌫¯µ ) =↵[J1 + J2 ],
8⇡

(2.9)

z 4 1 1
z
) +
( )8
2
2
3 ( y) 2(2.10)

z 4 8 z 6
) + ( )
2
3 2

(2.11)

Where J1 and J2 are given by,
8z
J1 = ( x)3 ( y)( )232
J2 = 8( x)2 ( y)2 (when z < 2p( x)2 (z 2)
28( x)( y)(x y.

For energy resolutions of order 1%, a box analysis with high signal efficiency issubject to radiative decay background at the level of merely 10 15 compared to thesignal sensitivity goal of order 10 13 .

Accidental Background
Another source of background stems from accidental occurrences of a positron comingfrom Michel decay (µ+ ! e+ ⌫e ⌫µ ) and a photon coming either from radiative muon

decay, positron annihilation in flight (AIF) in the stopping target or materials in thedrift chamber, or bremsstrahlung.
The e↵ective branching ratio of the accidental background can be estimated by!e) · (2 te )(2.12)
4⇡is the half-width of signal box for time

Bacc = Rµ · fe0 · f 0 · (where Rµ is the instant beam rate;tecoincidence, !e is that for opening angle, and fe0 and f 0 are fractions of the spectrumin Michel decay, and in gamma, respectively.
The positron spectrum (Michel decay) is shown in Figure 2.6. fe0 can be estimatedby integrating the Michel spectrum over 1
flat at x ⇡ 1, we can get fe0 ⇡ 2( x).

16x  x  1. Since the spectrum is almost

2.2 Experimental Search

Figure 2.6: Positron energy spectrum of unpolarized µ+ ! e+ ⌫e ⌫µ decay (Michel spectrum).

To estimate f 0 , the di↵erential branching ratio of radiative decay is integrated overpositron energy (x) and the angle between positron and gamma (⇥e ). The photon
spectrum is shown in Figure 2.7. By neglecting the terms suppressed by me /mµ , Theyy1r) can be

dBRD (µ+ ! e+ ⌫e ⌫¯µ )↵⇡( y)2 [ln( y) + 7.33]
dy2⇡(2.13)

partial branching ratio integrated over the signal region (1calculated as

0

f =

Z 1 r
1

y

dy

From the above, the e↵ective branching ratio of accidental background is

Bacc ⇡ Rµ · (2 x) ·



alpha
( z)4
( y)2 (ln( y) + 7.33) ·· (2 te )2⇡4(2.14)

With the proposed beam rate and resolutions of MEG[33], the e↵ective accidental
background is estimated to be of the order 10 14 .

17

2. µ ! E

DECAY

Figure 2.7: Photon energy spectrum of unpolarized mu+ ! e+ ⌫e ⌫¯µ ) decay.

2.2.3

Requirements of µ ! e Search

As discussed above, accidental background is the dominant background source, and it
will limit the experiment.
To search for µ+ ! e+

we need a large number of muons. Since the accidental

background is proportional to the intensity of muon beam, a direct current (DC) muon
beam is the best in order to suppress backgrounds. MEG experiment uses the worlds
most intense DC muon beam from 590 MeV proton ring cyclotron facility of 2 mA
intensity at Paul Sherrer Institut (PSI) in Switzerland.

There is little background of gamma ray near the signal region. We can reduce
background efficiently with a good gamma-ray energy resolution.

In contrast, there are many positrons in the signal region. Therefore it is difficult to
reduce background by improving positron energy measurement. It is more importantto efficiently measure a large amount of positrons generated by the high rate muon
beam.

18

2.2 Experimental Search

In summary, in order to search for µ+ ! e+intense DC µ+

at a high sensitivity, we need anbeam, a positron detector that can track high rate positrons, anddetectors with good resolutions.

19

2. µ ! E

DECAY

20

3

MEG Experiment
The MEG experiment takes place at Paul Scherrer Institut (PSI) in Switzerland. Inorder to reduce background events, precise measurement of energy, emission angle andtime of positron and gamma are very important. This chapter describes the hardware
components, calibration apparatus, DAQ and analysis software of MEG experiment.

3.1

Detectors Overview

The detectors of MEG consist of positron detector and liquid xenon(LXe) gamma-ray
detector, for the detection of positrons and photons from µ+ ! e+ + , respectively,
as shown in Figure 3.1.

Coordinate System
We define an orthogonal coordinate system by (x, y, z) as a global coordinate system(Figure 3.2). The origin is set as the center of COBRA magnet and also the center ofthe MEG target. The positive z-axis is defined to be parallel to the beam direction,
pointing downstream. The y-axis points upwards so the x-axis is perpendicular to theface of the LXe detector, which is confined to x < 0. For particle emission angle, ✓is the zenith angle with respect to positive z-axis, so the beam direction is presented
with ✓ = 0, andis an azimuthal angle made in the x

positive x-axis.

21

y plane with respect to the3. MEG EXPERIMENT

Figure 3.1: Overview of the MEG experiment.

Figure 3.2: Layout of detectors in the MEG experiment.

22

3.2 Beam

3.2

Beam

In order to detect µ+ ! e+ +with a great sensitivity, MEG experiment requires anextremely intense source of µ+ /s. The ⇡E5 beam line at PSI has the most intense
DC beam in the world with maximum intensity of 108 µ+ /s. A surface muon beam is
produced from pion decays on the surface of the production target. MEG experimentuses a reduced beam intensity of 3 ⇥ 107 µ+ /s to optimize the sensitivity of theexperiment.

3.2.1

Proton Accelerators

The facility to provide protons consists of three accelerators (Figure 3.3). At first the
Cockcroft-Walton accelerator injects 870 keV protons into Injector 2 cyclotron, which
provides 72 MeV protons with 0.2% FWHM spread, and 50.63 MHz frequency. Thenthe Proton Ring Cyclotron (Figure 3.4) accelerates the 72MeV proton beam up to
590MeV energies to form a beam current of 2.2 mA. They are then directed througha series of meson production targets, one of which, dubbed target E, is a 4cm thickgraphite target.

3.2.2

⇡E5

⇡E5, one of the 5 beam lines sharing target E as a source, supplies low energy pions
and muons. The ⇡E5 area has a suitable channel to obtain surface muons. Figure 3.5
shows the components of ⇡E5 beam line. The measured flux of pions and muons in E5
at the end of the last bending magnet (AST) is presented in Figure 3.6.

3.2.3

Beam Transport System

Figure 3.7 shows the beam transport system in the ⇡E5 area to bring the muon beam
to stopping target. The muon beam is directed through two sets of quadrupole triplet
magnets with a Wien filter in-between. The Wien filter applies perpendicular electricand magnetic fields, separating positive muons from positron contamination by 7.5 .
Next, the beam is focused by a superconducting beam transport solenoid magnet (BTS).
A momentum degrader at the center of the BTS is made of Mylar with a thickness23

3. MEG EXPERIMENT

Figure 3.3: Proton Accelerators at PSI.

between 200 and 450µm which can be optimized to maximize stopping efficiency withless backgrounds. The final spot size on the stopping target is24

x ⇡ 1cm,y ⇡ 1cm.

3.2 Beam

Figure 3.4: 590MeV ring cyclotron at PSI.

Figure 3.5: ⇡E5 beamline components.

25

3. MEG EXPERIMENT

Figure 3.6: Pion and muon beam intensity at ⇡E5.

Figure 3.7: Schematic view of MEG beam transport system.

26

3.3 Stopping Target

3.3

Stopping Target

The muon stopping target is a elliptical, polyethylene/polyester sandwich foil supported
by a Rohacell frame[34] (Figure 3.8).
The target material and thickness are optimized to minimize scattering of positrons
in the target and annihilation in flight. The dimension of the target is 79.8 mm along
vertical axis and 200.5 mm along the long axis, with a thickness of 205µ m. It has six
holes of 10 mm diameter to check its positioning using the vertex position reconstructedwith the positron tracking.
The target is put at a slant angle of 20.5 in the middle of the COBRA magnet (Figure 3.9) to increase muon stopping power without significantly increasing the amountof target material typically traversed by exiting positrons.

Figure 3.8: Muon stopping target.

Figure 3.9: Mounting position of the target.

27

3. MEG EXPERIMENT

3.4

Liquid Xenon Gamma-ray Detector

3.4.1

Liquid Xenon

For MEG experiment, a liquid xenon detector of 900 liter volume is developed[35].
The characteristics of liquid xenon as scintillator has been studied for a long time,
but rarely has it been used in such a great volume before due to the difficulties ofhandling. There are many advantages of using liquid xenon as a scintillator:• High density 2.95 g/cm3 and short radiation length X0 = 2.77 cm• High light yield, 80% of NaI• Fast response time of 45 ns for gamma rays
• No self absorption of scintillation light in liquid xenon• Uniformity• Particle discrimination
There are some difficulties such as cost, requirement of low temperature, highpurity to avoid absorption of scintillation light, and detection of vacuum ultraviolet(VUV) scintillation light. These concerns were all solved in MEG LXe detector.
The properties of liquid xenon are summarized in Table 3.1.

Mechanism of Scintillation Light
The origin of scintillation light from liquid xenon is de-excitation process of excited
dimers of xenon, Xe⇤2 . Figure 3.10 shows scintillation signals of liquid xenon by various
particles.
There are two di↵erent processes[44] for the de-excitation. One is a self-trapping
process of excited xenon atom,

Xe⇤ + Xe + Xe ! Xe⇤2 + Xe,
Xe⇤2 ! 2Xe + h⌫,

28

(3.1)
(3.2)

3.4 Liquid Xenon Gamma-ray Detector

Figure 3.10: Photon reaction in liquid xenon.

29

3. MEG EXPERIMENT

Table 3.1: Properties of LXeMaterial Properties

Value

Ref

Atomic Number
Atomic Weight
Density at 161.4K
Boiling point
Melting point
Triple point (temperature)
Triple point (pressure)
Radiation length
Critical Energy
Mollier radius

54
131.2932.978g/cm3
165.1K
161.4K
161.3K
0.805 atm
2.77 cm
14.5 MeV
4.2cm

[15]
[36]
[15]
[15]
[37]
[37]
[15]
[38]
[38]

Scinti. wavelength (peak±FWHM)
Rafractive index at 175nm
Wph for electron
Wph for ↵ particles
Decay time (recombination)
Decay time (fast components)
Decay time (slow components)
Absorption length
Scattering length(178 ± 14 nm)
1.57 to 1.72
21.6 eV17.9 eV
45 ns
4.2 ns22 ns> 100 cm
29cm to 50 cm

[39]
[40]
[41]
[41]
[42]
[42]
[42]
[35]
[43]

where h⌫ is vacuum ultraviolet (VUV) scintillation light. The two states of excited
dimer Xe⇤2 , singlet and triplet, correspond to fast and slow components respectively.
The other process is a recombination process originating from a xenon ion,

Xe+ + Xe + Xe ! Xe+
2,

⇤⇤
Xe+
2 + e ! Xe + Xe,

Xe⇤⇤ ! Xe⇤ + heat,Xe⇤ + Xe + Xe ! Xe⇤2 + Xe,
Xe⇤2 ! 2Xe + h⌫,

(3.3)
(3.4)
(3.5)
(3.6)
(3.7)

where the time dependence of the scintillation is dominated by the kinematics of30

3.4 Liquid Xenon Gamma-ray Detector

the recombination process.

The wave length of the scintillation light emitted from both of these two processes
are in the vacuum ultra-violet(VUV), which is 178 ± 14nm for peak and FWHM, andthe decay time constant is relatively short, 45 nsec in the recombination process.
The liquid state has the benefit of homogeneity. Figure 3.11 shows the diagram ofxenon and liquid phase should be kept between 161 and 165 K at 1 atm.

Figure 3.11: Diagram of xenon phase[7].

The impurities in the liquid xenon should be removed with a purification system
to avoid absorption. There is no self absorption in xenon, but contaminations suchas oxygen, nitrogen and water decrease a light yield of short-wavelength scintillation
light.

A purification system as well as a light yield monitor is needed for stable

operation of the liquid xenon detector.

31

3. MEG EXPERIMENT

3.4.2

Photomultiplier

We cooperated with Hamamatsu Photonics in developing UV-sensitive PMT R9869[45](Figure 3.12). The photo-cathode material is K-Cs-Sb. This is designed so that it canobserve scintillation light of liquid xenon directly while immersed in it.
For MEG experiment, the PMTs must work in low-temperature LXe and be sensitive
to the VUV LXe scintillation light. The window of PMT is made of synthetic quartz
with an 80% transparency around 178nm wavelength. We chose Bialkali (K-Cs-Sb) for
the VUV sensitive photo-cathode. Aluminum strips are attached on the photo-cathodeto avoid from the increase in the sheet resistance at a low temperature. The propertiesof the PMT are shown in Table 3.2.

Figure 3.12: PMT for the LXe detector.

3.4.3

Design of LXe Detector

The schematic view of MEG LXe gamma-ray detector is shown in Figure 3.13. It is
shaped to fit the outer radius of COBRA. The LXe detector has 900 liters of liquid
xenon and 846 PMTs (Figure 3.15). The PMTs are placed on all six faces of thedetector and are immersed in liquid xenon (Figure 3.16). Definition of six faces of
the PMT holders (inner, outer, upstream, downstream, top, bottom) are shown in the
Figure 3.13. PMTs are most closely arranged on the inner face and its coverage ofactive photo-cathode is about 35%. The arrangement and density of PMTs is shownin Figure 3.14. The active volume of the detector is 800 liters and it covers 11% of the32

3.4 Liquid Xenon Gamma-ray Detector

Table 3.2: Properties of the PMT.

Size
Active area size
PMT length
Photon-cathode material
Dynode type
Number of dynode
Typical Hv
Typical gain
Typical QE
Rise time
Typical transit time
Typical transit time spread

57 mm
45 mm
32 mm
K-Cs-Sb
Metal channel
12900 V
1 ⇥ 10615%
2 nsec12.5 nsec
0.75 nsec

solid angle from the stopping target.

Figure 3.13: Schematic view of the liquid xenon detector.

33

3. MEG EXPERIMENT

Figure 3.14: Development view of the faces of liquid xenon detector. Each face isequipped with 216, 234, 144 ⇥ 2 and 54 ⇥ 2 PMTs for inner, outer, lateral and top
or bottom.

3.4.4

Cryogenic System

A powerful and stable cryogenic system is needed for the operation of the LXe detector.
LXe is filled in a cryostat consisting of two layers of vacuum-tight vessels. The outerlayer makes a thermal insulation. The gamma-ray entrance window is designed to be
as thin as possible to maximize light penetration. The window of the outer vessel ismade of a 0.7 mm thick stainless steel plate, while that of the inner vessel is made ofaluminum honeycomb panels covered with carbon fiber plates. The total thickness of
the window is 0.075 X0 . A turbo-molecular pump is directly attached to each vesselto evacuate with high conductance. In addition, a cryo-pump is installed to the inner
vessel to efficiently remove water. A 200 W pulse-tube refrigerator[46], which wasdeveloped for this LXe detector, is mounted on top of the cryostat and controls thetemperature of LXe. In addition, cooling pipes of LN2 are also available when necessary.
There are two storages for xenon outside the detector (Figure 3.17). One is a 1000liter dewar (Figure 3.18) with a refrigerator to keep xenon as liquid. The other storage
system consists eight high-pressure gas tanks (Figure 3.19).

34

3.4 Liquid Xenon Gamma-ray Detector

Figure 3.15: Liquid xenon detector

Figure 3.16: PMTs mounted inside the LXe detector.

35

3. MEG EXPERIMENT

Figure 3.17: Cryogenic system of the LXe detector.

3.4.5

Purification System

In order to purify xenon, two types of purification system[47] in liquid and gaseous
phase were developed. The liquid purifier system has oxidization reduction filter madeof copper and molecular sieves, which removes oxygen and water with 180 liter/h
circulation by piston-type liquid pump. Another purification system in gas phaseuses metal-heated getter to remove H2 O, O2 , CO, CO2 , N2 , H2 and hydro-carbon
molecules from gaseous xenon.

36

3.4 Liquid Xenon Gamma-ray Detector

Figure 3.18: Liquid xenon tank and gaseous purification system.

Figure 3.19: Gas xenon storage tanks.

37

3. MEG EXPERIMENT

3.5

Positron Spectrometer

The positron spectrometer consists of a magnet specially designed to form a gradient
field, a drift chamber system to measure the positron track and scintillation counters
to measure the positron timing.

3.5.1

COBRA Magnet

We constructed a superconducting magnet specially designed to form a highly gradedmagnetic field[48]. It consists of five coils with three di↵erent radii: one central coil,
two gradient coils and two end coils. (Pictured in Figure 3.20). The magnetic field
ranges from 1.27 T at z = 0 to 0.49 T at |z| = 1.25 m, as shown in Figure 3.23.

Figure 3.20: A photograph of COBRA magnet.

As shown in Figure 3.21, compared with a simple uniform solenoidal field, it has
the following advantages:• Positrons are swept away much more quickly, which contributes to the minimization of backgrounds.(Figure 3.22)

38

3.5 Positron Spectrometer

Figure 3.21: Comparison between COBRA spectrometer and uniform magnetic field. (a)and (c) show trajectories of positrons emitted transverse to the field. The uniform field
makes many turns inside the detector, whereas the gradient field sweep the positron out ofthe detector much more quickly. (b) and (d) show trajectories of mono-energetic positrons
emitted at various angles. In the uniform field, the bending radius depends on the emission
angle, whereas it is independent in the gradient field.

• The positrons follow trajectories with a constant projected bending radius inde-

pendent of the emission angle. This allows us to define the absolute momentum
window of positrons to be detected.

Gamma rays must travel through the magnet structure before reaching the LXedetector and it causes signal inefficient when they interact with materials in front of
the detector. Therefore the cable and wall of the magnet are made very thin. The
total thickness of the superconducting coil, support structure, and cryostat amounts
to 0.197X0 .

39

3. MEG EXPERIMENT

Figure 3.22: Rate of Michel positrons per cm2 per second as a function of radius assuming
muon decay rate of 3 ⇥ 107 /sec.

Figure 3.23: Magnetic field (Bz ) along beam axis (x = y = 0).

40

3.5 Positron Spectrometer

The magnetic field produced by COBRA can deteriorate the performance of
PMTs in the LXe detector. In order to reduce the fringe field at the position of the
LXe detector, compensation coils are placed at both ends of the solenoid. With thecompensation coils we successfully reduced the magnetic field at the position of the
gamma-ray detector to 50 Gauss, as shown in Figure 3.24.

Figure 3.24: Map of COBRA magnetic field.

41

3. MEG EXPERIMENT

3.5.2

Drift Chamber

The drift chamber (DCH) measures a momentum, a decay vertex and an emission
angle of a positron. It consists of 16 chamber sectors aligned radially at 10.5 intervals
in azimuthal angle.

The radial position ranges from 19.3 to 27.9 cm to measure

only high momentum positron (> 40 MeV). Figure 3.25 shows a fully installed drift
chamber system.

Figure 3.25: A photograph of the 16 drift chamber modules mounted on the support
structure.

Each module of the chamber consists of a support frame, a foil with a cathodepattern and anode wires as shown in Figure 3.26, and is made up from 2 staggered
arrays of drift cells which measure the time and r-coordinate of positrons simultaneously (Figure 3.27). Each layer containing nine drift cells is shifted by one-half celleach other to resolve left-right ambiguity. The two layers are separated by two inner
cathode foils and also enclosed by a outer one.
The chamber walls are made of thin plastic foils. A thin layer of aluminum deposit on
the four cathode foils is shaped to make a Vernier pattern, as shown in Figure 3.28.

42

3.5 Positron SpectrometerThe z position is first determined by the charge division on the anode wire at an
accuracy of 1 cm, and later with more accuracy from the Vernier pads.
The chamber sectors and the volumes between them are filled with mixture of equalparts helium (He2 ) and ethane (C2 H6 ) by volume, which provides a substantial amountof ionization with little additional scattering[49]. A support frame made of carbon
fiber supports the wires and foils. This structure helps to reduce amount of material
in positron trajectories at a cost of a difficulty in the construction. The mean of
total radiation length along the signal positron inside the tracking volume is 2⇥10 3 X0 .

Figure 3.26: Drift chamber module.

Figure 3.27: Section of wires in two layers.

A map of the field lines within a cell and the resulting arrival times are simulatedby the GARFIELD[50] program. Figure 3.29 displays an example Garfield simulation.

43

3. MEG EXPERIMENT

Figure 3.28: Schematic of vernier pattern on the cathode pads.

44

3.5 Positron Spectrometer(a) Contour plot of the potential.

(b) Drift lines with isochrone map.
Figure 3.29: Field map and drift lines of drift chamber calculated by the GARFIELD
simulation

45

3. MEG EXPERIMENT

3.5.3

Timing Counter

A pair of timing counters (TIC) are placed at the both sides of the drift chambersystem. Each timing counter comprises two layers of plastic scintillators along z anddirection. The outer layer along z direction, called -counter (TICP), gives the timing
information and fast information of positronemission angle. The inner layer along

direction, called z-counter (TICZ), gives precise measurement of z impact position
and fast information of positron ✓ emission angle.
Thecounter consists of 15 straight plastic scintillator bars (4 ⇥ 4 ⇥ 80 cm3 , Bicron

BC-404[51]) lying along z direction as shown in Figure 3.30. They are placed at 10.5
intervals in

direction at a radius of 32 cm and they cover150 << 10 . Two

2-inch fine-mesh PMTs are attached to both ends of each bar. Details of the design isshown in Figure 3.31.
The z counter consists of 128 scintillating fibers (6 ⇥ 6 mm2 , Saint-Gobain BCF-20[51])put perpendicular tocounters as shown in Figure 3.32 . Each fiber is separated

optically at the center and read out independently at the both ends by a 5 ⇥ 5 mm2
silicon avalanche photo-diode (APD)[45].

46

3.5 Positron Spectrometer

Figure 3.30: Picture of timing -counter.

47

3. MEG EXPERIMENT

Figure 3.31: Design of timing

48

counter.

3.5 Positron Spectrometer

Figure 3.32: Timing z-counter and APDs.

