Doctral Dissertation
博士論文

Search for µ → eγ with the first year data of
the MEG II experiment(MEG II実験初年データを用いたµ → eγ探索)

A Dissertation Submitted for the Degree of Doctor of Philosophy
December 2023
令和 5年 12月博士 (理学) 申請

Department of Physics, Graduate School of Science,
The University of Tokyo東京大学大学院理学系研究科物理学専攻

Atsushi Oya
大矢 淳史

Abstract
A search for µ → eγ was performed at a sensitivity of 8.8 × 10−13 by observing 2.64 × 1012 muon
decays with the first MEG II data taken for effectively four weeks. No signal excess was observedin this analysis and we set an upper limit ofB(µ → eγ) < 7.5 × 10−13
at 90 % confidence level. This result indicates that the MEG II experiment, even with the very
limited data statistics, approached the previous search, which gave B(µ → eγ) < 4.2 × 10−13 as aresult of a search with the sensitivity of 5.3 × 10−13 . This is a firm demonstration that a few years
of MEG II experiment will reach its initial goal of searching for µ → eγ with an order of magnitude
better sensitivity.
In addition, a combined analysis was performed using the previous experiment and the MEG II
2021 data. The combined sensitivity was 4.3 × 10−13 , and we set an upper limit ofB(µ → eγ) < 3.1 × 10−13
at 90 % confidence level. This is the most stringent limit ever for the µ → eγ branching ratio.

Contents
Prefacevii

1

Introduction to µ → eγ
1.1 µ → eγ search as a probe of new physics . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Charged leptons in the Standard Model . . . . . . . . . . . . . . . . . . .
1.1.2 Model independent framework of µ → eγ phenomenology . . . . . . . . .
1.1.3 Predictions of µ → eγ in theories beyond the Standard Model . . . . . . .
1.2 Principle of experimental searches for µ → eγ decay . . . . . . . . . . . . . . . .
1.2.1 Signal signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Background from accidental coincidence . . . . . . . . . . . . . . . . . .
1.2.3 Background of physical origin . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Sign of muon charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Result of previous experiment and its limitations . . . . . . . . . . . . . . . . . . .
1.4 Previous MEG II upgrade works . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Original works of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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MEG II apparatus
2.1 Muon beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Muon stopping target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Positron spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 COBRA superconducting magnet . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Cylindrical drift chamber . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Pixelated timing counter . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 LXe gamma-ray detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Liquid xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Scintillation photon sensor . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Internal calibration sources — LED and alpha ray — . . . . . . . . . . . .
2.4.4 17.6 MeV gamma-ray calibration with proton beam and Li2 B4 O7 target . .
2.4.5 55 MeV gamma-ray calibration with π − beam and liquid hydrogen target .
2.5 Downstream radiative decay counter . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Electronics and data acquisition system . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Integrated waveform data-taking system . . . . . . . . . . . . . . . . . . .
2.6.2 Trigger system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Data management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Reduction of waveform data size . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Data storage and blinding scheme . . . . . . . . . . . . . . . . . . . . . .
2.8 Detector simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Data taking
3.1 Physics data taking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Detector condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Daily calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 RMD-enhanced calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 DAQ setting and performance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 π 0 → γγ data taking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Cosmic ray data taking for detector alignment . . . . . . . . . . . . . . . . . . . .
3.8 Optical scanning for detector alignment . . . . . . . . . . . . . . . . . . . . . . .
3.9 Target camera operation and photograph . . . . . . . . . . . . . . . . . . . . . . .

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Event reconstruction and selection
4.1 Positron reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Hit reconstruction and clustering of pTC . . . . . . . . . . . . . . . . . .
4.1.2 Hit reconstruction of CDCH . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Track finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Track fitting and matching with pTC . . . . . . . . . . . . . . . . . . . . .
4.1.5 Tracking refinement and missing hit recovery . . . . . . . . . . . . . . . .
4.1.6 Vertex and angle reconstruction . . . . . . . . . . . . . . . . . . . . . . .
4.1.7 Time reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.8 Track selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.9 Dedicated double turn reconstruction . . . . . . . . . . . . . . . . . . . .
4.2 Gamma-ray reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Waveform analysis and sensor calibration . . . . . . . . . . . . . . . . . .
4.2.2 Position reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Time reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Pileup analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Energy reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.6 Gamma-ray selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 RDC reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Selection with LXe time . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Reconstruction of combined kinematics . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Reconstruction of combined kinematics . . . . . . . . . . . . . . . . . . .
4.4.2 Fiducial volume of the detector . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Pair selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Calibration and performance evaluation
5.1 DRS calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Time calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Voltage calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 LXe sensor alignment and position resolution . . . . . . . . . . . . . . . . . . . .
5.2.1 MPPC alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 LXe position resolution evaluation . . . . . . . . . . . . . . . . . . . . . .
5.3 LXe energy calibration and resolution . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Sensor gain and excess charge factor . . . . . . . . . . . . . . . . . . . . .
5.3.2 QE and PDE of sensors and liquid xenon light yield . . . . . . . . . . . . .
5.3.3 Energy scale calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.3.4 Energy resolution evaluation with π 0 → γγ data . . . . . . . . . . . . . .
5.4 LXe time calibration and resolution . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Time walk and offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Effective velocity of scintillation light . . . . . . . . . . . . . . . . . . . .
5.4.3 Position dependence of time offset . . . . . . . . . . . . . . . . . . . . . .
5.4.4 Time resolution in π 0 → γγ events . . . . . . . . . . . . . . . . . . . . .
5.5 pTC time calibration and resolution . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Internal time calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Track-based time calibration . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Laser based calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.4 Performance evaluation with even-odd method . . . . . . . . . . . . . . .
5.6 Combined teγ calibration and its resolution . . . . . . . . . . . . . . . . . . . . .
5.6.1 Offset calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Calibration for trigger and data pre-selection . . . . . . . . . . . . . . . .
5.6.3 Full evaluation of teγ resolution . . . . . . . . . . . . . . . . . . . . . . .
5.7 RDC LYSO energy scale calibration . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 CDCH calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.1 CDCH Z measurement calibration . . . . . . . . . . . . . . . . . . . . . .
5.8.2 CDCH wire alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.3 CDCH drift time to distance calibration . . . . . . . . . . . . . . . . . . .
5.9 Positron momentum resolution and calibration . . . . . . . . . . . . . . . . . . . .
5.9.1 Energy scale and response evaluation . . . . . . . . . . . . . . . . . . . .
5.9.2 Alignment of magnetic field . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Positron resolution in double turn analysis . . . . . . . . . . . . . . . . . . . . . .
5.11 Global detector alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11.1 Target deformation and time shift of position . . . . . . . . . . . . . . . .
5.11.2 Alignment between CDCH and target . . . . . . . . . . . . . . . . . . . .
5.11.3 Alignment between CDCH and LXe . . . . . . . . . . . . . . . . . . . . .
5.11.4 Alignment summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 LXe efficiency evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Positron efficiency evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13.1 CDCH hit efficiency evaluation . . . . . . . . . . . . . . . . . . . . . . .
5.13.2 Positron tracking efficiency evaluation . . . . . . . . . . . . . . . . . . . .
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Analysis99
6.1 Analysis framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.1 Likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.2 Observables of PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.3 Toy MC simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1.4 Frequentists-based statistical method . . . . . . . . . . . . . . . . . . . . . 101
6.1.5 Theory independence of the analysis . . . . . . . . . . . . . . . . . . . . . 102
6.1.6 Incorporation method of systematic uncertainties . . . . . . . . . . . . . . 102
6.2 Analysis window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3.1 Michel positron counting method . . . . . . . . . . . . . . . . . . . . . . 103
6.3.2 RMD normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.3 Combined result and uncertainty . . . . . . . . . . . . . . . . . . . . . . . 108
6.4 Background estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Contentsvi

6.4.1 Number of accidental background . . . . . . . . . . . . . . . . . . . . . . 108
6.4.2 Number of RMD background . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5 Period-dependent event weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6 Accidental background modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6.1 Accidental gamma-ray energy PDF: A1 (Eγ |vγ , wγ ) . . . . . . . . . . . . . 109
6.6.2 Accidental positron momentum PDF: A2 (Ee |σe ) . . . . . . . . . . . . . . 111
6.6.3 Accidental time PDF: A3 (teγ , npTC |Ee , wγ ) . . . . . . . . . . . . . . . . . 111
6.6.4 Accidental angle PDF: A4 (θeγ ) and A5 (φeγ |vγ ) . . . . . . . . . . . . . . . 111
6.6.5 Accidental RDC PDF: A6 (tRDC , ERDC |Eγ ) . . . . . . . . . . . . . . . . . 113
6.7 Signal modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.7.1 Signal gamma-ray energy PDF: S1 (Eγ |vγ , wγ ) . . . . . . . . . . . . . . . 117
6.7.2 Signal positron momentum PDF: S2 (Ee |σe ) . . . . . . . . . . . . . . . . . 119
6.7.3 Signal time PDF: S3 (teγ , npTC |Eγ , Ee ) . . . . . . . . . . . . . . . . . . . 119
6.7.4 Signal angle PDF: S4 and S5 . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.7.5 Signal RDC PDF: S6 (tRDC , ERDC ) . . . . . . . . . . . . . . . . . . . . . . 125
6.8 RMD background modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.8.1 Incorporation of RMD kinematics in R1 . . . . . . . . . . . . . . . . . . . 126
6.8.2 RMD timing PDF: R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.9 Properties of PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.10 Summary of systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.11 Reliability check of analysis with full detector simulation . . . . . . . . . . . . . . 134
7

Result and Discussion
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7.1 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1.1 Sensitivity cross-check in timing sideband data fitting . . . . . . . . . . . . 137
7.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2.1 Combined result with MEG I final analysis . . . . . . . . . . . . . . . . . 139
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.4 Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8Conclusion

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A Radiative Muon Decay

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B Transformation of positron correlation to θeγ vs φeγ correlation

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C Analysis cross check

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D Full detail of the target hole analysis

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Acknowledgements

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References

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Preface
Background of this thesis
The standard model (SM) of particle physics describes the fundamental elements of matter andthe laws behind them. Its validity has experimentally been confirmed up to the electroweak scalearound O(100 GeV). However, several fundamental problems are addressed on the SM, and thus,
the particle physics research is further continued to better understand them.
As candidate solutions for the problems of the SM, theorists have proposed various new physics
models. In response to them, several experiments have been conducted in pursuit of their signature.
An important branch among such efforts is charged Lepton Flavor Voilation (cLFV), mixing
phenomena between charged leptons from three different generations, electron, muon, and tau.
Searches for cLFV are intriguing because of the experimental cleanness; the SM strictly forbids
cLFV, and thus, a discovery of such phenomena is clear evidence of new physics. The search
is valuable also because cLFV-based tests of new physics models can likely be experimentallyreachable, according to several predictions in theories beyond the SM.
The forerunner among the cLFV searches is µ → eγ, on which, the branching ratio is currently
limited as B(µ → eγ) < 4.2 × 10−13 (90 % C.L.) by the MEG experiment [1]. This limit has alreadystarted to exclude some models beyond the SM. In some predictions, however, B(µ → eγ) still lies
just below the current upper limit, being 10−14 – 10−12 . Accordingly, a new experiment — theMEG II experiment — is in progress to search for the µ → eγ decay with an order of magnitude
higher sensitivity.

Theme of this thesis
The main theme of this thesis is a search for µ → eγ decay with the MEG II 2021 data as apioneering work for the MEG II experiment. As well as building the analysis framework, the author
also established the detector calibration and event reconstruction, which is also a subject of thisthesis. The details of the analysis procedure, reconstruction, and calibration methods, and the resultof the 2021 data analysis are described in this thesis.
The background suppression is the key ingredient to finally achieve an order of magnitudesensitivity improvement in the MEG II experiment, as is evident from the experience in the previous
experiment. This requires improvements in the detector’s resolution, which is demonstrated inthis thesis. This thesis also presents that the detector’s efficiency is improved by a factor of ∼ 2.
Consequently, this thesis confirms that an order of magnitude improvement can be realized in a few
years by a continuous data taking of the MEG II experiment.
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Outline of this thesis
This thesis is organized as follows. Chap.1 gives up-to-date knowledge about the µ → eγ searchwith reviews of the theoretical motivations, experimental principles, and results of the previous
experiment. The situation of the MEG II experiment before this thesis is then summarized, whichis to clarify the original studies of this thesis that are briefly listed at the end of this chapter. Chap.2
describes the MEG II apparatus in detail, and then, Chap.3 summarizes the data-taking conditionin 2021. Chap.4 explains the first step of the data analysis; the event reconstruction and selection.
Chap.5 then describes the calibration of the detectors and evaluates their performances. Chap.6
gives the full methodologies of the analysis searching for µ → eγ and discusses all the propertiesof the analysis. Chap.7 presents the result of the analysis with its interpretation and discusses theoutlook of the MEG II experiment. Finally, this thesis is concluded in Chap.8.