49

3. MEG EXPERIMENT

3.6

Calibration Apparatus

Various apparatus are used to calibrate gamma energy scale; measure gamma energy,
time and position resolutions. Back-to-back gamma pairs from ⇡ 0 decay, as describedin section 3.7.1, are used to calibrate the energy scale and measure the gamma energy,
time resolutions, and efficiency. Light yield is monitored during data taking using
monochromatic photons from reactions with Li and B, as described in section 3.7.2.
Position resolution and relative timing between the LXe detector and timing counterare also checked from those reactions.

3.6.1

Setup For ⇡ 0 Run

The gamma ray near signal energy from ⇡ 0 decay is used to calibrate the absolute energy
scale. It is obtained by using the reaction ⇡ p ! ⇡ 0 n at rest followed by ⇡ 0 !
decay and selecting events where two gamma rays are in anti-parallel direction.
As shown in Figure 3.33, the ⇡ 0 momentum is fixed at 28 MeV/c. When thephotons are anti-parallel along the ⇡ 0 flight direction, they have energies of 54.9 and82.9 MeV. The 54.9 MeV gamma is close to the signal energy of 52.8 MeV and is used
to determine energy scale and measure gamma resolution. In addition, a 129.4 MeV
gamma from the radiative capture of negative pion (⇡ n !
Dalitz decay of neutral pion (⇡ 0 !

e+ en) is available. The) is also used to calibrate drift chamber.

Figure 3.33: Kinematics of the ⇡ 0 !

50

decay.

3.6 Calibration Apparatus

Figure 3.34: Overview of the setup in ⇡ 0 run.

Beam and Target Setup
The ⇡E5 channel provides a 70.5 MeV/c ⇡beam. During ⇡ 0 run, the traditionalmuon target is replaced with a cylindrical target filled with about 150 cc of liquid hydrogen, pictured in Figure 3.35.

It is 50 mm in diameter, and 75 mm inlength. The window of the target cell is made of a thin 135 µm Mylar film. The
target is inserted from down stream side with a two meter long pipe. The setup of
the LH2 target takes about three days, and the ⇡ 0 run is usually only taken once a year.

NaI Detector
To obtain monochromatic gamma rays from the back-to-back decays, anothercalorimeter is installed at opposite side of the LXe detector as pictured in Figure 3.36.
It consists of nine crystals of NaI(Tl) scintillator with each coupled to an APD forenergy measurement. To measure the time, a 5 mm-thick lead plate and two plastic
scintillators are placed in front of the central NaI crystal. The e+ e pairs produced in
the lead plate are collected by the two plastic scintillators. In order to scan the whole
acceptance of the LXe detector, the NaI detector is mounted on a mover that can get

51

3. MEG EXPERIMENT

Figure 3.35: Liquid hydrogen target.

the same coverage of the LXe detector.

Figure 3.36: NaI detector.

52

3.6 Calibration Apparatus

3.6.2

Cockcroft-Walton Proton Accelerator

A Cockcroft-Walton (CW) proton accelerator (Figure 3.37) is installed at the downstream side of the MEG detector for the calibration of the LXe detector. Gamma rays
in various targets are used to monitor the light yield of the LXe detector.
The normal target made of lithium tetraborate (Li2 B4 O7 ) provides reactions fromlithium and boron. The peak at 17.7 MeV is obtained from 73 Li(p, )84 Be and can beused as a precise light yield monitor within 1%. CW calibration runs are taken twicea week to monitor the light yield of the LXe detector throughout the MEG data taking.
12
Another reaction 11
5 B(p, )6 C, with a resonance of 163 keV, produces a 16.1 MeV
12photon when the 12
6 C is in ground state, or an 11.7 MeV photon when the 6 C is inexcited state, and a 4.4 MeV photon from the transition to the ground state. It allows
the timing measurement between timing counter and the LXe detector, as well as theconfirmation of the energy scale or the light yield monitor. Figure 3.38 shows thepeaks in CW run.

Figure 3.37: CW accelerator.

53

3. MEG EXPERIMENT

Figure 3.38: Energy peaks in the LXe detector with CW accelerator.

3.7

Electronics and Data Acquisition

3.7.1

Electronics Chain

The data flow from detectors to digitizers is schematically summarized in Figure 3.39.
The outputs from detectors pass through several devices and finally acquired as
waveform digitized by a fast waveform digitizer. Signals are also sent to a trigger
system to decide whether the event is of interest or not, and those that fit the trigger
criteria would be stored in software file.
PMT signals from the LXe detector are split into three outputs. One is a wide-band(1.9 GHz) fully di↵erential output that goes to the waveform digitizer, the Domino
Ring Sampler (DRS). The other outputs are a 320 MHz-bandwidth di↵erential output,
which is used for inner PMTs, and a four-to-one sum output that is used for the other
PMTs. Both go to trigger through a 2 m-long low-density cable.
Signals from the timing counter bars are divided into three outputs in an 8:1:1
ratio. The largest output goes through a double threshold discriminator (DTD), todiscriminate a positron hit from noise or low energy delta ray hits. The discriminatoroutputs 50 ns-wide standard NIM pulses that go to waveform digitizers. One of thesmaller outputs is sent through another splitter where one output is coupled to the
waveform digitizer and the other is coupled to the trigger system. The remaining 10%
output is used for online current monitoring.

54

3.7 Electronics and Data Acquisition

Figure 3.39: Schematic of data flow and electronics.

We get six waveforms for each cell of the drift chamber: two from the both ends of aanode wire and four from the vernier-pattern cathode pads. Signals from the anodewires are divided into ratio 9:1 with the largest output going to the waveform digitizer
and the small output amplified then used in trigger. Signals from vernier pads are sent
to waveform digitizers.

3.7.2

Data Acquisition System with MIDAS

An overview of the data acquisition (DAQ) system is shown in Figure 3.40. The
MEG experiment uses MIDAS[52] (Maximum Integration Data Acquisition System)
developed at PSI as a DAQ system. There are mainly nine frontend sub-systems:
four for trigger and five for DRS. The MIDAS system provides the control of frontend
processes as well as the logging system, the online database system for the parametersof DAQ and trigger, the slow control system, the alarm system, the history monitoring55

3. MEG EXPERIMENT
48

CHAPTER 3. Experimental Apparatus

system, and web interfaces.

Figure 3.40: Schematic of DAQ system.
Figure 3.34: Schematic of DAQ system.

3.7.3 Waveform Digitization with DRS
3.3.3 Trigger

All waveform digitization in the MEG experiment is performed by a switched capacitor
The trigger system is based on a coupled use of flash analog to digital converters (FADC)

array
called DRS(DominoSampler)developed
at PSI[53,55]. A isDRS chipand FPGA.
The input
signalsRingare sampled
by FADCs
and thedigitized 54,
information
analyzed by FPGAs.
haseight sampling channels, a channel to accept trigger signals, and a channel to
The whole system is arranged in a tree structure on three layers with two different

acceptEachof consists
the eight
samplingchannels
can store
in atypes aof clockboards.signal.
The first
layerof theType1 boards
compliantwith waveforms
the 6U
VME standard. The Type1 board receives and digitizes analog input signals with FADCs(AD9218 [73]) at 100 MHz with 10 bits resolution , implements some reconstruction
algorithms
on a large(Xilinx
Virtex-IIpro[74]), andfashion
sends the(domino
information
seriesof inverters
andsize
is FPGArunning
continuously
in circularwave) until
to the successive trigger layer. The two remaining trigger layers consist of a second typea oftriggerit to
stop
the standard.
sampled The
waveformis readout by a shiftboardssignal(Type2)causes
compliantwiththe and
9U VME
second layer
determines
trigger conditionsof sub-systems,
and finallyFADC
the third
layeranalog-to-digitalmakes a trigger decision.
registerand digitizedby a commercial(fast
converter). ThisIn addition, an ancillary system was developed to ensure synchronous operation of theprocess
illustrated
in Figuretree. Itisconsistsof ancillaryboards3.41.
of 9U VME boards, and distributes a reference clock
and controls signals such as start and stop of the DAQ system. A master board hosts
the reference clock oscillator (SaRonix SEL3935 [75]) generating a 19.44 MHz squared
For andthe receives
2010 run,all signals
detectors
versionwithrate of 1.6
clock,
controlfromused
the third
layerDRS4[55]of the trigger
tree.a samplingThose signals
are fanned-out by three slave boards and distributed to all boards of the trigger and DRS.
GHz, except for the timing counter, which used DRS3[54].
The jitter of distributed clock is measured to be less than 30 ps.
The trigger for the µ+ ! e+ event candidate (MEG trigger) is decided by the following three conditions: the gamma energy, positron-gamma time coincidence, and positrongamma direction match. Those kinematic conditions are checked with fast reconstruction
3.7.4
Trigger
Systemalgorithms
implemented
in the FPGA. A requirement of global trigger latency to be lessthan ⇠500 ns prevents us from using information of the drift chambers that have slowerThe
triggerof flashanalogof toconverters(FADC)[56],signaldue tosystem
the driftconsists
time. The
pulse heightsumdigital
waveformof all PMTs
gives an which
estimation of a gamma energy. A set of PMT-calibration factors can be incorporated

ring of 1024 capacitors. During operation, a sampling frequency is generated on adigitize the waveforms received from the detectors, and field programmable gate arrays(FPGA)[57], which analyze the digitized waveforms.

56

3.7 Electronics and Data Acquisition

Figure 3.41: Schematic of DRS principle.

The whole system is arranged in a tree structure on three layers with two di↵erent
types of boards. The first layer consists of the Type1 boards compliant with the 6U
VME standard. The Type1 board receives and digitizes analog input signals with
FADCs, implements some reconstruction algorithms on a large size FPGA, and sends
the information to the successive trigger layer.

The two remaining trigger layers

consist of a second type of boards (Type2) compliant with the 9U VME standard.
The second layer determines trigger conditions of sub-systems, and finally the thirdlayer makes a trigger decision.
The trigger for µ+ ! e+ event (MEG trigger) is determined by three conditions:the gamma energy, positron-gamma time coincidence, and positron-gamma direction
match. The online algorithm estimates the gamma energy by a global sum of PMT
waveforms, and the threshold is set to 44 MeV. The online resolution is measured to
be 8% in FWHM. The time coincidence is checked using the time di↵erence between
the PMTs from the gamma-ray detector and the timing -counter. Both the positronand gamma emission times are taken from a linear interpolation of PMT pulse leading57

3. MEG EXPERIMENT

edges. The resolution of the positron-gamma time di↵erence is 3.3 ns ( ), and the
MEG trigger requires this to be within 10 ns of zero. The direction match is checkedwith the most illuminated inner face PMT. There is a correlation between the emission
angle of a positron on the target and the hit position on the timing counter for thesignal energy positron. This correlation is investigated using Monte Carlo simulation
and a lookup table is formed in advance. The trigger rate for events that satisfy thesethree conditions was 6Hz.

3.8

Simulation and Analysis

The MEG software is composed of three packages; MEGMC, MEGBartender, andMEGAnalyzer. The structure is shown in Figure 3.42. MEGMC and MEGBartender
generate simulation data and MEGAnalyzer analyzes data from both experiment andsimulation.

Figure 3.42: Structure of MEG software.

Event Generation and Detector Response
MEGMC is a GEANT3.21[58] based Monte Carlo simulation (MC). It processes theevent generation and the detector simulation. The event can be generated for the58

3.8 Simulation and Analysis

signal, Michel decay, radiative muon decay and separated positron or gamma ray ofeach decay as well as other backgrounds such as annihilation in flight of positron.
Various calibration events are also simulated. Detector geometry and materials are
implemented in the standard GEANT3 manner.

Event Mixing and Electronics Simulation
After all event kinematics and response of the detector are simulated, the MEGBartender, can read multiple events generated by MEGMC and mix simulated events.
Events are randomly or sequentially picked up from the input files and placed randomly
in time according to the Poisson distribution.
Electronics simulation is also done by MEGBartender. To generate waveforms of
the DCH and LXe detector, actual measured responses are used. The signal in timing
counter is generated with digital filters in software and random fluctuations.

Analysis Framework
We analyze all processes from raw data to physics analysis in MEGAnalyzer.

It

can handle both experimental and simulated data. Analysis and monitor frameworkin the MEG experiment base on the Root based Object oriented Midas Extension(ROME)[59] developed at PSI.

59

3. MEG EXPERIMENT60

4

Run 2010
MEG experiment was approved at Paul Scherrer Institut in 1999 and the construction
and installation of detectors was finished at the end of 2007. In 2008 we took the firstdata for three months from September.
Before the physics run started in 2009, the waveform digitizer chip (DRS) was
upgraded and light yield of liquid xenon scintillation light recovered. The physics datataking was performed for two months in 2009, 55% shorter than 2008, but with twice
the data thanks to improved positron efficiency. Two setups of muon momentum degraders of 200µ m and 300µ m were used for the first and second half of the data taking.
In 2009, to shift the stopping distribution inside of target to the center, thethickness of the Mylar degrader was changed to 200µ m, but later reverted to 300µm since it was found that the event distribution had a larger asymmetry along beamaxis. The stopping efficiency of the muon target is estimated to be about 56% and
82% for 200 µ m and 300 µ m degrader, respectively. New version of DRS waveform
digitizer was installed for the LXe detector and drift chamber. The new version hasless e↵ect of temperature drifts, a good linearity up to 1 Volt, no ghost pulses due toresidual charge, better timing accuracy with 20 MHz global synchronization between
boards and ability of faster sampling up to 6 GHz.
Before the 2010 run, we upgraded the waveform digitizer to improve timing resolution. There was some ambiguity in the alignment among the LXe detector, magnet,

61

4. RUN 2010

and target, therefore alignment measurement with cosmic rays was conducted. The
physics data taking started in August and ended in November, more than one monthearlier than scheduled, due to the BTS problem (see Section 4.5). Evolution of thenumber of muon stops on target is shown in Figure 4.1.

Figure 4.1: Integrated muons during the MEG data taking in 2010.

4.1

Physics Run

Muon stopping rate at proton current of 2.2 mA is 3 ⇥ 107 µ+ s 1 . Eleven types oftrigger are mixed in the physics run. The total data taking time was 67 days. LED

monitoring was performed everyday during data taking and full sets of calibration
data including LED data with beam on and o↵, LED gain calibration, alpha and CW
data were taken twice a week. 12169 runs were taken and in total 345 TB data wascollected by November.

4.2⇡ 0 Run

We conducted a ⇡ 0 run for 12 days in August. Details of ⇡ 0 run setup are describedin Secton 3.6.1. We took several kinds of data in order to evaluate energy, time andposition resolutions of the liquid xenon detector as well as calibrate energy scale.

624.3 Calibration Runs

⇡0 !

patch scan We did a full scan of the inner face with ⇡ 0 !

reaction inorder to evaluate energy and time response. The inner face of the LXe detector
was divided into 24 parts and trigger was prepared for each part. NaI was moved
to the opposite of the patch to collect back-to-back events. Coincidence trigger
between the LXe patch and reference counter was used for time performance andthat between LXe patch and NaI was used for energy performance.

⇡0 !

reference run For reference we took data everyday at the same patch andthe result is used to determine energy scale.

Daliz decay We collected ⇡ 0 Daliz decay data for relative time measurement between
the timing counter and the LXe detector.

4.3

Calibration Runs

The full set of calibration runs include LED data with beam on and o↵, LED gain
calibration, alpha and CW data. They were taken three times a week. The details of
calibration are described in Chapter 6.

4.4

Run Condition

The target was set at a similar slanting angle

as 2009 ( ⇡ 21 ). The degrader was

set to 300 µm Mylar and in air doped at 5% He to ensure the stable operation of thedrift chamber. Beam rate was set to 3.7 ⇥ 107 µ+ s 1 at the COBRA center whichcorresponds to a stopping rate of 3 ⇥ 107 µ+ s 1 .

Degrader Optimization
The Mylar degrader (pictured at Figure 4.2) at the center of the BTS magnet hasa thickness range of 200-400 µm. It was changed from 200 µm to 300 µm during
the 2009 run due to larger asymmetry in event distribution along the beam axis. A
study on degrader optimization was carried out before the start of 2010 run in orderto maximize the number of good positron tracks while minimizing backgrounds. In

63

4. RUN 2010conclusion the thickness of the degrader was set to 300 µm.

Figure 4.2: Degrader system.

The LXe detector
Before the 2010 run, we did calibrations of liquid xenon detector to confirm light yield
to be the same level as 2009. Bad channels of PMTs were also checked and marked.
In total 8 bad channels were found, but the e↵ect was minimal since none of them wason the inner face.

Waveform digitizer upgrade
DRS4 (Figure 4.3) was introduced in 2009 for faster sampling rate and better linearity.
However it was found that the timing resolution worsened due to noise from electronics.

64

4.4 Run Condition

Before the start of 2010 run we applied fine tuning of DRS4 to reduce noise andminimize time jitter. As a result contribution to timing resolution from electronics
improved to 50ps compared to 130ps in 2009.

Figure 4.3: DRS4 board.

DAQ
DAQ live time in 2010 was ⇠ 84 % and dead time was 6% less than 2009.

Alignment
Optical survey was carried out to determine all DCH modules positions anddirections[60].

Relative alignment between LXe detector and drift chamber was

calculated using cosmic ray, CW and AmBe data[61]. Combining all the results, theaverage di↵erence between measured and reconstructed position wasz=

6.2 ± 2.3mm, where z is the position along the beam line. This result was applied in physics
analysis.

65

4. RUN 2010

4.5

BTS problem

In November there was a BTS quench which ended the run prematurely. We still
managed 67 real time days for data taking and gathered 1.9 times more data than2009. The incident was investigated and repair work was done to ensure the runningof experiment in the next year.

66

5

Event Reconstruction
An event recorded by the experiment consists of waveforms that are analyzed toreconstruct the kinematics of the detected particles.

This chapter describes themethods to reconstruct energy, time and angle of the photon and positron. We firstlook at each detector separately and then discuss the combined analysis to reconstruct
the angle and time between positron and gamma.

5.1

Drift Chamber

The waveform of a charged particle passing through the drift chamber is used to calculate the position and momentum of the particle, a decay vertex and set of emission
angles from the projection to the target, a projected impact location at the timing
counter, and the overall path length of the trajectory.
We measure the arrival time and charge on each wire and pad on the passage of theparticle through a cell. The information from each cell, called a hit, is checked withother hits on the same chamber and groups of hits from the same particle are collected
into clusters, which are then collected into tracks. A global fit is made to the track
trajectory to estimate the track momentum, the interception at the timing counterand target plane, and the total path length from the target to the timing counter.

67

5. EVENT RECONSTRUCTION

5.1.1

Waveform Analysis

We get six waveforms from a drift cell: two from both ends of an anode wire and four
from cathode pads, as shown in Figure 5.1. We begin by searching for pulses. A pulse
and its width are identified by a maximum peak over a threshold and pursuing thepulse in both directions above another given threshold. We extract time of each pulse
by a single-threshold crossing time and charge of the pulse is integrated over 50 ns inorder to optimize the signal-to-noise ratio.

Chamber 12 plane 1 cell 3

Chamber 12 plane 1 cell 3

50
50
40

40

30

30
20

20

10
10
0
0

-1200

-1000

-800

-600

-400

-200

0

-1200

Chamber 12 plane 1 cell 3

-1000

-800

-600

-400

-200

-800

-600

-400

-200

-800

-600

-400

-200

Chamber 12 plane 1 cell 3

30

18
16

25

14
12

20

10
8

15

6
4

10

2
5

0
-2

0

-1200

-1000

-800

-600

-400

-200

0

-1200

Chamber 12 plane 1 cell 3-1000

Chamber 12 plane 1 cell 3
35

20
18

30

16

25

14

20

12
10

15

8

10

6

5

4
0

2
0

-1200

-1000

-800

-600

-400

-5

-200

-1200

-1000

0

Figure5.6:5.1:
TheThesix six
waveforms
associatedleft are red
waveforms
Figurewaveforms
associatedwith
witha ahit.
hit. On
Thethehorizontal
lines are fromthe upstream
readouts,
and theon therightwaveforms
from edge
downstream
The top two
calculatedbaselines,
vertical
redarelines
are the leading
time, andreadouts.
the region between
arethefromtheanode,
themiddle
two
are
from
the
hood
pads,
and
the
bottomtwo
are fromtwo dashed blue lines is the expected region where triggered signal appear.
the cathode pads. The horizontal red lines are the calculated baselines, the vertical red lines
are the leading edge time, and the region enclosed by the two dashed blue lines is the allowed
window to search for hits.

685.1 Drift Chamber

5.1.2

Hit Reconstruction

We measure the z-coordinate of drift-chamber hit by a combination of anode charge
division and cathode-vernier-pad. Assuming equal preamplifier input impedance, R, ateach end of the anode, and a uniform resistance per unit length, ⇢, the z coordinate
along a wire of length L can be roughly calculated asz=(

R L+ ) · ✏a
⇢
2

(5.1)

Qu Qd
Qu + Qd(5.2)

where charge division ✏a is defined as✏a ⌘Here Qu and Qd are the charges on upstream and downstream end respectively.
Then we reconstruct is more precisely using vernier pattern. The charge division
for the vernier pads can be similarly defined for inner side cathode (✏i ) and outer sidecathode (✏o ) respectively. The phase of the vernier pattern is defined as↵ = tan 1

✏o
✏i

(5.3)

Then the z coordinate can be calculated asz=l·↵+n·l2⇡(5.4)

where l is the the length of a pad cycle, equal to 5 cm. Figure 5.2 shows the vernier
circle and relation between anode charge division and vernier circle.

5.1.3

Track Reconstruction

To find tracks, we first group hits into clusters if they are consistent with coming fromthe same positron, using the z and time information. The clusters are selected withrequirements on number of cells between hits, and their deviations from z.
Next, we connect the clusters that resemble one turn of a positron trajectory. If wefind a combination of three clusters, called seeds, we roughly reconstruct the trajectory69

5. EVENT RECONSTRUCTION

5.3. Positron Reconstruction

(a)

71

(b)

Figure 5.2: z reconstruction of the drift chamber hit. (a) shows the vernier circle. One
turn corresponds
to a period ofpattern (5 cm).
relationof the circle.
vernier One
Figure5.9: z reconstructionof vernier
the drift-chamberhit.(b)(a)showsshows
the vernier
turncorresponds
toaperiodofvernierpattern
(5
cm).
(b)
shows
relationofthe vernierphase with anode charge division.

phase with anode-charge-division.
with a circle. Once we find a track candidate by connecting clusters associated with apositron,
we can estimate the time of the track by using the timing of all hits belonging
Track
Finding
to the track.
To find tracks, we apply topological pattern recognition. First, using the z and time
The drift timeof eachhits
hit,associatedwhich is withthe time
elapsedbetweenprimary
ionization
information,
we cluster
a positron
pass
within aeach
module.
Since thetwolayersofamoduleisstaggered
by
a
half
cell,
we
can
solve
the
“left-rightambiguity”
event and the arrival of the earliest part of the subsequent avalanche, is calculated by inmostcase. Next, we connect the clusters. If we find a “seed” of track with three clusters,
tidrif t = tihit ttrack , where tihit is arrival time of i th hit, and ttrack is the track time,then
we roughly reconstruct the trajectory with a circle. Using this swim function andwhich
is calculated
by correcting
TIC
time by are
the connected
average timeof flight between
hitcoordinates,
clusters
associatedthe
witha positronprogressively,
resultinginthe
a track
candidate.
Duringthis
process,wecanrefinetheleft-rightambiguity.
We can
DCH and the TIC.
also get information of incident angle to each cell.
The drift distance can be calculated from drift time, incident angle, left-right solution,
For a found track candidate, we can estimate the time of the track, ttrack , by using
andtiming
B-fieldofstrength.
theall hits belonging to the track. Using this drift-chamber self-contained ttrack
and the track direction, the timing-counter hit associated with the track is searched for,and the ttrack is refined by the timing-counter hit time. The drift time of each hit is given
by twire,i ttrack , where twire,i is a time extracted from the i-th wire waveforms.
5.1.4
Fitting
Given Trackdrift time,
incident angle, left-right solution, and B-field strength, we can calculate the drift distance uniquely. Since the relation has strong angular dependence,
Allprepared
track candidates
with at least
4 clusters
sent toa Kalman[62] toin fit
wethe time-to-distancefunctionsfor aredifferent
incident
anglefilter
as shown
Figa trajectory
to thesethe setof hits. precise
The Kalmanfilter isise↵ective
for tracks with smallure
5.10. Using
functions,hit coordinate
determined.
number of hits or noise, thus suited to tracking in MEG experiment.

5.3.2

Tracking

TheFitting
Kalman filter recursively estimates parameters by each measurement in order.
Track
The trackof a positrondescribedisbya vectoranalyzed
of five parameters:
two forand
thetheNext,
a trajectoryof the cantrackbecandidate
precisely
by a track fitting
state of positron on the target is reconstructed. We use the Kalman filter technique intrack fitting. The Kalman filter [88] was originally developed as a linear estimation forthe state of a dynamic system from a series of incomplete and noisy measurements. In
this 20 years, it has been extensively used for track fitting in high-energy physics (for70

5.2 Timing Counter

position, two for the direction and one for the momentum. A fit example is shown in
Figure 5.3.

Figure 5.3: Event display of positron tracking. Fitting is performed with Kalman filter.

5.2

Timing Counter

After passing through the drift chamber, positrons reach the timing counter anddeposit energy in scintillating bars. The waveforms read out at the end of each bar isused to measure impact time and position, which are used to match a TIC hit with a
DCH hit.

5.2.1

Waveform Analysis

The output of a PMT is divided into three outputs with fraction of 1:1:8. 10% of theoutput is digitized by DRS directly (PMT pulse) while 80% of the output goes to the
double-threshold-discriminator (DTD) and digitized (NIM pulse). Figure 5.4 shows anexample of a PMT waveform.
A template waveform is prepared by averaging many pulses channel-by-channel. Thisis then fitted to the NIM pulse to get the baseline and leading edge time.
Charge and amplitude are measured with PMT pulses. The charge is estimated byintegrating the pulse over 30 ns, and the amplitude is measured as the di↵erence715. EVENT RECONSTRUCTION

between the estimated baseline and the peak voltage of the pulse.

Figure 5.4: Figure 5.10: An example of the waveforms used in the timing counter reconstruction. The black line is a DRS output, which is fitted to a template (red). The blueline is the NIM pulse output, which is fitted to another template (green). The time delayof the NIM pulse with respect to the DRS waveform is due to an electronic delay in the
double threshold discriminator.

5.2.2

Hit Reconstruction

The leading edge time of each PMT is corrected for time walk e↵ects, and the di↵erence
between the two times is used to calculate the z coordinate. We reconstruct the hit
time by averaging two PMT times corrected for photon propagation time in the bar
based on the z coordinate.
High momentum positrons often penetrate a few bars. For these events, hits associalted
with one positron are clustered by closeness in time and z location. Multi-hit clusters
take the time of the first hit bar as the cluster time.

5.3

DCH-TIC Interconnection

First reconstructions of the tracking and timing counter are done independently. Then
DCH tracks and TIC clusters that belong to a single trajectory are selected.
The time of flight of the trajectory from the muon decay vertex to the impact location72

5.4 LXe Detector

in the TIC is computed by dividing the total path length by the speed of light. The
projection from the last chamber to TIC is complicated by the length and materials
such as cables and preamplifiers in its path. The accuracy is thus limited by scatteringand energy loss. We use an additional correction, | zDCH T IC | to time of flight. It ismeasured with Daliz data from ⇡ 0 runs.

In the events with more than two hits, the trajectory length and hit time measurements
are used to estimate the impact time at the first bar. In this case the TIC time
estimator is the average of the two measurements. The best estimate of the hit timeat the first bar is then corrected by the best estimate of the time of flight to get thetime of emission from the decay vertex.

5.4

LXe Detector

Waveforms measured by the PMTs inside the liquid xenon detector gives the information of photon energy, time, and position. We first reconstruct the number of detected
scintillation photons and arrival time in each PMT, then the energy, time, and first
conversion position of the detected photon.

5.4.1

Waveform Analysis

Typical waveform of LXe interacting with a photon is shown in Figure 5.5. The timeof each PMT waveform is extracted with the digital-constant-fraction method, namely,
the time at which the waveform reaches 30% of full pulse height. By this method, we
can determine the pulse time independently of the amplitude (no time walk e↵ect).
In order to reduce noise and statistical fluctuations, the pulse height is measured
indirectly from the charge assuming a linear scaling relationship.
The pulse charge is estimated by integrating the waveform, and is later converted
to number of photoelectrons observed by the PMT. In order to reduce noise of ⇠ 1

MHz, we apply a high-pass filter. We do not need to filter the high frequency noise asintegration itself works as a low pass filter. A high-pass filter is applied by subtracting
a moving average of the waveform. The number of points in the moving average is set

73

5. EVENT RECONSTRUCTION

Figure 5.5: Typical waveform of a PMT in LXe detector.

to 89, corresponding to a cuto↵ frequency of 11 MHz. A filtered waveform is shown in
Figure 5.6. The integration range is determined by the zero-crossing time of the sumof waveform and is applied to all PMTs. The range of the integration is 48 ns, whichacts as a low-pass filter with a 21MHz cut-o↵ frequency.

Figure 5.6: A high-pass filtered waveform.

Sometimes when a photon interacts at a close point to a PMT, the signal is large
enough to saturate the electronics. Figure 5.7 shows a saturated waveform. In thesecases, the charge is estimated with a time-over-threshold (ToT) method. The timespan during which the waveform is above a threshold, 150 mV, is related to the
expected charge based on the average pulse shape of photon interaction events (shownin Figure 5.8 as ”template waveform”). In cases of PMT saturation, the charge is74

5.4 LXe Detector

calculated directly from the ToT value. The recovery is important because about 15%of the photonsinteract within 1 cm of a PMT and most of them have at least one
saturated channel.

Figure 5.7: A saturated waveform.

5.4.2

Photon Reconstruction

We reconstruct the position of the shower by treating it like a point-like interaction.
First we convert PMT charge to number of photoelectrons with

Npe,i = Qi /(e · Gi )(5.5)

where Qi is a pulse charge of the i-th PMT, e is the elementary charge, and Gi is thegain. The number of scintillation photons can also be written as

Npho,i = Npe,i /QEi

(5.6)

where QEi is the quantum efficiency of the PMT. The calibration of PMTs is describedin Section 6.4.1.

75

nstruction
5. EVENT RECONSTRUCTION
Pusle height (V)Amplitude (a.u.)

0
-0.2
-0.4
-0.6

0
-0.5
-1
-1.5

-0.8

-2

-1

-2.5

-1.2

-3
0

100

0

200

50

100

150

200

ToT (nsec)

Time (nsec)

Charge (nC)

Charge : ToT @ 150mV
0
-0.5
-1
-1.5
0

e (nsec)

el.

Pulse height : ToT @ 150mV

Template Waveform
Waveform

-2
-2.5
-3
0

50

100

150

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

Figure 5.8: Charge estimation with ToT. Pulse shape of template waveform is used to
get the conversion factor from ToT to charge.

arge estimation
with ToT. When a pulse saturates,
Positioninteraction
is reconstructedby fitting the PMT output
distribution.
At
timationA photon
insteadofpointcharge
integration.
Pulse
shape
each point (x , y , z ), the solid angle of a photon-cathode, ⌦ (x , y , z ), is calculated
he conversion
factor from ToT to charge.
numerically. The interaction position (x , y , z ) is calculated by minimizing
iNpho,i2
pos = ⌃ic ⇥ ⌦i (x , y , z )pho,i (Npho,i )where c is a free parameter of the fitting, andat least one
saturated channel.

(5.7)

pho,i (Npho,i ) is the statistical uncertaintyof the PMT charge and is calculated byruction1⇥
pho,i (Npho,i ) =QEi

and Track Finding

s
p
Npe,i
Npho,i=pe,i (Npe,i ) =
QEi
QEi

76

(5.8)

5.4 LXe Detector

The position fit is performed twice to smooth e↵ects of non-point like shower development. Only inner face PMTs are used because they are closest to most of the energy
deposition and thus the least sensitive to the scattering and absorption. First fitting
uses PMTs whose center lie within a 3.5-PMT radius around one with maximum light.
Second fitting is then performed with PMTs whose whose center lie within a 2-PMT
radius around the result of the first fitting. Typically, 45 and 15 PMTs are used in thefirst and second fit, respectively.
The final fitted position can be written in the (u, v, w) coordinate system as
u = z, v = r0 , tan 1 ( y/x), w =where r0 = 67.85 cm, the radius of the inner face.

px2 + y 2

r0

(5.9)

Based on Monte Carlo, no bias is observed in v, while |u| and w have larger values

than the true position. The photon comes into the detector with incident angle withrespect to the PMT direction, which causes a bias in u towards outside direction
because the shower usually develops in the original photon direction. The bias in w
is towards deeper direction for the same reason. Corrections deduced from MC are
applied to u and w. In addition, corrections for fluctuations in the shape of the shower
are applied, based on the di↵erence between two fitting results.

EnergyGamma rays around 50 MeV converted in LXe detector usually deposit all of theirenergy in the active volume. Our energy reconstruction is on the idea that the total
number of collected photons is proportional to the original gamma energy regardlessof position and shape of the shower. In that case, the deposited photon energy is
proportional to the weighted sum of the scintillation photons:

Nsum = ⌃i wi ⇥ Npho,iwhere wi is the inverse photocathode coverage of the i-th PMT.

77

(5.10)

5. EVENT RECONSTRUCTION

This estimation works well when the first interaction point is deep enough, i.e.
w>2

3 cm. For shallow events however the solid angle of PMT varies strongly withshower position and the coverage can no longer assumed to be uniform. 55 MeV photons from ⇡ 0 decay are used to study the dependence of Nsum on the conversion depth
and solid angle of the PMT with maximum light. The result is shown in Figure 5.9
and it is used to correct for events with w < 3 cm. We apply a correction of Nsum with⌃in,max for such events. The correction function is a linear function, whose coefficientsare extracted by the 55 MeV photons in the ⇡ 0 calibration.
The remaining position dependence is corrected globally after the reconstruction. The
details of calibration of position dependence and correction are described in Section 6.4.2.
A position-dependent correction factor, F (u, v, w), is estimated from the 17.6 MeV
peaks in the CW- Li data. Thus the photon energy is given by

E = ↵F (u, v, w)Nsum(5.11)

where ↵ is the correction factor from the 55 MeV calibration. The linearity of Nsum
to gamma energy E was checked with several calibration methods[6] as shown in
Figure 5.10.

Time
Each PMT has a first interaction time ofthit,i = tP M T,itdelay,i

tof f set,i(5.12)

where tP M T,i is the leading edge time of the i-th PMT. tdelay,i is the time delay during
scintillation light propagation in LXe, and constant electronic delays are in the lastterm.
The second term tdelay,i comes from three contributions:tdelay = tprop (d, vef f )tindir ( )78twalk (Npe )(5.13)

5.4 LXe Detector

Figure 5.9: Correlation between in, max and sum of inner PMT Npho for w < 3 cm.

tprop is the propagation time in LXe, and it depends on distance between the source
and PMT, d, and the speed of scintillation light, vef f . tindir comes from photons
reaching a PMT after reflection o↵ walls or scattering in xenon. It depends on theincident angle . And twalk is the time walk e↵ect of time extraction method, which iscorrelated to the number of observed photoelectrons, Npe .
Then we can determine the hit time, tLXe by minimizing(thit,i tLXe )2time = ⌃i2t,i (Npe )
where

2

t,i (Npe ) is the time resolution of i-th PMT as a function of Npe .

(5.14)t,i (Npe ) is

calculated with PMTs that collect over 50 photoelectrons, typically about 150 of them.

5.4.3

PileupWith the high muon rates in the experiment, LXe detector is susceptible to events withmultiple gamma rays. Such pileup events are identified spatially by light distribution

79

The number of scintillation photons

5. EVENT RECONSTRUCTION

×103
400
350
300
250
200
150
100
50
20

40

60
80
Energy (MeV)

Figure 5.10: Linearity plot of E vs Nsum . The five points are given from 4.4 and 11.7
MeV from CW-B run, 17.6 MeV from CW-Li
run, and 54.9 and 82.9 MeV from ⇡ 0 run.
sum
Red line is the best-fit linear function whose intercept is fixed to zero.

7: Linearity plot of E v.s. N . The five points are given fromom CW-B run, 17.6 MeV from CW-Li run, and 54.9 and 82.9 MeV
e is the best-fit
linearfunction
whoseby theintercept
fixed
in both inner andouter face
PMTs, and temporally
distribution inis
PMT
times. to zero.
For the first method, we perform a peak search. More than one peak in the PMTs
would indicate pileup. Secondary peaks are distinguished by locating PMTs with
Npho > 200 at positions isolated from the primary peak.

For the second method, we use the normalized

80

2 value of the time fitting, ˆ2time .

5.4 LXe Detector

Usually large ˆ2time occurs when di↵erent groups of PMTs have di↵erent interaction
times of di↵erent photons. The threshold for pileup recognition is ˆ2time > 3, as
deduced from MC (Figure 5.11).

Figure 5.11: Distribution of normalized
signal MC without pileup (solid line).

2of the time fitting on data (dotted line) and

Since the probability of pileup is as large as 15%, it is important to reconstructpileup events rather than simply discarding them. First a look-up table of expected
PMT outputs is prepared with the 17.7 MeV gamma-ray in CW run. The table has
average output of each PMT for each mesh (1.55⇥1.55⇥1.55cm3 ) for the reconstructed
position. If a pileup is found, the elimination is done in the following procedure:

1. Estimate energy by fitting PMT charges with the expected distribution stored in
the table except for those around the pileup gamma.
2. Calculate the expectation of PMT outputs around the pileup region using thetable and replace them with the expectation.

81

68

4. Method of Event Reconstruction

average output of each PMT for each mesh (1.55 ⇥ 1.55 ⇥ 1.55 cm3 ) for the reconstructed
position. If the pileup is found, the elimination is performed in the following procedure.
1. Reconstruct energy in the usual way.
5.2. EVENTFit PMT RECONSTRUCTION
scintillation-photon distribution with the expected distribution stored in
the table except for pileup region.

3. Calculate the expectation of outputs around pileup region based on the first recon3. Reconstruct energy again with the replaced PMT outputs.
structed energy.
showsof light
beforeandafterwitheliminating
pileupevents.
4. Figure
Replace5.12
photons
somedistribution
PMTs aroundpileupregionthose of the
expectation.
Pileup events identified only by time distribution but not spastically separated are not
5. Reconstruct energy again with replaced outputs of PMTs.
recovered.
The light distribution in Figure 4.3 shows the elimination effect on 2D PMT development
view.
100
80

100

21999

100

21999

80

100

60

80

40

60

20

40

0

20

6304
60

80

40

60

20

40

6304

1806

1806

517
0

517

20
0

0

148

-20

148

-20
42

42

-40

-40

-60

-60

12

-80

12

-80
3

-100
-100
z

-50

0

50

100

150

200

250

3

-100
-100
z

300
1

-50

0

50

100

150

200

250

300
1

(a) Before pileup elimination.
(b) After pileup elimination.
Figure 4.3: Photon distribution in PMTs with eliminating pileup.
Figure 5.12: Photon distribution before and after pileup elimination.

With
using
the
same table,
dead
PMT channels
aresamealso recovered.
Replacednumber
the numDead
PMTchannels
are also
recoveredusing thetable. Replacement
ber of scintillation photons of the dead channel are estimated from the photon averagedof photons are estimated from the average of three PMTs at symmetric locations alongover the three PMTs at the symmetric locations along u and v direction.
u The
and replacement
v direction. depends on the table, and it is not appropriate for cosmic ray, 241 Amalpha sources on wires and LED runs. This is because the light distribution of gamma
rays from central targets, such as muon or proton target, differs from that for the special
calibration sources. The monitor of the light yield is performed without the elimination to
avoid uncertainties such as the different energy region and light distribution by calibrationand the change of non-uniformity.
Di↵erent pileup depending on beam status The probability density function (PDF)
of gamma-ray energy for the accidental background is prepared for pile-up and non-pileup
events separately.
In the case of ⇡ 0 run, the source of pileup is different from that for the muons because
there are many high energy and low energy backgrounds. The background in the ⇡ 0
run comes from the ⇡ 0 decay (a flat distribution between 54.9 and 82.9 MeV) and fromthe radiative capture of the charged pion (129 MeV gamma-ray and 8.9 MeV neutron).
Additionally many positrons, electrons and muons are also expected because the Wien
filter is not used in ⇡ 0 run. The difference of the background condition between the pionand the muon beam is taken into account in the estimation of the energy resolution in ⇡ 0
run.

82

6

Calibration
Calibrations are crucial to the precise measurement required by MEG experiment. In
addition, one needs a considerable running stability of all detectors. We developed
various calibration and monitoring methods to fulfill these requirements. Some of them
are complementary and some are redundant in order to obtain optimized performanceof the experiment. Some instabilities and deviations are corrected by calibration data.

6.1

DRS

6.1.1

Voltage Calibration

Every cell of every DRS chip has a di↵erent and non-linear response to voltage. We
prepare detailed response curves in advance by looking at the response for di↵erent
input voltages from internal DC voltage generator on DRS boards. The calibration isdone online so the data written in the disk is already calibrated for voltage response.

6.1.2

Time Calibration

A sampling frequency is generated by a series of inverters in a DRS chip. The samplinginterval of each cell in a DRS chip can vary due to characteristics of each individual
inverter. Sampling intervals of cells between the zero-crossing period were adjusted sothat the measured zero-crossing period become that of input sine wave. This regulations
were repeated until sampling intervals of all cells got converged. A global common clock83

6. CALIBRATION

is distributed to all DRS boards from the trigger system and used to synchronize thephase among chips.

6.2

Drift Chamber

6.2.1

z-coordinate Calibration

The z-coordinate calibration of drift chamber means the relative gain correction oneach ends and is done by using the known pattern of vernier pad. We calibrate therelative gains so that the reconstructed length of a vernier circle is the pitch lengthof 5cm. We calibrate the relative gain correction for cathode pad based on the factthat the mean charge collected on one cathode pad should be a quarter of total charge
induced on the anode.

6.2.2

Time O↵set Calibration

It is necessary to determine the time o↵set for each channel in order to measure thedrift time precisely. Several sources of di↵erence, such as di↵erences of cable lengthof both the signal and the trigger electronics, transition-time spread of electronics
devices contribute to the intrinsic timing jitter and it can be determined by fitting the
distribution with a polynomial function, empirically given as fifth-order one. The o↵set
is determined by the time that the height of distribution reaches 15% of maximum.

6.2.3

Time-to-Distance Calibration

Time-to-distance functions are calculated by GARFIELD[50] and the calibration isdone with simple iterative procedure[6].

1. Calculate a di↵erence between obtained drift distance and closest approach fromthe wire to the track, called residual
2. Put a corrective o↵set to the drift distance so that the residual is minimized.
3. Perform above for many events with similar incident angle and closed fieldstrength, and build a new time-to-distance plot.

846.3 Timing Counter

4. Fit the plot and form a new time-to-distance function.
The procedure is repeated until convergence is achieved.

6.3

Timing Counter6.3.1

z-coordinate Calibration

The time o↵set between two PMTs is adjusted because z hit position is reconstructedby the time di↵erence between the two. Reconstructed positron tracks are used as areference of z hit position. The relative time o↵set is calibrated so that the mean ofzDCH T IC distribution is zero.

6.3.2

Inter-Bar Time O↵set Calibration

The relative time o↵sets between bars is calibrated with CW-B data.

11 B(p,5)12
6 C

reaction produces gamma rays at 11.7 and 4.4 MeV. We use the 4.4 MeV gamma ray
at xenon detector as a time reference and measure the 11.7 MeV one on each bar.
Relative o↵sets are determined by the distribution of time di↵erence after time-of-flightcorrection.