Chapter 1
Introduction to µ → eγ

1.1µ → eγ search as a probe of new physics

Despite its success up to O(100 GeV), the Standard Model (SM) is not considered as an ultimate
theory, but a low energy approximation of a more fundamental physics. This is because the SM
faces several problems, among which the following three are especially important as the backgroundof µ → eγ search. Firstly, the SM does not provide candidate particles for the dark matter and darkenergy, which are observationally evaluated to account for 26 % and 69 % of the universe [2, 3].
Secondly, the observed neutrino oscillation [4, 5] is not viable with the massless description ofneutrinos in the SM. The final one is called the hierarchy problem; no natural explanation is given as
to why the measured Higgs mass of 125 GeV lies in the electroweak scale, which is several ordersof magnitude smaller than the Plank scale of O(1019 GeV) [6, 7, 8].
These problems can be solved by introducing new physics beyond the SM, such as supersymmetry(SUSY) [9], grand unified theory (GUT), and see-saw mechanism [10, 11, 12]. The SUSY introduces
new partner particles for those in the SM, which can provide a candidate for the dark matter and solve
the hierarchy problem. The see-saw mechanism, on the other hand, grows our attention with its
ability to explain the tiny neutrino mass (< 0.151 eV [13]) and their mixing. This model introduces
right-handed neutrinos with large Majorana mass terms, and the mass of the left-handed neutrino
becomes MD2 /MR , where MD is the Dirac mass and MR is the Majorana mass. The GUT, which
unifies interactions and matter particles, also motivates us because the gauge coupling constants get
close to each other around O(1016 GeV) [9].
These new models can be probed by a µ → eγ search, and this section aims to explain why.
In the following sections, it is firstly clarified that µ → eγ decay is strictly prohibited in the SM(Sec.1.1.1). Then, we review the predictions of µ → eγ in the above models, which suggest thepossibility of an experimental detection (Sec.1.1.3). In short, model probing power is given toµ → eγ search by the cleanness in the SM and an enhanced rate in new physics predictions.
1

Chapter 1. Introduction to µ → eγ

2

W

µ

νµ

γ
e

νe

Figure 1.1: µ → eγ decay mediated directly by the neutrino mixing

1.1.1

Charged leptons in the Standard ModelIn the SM, the charged leptons have gauge coupling to the electromagnetic and the weak interactions,and the Higgs coupling for the mass generation. They are written as¯ µ `Aµ
L = e`γg− √ (ν̄`L γ µ `L Wµ+ + `¯L γ µ ν`L Wµ− )
2




p
(1.1)
1
2
µ
2
µ
2
02
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¯
− g +g`L γ − + sin θW `L + `R γ sin θW `R Zµ
2
m` ¯
−
``H.
v
Here, any reactions of charged leptons preserve the lepton flavor, which is one of the most important
properties of the SM. Therefore, µ → eγ decay, which violates the lepton flavor, is strictly prohibitedin the SM. However, this conservation law is not considered as a fundamental one that comes froma guiding principle. Rather, it has been thought to be an accidental conservation.
The observation of the neutrino oscillation revealed that the accidental conservation no longer
holds. This phenomenon itself, however, cannot produce an observable µ → eγ branching ratio. In
fact, its contribution to the µ → eγ branching ratio is given as [14, 15, 16]∆m2
3α X
Uµi Uei 2 i
B(µ → eγ) =
32π i
MW

2

∼ 10−54 ,(1.2)

where i runs over the neutrino generations. The non-zero branching ratio arises from the process
shown in Fig.1.1. Hence, µ → eγ decay search remains to be a clean channel to probe a new physics
that goes even beyond the neutrino oscillation.

1.1.2

Model independent framework of µ → eγ phenomenology

It is convenient to introduce a model-independent notation that characterizes µ → eγ decay, which
will be used hereafter in this thesis. The model-independent effective Lagrangian for µ → eγ decay
can be expressed by dipole operators,
4GF
Lµ→eγ = − √ [mµ AR µ̄R σ µν eL Fµν + mµ AL µ̄L σ µν eR Fµν + h.c] ,
2

(1.3)

where GF is the Fermi coupling constant, and the AR(L) is the polarization-dependent coupling thatinduces µ → eR(L) γ. When µ → eγ decay is observed from a polarized muon at rest, the angular
distribution of B(µ → eγ) is given asdB(µ → eγ)= 192π 2 |AR |2 (1 − Pµ cos θe ) + |AL |2 (1 + Pµ cos θe ) ,(1.4)
d cos θe

1.1. µ → eγ search as a probe of new physics

3∆m2µ̃ẽ
µ̃

ẽ

µ
χ̃

0

γ
e

Figure 1.2: Diagram for µ → eγ decay induced by slepton mass mixing [28].
where θe is the angle between the muon polarization vector and the positron emission angle, andPµ corresponds to the magnitude of the polarization.
The total branching ratio of µ → eγ behaves proportionally to |AR |2 + |AL |2 . On the other
hand, the above relation means that the angular distribution is sensitive to their ratio AR /AL .
This ratio depends on the detail of assumed new physics models as will be discussed later in
Sec.1.1.3. Therefore, experimental searches must be carefully designed with considerations about
their sensitivity to each polarization. We will get back to this discussion later in Sec.6.1.5, where
the main analysis for µ → eγ search is described.

1.1.3

Predictions of µ → eγ in theories beyond the Standard Model

This section focuses on the prediction of µ → eγ decay in supersymmetric GUT [17, 18, 19] orsupersymmetric see-saw models [20, 21, 22]. When the SUSY is exact, the particle masses mustdegenerate with those of SUSY partners, which is experimentally not the case. This contradiction
can be explained by SUSY breaking, which gives larger masses to the SUSY partners so that theycan escape from experimental detection. Various mechanisms are proposed for the SUSY breaking:gravity-mediated SUSY breaking [23, 24], gauge mediated SUSY breaking [25, 26, 27], and soon. At the Plank scale, most of these models predict a degenerate mass spectrum for sleptons (theSUSY partners of the leptons). This degeneracy results in a complete alignment between the sleptonand lepton mass matrix, where no flavor mixing appears. When this is renormalized down to theelectroweak scale, on the other hand, a finite sleptons’ flavor mixing can be induced by radiative
corrections; in particular, the GUT or see-saw can give rise to a sizeable effect. Hence, we expect arare but finite B(µ → eγ), which is the key concept in theoretical calculations.
The flavor mixing in slepton induces µ → eγ via the process shown in Fig.1.2, where the flavor
transition takes place at the cross marker. This contribution is generally calculated as!2 4m2µ̃ẽ
100 GeV10−6 ,(1.5)
B(µ → eγ) ∼m2`˜m`˜where m`˜ and m2µ̃ẽ correspond to the diagonal and off-diagonal element of the slepton mass matrix,
respectively [9, 18]. Though m2µ̃ẽ is zero at the Plank scale, it becomes finite by radiative correctionsin the GUT or see-saw mechanism. The diagram for such correction is drawn in Fig.1.3, where the
GUT case assumes SU(5) unification and the see-saw case assumes the type-I model. The SU(5)SUSY-GUT gives rise to the flavor mixing in the right-handed sleptons as


MP
3
∗
2 2
2
2
,
(1.6)
(mẽR )ij ∼ − 2 Vi3 Vj3 |(yu )33 | m0 (3 + |A0 | ) log
8π
MGU T
in a rough approximation. Here, V is the CKM matrix at the GUT scale, y is the Yukawa coupling,Chapter 1. Introduction to µ → eγ

4T̃3 , H (ÑR , H)
T3 (NR )T̃3 (ℓ̃L )T̃3 (ℓ̃L )T̃3 (ℓ̃L )T̃3 (ℓ̃L )H̃ (H̃)Figure 1.3: Radiative correction that gives rise to finite off-diagonal elements of slepton mass
matrix[28]. Note that the Higgs in GUT is different from the Higgs in see-saw.
m0 and A0 are the SUSY breaking mass parameter and their coupling, and MGU T is the GUT energy
scale. In this equation, the flavor mixing in the quark sector affects the slepton mass matrix because
the leptons and quarks are unified to the same gause multiplets. In SU(5), only the right-handed
sleptons are hosted in 10 representation together with the top quark, which gives the largest radiative
correction thanks to the largest (yu )33 . This is why only the right-handed slepton can have a large
flavor mixing in SU(5) model. Similarly, in the SUSY see-saw model, the left-handed sleptons
acquire a finite flavor mixing as


MP
1 ∗
2
2
2
.
(1.7)
(∆mL̃ )ij ∼ − 2 (yν )ik (yν )kj m0 (3 + |A0 | ) log
8πMN
In models that incorporate both the GUT and see-saw [29, 30, 31, 32], both the left and right-handed
slepton have flavor mixing, and even a larger contribution to µ → eγ can be expected.
The µ → eγ decay polarization appears differently between the following categories of models;• Models with mixing only in right-handed sleptons,• Models with mixing only in left-handed sleptons,• Models with mixing both in right-handed and left-handed sleptons.
When only the right-handed sleptons are mixed, only the AL term in Eq.(1.3) and Eq.(1.4) is sizeable
and AR is suppressed, whereas the opposite happens when only the left-handed slepton is mixed.
When both the left and right-handed sleptons are mixed, both the AR and AL can have observablevalues. Therefore, when µ → eγ decay is found, the angular distribution of the decay product will
also be an important observable to discriminate the models from each other.
As an example of theoretical calculations for B(µ → eγ), Ref.[32] presents a calculation for a
SUSY model that both includes the GUT and see-saw mechanism (Fig.1.4). Here, the GUT happens
in SU(5) group and three generation right-handed neutrinos are introduced as singlets. The result
indicates that the current upper limit on B(µ → eγ) is starting to exclude the large tan β region. At
the same time, we can see that a search for a higher sensitivity around 10−14 is of high value forfurther exploration of the new physics parameter space.

1.2

Principle of experimental searches for µ → eγ decay

1.2.1

Signal signature

Observed in the muon rest frame, µ → eγ can be identified by the unique two-body kinematics ofthe decay products. The event signatures are1.2. Principle of experimental searches for µ → eγ decay

5

Figure 1.4: An expectation on B(µ → eγ) (orange contour lines) on as a function of m0 (theuniversal scalar mass) and tan β (the VEV ratio between the two Higgs doublets) [32]. The blue
band indicates the region where the Higgs mass is consistent with the experimentally measured one
within the uncertainties.
1. teγ = 0 (the time coincidence between positron and gamma)
2. Θeγ = 180° (the back-to-back movement)
3. Ee = Eγ = m2µ ≈ 52.8 MeV (monochromatic at half the muon mass).
The expected number of signal events can be expressed as
Nsig = k × B(µ → eγ)
k = Rµ × T × Ω × e × γ × sel ,(1.8)

where k corresponds to the number of effectively measured muon decay in an experiment. This isequivalent to the inverse of the single event sensitivity and is also called the normalization factor inthis thesis. Rµ × T , the product of the muon rate and the total data taking time, corresponds to thetotal number of muons stopped during the experiment. The other parameters on the right hand sideof k are the efficiency factors; Ω is the geometrical acceptance, e(γ) is the detector’s efficiency for
positron (gamma), and sel is the efficiency of the analysis.
To have a high statistical yield, we need to build an experiment with a high muon rate. At thesame time, high efficiency of the detector is mandatory, which often gets more difficult with theincreasing beam rate. Therefore, one experimental requirement is given in this respect, to keep ahigh efficiency even in a high-rate experiment.

1.2.2

Background from accidental coincidence

The main source of the background events is the coincidence of background positron and backgroundgamma that accidentally mimics the signal signature. The former one comes from the normal muon
decay µ+ → e+ ν ν̄, which we often call Michel decay. The latter mostly comes either from theChapter 1. Introduction to µ → eγ

Relative rate

6

1




0.8
0.6



0.4
0.2
0
0

10

20

30

40

50
E e [MeV](a) Ee background spectrum.

(b) Eγ background spectrum.

Figure 1.5: Ee and Eγ spectrum in accidental background events [28]. (a) Ee background, fromMichel decay, is calculated in [33]. (b) In the Eγ spectrum, two major sources are considered; thedashed line is that from radiative decay, the dotted line is that from annihilation in flight, and thesolid line is their sum.
radiative decay µ+ → e+ ν ν̄γ or annihilation-in-flight of positrons from the normal decay.
The expected number of backgrounds follow
NAcc ∝ Rµ2 × (σEγ )2 × σEe × (σΘeγ )2 × σteγ × T,

(1.9)

where σ-symbols in front of the variables stand for each resolution. The Rµ × σteγ term corresponds
2
term corresponds to that in theto the rate of having a coincidence in the time axis, and the σΘeγ
angular axis. The σEe and σEγ contribution to the background can be evaluated by integrating theirspectrum within the resolution, namely an integration over [52.8 MeV−σ, 52.8 MeV]. The spectrum
is shown in Fig.1.5a and Fig.1.5b, and some formulas for them are given in Appendix.A. The linear
scaling on σEe in Eq.(1.9) comes from the integration of the finite spectrum around 52.8 MeV. On theother hand, the gamma spectrum has a vanishing rate there, which results in the quadratic dependence
when Fig.1.5b is linearly approximated in the integration over [52.8 MeV − σ, 52.8 MeV].
As the accidental background has been the limiting factor of µ → eγ sensitivity in experiments,
its suppression is important. Eq.(1.9) gives several implications in this regard. Firstly, a continuous
muon beam is better than a pulsed one, which is because the latter results in larger NAcc with ahigher instantaneous rate. In addition, four resolution parameters appear in Eq.(1.9) with the sixth
power of total impact to NAcc . Therefore, when experiments are designed, the resolution must be
in consideration as well as the muon statistics.

1.2.3

Background of physical origin

Another source of background events is the radiative muon decay (RMD) with the two neutrinos
carrying small energies. The detail of its differential branching ratio is given in Appendix.A, andthe effective branching ratio is shown in Fig.1.6. Here, the effective branching ratio is defined as
Z mµ /2

Z mµ /2
dEe

Ee

dEγ
Eγ

dB(µ → eν ν̄γ),
dEe dEγ(1.10)

E γ [MeV]

1.3. Result of previous experiment and its limitations

7

10-16

52.5

10-15
10-14

52
10-13
51.5
10-12
51
50.5

10-11

50
50

50.5

51

51.5

52

52.5
E e [MeV]

Figure 1.6: Effective branching ratio of the radiative decay when integrated over the phase space
above the corresponding Ee and Eγ values. Given an experimental resolution of σEe ∼ 100 keV
and σEγ ∼ 1 MeV, the effective branching ratio of RMD background events is bounded below
O(10−15 ).
which represents the integrated branching ratio of RMD decay above the corresponding Ee andEγ thresholds. With a consideration that RMD events with 52.8 MeV − Ee(γ) < σEe(γ) become
background, realistic resolution of σEe ∼ 100 keV and σEγ ∼ 1 MeV gives an effective branchingratio below O(10−15 ). This contribution is much smaller than the accidental backgrounds and thus
has only a minor impact on the µ → eγ search.

1.2.4

Sign of muon charge

Experimentally, muons are stopped in a material and their decay products are measured. Here, thechoice of negative muon (µ− ) is not favored because they form bound states in muonic atoms when
captured by a nucleus in material. This induces the nuclear recoil effect in muon decays and spoils
the simplicity of the two body µ → eγ kinematics. Therefore, the use of positive muons is requiredin an experiment, namely µ+ → e+ γ is searched specifically.