6.4

LXe Gamma-ray Detector

6.4.1

PMT calibration

Alpha sources and LEDs are used for the monitor and calibration of PMTs. Figure 6.1
shows the placement of LED and ↵ wires inside the LXe detector and Figure 6.2 shows
the closeup of 241 Am source on wire.

Gain
LEDs attached to the lateral faces of the LXe detector are used as light sources tomonitor PMT outputs and calculate PMT gains.
We calculate gains by means of statistic number of photoelectrons, Npe . Observed85

6. CALIBRATION

Figure 6.1: Inside view of LXe detector shows 241 Am source on wire and LED on thewall. Three LEDs with di↵erent attenuations are mounted on lateral faces.

Figure 6.2: Closeup of 241 Am source on wire. Diameter is 100 µm and length is about 2
mm.

charge, q, can be expressed in an equation with gain G, and a conversion constant e,86

6.4 LXe Gamma-ray Detector

q = G · e · Npe(6.1)

We assume the output of LED is constant and follows the Poissson distribution. Let
Npho be the number of photon from LED to the PMT, which is constant in our assumption. The average number of photoneletrons is given by N̄pe = Npho · QE where
QE is quantum efficiency.

If we take many LED events, the mean and variance of the distribution becomesq̄ = G · e · N̄pe ,

(6.2)

2
2
2
q = (Ge · pe ) + noise

(6.3)

and

respectively. And we obtain the relation,22
q = Ge · q̄ + noise(6.4)

By taking LED events at di↵erent intensities we can calculate the gain as a slopeof variance-mean plot and it is insensitive to contribution from noise. We take nine
steps with di↵erent LED intensities without beam. Each step has 3000 events and we
can obtain a relation between charge and variation as shown in Figure 6.3. The slopeof the plot is then calculated as gain.
Gain calibration is taken everyday or twice a week depending on the detector
situation. Additionally the most intense step was taken everyday and every time when
we start or stop using beam to monitor the variation of PMT gains. LEDs are flashedwith 100 Hz during the usual calibration and LED data with 1 Hz are mixed in MEG
physics data taking.

Gain Shift
The long term stability of PMT gains is checked during the experiment. We noticed
that there is temporary gain change depending on beam intensity. This is called876. CALIBRATION(a) Nine LED peaks by various intensities.

(b) Gain estimation from the slope on chargemean and variance plane.
Figure 6.3: Gain calculation.

’gain shift’ and it has been confirmed using di↵erent methods such as LED and alphapeaks. This comes from the PMT themselves since our calculation relies on statisticsof photoelectrons. Shift values are di↵erent for each PMT, and are dependent on the
beam rate.
Since the shift is correlated to beam intensity we can correct it by constant monitoring. In normal beam rate condition the shift value is ⇠ 2%. LED data shownin Figure 6.4 are used for the correction[3]. Gains are corrected using the variationof the peak value of LED events.

Results indicate that gain shift is stabilized

within 5 minutes of opening beam blocker and 20 minutes of closing beam blocker.
Therefore during the experiment we wait 15 minutes after opening or closing beam88

6.4 LXe Gamma-ray Detector

blocker before taking regular physics runs, and we wait 30 minutes to take calibrations.
LED flashing runs are taken during the waiting period to monitor and correct gain shift.

Figure 6.4: Gain shift of a worst PMT after beam blocker opened in upper figure andclosed in lower.

Gain Adjustment
Before physics data taking start, gains are adjusted to equalize each PMT performanceand to keep the pulse height within a certain range.
Normally the gain of a PMT with same resistive divider is proportional to V kn , where
V is a bias voltage, n is the number of dynode stage and k is a factor determined bymaterial of dynodes. The PMTs for MEG experiment however use two Zener diodes,
hence the equation is approximately modified as,

Gain / (V

0.95Z

89

V0 )11k ,(6.5)

6. CALIBRATIONZ = Z1 + Z2 = 130Volt,(6.6)

where Z is the total voltage at two diodes, and V0 is an o↵set of loss in cables. Figure 6.5
shows the relation between HV and PMT gain as well as fit result with Equation 6.5.

Figure 6.5: Gain - HV curve and fit.

Gain Monitor
We noticed that gains are in a slow decrease over a large time scale, as shown in
Figure 6.6. This is due to the e↵ect of aging PMTs in large current. On average during
MEG run we observe 0.1

0.2% decrease per day. However we can adjust the gain byincreasing HV for the next several years since the applied HV is still sufficiently lower
than maximum value.

Quantum Efficiency
The quantum efficiency of PMTs can be calculated using alpha events. The point
like alpha sources on wires are mounted in the detector. Since we know the positionsof alpha sources and spectrum of the alpha particles, we can estimate the amount of

90







Gain was calculated with statistical fluctuation of detected photon electrons frommultiple LEDs with different attenuations
LED runs were taken regularly to monitor the decrease of gains
Gain adjusted at beginning of physics run
Decrease rate in MEG runs was similar to 20096.4(~ 1.5%per week) Detector
LXe Gamma-ray
Gains can be restored by raising HV. Not a problem for the experimentMEG
0

Xue Bai

Figure 6.6: History of gain during MEG and ⇡ 0 run.

JPS meeting Sep 11-14, 201

light observed by a given PMT. We can then calculate QE by comparing the measured
number of photoelectrons with the estimation from MC, since gains are calculated
from LED data.
In liquid xenon the reconstructed positions from alpha sources make rings aroundthe wires as shown in Figure 6.7. This is because the range of alpha particle of about
40µm from the wire is close to the 100µm diameter of the wire and the wire makes ashadow.
The MC-data correlation of peaks from observed photons by each source are shownin Figure 6.8 and the slope indicates QE. The events are selected for each PMT toavoid shadow positions behind the wires and sources with a large incident angle to
the PMT. QE is calculated by averaging some sets of estimations to reduce statisticalfluctuations. The calculated QE is then normalized so the mean of all QE is equal to
0.16.
A fine tuning for the MC such as reflection, attenuation lengths and light speed
improved agreement with actual data, however the MC for the QE estimation is91

! éǞǩǷȑ1ơȐǖȈ5ÈǁU5ƺ
" QE!=!αīMCƾǱȔǫƽǂPMT5@áșMCÆ&(1PMTƪǍł

αīî25#ƪǍǂÒ5^1Ŕį6ǱȔǫ

6. CALIBRATION! QE!=!Npe/(Np" gainǃLEDƽł

•Figure 6.7: Reconstructed positions of alpha wires.

2012/3/24

.

Ě67_ÙmƑ(2012)Ơķ¦nCŕR

Figure 6.8: Comparison of alpha peaks between MC and data.

sensitive to these parameters and a small discrepancy between the data and the MCremains and makes it difficult to estimate QE. The e↵ect of QE estimation on energy

92

PMT calibrations

6.4 LXe Gamma-ray Detector

Frequent& precisecalibration
using
LED
& the
alpha
resolution
was studied
by randomly
fluctuatingQE
frommeasured value. The−−

PMT gains are continuously decreasing.
Rate-dependent gain shift (Beam On ↔ Off)e↵ect is found to be small compared to the current energy resolution.
In addition a global bias is observed as di↵erence of non-uniformity between data andAn example of typical PMT
MC. We use 17.7 MeV gamma ray from 73 Li84 Be to correct the asymmetries observed
Beam on
HV changed
from PMTs at symmetric positions. After correction the stabilityof QE is within±10%⇠ 0.2% (Figure 6.9).

Beam off

2010

Q.E.

CEX

Figure 6.9: Stability of averaged QE.

6.4.2

Non-Uniformity

After gain calibration,
stability of LED peak mixed in MEG run< 0.1% for all period

16/Feb/2011

Yusuke UCHIYAMA

The reconstructed scintillation photons has a non-uniform response due to geometricale↵ect of di↵erent e↵ective coverage and finite absorption length.

Such position

dependent response can be corrected by non-uniform correction based on measurementin calibration runs. Correction factors along u and v are made with 17.7 MeV peaks in
CW runs while depth correction alongside w is made with 55 MeV -ray peaks in ⇡ 0
runs. The correction factor is estimated with two dimensional relation on uv planeand one dimensional relation along depth, w. The peak transition along w depends
both on the energy scale and on the position on (u, v). Therefore, the depth correction
is made with three separated u positions. The lateral walls are closer to each otherthan the top and bottom faces, hence peaks have larger dependence along u direction
than v direction. Figure 6.10 shows factors of depth correction.
The remained uv non-uniformity is estimated after the depth dependence iscorrected. Figure 6.11 shows the correction on u
after correction is less than 0.3%.

93v plane. The total non-uniformity

56. CALIBRATION

6.4.3

Figure 6.10: Non-uniformity correction on uv plane.

Figure 6.11: Non-uniformity correction on uv plane.

Energy Scale and Light yield

Light yield monitoring is important because it relates to energy scale that is determined by the 54.9 MeV peak in ⇡ 0 runs. We monitor light yield with various
sources during beam-o↵ periods. Details on the energy scale is described in Section 7.2.

94

6.4 LXe Gamma-ray Detector⇡ 0 run
We use the 54.9 MeV near signal energy from ⇡ 0 run to determine the energy scale.
Details of the reaction is described in Section3.6.1.

CW runThe CW calibration run around 30k events was taken to collect 17.7 MeV lithium
peaks once per a few days during physics run. Details or the CW accelerator was
described in Section 3.6.2.

Cosmic ray
The Landau distribution from cosmic rays is obtained by selecting muons penetratingouter face and inner face, as shown in Figure 6.12.

Figure 6.12: Cosmic ray peak.

95

6. CALIBRATIONAmBe source
The 241 Am9 Be source provides a 4.4MeV

from 12 C ⇤ that is produced in the

9 Be(↵, n)12 C ⇤ reaction. Peaks with AmBe source are shown in Figure 6.13.
4
6Figure 6.13: Gamma ray peaks from AmBe sources.

Alpha source
The alpha peaks are taken with LED runs to estimate QE. The monitor of alpha peaks
is important to monitor the purity of xenon and stability of the detector. Alpha peaks
are shown in Figure 6.14.

Neutron capture by Ni
Another calibration source is the 9 MeV gamma peak from thermal neutron capture
reaction Ni(n, )Ni⇤ . The neutron generator can be used when beam is on. Peak from
Ni is shown in Figure 6.15.

966.4 LXe Gamma-ray Detector

Figure 6.14: Alpha peak of all sources.

Figure 6.15: Ni peak during beam time.

97

6. CALIBRATION

98

7

Performance
7.1

Event Selection

Here we describe selection criteria for the performance evaluation. These criteria arealso applied in the final analysis. However we apply likelihood analysis with event-byevent PDFs, which take into account the quality of each event. Thus, our policy of
event selection is to select events with minimum quality cuts and evaluate performancein detail.

7.1.1

Photon Cuts

7.1.1.1

Fiducial Volume and Acceptance

The fiducial volume is defined as:|u| < 25, |v| < 71, 0 < w < 38.5cm(7.1)

The acceptance is shown in Figure 7.1.
7.1.1.2

Pileup Rejection

Details of pileup elimination is described in Section5.4.3. In case of pileup events where
two peaks of light distribution are identified, we apply an additional cut on these eventsso that the eliminated energy is positive and no larger than 10% of the total energy.

0<

E normalE pileupcorrected
E normal

99

< 0.1

(7.2)

7. PERFORMANCE

Figure 7.1: The accepted regions of the LXe detector are shown in red in the u-v planeon the left and the u-w plane on the right.

where E normal is the total deposited energy and E pileupcorrected is energy after removing
the secondary peak. Pileup events identified only by ˆ2time are also rejected. Figure 7.2
shows the rejected events in red-filled region. The inefficiency by this selection isevaluated to be ⇠ 8%.

7.1.1.3

Cosmic Ray Veto

Photons from the stopping target tend to interact close to the inner face and deeper
events are more likely to be cosmic rays entering through outer shell. Hence we canmake a cut on the ratio between inner face and outer face as well as conversion depth
to get rid of these events. The criteria to keep the event is set as(w < 15 ⇥Ninner+ 15) ^ (w >Nouter100

400 ⇥Ninner+ 120)
Nouter

(7.3)

7.1 Event Selection

Figure 7.2: Distribution of the subtracted energy in the pileup elimination. Red filled
entries are rejected by the selection criteria.

where w is the conversion depth. We apply another cut using information of pulse
shape,
55 < PulseWidth < 500 nsec

(7.4)

The remaining contribution of cosmic is about 1% of the total background in the signal
region and 56% of cosmic rays are rejected. Figure 7.3 shows the event distribution of
measured and MC events.

7.1.2

Positron Cuts

7.1.2.1

Fiducial Volume

The positron acceptance is defined so that the inverse of the positron direction goes
into the gamma-ray detector fiducial volume.

7.1.2.2

Drift Chamber Hit Pattern

To select positrons reconstructed with reliable tracking, we have the following requirements: the total number of hits

7; the number of chambers that have hits

101

3; the7. PERFORMANCE

Figure 7.3: Cosmic ray cut. Black plot shows the events in the acceptance taken withoutbeam. Red one shows MC signal events. Blue lines show the selection criteria.

span of the track

7.1.2.3

4 chambers; the number of multi-hit chambers

1.

Quality of Track Fit

We apply several criteria on the fitted tracks to select events with good fitting quality.
The Kalman filter provides several indicators on the quality of the track fit. One
requirement is that the normalized

2 is smaller than 12.

uncertainties are: Ee < 1.1 MeV, ✓e < 0.6 , and7.1.2.4

Cuts on energy and angle

e < 1.5 .

Vertex Cut

To guarantee that the positron comes from a muon decay on target, we require thereconstructed muon decay vertex to be within an ellipse that excludes the target frameby 5 mm.

7.1.2.5

DCH-TIC Matching Cut

To select well-matched events between the drift chamber and timing counter, we apply
cuts on the qualities | ZDCH T IC | < 12 cm and | rDC T C

102

1.8| < 5 cm.

7.2 Gamma Energy

7.1.2.6

Ghost Track Selection

Because the tracking algorithm attempts to reconstruct all possible tracks in a highrate environment, more than one manifestation of the same track may be identified ina single event. If there are two clusters of hits, it may be possible to place either ofthem and fit an otherwise identical track. We call these tracks associating with a single
positron ghost tracks. A procedure is developed to classify ghost tracks and to selectamong them. All of the fitted tracks in an event are compared with one another andany two are considered to be ghosts of the same track if the following conditions are met:

1. They both share a common timing counter hit.
2. The number of chambers with at least one hit in common is more than half thetotal number of hit chambers on the shorter of the two tracks.
Each track in a set of ghost tracks is then ranked to indicate its quality of reconstruction. The rank Grank is based on the chamber span (S) which is the number ofchambers that the span of a track covers, and the normalized
Grank =

2 of track fitting, ˆ2 :1
+ ↵ ˆ2S(7.5)↵ = 0.025 is chosen so that the first term dominates the ranking when tracks have
similar ˆ2 and the second term dominates when tracks have very di↵erent ˆ2 .
If more than one ghosts of the same track passes the aforementioned cut criteria, theone with the smallest ranking is selected.
The inefficiencies of each cut is summarized in Table 7.1.

7.2

Gamma Energy

The energy resolution and scale of the gamma-ray detector are determined at 54.9MeV
peak in ⇡ 0 data (Figure 7.4). The energy peak is estimated with a combination of103

7. PERFORMANCE

Selections

Inefficiency

Fiducial Volume

3.0%

Pileup Rejection

8%

Cosmic Ray Veto

1%e+ Fiducial Volume

26.1%

DCH Hit Pattern

8.6%

Track Fit Quality

15.9%

Vertex Cut

15.9%

DCH-TIC Matching

9.2%

Table 7.1: A table of inefficiencies for each cut. The inefficiencies are calculated as thefraction of events rejected by making only the cut with respect to making no cuts. The
acceptance cut inefficiency is calculated when positron cuts, pileup cut, and cosmic ray
cut are already applied.

Guassian function for higher energy part and exponential function for a tail. These
two functions are smoothly connected at transition point lower than the peak energy.

Figure 7.4: Energy peak of 54.9MeV in ⇡ 0 run.

Since the resolution is position dependent, the response was extracted for eachposition. Figure 7.5 shows resolution maps for various positions of the gamma ray
interaction for w

2 cm. The averaged resolution is

up = 1.9% for w

2 cm. The

relative uncertainties of resolution evaluation are estimated to be 10 ⇠ 14% from theerrors of fitting and the variation of the resolutions in close positions. The resolutions1047.2 Gamma Energy

are measured separately for the events with w < 2 cm because the resolution depends
heavily on the depth of interaction in shallow region. The averaged resolution forw < 2 cm is 2.4%.

Figure 7.5: Energy resolution at 54.9 MeV on inner face (w2 cm).

Systematic Uncertainty on Energy Scale
We calibrate the energy scale of the detector using peaks from ⇡ 0 decay. Energy
scale in 2010 using various calibration methods is shown in Figure 7.6. Sources of105

7. PERFORMANCE

uncertainties come from di↵erence between 55 MeV and 83 MeV peak (0.3%), position
dependence (0.3%), gain uncertainty (0.09%) and statistic uncertainty in 55 MeV peakfit (0.03%).

Figure 7.6: Energy Scale.

1067.3 Positron Energy

7.3

Positron Energy

We calculate the resolution of positron energy by fitting the energy distribution of
normal muon decay, µ+ ! e+ ⌫e ⌫ µ , with a convolved function of a theoretical predictionin the SM and three Gaussian functions as the resolution contribution of a core and atail:

Probability density(Eemeasured ) = (M ichel ⇤ Acceptance)(Eetrue ) ⌦ Resolution (7.6)
Both the acceptance and resolution functions are based on Monte Carlo simulation.
The acceptance function is taken astrue

Acceptance(Eetrue ) =µacc
1 + erf ( Eep2 acc
)

(7.7)
2
and the resolution function is assumed to be a double Gaussian. The parameters are
extracted from the fit, as shown in Figure 7.7. This gives an average resolution of
Ee = 330 ± 16 keV in the core (82% of the events) and

Ee = 1.13 ± 0.12 MeV in

the tail. There is also a 60 keV systematic underestimation of the energy, which is
associated with a 25 keV uncertainty from Monte Carlo.
A complementary approach to measure the positron energy resolution is using double turn events. Figure 7.8 shows the energy di↵erence between the two turns, which is
fit to the convolution of a double Gaussian function with itself. This gives an average
resolution of

Ee = 330 keV in the core (79% of the events) and

Ee = 1.56 MeV in

the tail, which are consistent with the results obtained from the fit of Michel spectrum.
There is a systematic o↵set of 108 keV between the energies of the two turns as theenergy of the first turn is systematically larger than the energy of the second turn.

7.4

Relative Angle

7.4.1

Positron Emission Angle

The resolutions of emission angle ✓e ande are taken from data by using events that

where a positron makes two turns in the drift chamber. The resolution in each turnis assumed to be the same and these distributions are fit to the convolution of a sum

107

7. PERFORMANCE

Figure 7.7: Fitting of a Michel spectrum.The dashed black line is the theoretical spectrum
and the dashed blue line is the resolution function. The acceptance curve is shown in thebottom plot.

of two Gaussians with itself. The resolutions are Vertex z/y = 2.0/1.1mm,
mrad, and

7.4.2

e

✓e = 11= 7.2 mrad.

Muon Decay Vertex

The position of the decay vertex is determined by the extrapolated positron tracking.
We can estimate the position resolution with holes on the target. The vertex resolutions in 2010 along minor axis(Z) and major axis on the plane(Y) arecore = 0.97 mm,Ytail = 3.8 mm.
Y108

Z = 3.78 mm,

7.4 Relative Angle

Figure 7.8: A fit to the distribution of energy di↵erence between two turns.

7.4.3

Gamma ray Position

We measured position with CW run in 2010. Several small lead cubes (approximately
15 ⇥ 15 ⇥ 13 mm) were put in front of the LXe detector, as pictured in Figure 7.9.
The resolutions areu,v = 5 mm,

w = 6mm.

Figure 7.9: Lead cubes and the support.

109

7. PERFORMANCE

7.4.4

Combined Resolution

The resolutions of relative angle in ✓ anddirections were then obtained by combining

the angular resolutions of the two particles. They are evaluated to bemrad ande✓e= 17.1= 14.0 mrad.

7.5

Relative Timing

7.5.1

Positron TimeWe measure the resolutions of the timing counter bars by looking at the distributionsof time di↵erence between two consecutive bar hits. The width of the time di↵erence
distribution of the two hits gives the time resolution. For a single bar resolution,
the width is divided by squared root of two assuming that the two have the same
resolution. The average resolution is 75 ps. Path length and DRS jitters also contributeto the performance.

7.5.2

Gamma-ray Time

Intrinsic Resolution
In order to measure the intrinsic time resolution of gamma-ray detector, we divide
PMTs into two groups, even and odd groups, as shown in figure 7.10. The intrinsic
time resolution is evaluated by the distribution of half of time di↵erence between thetwo reconstructed time,
tint
todd teven
=
(7.8)
2
2
p
The intrinsic resolution proportional to E and dominated by photoelectronic statistics. It is about 36.4 ps at 55 MeV and 28.4 ps at 83 MeV.

Practical Resolution
We measure the practical time resolution of gamma ray detector using ⇡ 0 !

datawith scintillation counters that were mounted on the NaI detector as a reference counter.

1107.5 Relative Timing

Figure 7.10: PMT grouping. Blue circles belong to the ’odd’ group while orange circles
belong to the ’even’ group.

Figure 7.11 shows the distribution of the time di↵erence between the 55MeV

ray atthe gamma-ray detector and the 83MeV one at the reference counter,tabs = t

tref

(7.9)

The resolutions are 119ps at 55 MeV, which includes contributions from beam
spread (58 ps), resolution of the reference counter (81 ps), and time resolution of LXe.
By subtracting the other elements, we can estimate the time resolution of LXe to be
XEC = 67 ps.

This includes intrinsic resolution, time-of-flight (20 ps), contribution

from DRS (24 ps) and e↵ects of position uncertainty and shower fluctuation.

7.5.3

Gamma-Positron Relative Time

Radiative muon decays provide a means to measure the relative timing resolution under
the same experimental conditions of a signal. The time resolution from radiative decay
includes all contributions from the LXe detector, positron tracking and electronics.
Figure 7.12 shows the peak in E sideband in a range of 50M eV  Ee  55M eV ,40M eV  E  48M eV and within 300 mrad opening angle. The peak is fitted withdouble Gaussian and the resolution ise= 122 ps.

111

7. PERFORMANCE

Figure 7.11: timing di↵erence between LXe datector and reference counter

Figure 7.12: Time di↵erence between positron and gamma ray in 2010.

112

7.6 Efficiency

7.6

Efficiency

7.6.1

Positron Detection Efficiency

For normalizing purpose, the absolute positron detection efficiency is not needed.
Only the ratio of signal positron to Michel positron detection efficiency is required.
This is calculated by evaluating the fitted acceptance function at the signal energy
in Michel spectrum and dividing it with weighted average of the acceptance functionin the interval 50 < Ee < 52.8 MeV. The uncertainty is estimated by refitting theMichel spectrum with Gaussian and recalculating the efficiency ratio. The result is✏e+ = 34%.
The positron detection efficiency is actually the product of the drift chamber detection efficiency and the conditional timing counter detection efficiency given a reconstructed positron. The conditional timing counter efficiency can be measured using
data from DCH-self trigger and selecting events with a positron in the drift chamber
passing the drift chamber-related selection cuts.

7.6.2

Gamma ray Efficiency

Gamma ray detection efficiency is a crucial part of the LXe detector performanceand is required for normalizing the experiment.

Inefficiency comes from several

sources: failure to penetrate the magnet wall and cryostat, pile-up cuts and energy cuts.

The detection efficiency is estimated with signal Monte Carlo.

It isdefined as the conditional probability to reconstruct a signal photon that passes
energy and pile-up cuts, given a reconstructed signal positron. The uncertainty ismeasured using NaI self trigger data in ⇡ 0 run. We can get the condition that a photon around 55 MeV is emitted to the LXe detector by tagging a photon around 83 MeV.
We start by counting number of events in low energy part, where e↵ects of pileup
and AIF is smaller. We calculate the efficiency by dividing the number of signal events
mixed with pileup gamma rays with the number of radiative decay events mixed withpileup gamma rays. Energy cuts 48 MeV< E <58 MeV are applied to both counts
as well as acceptance cuts. The analysis efficiency is estimated to be 91.1%. The
detection efficiency which is the weighted average of efficiencies taking into account113

7. PERFORMANCE

the angle distribution of positron detection, is estimated to be 64.7%.

Thus theconditional probability to detect and reconstruct a photon once a positron is detected,
is 64.7% ⇥ 91.1% = (59 ± 3)%.

7.7

Summary of Performance

The detector performance in 2010 is summarized in Table 7.2.

Gamma energy (%)

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

Gamma timing (ps)67Gamma position (mm)5 (u,v)/6(w)Positron timing (ps)107

Positron energy (MeV)

0.32 (core 79%)

Positron angle (mrad)

7.2 ( , core), 11.0 (✓)

Vertex position (mm)2.0 (Z), 1.1 (Y)Positron-gamma timing (ps)122(core)

Gamma efficiency (%)

59

Positron efficiency (%)

34

Trigger efficiency(%)

92

Positron-gamma angle (mrad)14.0 ( , core), 17.1 (✓)

Stopping µ rate

2.9 ⇥ 10 7 s 1

DAQ time/Real time

56 days/67days

Total µ stops on target

1.1 ⇥ 1014
2.1 ⇥ 1012

Data statistics (k-factor)

Table 7.2: Performance in 2010.

114

8

Improvements for 2010 Data
For the analysis of 2010 data, we improved several aspects of calibration and analysis
such as detector alignment, implementation of correlations in position observables,
improved magnetic field map and improved likelihood analysis. Such improvements
were subsequently also applied to the 2009 data.

8.1

Alignment

8.1.1

DC Alignment

The alignment procedure for drift chamber consists of two parts, a geometricalalignment and a software alignment using cosmic rays and Michel positrons. First,
the position of the drift chamber is measured by an optical survey with the helpof reference markers on each module with a precision of 0.2 mm. This allows thedetermination of the wire positions in the absolute coordinate system. For a better
accuracy, a track-based alignment procedure consisting of three steps was developed,
based on the Millipede method[63].

The first step is the internal alignment ofdrift chamber modules, which is done by recording cosmic ray events without the
COBRA magnetic field. The displacement of each module from the initial position are
obtained with an accuracy of 150µm. The second step is placing the obtained framein the spectrometer with a fine-tuned DC-target-COBRA position, using a sample
of double-turn Michel positron tracks in the magnet volume. The third step is the115

8. IMPROVEMENTS FOR 2010 DATAalignment of the spectrometer with respect to the LXe detector.

8.1.2

Relative Alignment Between DC and LXe

The relative alignment of the sub-detectors is based on the initial alignment of
the LXe detector on the detector platform and the position measurements using
cosmic ray, CW and AmBe data.

Two plates with a reference hole are installedon the inner support structure of the LXe detector and the detector position isadjusted through the position of the hole center. The LXe detector is installed at(x, y, z) = (724mm, 1505 ± 1.5mm, 0mm), where x is the distance from the beamline

in parallel, y is the height of the detector, and z is the axis along the beamline. Thereis a systematic uncertainty of ± 1.5 mm for the placed position.

Cosmic Ray
Cosmic rays penetrating both LXe detector and the drift chamber were used tomeasure their relative position. The data are taken without COBRA magnetic field.
By extrapolating the trajectory in drift chamber to the inner face of XEC, we canreconstruct the entrance point, which is then compared with the reconstructed position
measured by LXe.
The trajectory of a cosmic ray is reconstructed using hits in drift chamber. The
reconstructed incident point is defined as (ztrack , vtrack ) and the incident angles are⌘xy and ⌘zx , as shown in Figure 8.1. The results aremm,v=

1.9 ± 0.7 mm.

z = zreczexp = ( 6.1 ± 0.7)

CW
CW data are taken with several small lead cubes in front of the LXe detector. CW datawith a LiB target were taken with dedicated trigger patches as shown in Figure 8.2.
An average of di↵erences between measured and reconstructed z,
using six bars is

7.3 ± 1.4 mm. An average of116

z = zreczexp ,

using Bar-C is 0.3 ± 1.1 mrad.

8.1 Alignment

Figure 8.1: Definition of variables used in CR analysis. Blue lines are the reconstructedtrajectory by DCH.

Figure 8.2: Trigger patches used in CW runs.

AmBe
An AmBe source is placed in front of the LXe detector to calculate its absolute
position. The reconstructed z-position of 4.4 MeV

event is compared with the source

position which is measured by an optical survey. The gamma ray from the AmBe117

8. IMPROVEMENTS FOR 2010 DATA

source is collimated with a 20 mm-thick lead collimator with a 10mm

hole. The

measurements are taken in front of two PMTs at z = 0 and di↵erent v-positions. The
source is moved along z direction for each measurement. The di↵erence between thereconstructed and the source z positions is

5.5 ± 1.8 mm.

Combining the results
A weighted mean was taken to get a combination of the results. The uncertaintyof the drift chamber position is added to that of CW and AmBe results. There isno significant shift inhowever the same uncertainty is applied. The results of therelative alignment arez=

6.2 ± 2.3 mm and= 0 ± 3.4 mrad. The uncertainty

of relative angle(3.4 mrad) is improved compared to 2009 (7.5 mrad).

8.1.3

Target Alignment

The alignment of the target is confirmed by the reconstructed positions of several holeson the target, as shown in Figure 8.3. When misalignment happens, the reconstructed
position would depend on the incident angle of the positron track as illustrated in
Figure 8.4.

The result confirmed the position of optical survey is correct.

Figure 8.3: Target markers.

118

8.2 Correlations

Figure 8.4: Position of the holes when misalignment happens. Dotted line shows theassumed target and solid line is the actual target.

8.2

Correlations

In principle no strong correlations among the positron variables are expected inside
the spectrometer. However since all positrons come from the target plane and thetarget thickness is 205 µm, it introduces a constraint in track reconstruction, thus
creates correlations among all signal positron variables.
To evaluate the correlations introduced by the target plane, we studied how
infinitesimal changes of each variable influence the others. We assume the positron
trace is helicoidal. For the extraction of thee vsE and Y vs E correlations, thetrack is assumed to be a circle when projected on the transverse plane, as shown in
Figure 8.5. Target intersection with the transverse plane passing from the muon decay
point V = (X, Y ) is shown in red. For the extraction of the Z vse , the track canbe projected on a plane that is parallel to the Z-axis and tangent to the track helix at
the muon vertex, as shown in Fig 8.6.
From this geometrical model we can establish the following relations among infinitesimal changes at the target.
e =

2 tan

R
=
R

119

2 tan

E
E

(8.1)

8. IMPROVEMENTS FOR 2010 DATA

Figure 8.5: Extraction of thee vsE and Y vs E correlations.

Figure 8.6: Extraction of the Z vs

Y = 2 R cos

e + R sin

120

e and

e

e =

Z vs E correlations.

2R E
cos e E

(8.2)

8.2 Correlations

Z=

2R
sin2 ee

2R cot

e

E
E

(8.3)

In order to evaluate resolutions and correlations, we use a double turn tracks
method where tracks which make two turns inside the drift chambers are used and eachturn is reconstructed as an independent pseudo-track. Each pseudo-track is projected
onto a plane with the same inclination as the target, as shown in Figure 8.7. The
di↵erence of their spatial and directional components is used to measure resolutions.

Figure 8.7: Illustration of the double turn track method.

We definexT U RN where x = ✓e ,e , E, Y, Z asxT U RN = xT U RN0xT U RN1(8.4)

where xT U RN0 is the value of variable x for the first turn pseudo-track and xT U RN1 forthe second turn pseudo-track. Similarly in MC we can define the true resolution on x
asxT RU ExT RU E = xM EASU RED

121

xT RU E

(8.5)

8. IMPROVEMENTS FOR 2010 DATA

where xM EASU RED is the measured value of x and xT RU E is the true value of x
generated in MC.
For likelihood analysis, correlations are more accurately implemented in signal PDF.
In 2009,

E

and E correlation was incorporated as a systematic error of 50% due

to lack of method to measure the correlations from data. This corresponds to 0.09
in RMS of di↵erence of likelihood ratio compared to 0.02 in 2010. The details ofsystematics are described in Chapter 9.5.

8.3

Magnetic Field Map

A reconstructed field is used for 2010 data to minimize the uncertainty due to misalignment of Hall sensors.
The magnetic field was measured in 2006 with a commercial three-axis Hall probe.
It has three orthogonally aligned Hall sensors on a wagon which moves along z, r and. The probe was aligned so that it can measure Bz , Br , and B individually. Possible
measurement errors come from limited readout accuracy of the Hall sensor and thepositioning of the Hall probe, and misalignment of the field measuring machine, themoving wagon, or the Hall probe.
The Hall sensors were calibrated prior to measurement at a precision of 0.05%. The
planar Hall e↵ect is less than 0.2% and the e↵ect of the temperature coefficient of
the Hall sensor is estimated to be less than 0.6%. The field measuring machine, themoving wagon and the Hall probe are aligned at a precision of a few mrad using thelaser tracker. The precision of the relative angle alignment of the three Hall sensors is±9 mrad. Since a small misalignment could cause a large e↵ect on Br and B fromthe main component Bz , only Bz is used to minimize the influence of misalignment
while Br and B are calculated from Maxwell equation.
The global o↵set of the field measuring machine from the theoretical magnetic centeris measured by comparing the measured Bz with the calculation. By minimizing thedi↵erence at the grid points, the o↵set is estimated to be (x0 , y0 , z0 ) =(+0.5 mm, -2.9mm, -0.6 mm).
After the global o↵set correction, Br and B are calculated from the measured Bz

122

8.4 Likelihood Analysis

using Maxwell equations.

8.4

Likelihood Analysis

To improve likelihood analysis, constraints from sideband data are applied and profile
likelihood is used, which will be discussed in Chapter 9.

8.5

Update on 2009 Result

All the improvements described in the preceding sections were also studied and applied
to the 2009 data. The obtained performance in 2009 is summarized in Table 8.1.

E (%)(u,v,w) (mm)✏Ee (%)

2.1

5,6

0.58

0.74

e

(mrad)
7.1✓e (mrad)11.2

VertexZ,Y (mm)

3.4,3.4✏ete (ps)✏trigger

0.4

142

0.84

Table 8.1: Resolution in 2009.

The preliminary result of the maximum likelihood fit of the 2009 data before applying the improvements was+38(Nsig , NRD , NBG ) = (3.0+6.9 , 35+24
22 , 332 36 ), Nobs = 370(8.6)

Figure 8.8 shows the distribution of each observable with average PDFs scaled withthe best fit numbers of each type. The estimated sensitivity and the obtained upper
limit of branching ratio in the preliminary analysis are
S2009 = 6.1 ⇥ 10 12(8.7)

Br(µ+ ! e+ ) < 1.5 ⇥ 10 11 at 90% C.L.

(8.8)

The event distributions are shown in Figure 8.9. The events are labeled with theorder of the likelihood ratio defined as S/(S + R + B).

123

Events / ( 0.0004 GeV )Events / ( 0.00024 GeV )8. IMPROVEMENTS FOR 2010 DATA

10

1

10-1

0.052

1

0.051

0.052

0.053

0.054
0.055
Positron energy (GeV)

0.048

0.054

0.056
0.058
Gamma energy (GeV)(b) Positron energy
Events / ( 0.004 )Events / ( 0.004 )

0.05

141

10

(a) Gamma-ray energy
25
20

25
20

15

15

10

10

5

5
0

-0.04
-0.02
0
0.02
0.04
Relative phi angle of flipped positron direction wrt gamma direction

-0.04
-0.02
0
0.02
0.04
Relative theta angle of flipped positron direction wrt gamma direction(d) Relative angle
Number of RD events(c) Relative angle
Events / ( 5.6e-11 sec )Discussion

10-1
0.05

0

10.7

30
25
20
15

100
80
60
40
20

10

0
5
0

-0.5

0
0.5
Time difference between positron and gamma (sec)

-20

×10-9

0

(e) Relative time

5

10

15
20
Number of signal events(f) Contour plots on Nsig and NRD

Figure 10.3: Result of the maximum likelihood fit. Summed PDFs weighted with best fit

Figure8.8:
result of 2009
data before
applying
thePDFimprovements.
values of
eachPreliminary
types are superimposedon (a)-(e).
The notationof eachis the same (a)-(e)as intheFigure10.2. PDFs weighted with best fit values of each types. The blue lines are totalshow
summedsums of all PDFs. The green, red, and magenta lines show the signal, RD, and background
PDF respectively. The black dots are the best fit values. (f) shows the contour plot on
Nsig. The dashed, solid, and dotted lines show contours of the likelihood function
10.7NRD
Discussion
at 1, 1.645, 2 sigma respectively[1][2][3].
10.7.1

Event distribution

The eventafter theunblinding areandshown
in Figure10.5 where
the eventsabove are
After
the distributions
improvements
in calibration
analysis
methods
discussed
near the signal are labeled with the order of the likelihood ratio defined as S/(S + R + B).

12 , mainly thanks toapplied,estimated
improved
3.3
In thethe
figure,(a) and (b)sensitivity
show E andwas
Ee , while
relative totimebeandcos⇥ 10e are shown in (c)and (d). likelihood
Separately, analysis,
the events aswithshowna high-quality
positronthe profile
in Figure9.19.tracking are plotted in the

figures (b) and (d). The most signal-like events in the analysis region remain even after
high quality cut is applied in (b) and (d) out of all events without the quality selectionin (a) and (c).

The best fit result after the improvements is

+4.5
+12.3
(Nsig , NRD , NBG ) = (3.4+6.6
4.4 , 26.9 4.5 , 273.1 12.3 ), Nobs = 311(8.9)

Figure 9.22 shows the distribution of each observable with average PDFs scaled124Analysis of the MEG experiment to search for µ + ! e+ γ decays

Ryu Sawada

t eγ (nsec)

58
57
56

2
1.5
1

55
54

0.5 6

53

0

81
5
23

52
51

-0.5

7

-1

50

-1.5

49
48
50

5 1
3 10
2
8 9

51

52

53

54

-2
-1

55
56
Ee (MeV)-0.9995(a)-0.999-0.9985cosΘeγ(b)

Figure 1: Event distribution in the analysis region. The contours show 1, 1.64 and 2 σ regions of the signal

Figure8.9: coverEvent
the analysis
region.[2][1][3]The
contours
PDF, which
39,distributions
74 and 87 % of inprobability,
respectively.
(a) Photon andpositronenergy.show
Cuts 1,
to 1.64
about 90 of
% of
time
and angle
signal
PDF areappliedplot.
(b) Cosineof opening angle
andandselect
2 regionsthesignal
PDF,
which
cover
39, for
74the
and
87%
of probability,
respectively.
time difference. Cuts to select about 90 % of photon and positron energy signal PDF are applied for the plot.
In the left plot, selections in te and ✓e , each of which is 90% efficient on the signal, areHighly ranked events in terms of the relative signal likelihood are numbered correspondingly.
applied. In the right plot, a selection in Ee (90% on the signal) and a selection in E (73%on the signal) are applied.
Figure 2 shows projections of the fitting result to each variable. The fitting was done on 370
observed events, and the best estimates in the analysis window are Nsig = 3.0 and NRMD = 35+24
22 .
The best estimate of NRMD is consistent with the expectation estimated from the Eγ sideband to
be 32±2. The confidence interval was evaluated using the unified classical frequentist method[4].
Systematic uncertainties are included by fluctuating parameters of PDFs in fittings of Monte Carlosimulations to calculate confidence levels. The upper limit of the number of signals is 14.51 ,and Nsig = 0 is included in the interval. Using the norminalization factor, the upper limit of the
branching ratio is set as
B(µ + ! e+ γ )< 1.5 ⇥ 10 11 at 90 % C.L.
B(µ + ! e+ ν ν )1 The best estimate and the upper limit of N
sig range from 3 to 4.5 and from 12 to 14.5, respectively, depending on
the analysis methods.

3

Figure 8.10: Updated result of sensitivity in 2009.

with the best fit numbers of each type.

125

PoS(ICHEP 2010)263

Eγ (MeV)The sensitivity is estimated as an average upper limit at 90 % confidence level (C.L.) from anensemble of a large number of simulation experiments to be 6.1 ⇥ 10 12 . The upper limits in Teγ
sidebands is 4 ⇠ 6 ⇥ 10 12 , and is consistent with the sensitivity.
Events around the analysis region were unmasked after the calibration, the optimization of theanalysis algorithms and the background study in the sidebands are completed. Figure 1 shows a
distribution of events after unmasking.
8.5 Update on 2009 Result
A physics analysis was performed on events in 48  Eγ  58 MeV, 50  Ee  56 MeV, |Teγ | 
0.7 nsec, |φeγ |  50 mrad and |θeγ |  50 mrad.

8. IMPROVEMENTS FOR 2010 DATA

58

tea (nsec)

Ea (MeV)Figure 8.11: Updated result of 2009 data. The green, red, magenta and blue lines show
the signal, RD, background and total sums of PDF respectively.

57
56

1.5
1

55

0.5

54
53

0

1 2

52

3

2
3

1

-0.5

51

-1

50

-1.5

49
48
50

2

51

52

53

54

-2
-1

55
56
Ee (MeV)(a)-0.9995-0.999-0.9985-0.998
cosOea(b)

Figure 8.12: Updated event distributions in the analysis region for 2009 data. In theleft plot, selections in te and ✓e , each of which is 90% efficient on the signal, are applied(|te | < 0.278 nsec, |✓e | > 178.34 ). In the right plot, a selection in Ee (90% on the signal)
and a selection in E (73% on the signal) are applied (51 < E < 55 MeV, 52.34 < Ee < 55
MeV).

As shown in Figure 9.26, an excess of events around the signal region still remain.
However the sensitivity is much improved compared to the preliminary result. The
obtained confidence level curve is shown in Figure 8.13 and the lower and upper limit

1268.5 Update on 2009 Result

Confidence Levelin number of signals and branching ratio are summarized in Table 8.2.

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

5

10

15
20
Number of signals

Figure 8.13: Confidence level curve in 2009 data.

Best fit
Nsig
BR

LL(90% C.L.)

ULL(90% C.L.)

UL(95% C.L.)

CL@0

0.2(0.2)

10.4(10.1)

11.9(N/A)

0.92(0.92)

3.4
3.2 ⇥ 1012

1.7 ⇥ 1013 (1.7 ⇥ 10 13 )9.6 ⇥ 10

12 (9.4 ⇥ 10 12 )Table 8.2: Confidence intervals on 2009 data.

127

1.1 ⇥ 1011 (N/A)

0.92(0.92)

8. IMPROVEMENTS FOR 2010 DATA

128

9

Physics Analysis on Data 2010
9.1

Overview of Analysis

This chapter shows the analysis to calculate the branching ratio of µ+ ! e+from2010 data. We perform a maximum likelihood analysis to extract the number ofµ+ ! e+events in the data sample. A likelihood function is constructed fromprobability density functions (PDFs) of distributions in Ee , E , te , ✓e

andefor signal, radiative muon decay, and accidental background events. A confidence
interval of the measured branching ratio is then estimated with a frequentist approach.
The sensitivity with null-signal assumption is calculated by averaging the branchingratio upper limits of the generated experiments. To avoid introducing bias in analysis, we blind the region around µ+ ! e+ until calibration and analysis have been fixed.

9.1.1

Data Set

There are three data sets discussed in this chapter, the physics data taken in 2009,
the physics data taken in 2010, and the combined data of 2009 and 2010. For the
combined data set, likelihood fitting is performed at once, with di↵erent PDFs of theobservables by means of the event-by-event feature. Di↵erent normalizations in 2009
and 2010 are taken into account in the generation of toy MC experiments.
The MEG trigger condition is as described in Section 3.7.4. The signal event ofµ+ ! e+are back-to-back gamma ray and positron each with an energy of 52.8

129

9. PHYSICS ANALYSIS ON DATA 2010

MeV. To distinguish signal events from backgrounds, we use the following kinematicparameters: gamma energy E , positron energy Ee , relative azimuthal angle between
positron and photon ✓e , relative polar angle between positron and photon
relative timing between positron and photon te .