1.3

Result of previous experiment and its limitations

The history of cLFV search results with muon is shown in Fig.1.7, where the latest point for µ → eγ
is based on the publication in 2016 [1]. This is a result of a search with the data collected by the
MEG experiment during 2009–2013. As a result of this search with the sensitivity of 5.3 × 10−13 ,an upper limit was set to be B(µ → eγ) < 4.2 × 10−13 at 90 % confidence level.
The MEG experiment was constructed at πE5 beam line of Paul Scherrer Institute (PSI) to
exploit the world’s most intense DC muon beam (Fig.1.8). The beam muons were stopped inside
the target and then emitted the decay products. The trajectory of the decay positron was bent inside
the magnetic field and was measured by the drift chamber. The magnetic field was provided by
the Constant Bending Radius (COBRA) magnet, which produces a gradient magnetic field. The
timing of the positron was measured by the timing counters in combination with the time of flightChapter 1. Introduction to µ → eγ

90% C.L. Limit

History of CLFV experiments with muons
110−1µ → eγ10−2µN → eN10−3µ → 3eBranching ratio

8

MEG 2009
Obtained 90% UL

MEGA
90% UL sensitivity

−11

10

±1 σ
±2 σ

10−4
10−5

MEG 2011

10−6
10−7
10−8

10−12

−9

10

10−10

MEG 2013

10−11

MEG Final

−12

10

10−13
10−14
1940 1950 1960 1970 1980 1990 2000 2010 2020
Year

10−13
0

50

100

150

200
250
300
Accumulated DAQ days

Figure 1.7: History of cLFV searches with muons [34] (left), and results of the previous experiment[35] (right).

Figure 1.8: Schematic view of the MEG detector [36]. The red and blue lines are the simulated
positron and gamma trajectories in a signal event.

1.4. Previous MEG II upgrade works

9

Table 1.1: Resolutions and efficiencies achieved in the previous experiment and those achieved withthis work.
Resoluition

MEG performanceEe (keV)θe (mrad)φe (mrad)ze /ye (mm) coreEγ (%) (w<2 cm)/(w>2 cm)
uγ , vγ , wγ (mm)teγ (ps)

380
9.4
8.7
2.4/1.2
2.4/1.7
5/5/6
122

MEG II achieved value
with this work
90
7.2
4.1
2.0/0.7
2.0/1.8
2.5/2.5/5
84

Efficiency (%)
TriggerGamma-ray
Positron

≈ 99
63
30

∼ 80
62
67

(TOF) measurement by the drift chamber. The position, timing, and energy of the gamma-rays were
measured by the liquid Xenon scintillation detector (LXe). The performance achieved in the MEG
experiment is summarized in Tab.1.1.
As can be seen in Fig.1.7, the sensitivity improves rapidly at the beginning of an experimentwith the speed being proportional to the DAQ time. However, once a large statistics is accumulated
and the background becomes non-negligible, the improvement slows down and behaves as thesquare root of the DAQ time. At this stage, an experiment becomes less efficient in sensitivity gain
than at the beginning. In fact, the sensitivity improvement was limited between “MEG 2013” and“MEG Final” in Fig.1.7. Therefore, it was preferable to design an experiment with a smaller numberof backgrounds rather than continuous data taking.
The background suppression can be achieved by improving the detector resolution as discussed
in Sec.1.2. Ref.[34, 37] proposed an upgraded experiment with twice better resolution on average,
resulting in a background reduction by a factor of ∼ 30. The MEG experiment also suffered from thelimited efficiency (30 %) in the positron reconstruction, whose improvement is another key conceptof the upgrade. According to simulation studies and experiences in [1], these improvements canachieve an order of magnitude higher sensitivity. This improvement would never be feasible withthe continuous data taking of the MEG, which would take > 100 years to achieve this sensitivity
improvement.

1.4

Previous MEG II upgrade works

The MEG II experiment was first proposed in 2013 to realize the sensitivity improvement [37], andthe concept of the upgrade is listed below.
• The positron tracking detector is updated to the cylindrical drift chamber (CDCH),• The positron timing detector is updated to the pixelated timing counter (pTC),• The scintillation photon sensors for the liquid xenon detector (LXe) are replaced with novel
SiPMs on the inner face,Chapter 1. Introduction to µ → eγ

10

• The radiative decay counter (RDC) is added to reduce background gamma-rays from radiative
decay
Preparation works have been in place since the proposal, and the rest of this section reviews theprevious works. The original works of this thesis will be summarized later in Sec.1.5.
The CDCH was assembled during 2016–2017 and delivered to PSI in 2018 for commissioning[38]. The commissioning aimed to find optimal working conditions, which continued during 2018–2021. Despite the difficulties in achieving a stable operation in the high-intensity muon beam,a stable operation was finally achieved at the beginning of 2021. Another difficulty was a wirebreaking problem, which happened a few times till the end of 2019. The cause was identified to becorrosion, and all the potentially problematic wires were successfully removed. In the meantime,studies progressed about the positron track reconstruction based on detector simulation [39, 40].
This is because studies on data have been impossible because of the absence of a full readout.
Accordingly, the evaluated performance so far is also based on simulations, and its evaluation forthe data has been left to be investigated.
The pTC assembly was finished in 2017 and integrated into the MEG II detector. Its calibrationis already established [41] and the performance is presented in [40, 42]. The largest operational
concern is the radiation damage, which results in resolution degradation [43]. This can be mitigatedby installing a cooling system and its operation was also established by 2020 [40]. The estimatedpositron timing resolution was 35 ps before the radiation damage. It was also expected to become
41 ps after three years of damage accumulation [39].
The new SiPM development and its mass production for LXe was completed by 2016 [44],followed by installation works in 2017 [45]. SiPMs were aligned by laser scan and collimated X-ray
data taken 2017–2018 [46, 47]. Ref.[48] presents the position reconstruction and its resolution
evaluated on collimated 17.6 MeV gamma-rays. The energy reconstruction and calibration werealso developed in [45, 48], and the resolution is presented in Ref.[48]. The time calibration was
firstly developed in [45], and the resolution on π 0 → γγ data was firstly evaluated to be 81 ± 3 ps in[49]. This, however, included a large contribution of the vertex position spread of π 0 → γγ decay,
which was found to cause an additional 68 ps contribution [50, 51]. In addition, the above result
was obtained for a limited readout configuration. Therefore, the time resolution evaluation should
be revisited.
The RDC is planned to be installed in two places, both the downstream and upstream ends ofthe detector. The construction and installation of the downstream RDC were completed in 2016
[52, 53]. The upstream detector, on the other hand, is still under development, and the current status
can be found in [54, 55, 56]. The calibration and the performance evaluation of the downstream
detector were also established in [49].

1.5

Original works of this thesis

As the main theme of this thesis, this work establishes the analysis framework searching for µ → eγ
decay, which is discussed in Chap.6, and the result with the MEG II first year data is presented
in Chap.7. This work describes a wide range of studies for the µ → eγ analysis; formalismand evaluation of the probability density functions, the evaluation of the statistics of the dataset,validation of the analysis reliability, and estimation of the systematic uncertainties. This also
involves the optimization of the event selection discussed in Sec.4.4.3. Ref.[57] is the publication
based on these works.
In addition to the main analysis, this work is also devoted to the detector calibration and the
optimization of the reconstruction. This work includes most of the data-driven studies of the CDCH:

1.5. Original works of this thesis

11

the optimization of the track reconstruction (Sec.4.1.8, Sec.4.1.9), calibration (Sec.5.8, Sec.5.9),
and performance evaluation (Sec.5.13). As will be discussed in Sec.5.4.3, the LXe data taken withthe full readout revealed the necessity of further improvement of time calibration and optimizationof the time reconstruction. The LXe time resolution with the improved calibration is evaluatedboth with the π 0 → γγ data (Sec.5.4.4) and the radiative muon decay events (Sec.5.6.3). This also
includes works on the timing alignment between positrons and gamma-rays (Sec.5.6.1, Sec.5.6.2).
The alignment between the different sub-detectors is also discussed in Sec.5.11. Ref.[58] andRef.[59] are the publications based on these works.
With the above works, we achieved the detector performance presented in Tab.1.1. Given thisachieved performance, the projected sensitivity of the MEG II experiment is also calculated, whichis the first work based on a realistic scenario.

12

Chapter 1. Introduction to µ → eγ

Chapter 2
MEG II apparatus
Fig.2.1 shows a schematic of the MEG II apparatus. The muons come from the right in Fig.2.1 andthen stop inside the target. The decay positrons are bent inside the magnetic field produced by the
COBRA magnet. The CDCH measures their helical trajectories from the hits on wires, and the pTC
measures the timing from the hits on scintillation counters. The LXe detector absorbs the gammarays in the liquid Xenon, and the scintillation lights are measured by the photo-multiplier tubes(PMT) and silicon photo-multipliers (SiPM), which surround the whole volume. The downstream
RDC measures the energy and timing of 1 – 5MeV positrons from RMD with a set of plastic and
LYSO scintillation counters.
The global coordinate system of the MEG II is defined as the usual right-handed system, withthe origin located at the center of the COBRA magnet. The z-axis is in the downstream direction(the left direction in Fig.2.1), the y-axis is in the upright direction, and thus, the x-axis points to the
opposite of the LXe. The definition of r, θ and φ are also given aspr := x2 + y 2 + z 2 ,zθ := cos−1 p,
2
x + y2 + z2yφ := tan−1 .
x
In this thesis, r is often used to describe the detector geometry, and θ, φ is used for the emission
angle of positron and gamma.

2.1

Muon beam line

As discussed in Sec.1.2, a direct current (DC) positive muon beam is necessary for the experiment.
This is available at the πE5 beam line of PSI (Fig.2.2) with the highest DC intensity in the world.
The muon beam originates from protons accelerated to 590 MeV by the main ring cyclotron at PSI(Fig.2.3), where the frequency is 50.7 MHz, the bunch width is 0.3 ns, and the beam current is up to
2.2 mA. Protons are injected into a target of 4 cm-thick graphite to produce pions, and muons canbe obtained from the subsequent π + → µ+ νµ process. The beam is then generated by collecting themuons from the surface of the pion production target (within a few µm), and called surface muon
beam. This offers a good property of ∼ 28 MeV/c momentum with a good momentum-byte of 7 %(FWHM), which can later be stopped in a thin muon stopping target. The 50.7 MHz frequency of
the accelerator is much higher than the muon lifetime of 2.2 µs, and thus, the decay rate of muons(Rµ in Eq.(1.9)) behaves continuous as required. At 2.2 mA, the maximum available beam rate canbe set up to 108 µ+ /s, which is reduced to a smaller value in the experiment on demand.
13

Chapter 2. MEG II apparatus

14γ

Figure 2.1: A schematic of the MEG II apparatus [34]. Beam muons (coming from the right) are
stopped in the target (red) at the center. The decay positrons (blue helical line) are measured by the
CDCH (green) and the pTC (yellow). The decay gamma-rays are measured by the LXe detector(blue C-shaped detector). Background events are identified by the RDC placed at the left end.

=

γ γ −
γ

Figure 2.2: πE5 beam line in PSI [34]Figure 2.3: The main proton accelerator at PSI [60].

2.2. Muon stopping target

15

Figure 2.4: Muon stopping target for the MEG II experiment and a CCD monitoring camera [34, 61].
When projected in the global XY plane of the MEG II experiment, the ellipse shape results in acircle.
A Wien-filter (also called separator), two quadrupole triplet magnets, and a superconducting
beam transport solenoid (BTS) are installed to transport the beam from the πE5 to the detector. The
Wien filter and the triplet magnets separate positron contamination in the muon beam, which comesfrom the Michel decay from a muon or pair creation of a gamma-ray from π 0 → γγ process in thepion production target. Though the contamination initially accounts for eight times larger than themuons, the separation reduces it down to < 1 % of the muons. The BTS, with a 0.36 T magnetic
field, focuses the beam to the center of the muon stopping target. It also adjusts the momentum,with a 300 µm-thick Mylar degrader placed in the middle, to maximize the stopping efficiency inthe muon stopping target.

2.2

Muon stopping target

The muon beam is finally stopped in the muon stopping target at the center of the COBRA magnet.
The target is required to have a high muon stopping efficiency, which motivates a thicker material.
At the same time, it is required to be as thin as possible to suppress positrons’ multiple scattering,annihilation, and bremsstrahlung inside. A 174 µm-thick plastic scintillator is used as the stopping
target (Fig.2.4) and is placed with a 15° slant angle from the beam axis. This geometry results in along thickness seen from the muons while keeping a short path for the decay positrons. The target
is supported by a carbon frame, which ensures the mechanical stability of the target shape.
The target alignment is important according to the experiences in the MEG experiment, in which
the target uncertainties resulted in a 13 % degradation of the sensitivity. Four holes are on the longaxis and two holes are on the short axis of the ellipse, which can be used to align the tracking detector
and the target as will be presented later in Fig.5.11.2. Two CCD cameras are also introduced tomonitor the time variation of the position and the shape [61, 62], which make use of the dot-shaped
markers printed on the target.

2.3

Positron spectrometer

2.3.1

COBRA superconducting magnet

The COBRA magnet (Fig.2.5) is a superconducting magnet with a dedicated design for an efficientµ → eγ detection. This was developed and installed in the MEG experiment and is reused in
the MEG II. The magnetic field is provided by a composition of coils with three different radii:
the central coil, the gradient coil, and the end coils. It generates a gradient magnetic field of
0.5–1.7 T, which is stronger at the center and weaker at the ends (Fig.2.6). In comparison with

Chapter 2. MEG II apparatus

16

60

1.6
1.4

50

1.2
40

1
0.8

30

0.6
20

0.4
0.2

10

0
0
-100

-80

-60

-40

-20

0

20

40

60

80
100
Z coordinate (cm)Figure 2.6: Map of magnetic field inside the spectrometer [63].

B z (T)

Radius (cm)Figure 2.5: Schematic of the COBRA magnet (left for cross-section and middle for side view) andpicture (right) [36, 63].

2.3. Positron spectrometer

17

26.5mT11.5mT

Photon detector(LXe)

5mT

60.8mT

Compensation
coil

0.14T

0.32T
1.7T

End coil
Gradient coil
Central coilFigure 2.7: The compensated magnetic field inside the LXe [63].