The trigger for µ+

! e+

e

and
event

requires that E > 44 MeV, te < 10 ns and direction match between gamma ray andpositron.

9.1.2

Pre-Selection

To reduce data size and speed up analysis, a pre-selection is performed on all the datataken during the physics run. At first analysis the calibration is not done completely
but some fundamental kinematics are already reconstructed. The pre-selection is doneloose enough so not to lose any good events.
The selection criteria is defined as

1. 6.875 ns < t
2. |ttrack
where ttT IC < 4.375 ns

tT IC | < 50 ns

is the gamma ray emission time reconstructed with an assumption thatthe muon decay vertex is at the origin; tT IC is the timing counter hit time and thetime-of-flight is not subtracted. Tracking information is not used at pre-selection.
The reason for asymmetric window in the first selection cut is to acquire multiple
turn events while the second condition requires at least one track associated with thetrigger is found. Events are reduced by a factor of 5 with pre-selection

9.1.3

Blinding

For the pre-selected events, we apply a blinding process with the following definition
for the hidden box.

1. 48 MeV < E < 57.6 MeV

130

9.1 Overview of Analysis

2. |te | < 0.7ns
The box size was decided with the detector resolutions to fully cover the signal region.
The blind box is opened after all analysis and calibration is settled upon.

Figure 9.1: The blind box, the analysis window, both timing sidebands, and the low
gamma energy sideband are shown on the (E , te ) plane.

9.1.4

Definition of Analysis Window

We define the analysis window to which the likelihood analysis is applied as the following:
1. |te | < 0.7 ns
2. 50 MeV< Ee <56 MeV
3. 48 MeV< E <58 MeV
4. |✓e | <50 mrad and | e | <50 mrad
The analysis window is wider and includes background events so that we can estimate
the signal and backgrounds simultaneously. In addition cuts for acceptance and the1319. PHYSICS ANALYSIS ON DATA 2010

quality of reconstruction as described in Section 7.1 are applied.
As gamma ray position is reconstructed only using PMTs on the inner face of the
LXe detector, the acceptance selection is applied as |u| <25 cm, |v| <71 cm since theresolutions near the edges worsen quickly. The quality of positron tracking is separated
into two groups to apply di↵erent PDFs. The analysis region, as well as the blind box
and sidebands are shown in Figure 9.1.

9.2

Maximum Likelihood Analysis

9.2.1

Definition

The goal of maximum likelihood analysis is to estimate the best values of number ofµ+ ! e+ events, that of RD events (µ+ ! ⌫e ⌫¯µ e+ ) and that of accidental background
events (NBG ).
Let us define the following variables:
1. ~x: a set of observables~ a data set of N0 independent observations {~x1 , ~x2 , · · · ~xN }2. X:
0~ a set of unknown parameters of the likelihood
3. ✓:The likelihood function is defined as

~ =
~ ✓)
L(X|

N0
Y
i=1

~
p(~xi |✓)

(9.1)

~ is a conditional probability density function(PDF) which gives thewhere p(~xi |✓)~ It is a
probability density of the observation ~xi when the parameter set is equal to ✓.
~ is a function of ✓.
~ The parameters ✓~ can~ ✓)function of ~x and normalized to 1. L(X|~ , which is ✓~best .
be determined so that the likelihood is maximized with a given X
Since we do not fix the number of measurements in advance, the observed number of
events fluctuates statistically in a Poisson distribution. When we regard the observed
number of events as one of parameters to be estimated, the likelihood function is132

9.2 Maximum Likelihood Analysis

extended to

~ =N
~ ✓)
Lext (X|N0
N0 e N Y

N0 !

i=1

~
p(~xi |✓)

(9.2)

where N is the estimated number of events. This is called the extended likelihood
function. We use the extended maximum likelihood fit to estimate the number of
events of each event type.
In the MEG experiment we define ✓~ by the number of event:✓~ = (Nsig , NRD , NBG )(9.3)

N = Nsig + NRD + NBG

(9.4)

and

The number of each event type obeys the Poisson distribution and the errors are
independent one anther in the extended maximum likelihood fit.
From a kinematic point of view, four parameters, energies, opening angle and timedi↵erence, (Ee , E , ⇥e , te ) can discriminate signals from backgrounds. However different detector responses of the drift chamber and timing counter means a di↵erent
position performance between z andangle ⇥e to ✓e anddirection. Therefore we separate the opening

e . Thus, the set of kinematic variables becomes(Ee , E , ✓e ,e , te )(9.5)

PDF is di↵erent for each event as the performance of the detector is position dependent. Therefore we use event-by-event PDF so that we can use a good event witha large weight while also use worse measured events without discarding them.
Resolutions of the parameters are introduced to construct event-by-event PDFs, where
we define the observable as~xi = (Ee , E , ✓e ,e , te ,Ee , E , ✓e ,

133

e ,

te )i = (~xi , ~xi )(9.6)

9. PHYSICS ANALYSIS ON DATA 2010

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

(9.7)

The PDF can then be written asp(~xi |Nsig , NRD , NBG ) =

Nsig
NRD
NBG
· p(~xi |sig) +· p(~xi |RD) +· p(~xi |BG) (9.8)
NNN

If we rewrite PDFs for signal, radiative decay and accidental background defined below,p(~xi |sig) = S(~xi ) = s(~xi | ~xi )p(~xi ),(9.9)

p(~xi |RD) = R(~xi ) = r(~xi | ~xi )p(~xi ),(9.10)

p(~xi |BG) = B(~xi ) = b(~xi | ~xi )p(~xi ),(9.11)

then the likelihood function can be defined as

L(Nsig , NRD , NBG )==+N N0 e N
e
N0 !

(NRD

N N0 e N

(NRD

N0 !

e

<NRD >)2
2 2
RD<NRD >)2
2 2
RD

(NBG

<NBG >)2
2 2
BG

(NBG

<NBG >)2
2 2
BG

e

e

N0
Y
Nsig
NRD
NBG
(
· S(~xi ) +· R(~xi ) +· B(~xi ))
N
N
N
i=1
N0
Y
i=1

NBG
· b(~xi | ~xi )p(~xi ))
N

(

Nsig
NRD
· s(~xi | ~xi )p(~xi ) +· r(~xi | ~xi )p(~xi )
N
N

Q 0 ~
Since it is assumed that p(~xi ) is common to all event types, the term N
i=1 p( xi )is omitted in the likelihood fit because it is independent of the fit parameters, whilep(~xi ) is defined by the measured event distribution for the toy-MC simulation.
The maximization of the likelihood, or the minimization of the negative log likelihood(NLL) is done using the MINUIT package[8].
In the following sections, we discuss the measurement of S(~xi ), R(~xi ) and B(~xi ).

134

9.3 Probability Density Functions

9.3

Probability Density Functions

The Probability density functions (PDFs) are based on measured data and we estimate
three types of PDFs: S, R and B for a signal, a prompt background from radiative
muon decays and accidental background respectively. Figure 9.2 shows averaged PDFs

0.03 0.03
0.025

0.03
0.03 0.025
0.025
0.02

0.025
0.020.025
0.02
0.015

Probability
Density
Probability
Density
FunctionyFunction
Probabilit

0.07
0.06
0.06
0.05 0.07
0.06
0.05
0.05
0.04 0.06
0.05
0.04
0.04
0.03 0.05
0.04
0.03
0.03
0.02 0.04
0.03
0.02
0.02
0.01 0.03
0.02
0.010 0.02
0.01
0.05
0.010 0.010
0.05 0.05
000.05 0.05

0.052
0.052 0.052

0.052 0.052

0.054
0.056
E (GeV)
0.054 0.054 e 0.056 0.056
E (GeV)Ee (GeV)
0.056
0.054 0.054 e
0.056
Ee (GeV)
Ee (GeV)
e

(b)E
0.03
0.03 0.03
0.025
0.03
0.03 0.025
0.025
0.02

0.025
0.020.025
0.02
0.015

0.02
0.02 0.015
0.015
0.01

0.015
0.010.015
0.01
0.005

0.02
0.02 0.015
0.015
0.01

0.015
0.010.015
0.01
0.005

0.01
0.01 0.005
0.005
0
-0.04
-0.02
0
0.0050 0.0050-0.04 -0.04-0.02 -0.02 000-0.04 -0.04-0.02 -0.02 0
eγ

(c)φ

0.02

0.04
φeγ (rad)0 0.02 0.020.04
0.04
φ γ (rad)φeγ (rad)0 0.02 0.02e0.04
0.04
φeγ (rad)φ (rad)0.01
0.01 0.005
0.005
0
-0.04
-0.02
0
0.0050 0.0050-0.04 -0.04-0.02 -0.02 000-0.04 -0.04-0.02 -0.02 0

eγ

0.02
0.04
θeγ (rad)0 0.02 0.020.04
0.04
θ γ (rad)θeγ (rad)0 0.02 0.02e0.04
0.04
θeγ (rad)eγ
θeγ (rad)(d)θ

Probability
Density
Function
Probability
Density
Function

0.04
0.04 0.04
0.03 0.04
0.04

Probability

Probability
Density
Function
Probability
Density
Function
Probability
Density
Function

Probability
Density
Function
Probability
Density
Function
Probability
Density
Function

Probability
Density
Function
Probability
Density
Function

(a)E

0.07
0.07
0.06 0.07

Probability
Density
Function
Probability
Density
FunctionyProbabilit

Probability
Density
Function
Probability
Density
Function
Probability
Density
Function

0.05

0.05 0.05
0.04
0.05 0.05
0.04 0.04
0.03
0.04 0.04
0.03 0.03
0.02
0.03 0.03
0.02 0.02
0.01
0.02 0.02
0.01 0.01
0
0.05
0.052
0.054
0.056
0.058
0.01
0.010.04800
E (GeV)
0.048 0.048
0.05 0.05
0.052 0.052
0.054 0.054
0.056 γ0.056
0.058 0.058
00E (GeV)Eγ (GeV)
0.058
0.048 0.0480.05 0.050.052 0.052
0.054 0.054
0.056γ 0.056
0.058
Eγ (GeV)Eγ (GeV)γ
0.03
Probability
Density
Function
Probability
Density
Function

Probability
Density
Function
Probability
Density
Function
Probability
Densityy Function
Probabilit

Probability
Density
Function
Probability
Density
Function
Probability
Function
yDensity
Probabilit

by observables.

Signal0.03 0.03
0.02 0.03
0.03
0.02 0.02
0.01 0.02
0.02
0.01 0.01
0.01 0 0.01
00SignalSignal
Radiative Muon Decay
SignalSignalRadiative
Radiative
MuonMuon
DecayDecay
Accidental Background
Radiative
Radiative
MuonMuon
DecayDecay
Accidental
Accidental
Background
Background
-0.5

0

0
-0.5 -0.5

00-0.5 -0.5

(e)t

0

0

0
eγ

E , Ee ,

×10-9
0.5
-9
-9
t (sec)×10 ×10
0.5eγ 0.5 -9-9×10t (sec)t ×10(sec)
0.5eγ 0.5eγt(sec)teγ (sec)eγ
e , ✓ e , te

Accidental
Background
Accidental
BackgroundFigure 9.2: Probability(PDFs) of each observable (E , Ee , e , ✓e , te )E , Ee ,density, Eee ,, ✓e ,functions✓te , teaveraged over all events
region. The green, red and magenta lines show,, ✓teanalysis, te
E , E ewithin, Eee ,, ✓eethe
PDFs of signal, radiative muon decay and accidental background (S, R, B), respectively.

135
Wednesday, December 19, 12

9. PHYSICS ANALYSIS ON DATA 2010

9.3.1

SignalThere is no intrinsic correlation among the observables of the signal, however the
positron variables are calculated from a fitted track and the intersection with thetarget, which results in correlations among the measurement errors of the observables.
Such correlations are incorporated into PDF. The signal PDF, can be written as

S(Ee , E ,✓e ,e , te |u , v , w , xe , ye , ze ,e) =S(Ee | e )⇥S(E |u , v , w )⇥S(✓e |u , v , w , xe , ye , ze , Ee )⇥S( e |u , v , w , xe , ye , ze , ✓e , Ee ,e)S(te |E , Ee )Ee PDF
The positron energy response is evaluated by fitting the kinematic edge of the Michel
spectrum.

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 three Gaussians. We also apply an additional correction
for relative angle selection. The e↵ect is evaluated by toy MC. Finally we fit the
distribution with a sum of two Gaussians, and the result is used as signal PDF. The
details of the parameters are in Table 9.1, together with those for background andradiative muon decay.

The PDF uncertainty from the statistics of side-band sample is evaluated in the fit
as a covariance matrix. The global energy scale uncertainty is estimated to be 25 keVby the study with Monte Carlo; the accuracy of the determination of the energy scalewith the Michel fit on the MC sample.

1369.3 Probability Density Functions

Parameter

Sig

BG

RMD

fcore

0.96

0.95

0.93

µcore (keV)

79,-49,-21,1

93,-35,8,15

79,-49,-21,1.4

core (keV)

327

308

314

µtail (keV)

92,-36,-8,14-500

246,141,169,191

tail (keV)

1144

1076

1165

µAcc (MeV)

-

50.4

51.2

⌫Acc (MeV)

-

2.26

2.85

Table 9.1: Ee parameters.

Relative Angle PDF
The PDF of relative angles are are defined by the angular responses formed by combining the evaluated resolutions of gamma-ray position, positron emission angles andmuon decay vertex. The PDF of the relative angles are sum of several Gaussian functions. The means and sigmas of the Gaussian are calculated on an event-by-event basis.

te PDF
The time response is evaluated by the RMD peak in E -sideband, as discussed in
Section 7.5.3.
The conditions of the E -sideband data used for the extraction are the following:

1. 40< E <47 MeV
2. 45 < Ee < 53 MeV
3. | e | < 280 mrad and |✓e | < 280 mrad
4. kinematic cut of the reconstructed invariant mass < MµWe fit the distribution with a sum of two Gaussians and a flat distribution. The
results are listed in Table 9.2. The origin of the tail component is studied with the
MC and comes from the error of the tracking, mainly by the scatter on the material
between TC and DC, and the responses of XEC and TC are well approximated by a137

9. PHYSICS ANALYSIS ON DATA 2010

Gaussian.

fcore

0.75

core (psec)

124µtail (psec)-133

tail (psec)

250

Table 9.2: Fit results of time response extraction from E -sideband.

E PDF
E PDF is defined by gamma energy response which is evaluated by the 54.9 MeV
gamma ray in ⇡ 0 run.
To fit the ⇡ 0 spectrum, we use the following selection criteria:

1. Energy of the gamma ray collected by NaI detector has to be 65 < EN aI < 98
MeV for the 54.9 MeV peak at the LXe detector side and 40 < EN aI < 62 MeV
for the 83.9 MeV peak.
2. Concentration of energy deposition on the central crystal larger than 60 % of thetotal energy deposition.
3. Opening angle larger than 170 .
Since the selected events are not exactly monochromatic and there is a correlation
between opening angle and the two gamma energies, we will later correct the reconstructed energy using the reconstructed opening angle.
We fit the spectrum with the following function,f (x) = f0 (x)Og⇡0 (x)Oh⇡ (x)

x0 )⌘(9.12)

f0 (x) is the response function of the detector and can be written as⇣
8t< A exp 2t
(x
2
up0
⇣
⌘
f0 (x) =2: A exp (x x20 )
2
up0

138

x  x0 + t,
x > x0 + t,

(9.13)

9.3 Probability Density Functions

where A is a scale parameter; x0 is the peak position; t is a transition parameter andup0 is a resolution parameter which indicates the spread of distribution in the higher

side. The lower tail is because of interactions with material in front of detector andshower escapes mainly from the inner face.
MEG Technical note No.72

24g⇡0 (x) comes from the opening angle resolution and is studied with MC. The spreadof energy distribution is found to not negligible and asymmetric (Figure 9.3), which
results in an overestimation of energy scale (for the 54.9 MeV peak) and resolution.
It is found that this contribution is negligible should we set the opening angle cut as> 175 . However there is not enough data to use the higher threshold for all positions.
Therefore this correction is added and as a result the two energy scales obtained from54.9 MeV and 83.9 MeV agree within 0.1%.
h⇡ (x) comes from the di↵erent background condition in ⇡ 0 run. The backgroundunder the ⇡beam is higher than that under normal µ+ beam due to positrons

from the beamline (no separator) and neutrons from the ⇡reaction or the radiative

capture reaction in low energy region and gamma rays from these reactions in high
22: TheCorrelationof theevents⇡ 0 decay
kinematics
betweenopeningTheenergy Figure
region.
pedestaltaken
duringthe energy
run isandused
as aangle.
measurementofhorizontal axis is the true energy of a gamma ray, while the vertical one is the reconstructed

the background.
opening angle in the simulation where the position resolution in XEC and NaI as well as the
stopping distribution of ⇡beam are taken into account.

1000

1200

EGammaDiff_55MeV

800
600

Entries

9819

Mean

0.0829RMS

0.3485

EGammaDiff_83MeV

1000

Entries

800

Mean -0.08704
RMS

600

12848

0.3766

400
400
200
0
-4

200
-2

0

0
-4

2
4
Ecorr - Eedge (MeV)

(a)

-2

0

2
4
Ecorr - Eedge (MeV)(b)

Figure 23: Distribution of residual energy after the opening-angle correction in the simulation.

FigureResidual
distributionin ⇡is0 corrected
due to opening
angle resolution.
The true
The9.3:true energyof a energy
gamma rayfrom ⇡ 0 decay
for the energy-opening
angle correlationtothekinematic
edge
using
the
reconstructed
opening
angle.
A
finiteresolutionoftheenergy is corrected by the energy-opening angle correlation to the kinematic edge using
opening angle reconstruction makes the finite width of these distributions. (a) is for the lower
the reconstructed
opening
angle.
(a)edgeis foredge (55 MeV) sideand (b)is upper(83 theMeV)54.9
side.MeV side and (b) is for the 83.9 MeV
side.
5.3

Signal PDF extraction

To extract the signal PDFs from ⇡ 0 data, we divide
the acceptance into boxes in0
We extract the position-dependent signal PDFs from the ⇡ scan data for each dataset. It isverydifficultto parametrize
the position
dependencePDF
becausethe parameters
each PDF
are(u, v, w)
andextract
the position
dependent
separately
forofeachdivision.
The
correlated one another and not all of them can be expressed by some smooth functions with acontrol of their systematic errors. Thus, we divide the acceptance into boxes in (u, v, w), whose
size can be di↵erent, and extract the PDF separately for each division. The PDFs described in139

9. PHYSICS ANALYSIS ON DATA 2010

PDFs obtained this way are statistically correct. The divisions for (u, v) are shown in

MEG Technical note No.

Figure 9.4. For 2010 data we divide w into four regions (0 < w < 0.8 cm), (0.8 < w <vv (cm)

3 cm), (3 < w < 8 cm), (8 cm < w).

60

40

20

0

-20

-40

u

-60

Figure 9.4: Divisions in (u, v) for signal PDF extraction. Separate but same color regions
are combined.

140
25

(a)
hSigma1_0

Entries

EntriesFigure 24: Divisions of signal PDF ex-20
traction in (u, v)-plane. Signal PDF is
Examplesseparatelyof the fit are shownin Figure 9.5 andall thefit results
produce the actual
extracted
fromthe⇡ 0 -scan⇡ 0 spectra well.
data in each box in this figure. SepaFigure 25: Resolutio
rated but same color regions are also
sition in (u, v) for 3 <combined.
45
40
35

(b)

MEG Technical note No.72

28

Number of events

Number of events

9.3 Probability Density Functions

nsum2_4_11_0

400
350
300
250

sigma = 1.57 ± 0.09 %FWHM = 4.48 ± 0.28 %Entries

1920

Mean

6.338e+05 ± 1240

RMS

5.434e+04 ± 877.1

Underflow

1

Overflow0

nsum2_7_23_0

300
250
200

200

sigma = 2.62 ± 0.19 %FWHM = 8.65 ± 1.02 %Entries

2788

Mean

6.179e+05 ± 1169

RMS

6.172e+04 ± 826.7

Underflow

1

Overflow

0

150

150
100
100
50

50
0

×103
400

0

500
600
700
Total number of photons

×103
400

Number of events

100(b) 3 < w < 8, edge

Number of events(a) 3 < w < 8, central

70

nsum2_4_11_0
Entries

1084

Mean

5.657e+05 ± 2689

RMS

8.849e+04 ± 1901

60

FWHM = 9.60 ± 1.88 %Underflow

1

Overflow0

nsum2_7_23_0

6080sigma = 2.36 ± 0.24 %

500
600
700
Total number of photons

50

sigma = 4.98 ± 0.58 %FWHM = 66.39 ± 148.03 %Entries

1209

Mean

5.536e+05 ± 2401

RMS

8.344e+04 ± 1698

Underflow

1

Overflow

0

40
30

40

20
20
10
0

×103
400

0

500
600
700
Total number of photons(c) w < 0.8, central

×103
400

500
600
700
Total number of photons(d) w < 0.8, edge

0 -550MeV peak fit. Red lines are the best-fit functions and the green
Figure28: Examplesof the ⇡ofFigure9.5: Examples⇡ 54.9 MeV peak fit. Red lines are the best fit functions.
and Green
yellow and
bands
show
the
confidence
intervalsintervalsof the function
at 68 and95 %yellow bands show the confidenceof the function
at 68%
andC.L.
95%calculated
C.L.
withcalculated
the covariancematrixevaluatedin fitting.
(d)inisfitting.
the worst case where the uncertainty ofwith thecovariancematrixevaluatedthe fit as well as the resolution are largest.