Figure 2.8: Pictures of CDCH taken before the enclosure by the carbon support frame [34]. Left:
Full picture of the entire drift chamber. The hyperbolic shape is visible. Right: A zoom-up picture
showing the wire crossing in the stereo geometry.
a uniform magnetic field, this is advantageous because the radius (projected on XY plane) of
the decay positrons does not strongly depend on the θ-emission angle of positrons. In addition,
this configuration sweeps out decay positrons more quickly than a uniform field. Therefore, only
positrons with the energy of interest can leave a moderate number of hits on the detector, which isa good feature in association with the rate capability.
Co-existence with the LXe detector is another important requirement in its design. The material
budget in the direction of the LXe detector is kept as small as 0.197 X0 , in order not to lose the
gamma-ray detection efficiency. In addition, the compensation coil is introduced at the detector’send, which cancels the stray field in the LXe detector to keep the field strength to be < 5 mT
(Fig.2.7). Thanks to this compensation, the PMTs of the LXe detector can be operational.

2.3.2

Cylindrical drift chamber

A single-volume wire drift chamber is developed for the MEG II experiment (Fig.2.8), with theoriginal inspiration from the KLOE experiment [64]. It has a geometry of 191 cm in the length,

Chapter 2. MEG II apparatus

18

Y [cm]-14
Z = +0.0 [cm] **

-16

(a)

-18

(b)
-20
-22

(c)

-24
-26

(a)
-28
-30
-8

-6

-4

-2

0

2

4

6

8
X [cm]

Figure 2.9: Drift cell configuration in the full layer[37]. Red and blue markers correspond to ± stereo
angles (see Fig.2.10). The open circles are sense
wires and the filled circles are field wires.

(c)𝜃stereo

191 cmℓ = 𝑟wire(b)

Figure 2.10: Illustration of the wiring
scheme in the stereo geometry.

and 17 cm and 29 cm for the inner and outer radius (i.e. 17 cm < rCDCH < 29 cm). This uses
∼ 2000 sense wires and ∼ 10000 field wires in total, which forms a nine-layer structure eachwith 192 drift cells of 5.8 – 8.7 cm size (Fig.2.9). The drift cells are defined by sense wires ina stereo configuration (Fig.2.10), with the θstereo ranging from 6.0° (the innermost layer) to 8.5°
(the outermost layer). Two wires from neighboring layers have opposite directions, which enablesa precise track reconstruction in the z-coordinate. This configuration results in a hyperbolic shapeseen from the side, as is visible in Fig.2.8.
The sense wires are made of 20 µm diameter gold plated tungsten, and the field wires are
made of 40 µm diameter silver plated aluminum. A gas mixture of He/iC4 H10 /O2 /2-propanol witheach fraction of 88.5 %/9.8 %/0.5 %/1.2 % is used inside the active volume. The whole volume is
enclosed by an inner 20 µm-thick one-side aluminized Mylar foil and an outer support structure madeof 2 mm-thick carbon fiber. The whole material of the CDCH is highly transparent and positrons
pass 1.58 × 10−3 X0 in a single turn. This low-mass design reduces the multiple scattering and thegeneration of gamma-rays.
Only 2/3 of the sense wires at the bottom are read out because it is enough to cover the full
acceptance defined by the C-shaped LXe detector. A low-noise high-bandwidth front-end amplifierelectronics is specially designed and connected to both ends of the wires (Fig.2.11). The highvoltage, on the other hand, is applied to all of the sense wires (full 0 < φ < 2π) to have ahomogeneous and symmetric electric field for the gas amplification.

2.3.3

Pixelated timing counter

The pixelated timing counter, a group of highly segmented plastic scintillation counters, is developed
for the MEG II experiment for the positron timing measurement. It is placed just outside the CDCH
with a minimum material in-between, which suppresses multiple scattering between the CDCH andthe pTC. This ensures a high matching efficiency between the reconstructed track in the CDCH andthe hits on the pTC, which was low and resulted in the largest inefficiency in the MEG experiment.

2.3. Positron spectrometer

19

Figure 2.11: Circuit diagram of amplifier electronics for CDCH [34].

Figure 2.12: Picture (left) and schematic (right) of downstream pTC containing 256 scintillation
counters [34, 48].

The pTC is divided into two sectors, downstream and upstream, with a geometrical separation by
the central coil of the COBRA magnet. Fig.2.12 shows the picture and schematic of the downstream
part, which hosts 256 scintillation counters. The upstream also hosts 256 counters, and 512 counters
are used in total. Each counter is tilted by 45° to the beam, in order to be perpendicular to theincoming positron tracks. They are arranged to maximize the hit density, i.e. to maximize thenumber of hits, so that a good overall resolution can be achieved by a combination of multiple hits.
Fig.2.13 shows a picture of scintillation counters. Each counter is made of 120 × 40/50 × 5 mm3BC422 (Saint-Gobain) scintillator with both ends coupled to six SiPMs from AdvanSiD (ASDNUV3S-P-High-Gain). The six SiPMs are connected in series to have a reduced capacitance in thereadout, which results in a good resolution thanks to the fast response. Each counter is wrapped bya reflector film to have a high scintillation light collection efficiency.
An optical fibers system is introduced to a part of the counters as a time calibration tool (Fig.2.14),
which is complementary to another method using the reconstructed positron tracks (described laterin Sec.5.5). Pulsed laser light is divided by the optical splitters to generate a synchronized signal,
and it is then injected into the counters through optical fibers (Fig.2.13). With the optical length of
the system measured in advance, this can be exploited to calibrate the counter time offsets.
The local coordinate is defined to express the hit position in the counter as shown in Fig.2.15.
The w axis is defined in the direction between the two readout ends. The v axis is defined so that it
points to the direction of decreasing r in the global MEG II coordinate.

Chapter 2. MEG II apparatus

20

Figure 2.13: Picture for a 4 cm counter after wrapping (left) and a 5 cm counter before wrappingwith optical fiber introduced from the bottom (right) [34].

Figure 2.14: Schematic and picture of laser-based TC calibration system [41].

𝑣𝑤

Figure 2.15: The local coordinate system on each counter of the pTC.

2.4. LXe gamma-ray detector

21

Figure 2.16: Scnitillation photon sensor layout (left) and picture (right) of the LXe detector [34, 45].

2.4

LXe gamma-ray detector

The gamma-ray detector uses ∼ 900 L liquid Xenon with the scintillation readout by SiPMs and
PMTs (Fig.2.16). As the wavelength of the scintillation light lies in the vacuum ultraviolet (VUV)
regime, special scintillation photosensors are used to be sensitive to VUV light. The photosensorsof choice are VUV-MPPC (SiPMs) and R9869 (PMTs), both produced by Hamamatsu Photonics.
The major upgrade from the MEG experiment is that the sensors on the inner face (originally
PMTs) were replaced by VUV-MPPCs, while the PMTs in the other faces are reused from the MEG
experiment.
The LXe detector is placed just outside the COBRA magnet and has a C-shaped geometry
surrounded by six faces: inner, outer, upstream, downstream, top, and bottom face. The inner face
has rin = 64.97 cm, and the local coordinate of the LXe detector (Fig.2.17) is defined as 1u=z
 yv = rin · tan−1 −x
p22
w = x + y − rin .

2.4.1

(2.1)

Liquid xenon

When a gamma-ray around 52.8 MeV interacts with liquid xenon, an electromagnetic shower isformed inside. The elements contained in the showers ionize or excite the xenon atoms, which isfollowed by scintillation light emissions. The studies on its highly complex mechanism were initiatedin the 1970s [65, 66, 67, 68], and a comprehensive review is given in [69]. Two different processes

1Technical caution: This coordinate definition is for this analysis. It is planned to be changed in future publications.

Chapter 2. MEG II apparatus

22

top

100
80

downstream

upstream

100

60

80

40

60

20

40

0

20

v-axis0−20

u-axisw-axis
inner

−40
−60
−80
−100
−100
z

0

100

bottom

200

300
outer

Figure 2.17: The local coordinate definition of the LXe detector [49].
have been identified, a self-trapping process and a recombination process. The self-trapping is aprocess where excited xenon atoms are trapped,Xe∗ + Xe + Xe → Xe∗2 + Xe
Xe∗2 → 2Xe + hν,(2.2)

where hν has a VUV wavelength. There are singlet state and triplet state for the Xe∗2 , and they
respectively have decay time of 4.2 ns and 22.2 ns. The recombination is a process associated with
the ionized xenon,
Xe+ + Xe → Xe+2+−
Xe2 + e → Xe∗∗ + Xe
Xe∗∗ → Xe∗ + heatXe∗ + Xe + Xe → Xe∗2 + Xe
Xe∗2 → 2Xe + hν,(2.3)

where the decay time of this process is 45 ns. The authors of [68] found that the 45 ns decay timecomponent dominates when the liquid xenon is ionized by gamma-rays. This is not the case whenthe energy deposit density is larger, where 4 ns and 22 ns components become prominent.
As a scintillator, liquid xenon shows good properties for µ → eγ detection. Thanks to its large
atomic number and high density (2.98 g/cm2 ), it has a high stopping power of X0 = 2.77 cm.
This results in a high detection efficiency of gamma-rays. It also has a high light yield of
46 000 photon/MeV, and a good resolution can be expected. Its fast response with the decay
time of 45 ns is suitable in a high-rate environment, where the pileup becomes a severe problem.
In addition, the liquid material contributes to a higher uniformity of the scintillation response thancrystals.
In addition to the monetary cost, there exist three technical difficulties in the use of liquid xenon.
The first difficulty is the narrow temperature range (161–169 K) of the liquid state, which poses achallenge for maintaining such precise temperature control in a 900 L detector. The second difficulty is that high-quality liquid purification is demanded because the scintillation light can be easily

2.4. LXe gamma-ray detector

23

Figure 2.18: Picture and sketch of VUV-MPPC [44].
K

DY1

R1

DY2

R2

DY3

R3

DY4

R4

DY5

R5

DY6

R6

DY7

R7

DY8

R8

DY9

R9

DY10

R10

DY11

P

DY12

R17

R14

R15

R16

R11

R12

R13

R20

Z1

C4

R21

R19

Z2
C5

C1

R1,R12,R13 : 2MΩ 1%, 1/8W
R2-11 :
1MΩ 1%, 1/8W
R14-R16 :51Ω 1%, 1/8W
R17 :10kΩ 1%, 1/8W
R18 :10kΩ 1%, 1/4W
R19-R21 :100kΩ 1%, 1/8W

C1-C3 :
C4, C5 :
Z1 :
Z2 :

C2

0.022µF, 1kV
0.01µF, 2kV

RD68S
RD82S

C3

R18

+HV
(RG-188A/U)

SignalOutput(RG-196A/U)

Figure 2.19: Picture of PMT (left) and diagram (right) [73].
absorbed by contaminations such as oxygen and water. The final difficulty is the wavelength of thescintillation light lying in the VUV region (175 nm), which requires the scintillation photon sensors
to be VUV-sensitive. The MEG experiment has already solved the first two difficulties [70, 71, 72],
and the solution to the third difficulty is described in the next section.

2.4.2

Scintillation photon sensor

For the scintillation photon readout in the liquid xenon, the photosensors are required to be sensitive
to VUV-light in 165 K and 5 mT. These are fulfilled by specially developed VUV-MPPCs and R9869
PMTs, and 4092 VUV-MPPCs (Fig.2.18) are used on the inner face and 668 PMTs (Fig.2.19) areused in total on the other faces.
The PMTs use a VUV-sensitive photocathode made of Bialkali (K-Cs-Sb) and a quartz windowof ∼75 % transparency for VUV light. As a result, a good quantum efficiency of 16 % was achievedfor VUV light [34]. With an operation voltage of ∼ 900 V, an 106 gain can be obtained. Still, they
have a problem with their round shape with the diameter of 46 mm. This results in spatial gaps
between adjacent PMTs, and the scintillation light cannot be uniformly readout. The non-uniformity
results in a non-uniform light collection, which has a large impact on the energy and the position
resolution. This is crucial on the inner face because most of gamma-rays are converted to a showerin the shallow region. Therefore, the PMTs are used only for the other five faces.
The VUV-MPPCs have the p-on-n structure as a semiconductor to deal with the short absorption
length of VUV in silicon (Fig.2.20). To further suppress the VUV absorption, the protection layer isalso optimized; it is made of VUV transparent quartz instead of the epoxy resin of the conventional
SiPMs. As a result, the VUV-MPPCs have a sufficient photon detection efficiency (PDE) of 20 %.
Though a highly granular and uniform readout can be achieved thanks to its square shape, its use
results in an increase in the number of readout channels. Four sensor chips of 6 × 6 mm2 , hosted in
one package (Fig.2.18), are readout in series to mitigate this problem, which gives a 12 × 12 mm2coverage by each readout channel. This readout scheme is also advantageous because the smallerChapter 2. MEG II apparatus

24

Insulation layer

Depletion layer

Incident light

Positive electrode(anode)Negative electrode(cathode)

ShortwavelengthLong
wavelength
P-layer

N N+

N-layer
KPD

Figure 2.20: P-on-N structure of MPPC [74].
Top

Outer
old LED
from MEG
new LED
from MEG II

DS

BottomUSz axisFigure 2.21: LED light sources installed on the outer and lateral faces [48].
pulse width with the reduced capacitance helps to efficiently identify pileup pulses. Though SiPMs
are known to have non-linearity, its impact on the 52.8 MeV detection is not expected to be severe
thanks to the sufficient number of pixels (∼ 13000 on each 6 × 6 mm2 area).
Both sensors have operational difficulties and high calibration demands for long-term use in ahigh radiation environment. The PMT gain shows a decrease with a deterioration of dynodes in an
accumulated multiplication. This demands a good and constant calibration of gain and a periodical
adjustment of operation voltage. The MPPCs, on the other hand, are damaged during the operationin the muon beam, which results in a decrease of PDE [48]. This can be recovered by the thermal
annealing method, which was completed at the beginning of 2022 and 2023.

2.4.3Internal calibration sources — LED and alpha ray —Blue light LED sources, Toyoda Gosei E1L49–3B1A–02 [75] and Kingbright KA-3021QBS-D[76], are installed inside the detector to calibrate the gain of the sensors. They are placed on theouter and lateral faces as shown in Fig.2.21. A Teflon sheet covers the LEDs to diffuse the light,
which improves the overall stability of the light because the fluctuation of the light intensity reduces
when the LEDs flash strongly.
Alpha-ray sources of 241 Am are also installed to calibrate the light yield and the PDE of the
MPPCs. Five gold-plated tungsten wires of 100 µm diameter are stretched between the upstream
and downstream faces, each with five Am sources at 12.4 cm intervals (Fig.2.22) [77]. The activityof each Am source is ∼ 200 Bq, which is a negligible rate in the data-taking with the muon beam.