141

9. PHYSICS ANALYSIS ON DATA 2010

9.3.2

Radiative Muon Decay

The radiative muon decay has a kinematic correlation between observable parameters.
It is defined as

R(Ee , E ,✓e ,e , te |u , v , w ,R(Ee , E , ✓e ,e , ✓e ) =e |u , v , w ,e , ✓e )⇥R(te |E , Ee )Kinematically correlated term
The kinematically correlated term is formed with the theoretical correlation of radiative
decay folded with the detector response functions and acceptance functions. We startby building the first term of the PDF from the theoretical formula of the di↵erential
branching ratio of RMD[32], however we cannot ignore terms with polarization vector.
The PDF is formed by folding detector responses in four dimensions with the following
procedure: multiply the Ee acceptance curve; Convolve two dimensional angularresponse; convolve Ee response from Michel spectrum; Convolve E response whichis the same as signal PDF; multiply E efficiency curve, which is the error function
extracted from the gamma-ray BG spectrum fit for each position; multiply angular
efficiency estimated from BG distribution.

te PDFWe use the same time PDF as the one for signal except for the correlation. Dependenceof the time center t0 on Ee in both the data and MC is observed. We measure t0
values for three Ee ranges in the Eg -sideband data, and extract the dependence as alinear function. The coefficient is measured to be 4.6 psec/MeV, which is one ordersmaller than that of the signal correlation, and it is consistent with that from MC.
This dependence is applied to the RMD time PDF.

142

9.3 Probability Density Functions

9.3.3

Accidental Background

The accidental background PDF, B, can be written as

B(Ee , E , ✓e ,e , te |u , v , w ,e ) =B(te )⇥B(E |u , v , w )⇥B(✓e |v )⇥B( e |u )⇥B(Ee | e )The PDFs are extracted from sideband data except for te which is assumed to bea constant function.

Relative Angle PDF
The PDFs for relative angles are obtained from data in time sidebands with looser
cuts of |✓e | < 200 mrad, | e | < 200 mradand E > 46 MeV.
No strong correlation between ✓e andeis found as shown in Figure 9.6. However

Figure 9.7 shows that there is correlation between ✓e ( e ) distribution and u (v ).
To extract the angular PDF, we divide ✓e in five slices andein eight slices andfit the distribution with a third degree polynomial function. Figure 9.8 and Figure 9.9
show the angular distributions of BG.

143θeγ [rad]are necessary also for a toy MC study with a looser cut on the opening angle, e .
Figure 2 shows the distribution of e and e for the 2010 data where no strong
correlation between the two angles is observed. Figure 3 shows the dependence of theangle distribution on the calorimeter reconstructed position. The e ( e ) distribution
depends on the u (v ) as seen in Fig. 3(a) (Fig. 3(d)), whereas there is only weak
dependence on the v (u ) as seen in Fig. 3(b) (Fig. 3(c)). We note that the distribution
PHYSICS
ONfor
DATA
2010 2010. We include only the u -dependence
for the9.2009
data is ANALYSIS
similar to thatthe dataof e and the v -dependence of e in the BG PDF as shown in Eq. (11).
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15
0.2
θeγ [rad]Figure 9.6: BG distribution on (✓ , phi ) measured in time sidebands. The red box is

0.2

0.2

θeγ [rad]θeγ [rad]Figure 2: BG distribution on ( ee , e e ) measured in the te -sidebands for the 2010
the analysis region.
MEG-TN074
data. The red box shows the region for the likelihood analysis(| e | < 50 mrad,| e | < 50 mrad).
We dividein each of which the angle PDF is obtainede ( e ) in five (eight) slices, 0.15
0.15
by fitting0.1the distribution with a third degree
polynomial function. Figure 4 (Fig. 5)
0.1
0.05
0.05
shows BG e ( e ) distributions in u (v ) slices for the 2010 data where the best fitted0function-0.05
is0 also shown together with its 68%
and 95% confidence intervals. The error-0.05
in the fitting
is taken into account in the likelihood
analysis using the error covariance
-0.1
-0.1
-0.15
matrices-0.15
in the fitting.
-0.2
-25

3.4.2

-20

-15

Ee PDF

-10

-5

0

5

10

15

-0.2

20
25
uγ [cm](a) u vs. ✓e

-60

-40

-20

0

20

40

60
v γ [cm](b) v vs. ✓e
[rad]

eγ

φ

φ

eγ

[rad]

The Ee PDFs
are measured in the side-bands
separately for the tracking-quality cate0.2
0.2
0.15
0.15
gories. The Michel spectrum in the side-bands is fitted as shown in Fig 6 with a similar
0.1
0.1
function 0.05
used for the signal PDF extraction
but the response is modeled with a sum
0.05
of two Gaussians.
As described in Sec. 3.2.3,0 the e dependence of the energy scale is0-0.05-0.05 four regions. The condition of the sample
treated withdifferent mean parameters for the
-0.1
-0.1
used in the fit is relaxed to 150 (100) mrad cutson e and e for the 2009 (2010) data.
-0.15
-0.15
The fit regionis 49–56 MeV. The mean parameterof the tail component is floating in
-0.2
-0.2
-25 -20 -15 -10
-5
0
5
10
15
20
25
-60
-40
-20
0
20
40
60
u [cm]
v [cm]
contrast with zero-fix in signal extraction. The parameters are given in
Tab. 2–5. The(c) u vs. e(d) v vs. e
uncertainty is evaluated in the fitting as a covariance matrix including the errors on theacceptance
Figureparameters.
3: Dependence of the angle distribution on the calorimeter reconstructed positionγγ

Figure 9.7: Position dependence of angular distribution for 2010 data.

for the 2010 data. Note that the angular ranges are much wider than those for the
likelihood fit (| e | < 50 mrad, | e | < 50 mrad).

11

3.4.3

E PDF

It is described in detail in a technical note [8].
144MEG-TN074

500# of events /(11.0mrad)# of events /(11.0mrad)

9.3 Probability Density Functions

700-25 < uγ < -15 cm

600

400

-15 < uγ < -5 cm

500
400

300

300

200

200

100

100

700

0

0 -0.15 -0.1 -0.05

0.05 0.1 0.15
θe γ [rad]900# of events /(11.0mrad)# of events /(11.0mrad)

0 -0.15 -0.1 -0.05-5 < uγ < 5 cm

800

600

5 < uγ < 15 cm

600
500

400

400

300

300

200

200

100

100

0 -0.15 -0.1 -0.05# of events /(11.0mrad)0.05 0.1 0.15
θe γ [rad]

700

500

700

0

0

0 -0.15 -0.1 -0.05

0.05 0.1 0.15
θe γ [rad]00.05 0.1 0.15
θe γ [rad]15 < uγ < 25 cm

600
500
400
300
200
100
0 -0.15 -0.1 -0.05

00.05 0.1 0.15
θe γ [rad]BG ✓e distributions in u slices for 2010 data. The black line is the best
Figure 4: Figure
BG e9.8:distributions
in five u slices for the 2010 data. The black solid curve
fitted function. The green and yellow band show the 68% and 95% confidence intervals
shows therespectively.
3rd degree
polynomial function best fitted to the distribution. The green
The red arrow shows the analysis region in the likelihood fit..
(yellow) band shows the 68 (95)% confidence intervals for the fitted function. The redarrow shows the region used in the likelihood fit.

145

MEG-TN074

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

# of events /(12.6mrad)# of events /(12.6mrad)

9. PHYSICS ANALYSIS ON DATA 2010-71 < vγ < -60 cm

500-60 < vγ < -40 cm

400
300
200
100

-0.15 -0.1 -0.05

0

0.05

0

0.1 0.15
φ [rad]-0.15 -0.1 -0.05

0

0.05

500

500

-40 < vγ < -20 cm

0.1 0.15
φ [rad]eγ# of events /(12.6mrad)# of events /(12.6mrad)eγ-20 < vγ < 0 cm

400

400

300

300

200

200

100

100
0

-0.15 -0.1 -0.05

0

0.05

0

0.1 0.15
φ [rad]-0.15 -0.1 -0.05

0

0.05

500

500

0 < vγ < 20 cm

20 < vγ < 40 cm

400

400

300

300

200

200

100

100
0

-0.15 -0.1 -0.05

0

0.05

0

0.1 0.15
φ [rad]-0.15 -0.1 -0.05

0

0.05

180

40 < vγ < 60 cm

160

350

60 < vγ < 71 cm

140

300

120

250

100

200
150

80
60

100

40

50

20

0

0.1 0.15
φ [rad]eγ# of events /(12.6mrad)# of events /(12.6mrad)

eγ

400

0.1 0.15
φ [rad]eγ# of events /(12.6mrad)# of events /(12.6mrad)eγ-0.15 -0.1 -0.05

0

0.05

0

0.1 0.15
φ [rad]eγ-0.15 -0.1 -0.05

0

0.05

0.1 0.15
φ [rad]eγ

9.9: eBGdistributionsin v slices
for 2010
The blackis thebestsolide distributions
FigureFigure
5: BG
in eight
v slicesfor data.
the 2010
data.lineThe
black
fittedfunction.
The
green
and
yellow
band
show
the
68%
and
95%
confidence
intervals
curve shows the 3rd degree polynomial function best fitted to the distribution. The green
respectively.
The redshows confidence
the analysis intervals
region in thefit.function. The red(yellow)band shows
the arrow
68 (95)%forlikelihood
the fitted
arrow shows the region used in the likelihood fit.

146

14

9.3 Probability Density Functions

Ee PDF
Ee PDFs are measured in side-bands by fitting Michel spectrum, similarly to the signal
PDF extraction except that the response is modeled with a sum of two Gaussians.
The sideband data used here is cut with |✓e | < 100 mrad, | e | < 100 mrad and thefit region is 49-56 MeV. The fit parameters are given in Table 9.1.

E PDF
E

is position dependent and estimated by using the BG spectra observed in tesideband. The selection criteria is same as that for signal search except that there isno condition on relative angle. The time range for the sideband is 1 < |te | < 1000
nsec.

We extract the background PDFs according to position just like signal PDF extraction.
This is not only due to di↵erent resolutions in di↵erent positions but also the BG
spectrum itself may be di↵erent. We divide the depth into three parts: (0 < w < 1cm), (1 < w < 2 cm) and (w > 2 cm). The (u, v) divisions are shown in Figure 9.10.
Pileup events are treated separately.
We fit the BG spectra with expected smooth distributions. We start from the MC
distribution of energy deposition in LXe detector by a single event from RMD or AIF,
then convolve it with pedestal distribution and detector response. Then the cosmic
ray spectrum is added and the trigger curves are applied on the lower and higher side.
To take into account the e↵ect of pileup elimination, the pedestal distribution isshrunk by factors depending on the pieup identification result. The factors are adjusted
in advance and stay fixed in fitting. The detector response is given as a Gaussian withthe sigma for detector resolution and mean for energy scale. The cosmic ray spectrum
is obtained from CR runs and is given as position dependent templates. The trigger
efficiency curve is an error function with two free parameters at the lower side andat the higher side, it is measured in CR data by applying the trigger selection andcomparing the spectra with and without the selection. In total, six parameters are
determined in the fit. Some examples of the fit are shown in Figure 9.11. The total

147

MEG Technical note No.72

32

9. PHYSICS ANALYSIS ON DATA 2010

0 < w < 1, 1 < w < 2 cm

43

44,45

v

w > 2 cm

5 7 9 11 13 15 1719 21 2325 27 29

38,39

4 6 8 10 12 14 1618 20 2224 26 2830

44,45

42

u
48: pileup events

49: pileup events

Figure 29: Divisionsof 9.10:
background
PDFinextractionin (u, v)-plane(2010).
Background PDF is
FigureDivisions(u, v) for background
PDF
extraction.
separately extracted from the time-sideband data in each box in these figures. The numbers in
the figure show the indices of divisions which are used in other figures. For 2009 200µm-degrader
spectrum,whichis (5,7),
an average
PDF is (28,30),
shown inbut itareis combined,
not used in
the physicsto
data,
divisions(4,6),(23,25,27,29),
and(a)(44,45)
respectively,
increasestatistics. In particular, small statistics in top-downstream part are due to a problem
analysis.
of trigger in the first-part of 2009 run.

148

9.3No.72
Probability Density Functions
MEG Technical note33

All All104

hist17

Entries1125919

EntriesMean

0.04578Mean

0.0456RMS
χ2 / ndf

0.002572

RMSχ2 / ndf

0.002571324 / 284

AIF RD Con1.434e+06 ± 3.374e+03
CR
Sigma

103

0.006635 ± 0.000352
0.9978 ± 0.0002

TRGAcc0.04259 ± 0.00000

TRGRes

0.01826 ± 0.00007

53.89 / 66

AIF RD Con3.324e+05 ± 3.529e+03

103

CR

0.004103 ± 0.001986

Sigma

0.01776 ± 0.00022

EScale66092

0.01852 ± 0.00097

EScale2

100.9953 ± 0.0009

TRGAcc0.04241 ± 0.00001

TRGRes

0.01899 ± 0.00035

102
10
10

1

1
0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.04

(a) 2 < w, all

0.045

0.05

0.055

0.06

0.065

hist43 (top)hist30103

Entries

16915

EntriesMean

0.04592

Mean

0.04647RMS
χ2 / ndf

0.002442

RMS

0.002695

CR
Sigma

0.003459 ± 0.001972
0.9921 ± 0.0016

TRGAcc0.04293 ± 0.00003

TRGRes

0.01904 ± 0.00052

149.4 / 66

AIF RD Con3.306e+05 ± 4.038e+03

103

CR

0.02788 ± 0.00189

Sigma

0.01848 ± 0.00196

EScale53179

χ2 / ndf

36.08 / 66

AIF RD Con9.862e+04 ± 2.089e+03

102

0.07

(b) 2 < w, central (index 17)1020.01781 ± 0.00102

EScale0.9983 ± 0.0009

TRGAcc0.04358 ± 0.00002

TRGRes

0.02394 ± 0.00031

10
10

1
1
0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.04

0.045

(c) 2 < w, edge (index 30)

0.05

0.055

0.06

0.065

hist39 (1−2cm)104

hist38(0−1cm)
hist38(0−1cm)Entries

223046

EntriesMean

0.04596Mean

0.04618RMS

0.002782

RMS

0.002883

χ2 / ndf

103

0.07

(d) 2 < w, top (index 43)

130.9 / 60

104271331χ2 / ndf

242.1 / 66

AIF RD Con1.318e+06 ± 1.392e+04

AIF RD Con1.773e+06 ± 3.221e+04

CRCR

0.02897 ± 0.00212

Sigma

0.03702 ± 0.00083

0.02468 ± 0.00183

Sigma

0.0235 ± 0.0006

EScaleTRGAcc0.9946 ± 0.0007
0.0435 ± 0.0000

TRGRes

0.02911 ± 0.00114

103

EScale20.9912 ± 0.0013

TRGAcc0.0437 ± 0.0000

TRGRes

0.04597 ± 0.00047

10

2

10

10
10
1
1
0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.04

0.045

(e) 1 < w < 2 (index 39)

0.05

0.055

0.06

hist48 (pileup)
Entries205963Mean
0.0448RMS
0.00309χ2 / ndf
88.69 / 66
p0
3.354e+05 ± 3.259e+03
p1
0.02143 ± 0.00129
p2
0.01975 ± 0.00085
p3
0.9905 ± 0.0007
p4
0.04136 ± 0.00002
p5
0.03989 ± 0.00128

4

10

103

0.07

hist49 (pileup,0−2cm)
Entries135844Mean
0.04512RMS
0.003363
χ2 / ndf
144.7 / 66
p0
2.554e+05 ± 4.717e+03
p1
0.0295 ± 0.0014
p2
0.03479 ± 0.00142
p3
0.9854 ± 0.0016
p4
0.04201 ± 0.00004
p5
0.055 ± 0.000

103

102

102

10

10

0.04

0.065

(f) 0 < w < 1 (index 38)

0.045

0.05

0.055

0.06

0.065

0.07

0.04

(g) 2 < w, pileup (index 48)

0.045

0.05

0.055

0.06

0.065

0.07

(h) 0 < w < 2, pileup (index 49)

Figure30: 9.11:
Examplesof the backgroundfit (2010).
Black
are thebest-fit
FigureExamplesof backgroundfit for2010lines
data.
Black
linesfunctions
are the andbestbluefit
bands show the confidence intervals of the functions at 95 % C.L. calculated with the covariancefunctions and blue bands show the confidence intervals at 95% C. L. calculated with thematrix. The position indices are defined in Fig 29.
covariance matrix.

149

9. PHYSICS ANALYSIS ON DATA 2010

The functions fit well and we can describe the observed distributions with smoothfunctions to eliminate statistical fluctuation and estimate the uncertainties. It also
shows our understanding of background is correct. We can decompose the components
of background. Each component is shown in Figure 9.12 and their fractions are listed
in Table 9.3.

Figure 9.12:
of background.
specFigureComponents
61: -ray background
spectrum Data
duringpoints
physicsshow
run inthe
2010.-ray
Databackgroundpoints
show
the
-ray
background
spectrum
taken
from
the
sidebandphysics
data,
the
trum
ge ratio collected on thein- taken from the sideband physics data. Green line is the fitted spectrum from RMD
green line shows fitted spectrum from the RMD plus AIF, the blue line showsandhe data from the dedicatedAIF. Blue line is the green line plus e↵ects of pileup. Black line is the cosmic ray
the green line with taking into account the effect of pile-up, the black line shows
nte Carlo events. Twospectrum
blueand the red line is the fitted result of all background.
the cosmic-ray spectrum, and the rede line shows the fitting result taken intots.
account all the background spectra.

rum accounts for lessfrom pile-up, energy
ount for 3-5%.

Component

7. Trigger Single

from RMD or AIF

Cosmic ray

7.1. Requirements
All the rest

Fraction
94-96%<1%
3-5%

An experiment to search for ultra-rare events in a huge beam-

Table 9.3: Fractions of background components in the signal region. All the rest comerelated background needs a quick and efficient event selection,from pileup or reconstruction tail.

as |u| < 25 cm, |v| <71
detection efficiency istion, and is confirmed
ray background ratey region comes fromhe active volume andthe analysis window
efficiency is found to
unt the positron event
ured using NaI single
a of 83 MeV from
ection efficiency in the
7% requiring a reconistent with the Monte

which demands the combined use of high resolution detection
techniques and fast front-end, digitizing electronics and trigger.
The trigger system plays an essential role in processing the detector signals to find the signature of µ+e+ events in a highbackground environment.
The most stringent limitation to the trigger latency originatesfrom the matching with the DRS, whose digitization needs to be
stopped at latest 450 ns after the event occurrence. If the trigger
were later, the charge stored in part of the DRS cells might beoverwritten and a few bins of the sampled waveform (in particular those recording the leading edge of PMT signals) might belost. The speed objective must150
be achieved while keeping thesignal efficiency close to unity. On the other hand, the event selection needs to operate a significant background rejection; the9.3 Probability Density Functions

9.3.4

Confidence Interval

The confidence interval is calculated using the Feldman-Cousins unified approach withthe profile-likelihood ordering[64].
We define the following parametersq(Nsig ) =ln p (Nsig ) =2 ln

p (Nsig )ˆˆL(Nsig , N̂RD (Nsig ), N̂BG (Nsig ))L(Nsig , N̂RD (Nsig ), N̂BG (Nsig ))(9.14)(9.15)

ˆ
where N̂j (j = sig, RD, BG) is Nj which maximizes the likelihood and N̂ is Nj thatmaximizes the likelihood of a fixed Nsig .
We calculate the confidence interval with the following procedure:

1. Make a list of events from the data. Variables to decide detector response suchas positron track quality,e ,conversion point are associated with each event.

2. Generate many toy-MC experiments assuming a true B. q(Nsig ) is calculated ineach experiment. For the toy-MC experiments, an event is picked up from the list
made above and detector response is determined according to the list, generating
an event. Then calculate q(Nsig ) on the data and use the distribution of q(Nsig )obtained at the earlier step to find the probability to observe q(Nsig ) that is larger
than that observed on the data (q obs (Nsig )).
3. Repeat the second procedure and find the 90% C.L. where the probability is lessor equal to 0.9.
With this method when N̂sig is small only the upper limits are calculated and lower
limits appear when Nˆsig becomes large. The confidence intervals at 90% C.L. should
include the true Nsig in 90% probability. When the background-only hypothesis istrue, the lower limits should appear and Nsig = 0 should be excluded in 10% probability.

1519. PHYSICS ANALYSIS ON DATA 2010

9.4

Normalization

A branching ratio is defined as a ratio of the decay rate for a particular process, in
relation to the total one. For muon decay, we use Michel decay for normalization purpose to avoid influences from changes in instantaneous beam intensities or conditionsof DC detectors.
The total number of the Michel decay (MD), NM D , is estimated from the event
triggered with only timing

counter mixed in the MEG physics data taking simulta-

neously. It consists of some efficiencies and acceptances,E
NM D = Nµ ⇥ Te⌫ ⌫¯ ⇥ Be⌫ ⌫¯ ⇥ fe⌫⌫¯ ⇥1Pe⌫ ⌫¯T IC
DCH DCH⇥ ✏trige⌫ ⌫¯ ⇥ Ae⌫ ⌫¯ ⇥ ✏e⌫ ⌫¯ Ae⌫ ⌫¯(9.16)

where each factor has the following definitions:Nµ ⇥ Te⌫ ⌫¯ : Number of stopped muons during a time Te⌫ ⌫¯ .
Be⌫ ⌫¯ Branching ratio of Michel decay.
E Fraction of Michel spectrum above 50MeV.
fe⌫⌫¯

Pe⌫ ⌫¯ Pre-scale factor of TIC trigger.
✏trige⌫ ⌫¯ : Conditional trigger efficiency.
ATe⌫IC⌫¯ : Conditional acceptance of timing counter including DC-TC matching efficiency.
✏DCH: Conditional tracking efficiency including selection criteria.
e⌫ ⌫¯
ADCH
: Geometrical acceptance of drift chamber.
e⌫ ⌫¯A similar equation can be written for the number of signals:eeNsig = Nµ ⇥ Te ⇥ Be ⇥ ✏trig⇥ ATe IC ⇥ ✏DCH
ADCH
⇥ ✏LX
⇥ ALX
e
e
e
e
e

(9.17)

e is the gamma-ray detection and reconstruction efficiency and ALXe is thewhere ✏LXeeconditional acceptance of gamma-ray from µ+ ! e+ decay.