2.4. LXe gamma-ray detector

25

Top

Outer
α source
4

3

2

1

0

Wire0
9

8

7

6

5

Wire1
14 13 12 11 10

Wire2
19 18 17 16 15

Wire3
24 23 22 21 20

Wire4

DS

BottomUSz axisFigure 2.22: α-ray sources mounted in the LXe detector [48].

Figure 2.23: Layout of the apparatus for 17.6 MeV calibration gamma-ray data-taking [36]. The
CW accelerator downstream of the MEG II detector injects protons to Li2 B4 O7 target at the centerof the detector.

2.4.4 17.6 MeV gamma-ray calibration with proton beam and Li2 B4 O7 target
When 440 keV protons are injected to a Li2 B4 O7 target, gamma-rays are emitted from the 73 Li(p, γ)84 Be
reaction with a spectrum sharply peaked around 17.6 MeV. Protons are accelerated by Cockcroft–Walton (CW) and injected from the downstream to the target at the COBRA center (Fig.2.23).
A control system is implemented for the insertion and extraction of the muon stopping target andthe CW equipment. This allows the whole operation to be completed within a few minutes, andthus, it does not heavily interfere with the physics data-taking.

2.4.5 55 MeV gamma-ray calibration with π − beam and liquid hydrogentargetWhen π − beam is injected into a liquid hydrogen target and stopped inside, π 0 are generated by
π − + p → π 0 + n reaction — called charge exchange reaction (CEX) —, which is immediately (π 0lifetime is 8.5 × 10−17 s) followed by π 0 → γγ. Here, the energy of the emitted π 0 isEπ0 =(mπ− + mp )2 + m2π0 − m2n= 137.85 MeV,
2(mπ− + mp )

(2.4)

Chapter 2. MEG II apparatus

26

PMT
LXe

ɶ
COBRA

MPPC

Target insertion tube

ʋ- beamɶ

LH2 cellPre-shower counterz

BGO calorimeteryx

Figure 2.24: Setup of 55 MeV calibration data-taking [48].
where mπ− = 139.57 MeV is the π − mass, mπ0 = 134.97 MeV is the π 0 mass, mp = 938.27 MeV
is the proton mass, and mn = 939.57 MeV is the neutron mass. Given the boosted π 0 , the opening
angle of the gammas (Θγγ ) is correlated with the energy,
s
Eπ20
m2π0
Eπ0
Eγ =
±
−
.
(2.5)
2
4
2(1 − cos Θγγ )In particular, when the decay gammas are completely back-to-back, the low energy gamma has
55 MeV and the high energy one has 83 MeV. This quasi-monochromatic gamma-ray close to thesignal gamma energy is also used to calibrate the energy and evaluate the detector response.
As a by-product of this data-taking, a radiative capture reaction, π − + p → γ + n, is also
expected with the cross-section being [78]σ(π − + p → π 0 + n)
= 1.533 ± 0.021,σ(π − + p → γ + n)

(2.6)

which is called “Panofsky ratio”. The 129 MeV gamma-rays from this reaction can be used tounderstand the linearity of the gamma-ray energy measurement.
The setup for the calibration data-taking is shown in Fig.2.24. The liquid hydrogen target isinserted from the downstream together with a cooling circuit. As we are interested in events close to
the complete back-to-back kinematics, the opposite side is covered by another gamma-ray detector;
the BGO calorimeter + pre-shower timing counter (Fig.2.25). The BGO calorimeter consists of
16 BGO crystals coupled to PMTs. The pre-shower counter consists of two plastic scintillation
counters coupled to SiPMs.
As a strong constraint on its calibration use, it takes a few days to install the special target
and beam systems, which interrupts the physics data-taking. This gives the largest limitation to
55 MeV gamma calibration; we only have once-a-year opportunities with a limited time, and thus,
the statistics are also limited.

2.5

Downstream radiative decay counter

The radiative decay counter is newly developed for the MEG II, which aims to identify backgroundgamma-rays from the RMD. Most RMD gamma-rays close to 52.8 MeV are accompanied by a

2.6. Electronics and data acquisition system

27
第

図

章

液体キセノンガンマ線検出器時間分解能評価原理

プレシャワーカウンター

Figure 2.25: BGO crystal (left) + pre-shower timing counter (right) [35, 50].

Figure 2.26: Downstream radiative decay
counterat the endcap [34].
図
検出器前面に取り付けられたプレシャワーカウンターターゲットは安定ではなかった。また、ターゲットは完全に液体水素で充填されている状態ではなかった。

coincident low energy positron (typically 1 – 5 MeV), which flies towards the detector ends along
the beam axis in the COBRA field (Fig.2.26). About half of the RMD-associated low energy
positrons are emitted in the downstream direction, and ∼ 90 % acceptance for them can be covered
with a ∼ 20 cm size detector. Efficient identification of the RMD-associated low-energy positrons
can be realized by measuring both the hit timing and energy (Fig.2.27).
The downstream RDC has two parts, one for the timing measurement part and the other for thecalorimetry (Fig.2.28). The timing measurement part consists of 12 plastic scintillation counters(BC418, Saint Gobain) of different sizes; the thickness being 5 mm for all the counters, the height
being 1 cm (the central six counters) or 2 cm (the outer counters), and the width is 7, 11, 15, and 19 cm(wider for inner counters). Both ends of the 1 cm (2 cm) high counters are coupled to two (three)
SiPMs connected in series, where all the SiPMs are from Hamamatsu Photonics (S13360-3050PE).
The calorimeter is located behind the plastic counters and hosts 76 LYSO crystals (2 × 2 × 2 cm),
each of which is coupled to a SiPM from Hamamatsu Photonics (S12572-025P).

2.6

Electronics and data acquisition system

2.6.1

Integrated waveform data-taking system

In order to have a good resolution in the high pileup environment, the accessibility of the raw
waveform is desired in the reconstruction. For this purpose, the waveform of all the detectors isdigitized by the Domino Ring Sampler (DRS), which has a wide bandwidth of ∼ 1 GHz and ahigh sampling speed up to 5 GSPS [79, 80]. The DRS makes use of the switched capacitor arrays
technique, where one DRS channel contains 1024 sampling cells. The sampling signal is generated
by the inverter chain called “domino wave” (Fig.2.29), which is stopped when a trigger signal isChapter 2. MEG II apparatus

28

fraction (%)

1

10−1
10−2

10−1

10−3
10−2

10−4

10

10−5

−3
−9

×10
0

10

20

30

40

T RDC - T γ (s)

10

20

30

40
50
Energy (MeV)

Figure 2.27: Timing (left) and energy (right) distribution of RDC hits [34]. The red (blue) line
shows the distribution for the RMD-associated accidental gamma-rays (signal gamma-rays). In thetiming plot, the time difference between the RDC hit and the LXe hit is shown. Here, the 4 ns offset
corresponds to the TOF from the muon stopping target to the RDC.

Figure 2.28: The picture of the downstream RDC components [34].

2.6. Electronics and data acquisition system

0.5-2 ns

29

Rotating
singalInverter “Domino” chain

Input

Output40 MHz

Shift Register

Figure 2.29: Schematic explaining the principle of DRS (left) and picture (right) [79, 80].
MCX
TRG IN
Temperature sensors 1-Wire®

MCX
TRG OUT

previous channel

Attn.
:2

Input

Attn.
:1,2,4

x10

DAC
DAC

Comp
C

PHY

Trigger

offsetDAC

Shaper

x10

x2

DRS4

ADC

XTAL
calibrationPM

next channel

PLL
(LMK)

RAM

FPGA

Input

... 2 x 8 channels ...

RJ45

PoE optional
+5V
+3.3V
+2.5V
+1.8V
+1.2V
-2.5V

x2

DRS4

DC

DC in
24 V

DC
power on

DAC

24 V

RGB

BUS

SPI

ADC
SPI

DAC

HV Piggy-back
58-63V58-63V

5 V DAC

58 V

Comp
C

Trigger

Temp

PROM

CockroftWalton
TR BUS CLK

DC
+24V

DC

Osc.

En

HV
ON

LVDS

MCX
CLK INMCXCLK OUT

Figure 2.30: Structure of circuits in WaveDREAM board [34].
issued. When the sampling signal stops, the cell array keeps the waveform at the most recent 1024
sampling points, which is then read out by the shift resistor and digitized. One DRS chip shown in
Fig.2.29 contains eight digitizer channels.
Another requirement for the electronics is to be housed in a limited space. This is highly demanding with the necessity of pre-amplifiers and high-voltage application systems for the SiPMs aswell as the trigger system and the waveform digitizers. A new DAQ module — called WaveDREAM
Board (WDB) — is developed for the MEG II experiment (Fig.2.30), which integrates the above
functionalities. One WDB has two DRS chips and can host 16 readout channels in total. The DRSs
are operated in a way that it continuously samples the input waveform and sends its copy to the
output. The DRS outputs are connected to the ADCs, which continuously digitize the waveform
at 80 MSPS. The ADC outputs are fed to the FPGA to finally build the trigger logic. The highvoltage for the SiPMs (up to ∼ 240 V) is provided by the Cockcroft-Walton multipliers, which canbe further tuned on each channel by the 5 V DAC. The amplifiers before the DRS are combined with
programmable attenuators and a programmable pole-zero cancellation circuit, giving flexibility inthe gain (from ×0.5 to ×100) and the shaping.
The full communication between the ∼ 9000 readout channels are designed on a crate basis(Fig.2.31), whose global control is taken by a DAQ computer hosting the MIDAS system [81]. A
WaveDream crate hosts 16 WDBs, a trigger concentrator board (TCB), and a data concentrator

Chapter 2. MEG II apparatus

30

Figure 2.31: A WaveDREAM crate [34].
board (DCB). The DCBs are responsible for the configuration of boards, the distribution of themaster clock and trigger signals, merging and formatting the waveform data, and the interface to the
DAQ computer. The TCBs gather the low (board) level information, perform online reconstruction,and generate trigger signals.

2.6.2

Trigger system

The trigger system has a strong constraint on the latency, which is determined by the depth of
the waveform storage in DRS. Its typical value is a few hundred nanoseconds, which is 1024 (thenumber of cells) divided by the DRS sampling speed. Therefore, the trigger logic must be built
within such a limitation. In particular, the CDCH information usually cannot be included in the
logic with its drift time being up to ∼ 300 ns.
The logic for the µ → eγ is thus implemented based on the LXe and the pTC. Here, only Eγ , teγ ,and Θeγ can be reconstructed online. The online gamma energy reconstruction uses a weighted sumof the waveform of scintillation photon sensors. Here, the weights are calculated from the sensor
gain and PDE, and the overall energy scale is calculated from the 17.6 MeV gamma-ray calibration
data. The online teγ reconstruction uses the hit time difference between the pTC and the LXe,
whose resolution is dominated by the positron time of flight (TOF) due to the absence of the fulltrack reconstruction. It also received a non-negligible contribution from the tγ time walk effect inthe 2021 run, which has been mitigated in the 2022 run. This is because the online tγ reconstruction
relies on MPPCs (PMTs) in the 2021 (2022) run, where the time walk effect is stronger for the
MPPC waveform. The online Θeγ reconstruction uses the first hit counter of the pTC and the peak
position in the scintillation light distribution on the LXe inner face. The trigger judgment relies
on a look-up table created from a detector simulation for the signal events, which is tight enoughthanks to the highly segmented design of the pTC. The latency of the full trigger is ∼ 600 ns andthe sampling speed of the DRS is 1.4 GSPS for the LXe, pTC, and RDC detector and 1.2 GSPS for
the CDCH.
During the experiment, calibration data-taking is also required in addition to the main physics
data-taking. Using the outputs of the dedicated calibration detectors, several different logics arealso prepared for them as briefly summarized in Tab.2.1.

2.7

Data management

2.7.1

Reduction of waveform data size

When digitized waveform data is recorded by the DAQ system, the data size reduction is necessary
for two reasons. The first one is to reduce the data rate to keep it below the data bandwidth of

2.7. Data management

31

Table 2.1: List of trigger settings.

[mV]

Logic
Aim
Cuts on Eγ , teγ , ΘeγMain physics data for µ → eγ
Self trigger on the pTC
Michel positron data
LXe hit ∧ BGO + pre-shower counter hit π 0 → γγ calibration
Inputs from LXe LED
Sensor calibration
Inputs from pTC laserTime offset calibration of each tile
α particle identification with waveform
α-ray calibration
Self trigger on RDC LYSO
Energy scale calibration for RDC
Random timing
Electronics noise calibration
100

Region of interest

Reduced

50
0
−50
−100
−150
−200
−250
−300

−600

−400

−200

0
[ns]

Figure 2.32: Recorded waveform after region of interest cut. The first half of the window is theregion of interest and the waveform is stored, whereas the waveform in the later half of the window
is dropped.
the whole system. In case of any overflow, some of the triggered events are not recorded and they
become inefficient. Another purpose is to reduce the disk consumption of the storage system, which
finally is required to store years of the MEG II dataset.
In addition to the file compression with bzip2, the data size is reduced by dropping some
information from the recorded waveform. This scheme is optimized for each detector by combining
three methods: zero-suppression, region of interest cut, and rebinning. The zero-suppression applies
a threshold cut on the waveform amplitude, and channels below the threshold are dropped fromthe output file. The region of interest cut keeps the voltage information only in the important time
region (Fig.2.32). The rebinning re-samples the waveform to have a sparse recording, where only
averages of neighboring sampling points are stored (Fig.2.33). Here, the number of sampling points
can be changed dynamically depending on the amplitude of the channel waveform.

2.7.2

Data storage and blinding scheme

During the offline reconstruction, the raw waveform data files are re-formatted into a ROOT [82]format which contains both the waveform information and the reconstructed kinematics. The
blinding is also applied here according to the criteria described later in Sec.6.2, which is to avoid
psychological biases in the analysis [83]. After the blinding, the blinded events and the open events
are written into different re-formatted files, and users are deprived of permission to access either

Chapter 2. MEG II apparatus

Amplitude[mV]

32

0
− 100
− 200
− 300
− 400
− 500
− 600

Amplitude > 400 mV

Amplitude[mV]

− 700
− 800
20

−

−

−

−

−

−

−

0
− 20
− 40
− 60
− 80
− 100
− 120
− 140

Amplitude > 50 mV

Amplitude[mV]

− 160
− 180
5

−

0
−5
− 10
− 15
− 20

Amplitude < 50 mV

Amplitude[mV]

− 25
− 30

1.5
1
0.5
0
− 0.5
−1
− 1.5
−2
− 2.5

−

Amplitude < 10 mV
− 600

− 400

− 200

Time[ns]

Figure 2.33: Recorded waveform for the LXe detector after rebinning [48]. The waveform tailregion (after the vertical red dashed line) is rebinned, while the leading edge is fully preserved
to keep a good time resolution. The number of sampling points in the rebinning is dynamicallychanged depending on the waveform amplitude.

2.8. Detector simulation33

the blinded files or the original format waveform files. Data preselection is also applied here to
reduce uninteresting events in the re-formatted files. This is based on teγ and Eγ with a similar
logic as that used for the online trigger judgment, but makes use of the higher resolution in theoffline reconstruction. Once the re-formatting ends, all the original files are moved to tape storage
and fully preserved there.

2.8

Detector simulation

The final output of the detector simulation is a fully simulated waveform similar to the DRS data inthe experiment, which ensures compatibility with the reconstruction framework. The simulation canbe separated into three parts; the muon decay generation, the simulation of the particle transportationand ionization, and the simulation of pileup and waveform digitization.
In the event generation step, both the SM decay and the µ+ → e+ γ decay are simulated startingfrom muons. The muon polarization is also considered in the event generation according to themeasured one in the MEG experiment [84]. Here, the angular distribution of µ+ → e+ γ decay
is assumed to be isotropic, namely AR = AL is assumed in Eq.(1.3) notation. Two different
approaches are adopted regarding the simulation for the muon decay positions,• Simulation starts from muons already stopped in the target, and the position is based on anassumption about the stopping position distribution,• Simulation starts from muons during the transportation to the target in the beamline.
The latter case can simulate muons that decay in flight or the muons that are not stopped in thetarget but at the expense of the CPU time consumption. Though the choice of the approach changes
case by case, simulations for background events generally adopt the latter approach.
The particle transportation and ionization are simulated with the Monte Carlo (MC) method
based on the Geant4 package [85], which is a particle simulation framework to evaluate the particle
trajectories and the energy deposit along them. This incorporates all the relevant electromagnetic
reactions; pair creation and annihilation, bremsstrahlung, Compton scattering, photoelectric effect,
ionization, and multiple scattering. After the detector hits are simulated, the detector’s responseto them is simulated in dedicated codes. The LXe simulation includes the scintillation photon
absorption, scattering, and reflection in the detector material. The CDCH uses the Garfield++ [86]to simulate the gas ionization process in the drift cells.
The simulation of the full experiment requires simulations on the overlay of particle activities
in a given event. This reads the results of the ionization simulation in the previous step and mixes
them with the rate determined by the assumption on the beam. The timings of the mixed events are
simulated according to the exponential decay time distribution. The waveform in the mixed events
is simulated with a reliance on waveform templates, which are given as the impulse responses ofthe detectors. The templates of the pTC, LXe, and RDC are taken from the data samples while thetemplate of the CDCH is based on SPICE [87] simulations. The noise is then added to the simulated
waveform to include the observed noise situation in the data-taking. In the final step, the waveform
is digitized including the sampling and the discretization of the DRS.

34

Chapter 2. MEG II apparatus

Chapter 3
Data taking
All the MEG II detector hardware was delivered to PSI by March 2021, followed by a commissioning
period that lasted till the end of September. During the commissioning, a stable operation of thedetectors was achieved and the signal check was also completed. The physics data-taking started
at the end of September and continued till the middle of November. The π 0 → γγ calibration
data-taking system was then installed in late November, and its data-taking started at the beginningof December. During both the physics and π 0 → γγ data-taking period, there were also daily shortcalibration routines. This chapter describes the condition of the data-taking activities relevant tothis thesis.

3.1

Physics data taking

The physics data was taken during September 25th – November 18th, 2021, which is the first MEG II
dataset. The total live time of the 2021 physics data-taking was 2.9 × 106 s. This is 63 % of thetotal elapsed time during the whole period, and its day-by-day breakdown is shown in Fig.3.1. The
inefficiency came from the dedicated calibration data-taking — the difference between the black
hatched graph and the red graph in Fig.3.1 corresponds to the fraction of calibration DAQ time (seeSec.3.3) — and the deadtime from the transitions related to the changes in recorded data files. In
addition, there were a few occasions of short (up to 3 days) accelerator shutdowns, which entirelyinterrupted the physics data-taking.
During the physics data-taking, the beam intensity was risen step-by-step; we started from alower intensity to validate a stable detector operation and then moved to a higher value. Here,the major concerns were the CDCH discharge and the PDE of the LXe MPPCs. As the CDCH
discharge problem was expected to become severe at a higher intensity, detector conditioning works
were performed carefully at lower intensities. The minimum requirement on the PDE value of the
LXe MPPCs to keep a high energy resolution is 4 % as discussed in [45]. The PDE decrease speed
was measured during the 3 × 107 µ/s period, and finally it was concluded that the experiment cansurvive the 2021 data-taking even with 5 × 107 µ/s intensity. As a result, four different muon beam
intensities were adopted as summarized in Tab.3.1. The time evolution of the number of muons
stopped in the target is shown in Fig.3.2, where the final value is 1.04 × 1014 .

3.2

Detector condition

The CDCH was stably operated during the whole physics data taking with few occasions of discharges. Only ∼ 3 % of all the CDCH readouts were found not active (Fig.3.3), which does not
35

Chapter 3. Data taking

36

Efficiency

Efficiency of time use (every 24.0 hours)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
All DAQ fraction

0.2
0.1
0

Physics DAQ fraction

Sep/30

Oct/07

Oct/14

Oct/21

Oct/28Nov/04

Nov/11 Nov/18
date (UTC)

Figure 3.1: Efficiency of time use during the 2021 physics data-taking.

Table 3.1: Beam rate during 2021 physics data taking.
Period
Sep/25 – Oct/15
Oct/15 – Oct/28
Oct/28 – Nov/2
Nov/2 – Nov/10
Nov/10 – Nov/18

Number of stopped muon

×1012

Nµ

Beam rate
3 × 107 µ/s
2 × 107 µ/s
3 × 107 µ/s
4 × 107 µ/s
5 × 107 µ/s

100
80
60
40
20
30/Sep07/Oct14/Oct21/Oct28/Oct04/Nov

11/Nov

Figure 3.2: The number of muons stopped inside the target.

18/Novdate (UTC)3.3. Daily calibration

37

100
80
60
40
20
0
−20
−40
−60
−80
−100
−100

Figure 3.3: Map of active drift cells of
CDCH in 2021.

0

100

200

Figure 3.4: Map of active channels of LXe in 2021
[49].

have any critical impact on the performance. Some of them come from detachments of high-voltage
application systems and some others from problems in the front-end amplifier.
In the LXe detector, the liquid xenon was filled to y = 83 cm level. This is below the position ofsome PMTs in the top region, which means that such PMTs could not efficiently collect scintillation
lights. With a xenon purification system, the light yield was stable within a few percent [48]. The
readout of the LXe detector also had a few problematic channels; 23 from the MPPCs and 22
from the PMTs (Fig.3.4). They came from problems in the circuit, malfunctions of the sensors,
or malfunctions of the high voltage systems. Though the PDE of the VUV-MPPCs decreased as
expected in the exposure to the muon beam, the final PDF value (6 % [48]) was high enough.
The pTC was also operated in a stable condition, and only one counter was inactive during thedata-taking. The readout had a large electronic noise at the beginning, which was reduced during
the data-taking by removing problematic elements in the circuit.
The RDC detector kept a good condition for the data-taking with all the channels actively
readout. However, its installation into the data-taking was delayed till the end of October. This isbecause of a safety problem; An interlock system was missing to avoid collisions between the RDC
and the 17.6 MeV calibration system.
Two target cameras were installed during 2021 data-taking. One was in operation from the
beginning until its stall in the middle of October. The other started its operation in the middle of
October and ran till the end of the physics data-taking.

3.3

Daily calibration

The LXe and the pTC require daily calibration data-taking. Tab.3.2 shows the calibration routine in
the final period of the physics data-taking. The LXe detector calibration, which requires frequent andprecise calibration to correct for the time variations explained in Sec.2.4.2, dominates the calibration
time consumption. As the requirements for the quantity of each calibration data were not clear at the
beginning, we started with a scheme to collect conservatively large calibration statistics. Initially,
the whole calibration data-taking consumed 20 % of the time (Fig.3.1), which was continuously
reduced during the data-taking period. As a result, the calibration time consumption was finallyreduced to ∼ 10 % at the end of the 2021 data-taking.

Chapter 3. Data taking

38

Table 3.2: Calibration data taking scheme.
Data
Electronics noise data
LXe LED data (long duration)
LXe LED data (short duration)
LXe alpha ray data
LXe cosmic ray data
17.6 MeV gamma datapTC laser data

Frequency of data taking
2 / day
1 / day
1 / day
2 / day
1 / day
3 / week
1 / day

Time consumption
10 min40 min15 min20 min20 min1 hour10 minTable 3.3: Applied data reduction method (explained in Sec.2.7.1) for each detector.
Detector

MethodFull waveform rebinned by 8 (when amplitude < 10 mV)
Waveform tail rebinned by 32 (when 10 mV < amplitude < 50 mV)
LXe (also see Fig.2.33)
Waveform tail rebinned by 16 (when 50 mV < amplitude < 400 mV)
Waveform tail rebinned by 8 (when 400 mV < amplitude)
CDCH
pTC
RDC plastic
RDC LYSO

3.4

Full waveform rebinned by 2
Region of interest (first half) and zero-suppression
Region of interest (first half) and zero-suppression
Region of interest (first half)

RMD-enhanced calibration

The full teγ offset calibration requires RMD events, which are difficult to find in a high-rate condition.
Hence, dedicated data-taking was performed at ∼ 1 × 106 µ/s to collect RMD events, where thestatistics were enhanced by setting a low Eγ threshold of 10 – 20 MeV. The data-taking was carriedout twice; the first one on October 8th to optimize the trigger teγ threshold, and the second one
during November 18 – 21 (after the end of physics data-taking) to have sufficient statistics for thecalibration in the offline reconstruction.

3.5

DAQ setting and performance

The data-taking condition was tuned in the first month of the physics data-taking to have a good
DAQ performance. The data reduction scheme was optimized during this period as summarized in
Tab.3.3. The trigger teγ threshold was optimized based on the RMD-enhanced dataset taken in
the middle of October (Sec.3.4). The trigger Eγ threshold was optimized based on the 17.6 MeV
gamma-ray dataset.
The efficiency of the DAQ is defined as the number of recorded events divided by the numberof triggered events, which is shown in Fig.3.5. In the beginning, both the trigger rate and the data
size were too high (higher than the capacity) without the above optimization works. As a result, thetrigger pre-scaling was applied at the beginning to drop a part of the triggered events, which was
the main reason for the low efficiency until October 14th. The other inefficiencies came from theoverflow of the data rate. They were solved by tightening the trigger thresholds of both teγ and Eγ ,namely by reducing the trigger rate. The overall trigger efficiency is discussed later in Sec.6.3.1.

3.6. π 0 → γγ data taking

39

Efficiency

DAQ efficiency
1
0.9
0.8
0.7
0.6

5

:
ale
c
s
e
pr

0.0

:0

le
sca

.5

e
pr

0.5
0.4
0.3
0.2
0.1
0

30/Sep07/Oct14/Oct21/Oct28/Oct04/Nov

11/Nov
18/Novdate (UTC)

Figure 3.5: DAQ efficiency during the 2021 physics data-taking. The large inefficiency at the
beginning of the data-taking came from the pre-scaling of the trigger, which was applied to keep
the data rate below the capacity.

3.6

π 0 → γγ data taking

The π 0 → γγ data was taken during December 16th – December 22nd, after preparation works forthe liquid hydrogen target. Though the π 0 → γγ data with the MEG II LXe detector was first taken
in 2020 [49], there were two additional important goals in 2021. The first one was the full scan of
the whole LXe acceptance region, which was not possible in 2020 because of the limited numberof readout channels. The second goal was to evaluate the geometrical spread of the π 0 → γγ decay
vertex with a dedicated measurement as proposed in [49]. Its importance was recognized in theprevious time resolution evaluation with the 2020 data, which suffered from a large uncertainty
coming from our ignorance about the decay vertex.
The liquid hydrogen target had a problem with insufficient cooling power to stably keep thehydrogen in a liquid state. This resulted in a time limitation for the data-taking. As a result, thefull acceptance of the LXe detector was not covered as shown in Fig.3.6. On the other hand, themeasurement of the spread of π 0 → γγ decay vertex was successful though its statistics were close
to the minimum requirement also due to the problem of the liquid hydrogen target.

3.7

Cosmic ray data taking for detector alignment

The cosmic ray data for the alignment was taken without a magnetic field to have linear tracks,
which are triggered on energy deposits to the LXe detector. They are taken three times during
the scheduled short accelerator shutdown periods: September 16 – 17th, October 19 – 20, andNovember 23rd.

3.8

Optical scanning for detector alignment

The COBRA magnet has a few optical markers to perform laser scans, which define the origin of
the MEG II coordinate system. The CDCH wire scanning was performed before the chamber gas

Chapter 3. Data taking

40

175770

60556

×10
800

181475

60693

700

30336

157703

60522

600

30325

22775730303

500160520850419360412

18176030300

182421

30336

30337

18205430167

USCenter

DS

3

Top

30263

400
300
200
100

Bottom0

Figure 3.6: The number of acquired events for the π 0 → γγ calibration [48]. Each sub-range of theacceptance is the region that can be covered by the finite-size BGO detector, which was moved todifferent positions to fully cover the acceptance.
volume was closed [38]. Its global position in the MEG II apparatus was aligned by optical markers
on the outer structure. The alignment of pTC combined optical markers on the support structureof the scintillation counters and a full 3D scanning of all the counters. The LXe was aligned by
combining a full scan of photosensors discussed later in Sec.5.2.1 and the time variation followedby optical markers on the outer structure of the detector. The alignment of the target relied on a
combination of a CT scan and optical markers on the support frame. The CT scan data was taken
before the installation to estimate its shape or deformation, and the position in the global coordinate
was estimated from the optical markers.