1529.5 Systematics

Then the branching ratio on mu+ ! e+ decay can be written as
Be
Nsig
fE
✏trig⌫¯
ATe⌫IC
✏DCH
1
1
⌫¯
e⌫ ⌫¯
=
⇥ e⌫ ⌫¯ ⇥ e⌫⇥⇥
⇥ LXe ⇥ LXe
trig
T
IC
DCH
Be⌫ ⌫¯
NM DPe⌫ ⌫¯ ✏e✏eAe⌫ ⌫¯✏eAe(9.18)

Normalization factor k is defined as
1/k =

Be
1
·
Nsig Be⌫ ⌫¯(9.19)

The factors for calculation include number of Michel positrons (30670), pre-scale
factor (1 ⇥ 107 ), fraction of Michel spectrum (1.23), trigger efficiency ratio (0.94),

DC-TC matching ratio (0.922), gamma ray efficiency (0.59) and gamma ray acceptance(0.99), which give a normalization factor ofk2010 = (2.23 ± 0.16) ⇥ 1012

(9.20)For the combined data of 2009 and 2010, the normalization factor iskcombined = (3.31 ± 0.22) ⇥ 1012

9.5

(9.21)

Systematics

To incorporate systematics, we use two di↵erent methods. For NRD and NBG , profile
method is used. The best estimates for NRD and NBG are calculated for each Nsig .
Their uncertainties are incorporated in the Nsig interval through correlations between
Nsig and NRD , and Nsig and NBG .
The uncertainties of the parameters of the PDF of the observables (the resolutions,correlations, fraction of core and tail etc.) are incorporated by randomizing themin the generation of the toy-MC experiments. The signal normalization uncertainty
is incorporated by randomizing Nsig in the generation. The fittings on the toy-MC
experiments are done using the same parameters as those used in the fitting on thedata.
A random number is generated for each parameter once for each experiment andused to generate all the events in the experiment. Di↵erent random numbers are1539. PHYSICS ANALYSIS ON DATA 2010

generated for the statistically independent uncertainties, such as the time center,the parameters of Michel shape etc. For the common uncertainties, for example therelative alignment between the photon detector and the positron spectrometer, asingle random number is used to generate events. The correlations in the errors of theparameters are taken into account using the covariance matrices.
The uncertainty of the signal PDF is evaluated from the data statistics as acovariance matrix since the parameters are highly correlated to each other. The mean
parameter is fixed to the best-fit value, and the other PDF parameters are free.
The systematic uncertainties on the mean andof the signal te PDF are estimatedfrom the statistical error on those parameters from the fit to the radiative decay timing
peak. The mean of the signal Ee PDF is estimated by using it as a floating parameterin Michel fit to accidental positrons.
The systematic uncertainty in the signal Ee PDF shape is estimated with the uncertainty of the full RMS within the signal window.
The peak of the signal E PDF is obtained from the 55 MeV in the ⇡ 0 data. The
uncertainty of energy scale is 0.43%. The uncertainties of the shape of the signal E
PDF are taken from the statistical error on the fits to the 55 MeV peak.
The typical uncertainties of te PDF are ⇠ 0.05 on fcore , ⇠ 7psec onon µtail , and ⇠ 50psec on

tail .

core , ⇠ 60psec

The uncertainty of the mean, t0 , is independently

estimated to be 15 psec including systematic errors. The stability of t0 is confirmed to
be within this uncertainty by RD peaks during di↵erent run periods.
We estimated the e↵ect of each systematic uncertainty with the following procedure:

1. Generate 100 pseudo experiments with the nominal PDF, Nsig = P oisson(3.8).
3.8 is the upper limit obtained from the data.
2. Fit to the pseudo experiments with the nominal PDF. And calculate the likelihood
ratio at the best fit and Nsig = 3.8.
3. Do the same on the same 100 pseudo experiments using the alternative PDF. In
each experiment, the parameters in the fitting PDF are randomized according totheir uncertainties.

1549.6 Sensitivity

4. Make a distribution of the di↵erence of negative-log-likelihood-ratio at 2. and 3(NLL).
Largest contributions come from uncertainties of o↵sets of the relative angles,
correlation in the positron observables and normalization. Negative-log-likelihood-ratio
from each factor is summarized in Table 9.4.

Table 9.4: Systematic uncertainties.

FactorEstimated value

NLL

Center of ✓e and e
Positron correlations
Normalization
E scaleEe bias, core and tailte center
E BG shape
E signal shape
Positron angle resolutions (✓e , e , ze , ye )angle resolution(u , v , w )Ee BG shapeEe signal shape

3.9mrad

0.18
0.16
0.13
0.07
0.06
0.06
0.04
0.03
0.02
0.02
0.02
0.01

6.7%
0.31%
50keV(core)250(tail)15psec

10%
10%

9.6

Sensitivity

9.6.1

Background Evaluation By Sideband Analysis

Before analyzing the analysis window, we applied the maximum likelihood fit onside bands. We defined two windows in positive and negative te -sideband and four
windows in ✓e ,esideband.

Negative time sideband: 48  E  58 MeV, 50  Ee  56 MeV, 2.7  te 
1.3 ns, |✓e |  50 mrad, | e |  50 mrad

155

9. PHYSICS ANALYSIS ON DATA 2010

Positive time sideband: 48  E

 58 MeV, 50  Ee  56 MeV, 1.3  te



2.7 ⇥ 10 9 ns < te<

2.7 ns, |✓e |  50 mrad, | e |  50 mrad

For time sidebands, the negative sideband is defined as
1.3 ⇥ 109ns and the expected number of background events was scaled

from the actual value observed in 1 ns < te1.3 ⇥ 109 ns < te< 2.7 ⇥ 10

< 3.9 ns.

The positive sideband(9 ns) was fit with the expected number of background

events scaled from the observed number in

3.9 ns < te<

1 ns. The best fit

results and confidence intervals in number of signals and branching ratio are shownin Table 9.5 and Table 9.6. The upper limits obtained from sideband analysis are
consistent with the evaluated result from Toy MC (Section 9.6.2).

Nsig

NRD

NBG

Negative time sideband

2.3+7.7
4.8+7.6+21.5
18.8+619.8+25.3
25.2

Positive time sideband

4.1+5.5
2.2+14.0+24.4
21.9+595.1+24.6
24.4

Table 9.5: Best fit results of number of signal, radiative muon decay and background
events.

UL in Nsig

UL in B

Negative time sideband

5.2

Positive time sideband

2.7

2.3 ⇥ 10 12
1.2 ⇥ 10 12

Table 9.6: Upper limits at 90% C.L. in number of signal events and branching ratio B.

In time sidebands, RD background events are not be counted due to the time
shift. Such events however would be observed in angle sidebands, which would o↵er across check in addition to time sidebands. Four angle sidebands are defined as following:

Negative ✓e sideband: 48  E
0.7ns, 150  ✓e  58 MeV, 50  Ee  56 MeV, |te | 

50 mrad, | e |  50mrad

Positive ✓e sideband: 48  E  58 MeV, 50  Ee  56 MeV, |te |  0.7 ns, 50 ✓e  150 mrad, | e |  50 mrad

156

9.6 Sensitivity

Negativesideband: 48  Ee

0.7 ns, |✓e |  50 mrad, 150 
Positiveesideband: 48  E

0.7 ns, |✓e |  50 mrad, 50 e

 58 MeV, 50  Ee  56 MeV, |te | e

50 mrad 58 MeV, 50  Ee  56 MeV, |te |  150 mrad

The expected number of BG events are counted as described above. The numbers ofRD events are calculated using 10000 generated MC events and scaled with total number of RD in the wider angle region(NRD = 87.7). The fit results are shown in Table 9.7.

150  ✓e 

50mrad150 50mrad50  ✓e  150mrad

50 

e
e



 150mrad

Nsig

NRD

NBG

5.8+5.9
2.8
0.1+4.0
6.4
3.5+5.9
3.4+6.7
4.2 3.9+26.1+4.3
4.3+13.0+2.2
2.1+20.6+3.4
3.4+3.9+23.7 3.9

+556.5+23.2
23.0+446.8+20.4
20.2+568.3.5+23.5
23.3+558.6+22.9
22.7

Table 9.7: Fit results in angle sidebands.

Figrue 9.13 and Figrue 9.14 show the likelihood fits in positive and negative timesidebands. Figure 9.15 to Figure 9.18show the likelihood fit angle sidebands.

157

9. PHYSICS ANALYSIS ON DATA 2010

Figure 9.13: Likelihood fit in positive time sideband. Summed PDFs weighted with best
fit numbers of each types are superimposed on each plot. The green, red, magenta andblue lines show the signal, RD, background and total sums of PDF respectively.

Figure 9.14: Likelihood fit in negative time sideband. Summed PDFs weighted with best
fit numbers of each types are superimposed on each plot. The green, red, magenta andblue lines show the signal, RD, background and total sums of PDF respectively.

158

9.6 Sensitivity

Figure 9.15: Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red, magenta and bluelines show the signal, RD, background and total sums of PDF respectively.

Figure 9.16: Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red, magenta and bluelines show the signal, RD, background and total sums of PDF respectively.

159

9. PHYSICS ANALYSIS ON DATA 2010

Figure 9.17: Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red, magenta and bluelines show the signal, RD, background and total sums of PDF respectively.

Figure 9.18: Likelihood fit in angle sideband. Summed PDFs weighted with best fit
numbers of each types are superimposed on each plot. The green, red, magenta and bluelines show the signal, RD, background and total sums of PDF respectively.

160

9.6 Sensitivity

9.6.2

Sensitivity

The sensitivity of an experiment is a measure of the result that the experiment is
expected to give for the limit on a parameter in the absence of signal. It describes
accuracy of the experiment, and is independent of the actual data that could fluctuate.
We define the 90% confidence sensitivity by the median upper limit at 90% CL of thesimulated toy experiments. In each experiment the expected number of RD and BG
are given by the values in the signal region estimated with sideband data (NRD =50.2, NBG = 608.5) and Nsig is expected to be 0. The upper limit is calculated withthe method described in Section 9.3.4.
Fig 9.20 shows the distribution of the upper limit of the number of signal for 1000
experiments (systematic errors are not included). The median is 4.8 for 2010 data.
After normalization, it corresponds to a branching ratio of 2.2 ⇥ 10 12 .

Figure 9.19 shows the updated sensitivity of 2009 data, which is 3.3 ⇥ 10 12 .

The sensitivity for the combined data of 2009 and 2010 is 1.6 ⇥ 10 12 , which is shownin Figure 9.21.

Figure 9.19: Updated result of sensitivity in 2009.

161

9. PHYSICS ANALYSIS ON DATA 2010

Figure 9.20: Distribution of Nsig upper limit at 90% confidence level in toy experimentsfor 2010 data. A median of this distribution (4.8) is defined as a 90% confidence sensitivity.

Figure 9.21: Distribution of Nsig upper limit at 90% confidence level in toy experiments
for combined data of 2009 and 2010. A median of this distribution (5.2) is defined as a
90% confidence sensitivity.

162

9.7 Maximum Likelihood Fit on the Data

9.7

Maximum Likelihood Fit on the Data

When we finished the optimization of the analysis and the background studies in the
sidebands, we opened the blinding box and performed a likelihood analysis.
For 2009 data, the best fit results are
+4.5
+12.3
(Nsig , NRD , NBG ) = (3.4+6.6
4.4 , 26.9 4.5 , 273.1 12.3 ), Nobs = 311(9.22)

Figure 9.22 shows the distribution of each observable with average PDFs scaledwith the best fit numbers of each type.

Figure 9.22: Updated result of 2009 data. The green, red, magenta and blue lines show
the signal, RD, background and total sums of PDF respectively.

For 2010 data, the numbers of signal, radiative muon decay and accidental background are estimated to be,
+9.2
+18.7
(Nsig , NRD , NBG ) = ( 2.2+5.0
1.9 , +50.2 9.2 , +608.5 18.6 )(9.23)

where number of analyzed events is Nobs = 645 and asymmetric errors are taken fromMINUIT of fitting, which is 1.645 sigma MINOS errors.
Figure 9.23 shows the distribution projected onto each observable with averaged PDFs

163

9. PHYSICS ANALYSIS ON DATA 2010

45
40
35
30
25
20

Events / (0.4 MeV)

Events / (0.24 MeV)

Events / (56 psec)

scaled with the best fit number of each type. The blue line is a sum of all PDFs.

70
60
50
40

80
60
40

20

10
5

10

0

0
50

-0.5

0

Time (nsec)

0.5

Events / (4 mrad)

Events / (4 mrad)

100

30

15

40
35
30
25

20
51

52

53

54

-20

0

20

55

Positron Energy (MeV)

56

0
48

50

52

54

56

Gamma Energy (MeV)

58

45
40
35
30
25

20

20

15

15

10

10

5
0

120

5
-40

-20

0

θ (mrad)

20

40

0

-40

φ (mrad)40

Figure 9.23: Result of maximum likelihood fit for 2010 data. Summed PDFs weighted
with best fit numbers of each types are superimposed on each plot. The green, red, magenta
and blue lines show the signal, RD, background and total sums of PDF respectively.

For the combined data of 2009 and 2010, the best fit results are
+12.0
+22.4
(Nsig , NRD , NBG ) = ( 0.5+7.9
4.7 , +76.5 12.0 , +882.1 22.3 )(9.24)

where number of analyzed events is Nobs = 956 and asymmetric errors are taken fromMINUIT of fitting, which is 1.645 sigma MINOS errors.
Figure9.24 shows the distribution projected onto each observable with averaged

PDFs scaled with the best fit number of each type. The blue line is a sum of all PDFs.

9.7.1

Upper Limit on Branching Ratio

Since the fit result is zero-consistent, an upper limit on µ+ ! e+ +branching ratio isset. Together with the normalization factor, the upper limit is calculated to be B(µ+ !
e+ + ) < 1.7 ⇥ 10 12 at 90% C.L. for 2010 data and B(µ+ ! e+ + ) < 2.4 ⇥ 10 12

for combined data of 2009 and 2010. The total e↵ect of systematic uncertainties is 2%1649.7 Maximum Likelihood Fit on the Data

Figure 9.24: Result of maximum likelihood fit for combined data of 2009 and 2010.
Summed PDFs weighted with best fit numbers of each types are superimposed on eachplot. The green, red, magenta and blue lines show the signal, RD, background and totalsums of PDF respectively.

change in the upper limit of the combined data set.
The confidence intervals are shown in Table 9.8 to Table 9.10.

Best fit
Nsig
BR

3.4
3.2 ⇥ 10

12

LL(90% C.L.)

ULL(90% C.L.)

UL(95% C.L.)

CL@0

0.2

10.4

11.9

0.92

1.7 ⇥ 1013

9.6 ⇥ 1012

1.1 ⇥ 10

11

Table 9.8: Confidence intervals on 2009 data.

Best fit
Nsig
BR

-2.2
9.9 ⇥ 10

13

UL(90% C.L.)

UL(95% C.L.)

3.8

5.0

1.7 ⇥ 1012

2.3 ⇥ 10 12

Table 9.9: Confidence intervals on 2010 data.

165

0.92

9. PHYSICS ANALYSIS ON DATA 2010

Figure 9.25: Confidence level curve for 2009, 2010 and combined data.

Best fit
Nsig
BR

-0.5
1.5 ⇥ 10

13

UL(90% C.L.)

UL(95% C.L.)

7.8

9.8

2.4 ⇥ 1012

2.9 ⇥ 10 12

Table 9.10: Confidence intervals on combined data of 2009 and 2010.

9.7.2

Event Distribution

The event distributions after the unblinding are shown in Figure 9.26 to Figure ??.
The events are labeled with the order of the likelihood ratio defined as S/(S + R +B). Figure (a) shows E and Ee and (b) shows te and cos⇥e . The best fit resultand event distribution of 2009 data is largely unchanged compared to the preliminary
result. And an excess of events around the signal region still remain.

166

58

tea (nsec)

Ea (MeV)

9.7 Maximum Likelihood Fit on the Data

57
56

2
1.5
1

55
54

0.5

53

0

1 2

52

3

3

1

-0.5

51

-1

50

-1.5

49
48
50

2

51

52

53

54

-2
-1

55
56
Ee (MeV)(a)-0.9995-0.999-0.9985-0.998
cosOea(b)

Figure 9.26: Updated event distributions in the analysis region for 2009 data. In theleft plot, selections in te and ✓e , each of which is 90% efficient on the signal, are applied(|te | < 0.278 nsec, |✓e | > 178.34 ). In the right plot, a selection in Ee (90% on the signal)
and a selection in E (73% on the signal) are applied (51 < E < 55 MeV, 52.34 < Ee < 55
MeV).

(a)

(b)

Figure 9.27: Event distribution in the analysis region for 2010 data. The signal 2D PDFs
are superimposed as contours at 1, 1.64, 2 sigma as blue solid, dot-dashed, and dashed lines
respectively. The number shows the rank by S/(S + R + B). In plot (a), selections in teand ✓e , each of which is 90% efficient on the signal, are applied (|te | < 0.278 nsec,cos|✓e | < 0.9996). In plot (b), a selection in Ee (90% on the signal) and a selection in
E (73% on the signal) are applied (51 < E < 55 MeV, 52.34 < Ee < 55 MeV).

167

9. PHYSICS ANALYSIS ON DATA 2010

(a)

(b)

Figure 9.28: Event distribution in the analysis region for combined data of 2009 and2010. The signal 2D PDFs are superimposed as contours at 1, 1.64, 2 sigma as blue solid,
dot-dashed, and dashed lines respectively. The number shows the rank by S/(S + R +B). Events with rank (2,3,4) are from 2009, events with rank (1,5) are from 2010 data.. Inplot (a), selections in te and ✓e , each of which is 90% efficient on the signal, are applied(|te | < 0.278 nsec, cos|✓e | < 0.9996). In plot (b), a selection in Ee (90% on the signal)
and a selection in E (73% on the signal) are applied (51 < E < 55 MeV, 52.34 < Ee < 55
MeV).

168

9.8 Conclusion

9.8

Conclusion

The MEG experiment is a precise rare decay search designed to observe µ+ ! e+ orlack thereof as a sensitive low energy probe of new physics. In this thesis we presented
an updated result using the data taken in 2010, which correspond to 1.1 ⇥ 1014 muon
decays in the stopping target.

The previous preliminary result based on the 2009 data gave a higher than
expected upper limit of 1.5 ⇥ 10 11 at 90% C.L. with a few possible events in thesignal region[1][2][3]. To examine this result, we analyzed the 2010 data which has
twice higher statistical sensitivity than the 2009 data.

In addition, we improved

several aspects of calibration and analysis such as detector alignment, implementationof correlations in position observables, improved magnetic field map and improved
likelihood analysis. We applied these improvements to the 2009 data and confirmedthat the excess signals still remain but the sensitivity is improved.
We adopted a ”blind analysis” and a maximum likelihood fit. After unblinding thesignal region, the number of of µ+ ! e+ decay events in the data sample is extracted
by a maximum likelihood fit. A 90% confidence interval is then constructed using
the Feldman-Cousins technique. We evaluated an expected sensitivity of the 2010
data to be 2.2 ⇥ 10 12 by toy Monte Carlo experiment, which was also confirmed byanalysis of the side band data. All the analysis is done by hiding the signal region
until probability density functions for a likelihood fit are settled upon.

The obtained result is consistent with a null hypothesis and we set an upper limiton the branching ratio

B(µ+ ! e+ ) < 1.7 ⇥ 10 12 at 90% C.L.

(9.25)

B(µ+ ! e+ ) < 2.4 ⇥ 10 12 at 90% C.L.

(9.26)

for the 2010 data and169

9. PHYSICS ANALYSIS ON DATA 2010

for the combined data of 2009 and 2010. This result exceeds the previous world’s best
limit of 1.2 ⇥ 10 11 set by the MEGA experiment[4].

170

10

Prospects
MEG continued to take data in 2011 and the total data statistics is doubled with thisrun. The analysis on data 2011 is ongoing and we expect to continue the data-taking
in 2012 and possibly 2013. The total data statistics is expected to be tripled by 2013
and the branching ratio is expected to reach ⇠ 6 ⇥ 10 13 at 90 % C.L. as shown in

20
08
20
09
20
10

Figure 10.1.

2011

2012

Sensitivity Prospects

data UL (2009)

data UL (2009+2010)

Figure 10.1: Future prospects of sensitivity.

17110. PROSPECTS

We are also considering upgrades to the current detectors in order to further
improve sensitivity. One way to increase statistics is to use higher beam rate. However
the current positron efficiency is limited to less than 50% due to the material indrift chamber. Use of thinner cables is considered. In order to improve detector
performance, currently some ideas such as smaller photodetectors (MPPC) for the LXedetector, new tracking system for positron, timing counter system based on MPPC
instead of PMTs, with smaller scintillator cells, etc are under study.

172

Acknowledgements

This thesis would not have been possible without the help of many people
who contributed and extended their valuable assistance.

First and foremost I would like to express my utmost gratitude to mysupervisor, Prof. Toshinori Mori, who provided me opportunities for mydoctoral study and supported me throughout the writing of this thesis withhis patience and knowledge.

I would like to offer my sincere gratitude to Prof. Wataru Ootani, whogave me tremendous help and support during the writing of this thesis as
well as my study in Switzerland.

I would like to thank Prof. Toshiyuki Iwamoto, for his guidance during mystudy at PSI, and Dr Ryu Sawada, for answering so many of my questions
about analysis, and Dr Yusuke Uchiyama, who taught me a lot about
calibration. I am greatly indebted to Prof. Satoshi Mihara, Dr. Hajime
Nishiguchi, Dr Yasuhiro Nishimura, Mr.

Yuki Fujii and Mr.

Daisuke

Kaneko for their valuable work and advice.

I offer my great appreciation for all the members of the MEG collaboration,
without whose hard work the experiment would not have been possible. It
has been a great pleasure to work with them.

I am deeply grateful to all of ICEPP members and secretaries.

I would like to convey thanks to PSI for their financial support during mystay at the institute.

Finally, I would like to express my gratitude to my family and friends fortheir love and support through the duration of my study.

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