3.9

Target camera operation and photograph

The target cameras were not fully operational during the 2021 data-taking, and special care is neededin the analysis. Firstly, no photograph was taken when an optical survey (an alignment method
based on a laser scan) of the target position was performed, and the camera operation started after
its position was moved from the position in the survey. Secondly, there was a period in November
with several target movement operations for LXe calibration (Sec.2.4.4) but with no photograph tofollow it (Fig.3.7).

3.9. Target camera operation and photograph

41

Camera picture

Picture of camera 2

Picture of camera 1

Oct/07

Oct/14

Oct/21

Oct/28Nov/04

Nov/11

Nov/18Figure 3.7: List of photographs taken by the target cameras. The blue markers correspond tophotographs taken by the first camera and the red markers correspond to those by the second
camera. The vertical lines correspond to target movements.

42

Chapter 3. Data taking

Chapter 4
Event reconstruction and selection
Fig.4.1 shows the reconstruction overview. The positron energy, vertex, and emission angle are
estimated from reconstructed trajectories in the CDCH and their extrapolations to the target. The
positron time is reconstructed from a cluster of pTC measurements with time-of-flight (TOF)
corrections according to the measured trajectory. The gamma-ray energy, position, and timing at
the hit position are directly measured by the LXe. The gamma reconstruction at the vertex relies on
the reconstructed positron vertex, namely the gamma emission angle and the TOF are reconstructedby connecting the hit position in LXe with the positron vertex.

4.1

Positron reconstruction

The overview of the positron reconstruction algorithm is shown in Fig.4.2.

4.1.1

Hit reconstruction and clustering of pTC

The pTC hits are reconstructed by combining waveform analysis results from each edge of thecounters. In the waveform analysis of the pTC, the pulses are detected by setting a threshold, andthe time is evaluated from the constant fraction method. The reconstructed time at both ends, t1,2 ,are then combined to reconstruct the hit on the counter. The hit time on the counter is reconstructedby averaging them,t1 + t2− tcounter ,(4.1)
thit =
2
where tcounter is the offset of the counter. The w position in the local coordinate (Fig.2.15) isreconstructed from the time difference between the ends,
whit =

t1 − t2 − tchannel· veff ,2(4.2)

where tchannel is the offset between the two ends and veff is the velocity of the scintillation lightpropagation inside the scintillator.
The reconstructed hits are then grouped to separate those from pileup positrons. The clustering
algorithm groups the hits by combining the timing with the counter position, whose principle isshown in Fig.4.3. The hit positions are ordered according to the depth from the innermost counter,
which is used to roughly correct the TOF between the counters. After the rough TOF corrections,the hits can be clustered in the time coordinate with a 0.5 – 0.7 ns wide window. In addition, a
distance cut is also applied to separate hits further than 10 cm. This is to detect a small turn during
the hits in the cluster; an example is shown in Fig.4.4. Here, the positron makes hits in the green
43Chapter 4. Event reconstruction and selection

44

1st hit on CDCH

Hits on CDCH

Figure 4.1: An overview of reconstruction procedure.

Matching of wire ends
Momentum

CDCH

Waveform
analysis

Hit
reconstruction

Angle & vertex
TrackfindingTrackfitting

Tracking
refinement

Positronselection

A few iterationpTC

Waveform
analysis

Hit
reconstruction

Matching of counter ends

Hit
clustering

pTC – CDCH
matching

Double turn
analysis

Extrapolation
to target

Combine pTChits with TOFcorrection

Time

Figure 4.2: Flow chart of positron reconstruction.

Combinedwith gamma

4.1. Positron reconstruction

45

Figure 4.3: Principle of clustering of the pTC hits [42]. (Left) The counters are geometricallyordered according to the depth from the innermost counter. (Right) The correlation between the
TOF and the index of the geometrical order.

Projected timing distribution

Projected position distribution
Figure 4.4: Purpose of distance cut shown in an example event display [40]. A small turn in thepositron trajectory makes it difficult to separate two clusters only with timing. Instead, the counter
position can be used to separate them.

Chapter 4. Event reconstruction and selection

46

region, makes a small turn, and then comes back in the orange region. The bottom right plot in
Fig.4.4 shows that the hits in the green and the orange region can be successfully separated thanks
to the distance cut. Though the two peak structure in the timing is not visible at this stage with thelimited TOF precision, the small turn in-between is harmful to the final reconstruction aiming at
50 ps precision.

4.1.2

Hit reconstruction of CDCH

Two waveform analysis algorithms are adopted to detect hits on the CDCH. One method detects hitsby a simple threshold crossing after subtracting coherent noise, which appears on 16 adjacent wires
corresponding to one front-end readout electronics. This coherent noise is extracted by averaging
waveform on channels without pulses. Another method uses a more elaborate method based on a
convolutional neural network, which is trained both to efficiently subtract coherent noise and detectsignal pulses. The training samples are from simulated waveform; simulated signal waveform isoverlaid with data-driven noise waveform (from data taken without beam).
By combining these two methods, a higher hit efficiency is achieved, but at the cost of a higher
fake hit rate. As a result, it is found that the tracking efficiency is higher when two waveform
analyses are combined. As a drawback of the increased fake hits, it is also observed that theresolution gets worse when tracks include hits from the second (machine learning-based) waveform
analysis. Therefore, the positron track finding (discussed later in Sec.4.1.3) is first performed onlywith hits from the first method, and then additional track finding is repeated with all hits from bothmethods to recover missing tracks.
The waveform on both ends of wires for the detected hits is then proceeded to waveform
cross-fitting. The waveform cross-fitting compares the waveform between the two ends, which are
expected to be identical except for the difference in the amplitude and the timing. The waveformfrom the two ends, f (t) for one and g(t) for the other, are fitted to minimizeZ 

2f (t) − A × g(t + τ ) dt,(4.3)

where two fit parameters A, τ are the amplitude difference parameter and the time difference
parameter, respectively.
Once the cross-fit result is given, the hit position along the wire can be estimated by two methods:
the charge division method and the time difference method. The charge division method makes useof the difference in the charge of the two ends (Q1,2 ) as
Gdiff Q1 − Q2zQ =Gdiff Q1 + Q2



Z L
+
ρ
2


,

(4.4)

where Gdiff is the gain difference between the ends, Z = 360 Ω is the input impedance of theamplifier, ρ = 175 Ω/m is the wire resistivity, and L is the wire length. The time difference
method estimates the hit z position from the signal propagation speed on the wire (veff ) and the timedifference,
t1 − t2 − twireendszT =· veff ,(4.5)
2
where twireends is the time offset between the two ends. The z position is finally estimated as aweighted sum of the two. The hit time is also reconstructed as the average of the two hits, namely
thit = (t1 + t2 )/2.

47

0.4
0.3

300

0.2

250

0.1

Drift time [ns]Y [cm]

4.1. Positron reconstruction

200
0
150

−0.1
−0.2

100

−0.3

50

−0.4
−0.4

−0.2

0

0.2

0.4
X [cm]

0

Figure 4.5: Time to drift distance relation in a drift cell.

2nd layer hits

1st layer hits

wire

Drift circle

Found track seed
Figure 4.6: The seeding in the track finding. Different patterns of hit connections are tested and theconsistent one is picked up as a track candidate.

4.1.3

Track finding

The track finding is the first step of the track reconstruction, which starts from the drift distanceevaluation of CDCH hits. The drift distance is the track distance of the closest approach to the wire,
and thus, is essential as the basis of the track finding. It is estimated from the wire hit time (thit )and the pTC cluster time (tpTC ) with a TOF correction (tTOF ) as
ddrift = ζtxy (thit − tpTC + tTOF ),(4.6)

where ζtxy (t), shown in Fig.4.5, is the relation between the drift time and the drift distance and it.
As we have an ambiguity in tTOF at this stage because of the multiple-turn structure of the tracks,
three typically expected values (0.5, 4, and 8 ns) are used in parallel.
In the next step, track seeds are made by connecting four hits on the CDCH (Fig.4.6). Several
different left-right patterns are tested here to connect the drift circles, and the consistent ones are
selected as seeds. As there are many hits on the CDCH, it is impossible to try all the possiblecombinations. Therefore, the seeding attempts are made only for hits associated with the tpTC ,Chapter 4. Event reconstruction and selection

48

where the association is judged by the geometrical compatibility between the CDCH hit position
and the pTC cluster position.
The seed is then further propagated and hits on different layers are added to form a candidate
track. During this propagation, the hits belonging to the same turn are clustered, where the turnmeans a track segment from an out-moving point near the z = 0 axis to the returning point again
getting close to the z = 0 axis.

4.1.4

Track fitting and matching with pTC

The candidate tracks are then fitted with the measurement of the drift distance in Eq.(4.6) and the z
measurement in Eq.(4.4) and Eq.(4.5). The fitting algorithm uses the Kalman filter technique [88],which has widely been used in track fitting in high-energy physics [89, 90, 91]. In particular, thedeterministic annealing filter (DAF), an extension of the Kalman filter, is adopted because of its
high performance in a high noise and pileup environment [92, 93]. It removes outlier hits during
the fit iteration, and thus, shows a good performance. Its implementation in the track fitting makes
use of the GENFIT package [94]. Here, the overall track fitting uncertainty is provided including
the effect of scattering inside the material.
The fitted track, which includes only hits on a single turn at this stage, is then merged with thesegments belonging to the other turns. Each segment track is propagated in both back and forward
directions to evaluate the kinematics at the turn start and end. The propagated kinematics are thencompared between different segments, namely by trying to connect the turn-end of one and theturn-begin of the other. When the momentum and the position at the connection point agree withina pre-defined threshold, then two segments are merged into a single track.
The fitted tracks are then matched with a pTC cluster by extrapolating the tracks to each pTC
hit. When the extrapolated position on the pTC detector plane is inside the real counter volume
within a pre-defined margin, then the track and the pTC hit are matched. This is attempted from theinnermost counter and the first match pTC hit is defined as the matched point.
The matching efficiency cannot be well-defined due to a strong dependence on the track quality;
namely quality tracks have a large matching efficiency. Still, this is a clear indication that thematching inefficiency in the previous MEG experiment, which was due to the scattering effects, issignificantly reduced.

4.1.5

Tracking refinement and missing hit recovery

Once the trajectories are connected from the CDCH region to the pTC region, the measurement oneach hit should then be refined. Firstly, the TOF correction in Eq.(4.6) can be improved thanks to theconnected trajectory to the matched point. In addition, Fig.4.5 shows that the drift time isochrone is
not a complete circle, which can now be taken into account according to the estimated track angle
at each CDCH hit. After updating the drift distance measurements, the track is better fitted.
The hits missed in the track-finding stage can also be recovered once we have a well-fitted track.
All the possible wire crossing of the fitted trajectory are checked and some missed hits are found as
a result. Those hits are added to the track and the track fitting is performed again. The refinementof the drift distance measurement is also iterated after the missing hit recovery.

4.1.6

Vertex and angle reconstruction

The fully reconstructed track is extrapolated to the muon-stopping target, which gives the muon
decay vertex and the positron emission angle. The non-flat target shape and the time-varying target

Axis along long axis of target plane [cm]4.1. Positron reconstruction

49

15

2000
1800

10

1600
5

1400
1200

0

1000
800

−5

600
−10

400

−15
−0.1
−0.05
0
0.05
Axis perpendicular to nominal target plane [cm]2000Figure 4.7: The reconstructed position distribution considering the deformed target shape. The
horizontal axis of the plot corresponds to the coordinate along the axis perpendicular to the nominal(15°-slanted) target plane. The vertical axis of the plot corresponds to the coordinate along the longaxis of the nominal target ellipse. The coordinate along the target short-axis is projected out in this
plot.

𝑡 −𝑡

𝑤 = 1 2 ⋅ 𝑣eff2
𝑣 = 𝑓(hit pattern)
𝑡1

Hit pattern: 00111|10000• 00111 for forward 5 counters(Hits on last 3 counters)• 10000 for backward 5 counters𝑡2

Figure 4.8: The tracking with the pTC hits.
position (monitored by the cameras) are considered in this extrapolation. Fig.4.7 shows the impactof the non-flat target shape with bowing up to 1 mm.

4.1.7

Time reconstruction

In the time reconstruction, the TOF to each pTC hit must be estimated when the pTC hit measurements are combined. This is realized by fitting the track in the pTC region starting from theextrapolated kinematics at the matched point. Here, each counter is treated as a two-dimensional
detector of w and v, where the w estimation comes from Eq.(4.2) and the v estimation relies on
the hit pattern (Fig.4.8). The hit pattern is the hit information on the forward and backward fivecounters in the ordering shown in Fig.4.3, and the v estimation from the pattern is based on detector
simulation samples. By fitting the track to v, w measurements again with the DAF algorithm, theTOF from the vertex to each i-th counter (tTOFi ) is now reconstructed. The emission time at thetarget is then reconstructed by averaging the hit time on each counter with the TOF correction,

Chapter 4. Event reconstruction and selection50

namelyntdecay =1X(thiti − tTOFi ),
n i=1

(4.7)

where thiti is the measured time on i-th counter.

4.1.8

Track selection

In the reconstruction, multiple tracks are reconstructed from one physical positron, which is called
ghosts. To make one-to-one correspondence with the physical positron, track grouping is introduced.
This grouping is associated with the index of the matched TC cluster, and its quality and purity arestudied with MC samples and the results are as follows.
• . 1% positrons are lost by judging them as ghosts associated with different positrons, whichis caused by highly coincident accidentals and hence the inefficiency depends on the beamrate.
• . 0.1% fraction of positrons are split into different ghost groups.
Tracks in the same ghost group are ranked and the highest rank track is selected. This ranking isbased on a machine learning output making use of several input parameters: the number of hits usedin the track fitting, the χ2 value of the track fitting, the covariance matrix, and the track propagation
length from the CDCH to the stopping target and the matched pTC counter. This is trained withMichel positron data to classify tracks into good and bad ones, where the training data samples
are identified as good (bad) if the reconstructed momentum is larger (smaller) than 55 MeV. This
sample choice exploits the 52.8 MeV upper bound of the Michel positron spectrum; namely the“bad”-labelled training samples have Ee error of at least 2.2 MeV.
After the best ghost selection, a quality cut is applied again making use of the machine learning
output. This reduces low-quality background tracks, which account for 7 % of all, without introducing additional inefficiency for the signal positrons (evaluated to be < 1 %). This behavior is causedby the event selection for the physics analysis, where only samples with 52.2 MeV < Ee < 53.5 MeV
and |θ, φeγ | < 40 mrad are used (also see Sec.6.2). These kinematical cuts, requiring the error to besmall, cannot be satisfied by low-quality signal positrons even without the above quality cut. Therefore, this quality cut does not cause any additional inefficiency of the signal. This mechanism is not
the case for background positrons because the above cuts are not associated with the measurement
error at all.

4.1.9

Dedicated double turn reconstruction

About 10 % of the tracks make two turns during their propagation from the target to the pTC. In
addition to the full track reconstruction for the physics analysis, such tracks are also subject todouble-turn analysis, which is useful for evaluating the position and angle resolution.
In the double-turn analysis, the full track is separated into the first turn track segment and thesecond one, and each is separately fitted (Fig.4.9). The first (second) turn is then extrapolated in theforward (backward) direction to a middle point, where a plane identical to the target is used. This
choice of the middle point is to study the correlation between the errors, such as δφe vs δEe , which
is important in the physics analysis, as will be discussed later in Sec.6.7.4. The double-turn analysis
becomes sensitive to the correlations in this way because the error correlation is geometrically
produced by the track propagation to the muon stopping target plane.

4.2. Gamma-ray reconstruction

51

Turn 2

Turn 1Figure 4.9: An example double-turn track. The first turn track (the dashed part) and the second turn
track (the solid part) are fitted independently and compared at the border point. The green markers
are hits used in the track fitting.
Time at interaction point
Angle & time on target
Time
reconstruction

Waveform
analysis

Sensorcalibration

Position
reconstruction

Energyreconstruction

Selection andcombinedwith positron

Pileupanalysis

Figure 4.10: Flow chart of gamma-ray reconstruction.

4.2

Gamma-ray reconstruction

The overview of the gamma-ray reconstruction is shown in Fig.4.10.

4.2.1

Waveform analysis and sensor calibration

The first step of gamma-ray reconstruction is to evaluate the distribution of scintillation light andits arrival time measured by each scintillation photon sensor. The waveform analysis estimates
the pulse timing (tpulse ) with the constant fraction method and the pulse charge (Q) with pulse
integration over 150 ns.
The pulse charge is then converted into the light distribution according to calibrated sensor
parameters. The number of detected photoelectrons is evaluated from the sensor gain (G) and theexcess charge factor (FECF ) defined in [95, 96],
Nphe =Q.
G × FECF(4.8)

The excess charge factor is calibrated for the MPPCs to correct the contribution of the correlated
noise, cross-talk, and after-pulse. The number of scintillation photons (Npho ) is then evaluated bycorrecting Nphe with the quantum efficiency (QE) of the PMTs or the photon detection efficiency

Chapter 4. Event reconstruction and selection

52

Figure 4.11: The MPPCs used for the position reconstruction in an example event [48]. The MPPCs
in the circles (each color indicates Npho ) are used for the position fitting. The red cross marker at
the center is the result of the pre-fitting based on the uv-projected light distribution, which is used
to initiate the main fitting.
(PDE) of the MPPCs,
Npho =Nphe
QE(PDE).

(4.9)

The pulse timing is also converted to the scintillation light arrival time at the sensor (tpm ) withtwo calibration parameters, the time walk parameter (twalk ) and the time offset parameter (toffset ) of
each sensor,
tpm = tpulse − twalk − toffset .
(4.10)
Though the constant fraction method in the waveform analysis is a robust algorithm against the time
walk effect, further correction is applied here to correct observed remaining dependences on Nphe .

4.2.2

Position reconstruction

The position of the gamma-ray interaction point in the liquid xenon is reconstructed from the light
distribution measured by the MPPCs on the inner face. It starts from a pre-fitting based on the
uv-projected Npho distribution on the inner face. The estimated parameter in this step is then used
to initiate the subsequent main fitting step, which minimizes
X Npho − C × Ωi (~x) 2i2.
(4.11)
χpos (~x) =
2
σ
pho
i
i∈region
Here, C is the scale of the lightp
distribution, Ωi (~x) is the solid angle of the i-th sensor seen from theposition ~x, and σphoi = Nphoi / Nphei is the uncertainty of Nphoi assuming the Poisson fluctuation.
The set of MPPCs used in Eq.(4.11) is dynamically determined by the result of the pre-fitting with
uv-projected light distribution. This is because the peak width of the light distribution gets larger
with an increasing w, and the set of the used MPPCs needs to be optimized. The dynamical decisionuses the distance between the MPPC position and the uv projected pre-fitted result (Fig.4.11), where
the radius varies from 3 cm (shallow events) to 15 cm (deep events).
Though the electromagnetic shower has a finite size, the minimization in Eq.(4.11) corresponds
to a model where the scintillation light comes from a point source. This incomplete modelingbiases the fitted position in the direction of the shower development. This is solved by combining4.2. Gamma-ray reconstruction

53

outer faceHit
position

U_ﬁt - U_mc (cm)Fittedposition

Residual of U vs. U

inner face

5

45

4

40

3

35

2

30

1

25

0
20

1

15

2

γ

u

w

3

10

4

5

5
40

30

20

10

0

10

20

30
40
U_ﬁt (cm)0Figure 4.12: The global correction of the geometrical correlation between the shower incident
position and the shower direction [45].
outer face

Residual of U vs. Difference of U b/w different Fitting Range

U_ﬁt - U_mc (cm)1

FittedpositionHit
position

inner face

50

0.8
0.6

40

0.4
0.2

30

0
0.2

20

0.4

γ

0.6
u

w

10

0.8
1
0.2

0.15

0.1 0.05
0
0.05
0.1 0.15
0.2
Difference of U b/w different Fitting Range (cm)0Figure 4.13: The correction of the event-by-event fluctuation of the shower direction [45].
the following two corrections. The first one corrects the bias from the detector geometry; eventswith large |u| tend to have a large incident angle to the inner face (Fig.4.12), which results in aposition-dependent bias. This effect is evaluated on the detector simulations as a function of theposition, which is thus used for its correction. The second one corrects the bias caused by theevent-by-event fluctuation of the shower direction (Fig.4.13). As this bias has a larger impact witha larger fit range, it can be estimated by comparing the main fit result with those from different
ranges. This correction is also based on the detector simulations.

4.2.3

Time reconstruction

The gamma-ray conversion time in the liquid xenon scintillator is reconstructed from the measured
arrival time of the scintillation light by both the MPPCs and PMTs. The best-fit time (tfit ) is obtainedby minimizing
X (tpm − tprop − tfit )2ii
χ2time (tfit ) =,(4.12)
2
σ
i
N
>N
phei

th

where tpmi comes from Eq.(4.10), tpropi is the propagation time for scintillation light between
the interaction position and the i-th sensor, and σi is the estimated time resolution there. The

Chapter 4. Event reconstruction and selection

54

Number of sensors in time fitting

500
400
300
200
100
0
100

200

300

400

500

Figure 4.14: The number of sensors used in the time fitting.
propagation time, tprop , is estimated by dividing the distance from the conversion point to the sensor
by the effective velocity of scintillation light propagation in liquid xenon.
The fitting is performed for sensors with Nphei larger than 50. In order to have robustness
against the pileup gamma-rays, the fitting is iterated until it converges up to three times with outlier
removal. The number of sensors finally used in the fitting is typically a few hundred for ∼ 52.8 MeV
gamma-rays as shown in Fig.4.14.
As will be described later in Sec.5.4.3, calibration studies found that tfit here shows position
dependent biases. This is corrected to obtain the final estimation (tLXe ) astLXe = tfit − Ft (u, v, w).

4.2.4(4.13)

Pileup analysis

There are two types of gamma-ray pileup events: off-timing pileup and on-timing pileup. The offtiming pileups come from accidental gamma-rays from different parent muons, and the on-timing
pileup is from positron annihilation in flight (AIF) where both the created gamma-rays enter the
LXe. The impact of pileup is limited for the position and time reconstruction thanks to the use oflimited sensors in the fitting. The energy reconstruction, on the other hand, does receive a strong
effect, and it results in a tail in the background energy distribution around 52.8 MeV.
The purpose of the pileup analysis is firstly to identify the off-timing and on-timing pileups. In theoff-timing case, it aims to remove the energy deposit from the pileup to have a clean reconstruction
only for the main gamma-ray. When an event is identified to be the on-timing, on the other hand, it
is discarded in the analysis because it is not likely from a signal event. The analysis is composed of
two parts, pileup clustering and pileup unfolding.
The pileup clustering performs the pileup search in the spatial distribution of the scintillation
light on the inner and the outer faces. When local maxima are found with the peak Npho larger thana pre-defined threshold, they are identified as pileups. The on-timing pileup events can be identified
with this algorithm.
The pileup unfolding to search for the off-timing pileups firstly makes the waveform sums for
the MPPCs and the PMTs with channel-by-channel waveform normalization and time shifting asfollows. In the normalization, the i-th channel waveform is scaled with correction factors to have

4.2. Gamma-ray reconstruction

200
0
−200
−400
−600
−800
−1000
−1200
−1400
−1600
−1800

55

×103

Raw
Fit
Main Gamma

0

0.2

0.4

×10−6

Figure 4.15: Fitted waveform in the pile-up unfolding algorithm [48].
an integral value ofBi × Fface (u, v) × Nphoi .

(4.14)

Here, Bi is a correction factor of the light collection efficiency, which considers the insensitive areaaround each sensor including the dead channels. Fface (u, v) is a face factor, which is introduced to
improve the uniformity (see Sec.5.3.3 for detail). After the waveform normalization, the waveform isalso shifted in the time coordinate to correct the channel-by-channel time offset and the scintillation
propagation time.
The time differential of the summed waveform is searched for off-timing pileup pulses with an
optimized threshold. By matching the pileups found in the waveform search with those found in thespatial search, a full list of candidate pileups becomes available.
The sum waveforms are then fitted with a sum of multiple templates,
Npulse~ =g(t; ~τ, A)X

Ai · h(t − τi ),(4.15)

i=1

where τi , Ai are the timing and the amplitude of the overlaid waveforms and h(t) is the waveform
template, independently made for MPPCs and PMTs, describing a pulse that comes at t = 0.
~ runs over the number of overlaid waveforms, Npulse . The waveform fitting
The vectors, ~τ and A,
minimizes

Z
~ 2
f
(t)
−
g(t;
~
τ
,
A)
~ =
dt,
(4.16)
χ2PL (~τ, A)
σg2 (t)where f (t) is the observed sum of the waveform. The denominator in the integra, σg (t), is the~ summing up the uncertainty of the template waveform according
amplitude uncertainty of g(t; ~τ, A),
to Eq.(4.15).
The initial estimation of Npulse in this fitting is based on the list of candidate pileup pulses. Here,additional discrepancies between f (t) and g(t) are sometimes found as a result of the fitting, which
is identified according to the integral of f (t) − g(t). In such cases, Npulse is incremented and thefitting is iterated until it converges or it is iterated three times. An example of the template fitting isshown in Fig.4.15.

Chapter 4. Event reconstruction and selection

w [cm]

56

Rejected

35

Accepted

30
25
20

Gamma

15

Cosmic

10
5
0
0

0.5

1

1.5
2
R (NMPPC/N PMT)

Figure 4.16: The distribution of gamma-ray data and cosmic-ray data on wγ vs NMPPC /NPMT plane.
The black lines define the thresholds of the cosmic-ray rejection cut [48].

4.2.5Energy reconstruction

After the waveform fitting in the pileup analysis, the amplitude of the main waveform, Ai , can thenbe interpreted as the number of original scintillation photons with the waveform normalization inEq.(4.14). As the waveform sums are made separately for MPPCs and PMTs, we now have NMPPC
and NPMT . The energy is then reconstructed as
Eγ = SE (u, v, w) × Nsum ,(4.17)

where SE (u, v, w) is the position and period dependent energy scale and Nsum = RMPPC · NMPPC +NPMT with RMPPC correcting the time variation of NMPPC /NPMT .

4.2.6

Gamma-ray selection

The gamma-ray selection applies three cuts: the pileup rejection, the bad time fitting rejection, andthe cosmic rejection. The first one selects events only if the event is judged not to have pileup
gamma-rays or if it is judged to have only off-timing pileups that are successfully unfolded. Eventsare dropped when identified in the pileup clustering, but not unfolded in the waveform fitting, which
are interpreted as on-timing pileups. Events are also dropped when the waveform fitting is badin the unfolding (large χ2PL in Eq.(4.16)). The time-fitting cut checks the convergence of the time
fitting during the iteration, and ill-converged events are dropped from the analysis, whose fraction is< 1 %. The convergence is judged by χ2time (tfit ) value in Eq.(4.12), where the threshold is optimized
to ensure a good time resolution according to a data-driven resolution evaluation with the π 0 → γγ
events. The cosmic rejection applies cuts on two dimensional wγ vs NMPPC /NPMT plane (Fig.4.16),
which drops ∼ 50 % of cosmic-ray events while suppressing the signal inefficiency at 0.4 %.

4.3

RDC reconstruction

The overview of the RDC reconstruction is shown in Fig.4.17.

4.4. Reconstruction of combined kinematics

Plastic

57

Match successWaveform
analysis

Time andenergyMatching
andClustering

LYSO

Waveform
analysis

LXe

Lxe timing

Selection
Only time

Match failFigure 4.17: Flow chart of RDC reconstruction.

4.3.1

Reconstruction

The RDC reconstruction is initiated by the waveform analysis for the plastic counters, which detects
pulses by the threshold crossing. The pulse time of the plastic counter is then input to the subsequent
LYSO waveform analysis. This procedure is beneficial when searching for associated LYSO pulses,
integrating its charge for energy measurement, and then clustering the hits from the same positron.
There exist two inefficiencies of LYSO pulse finding in association with the plastic pulses. The
first inefficiency is caused by the positrons leaving hits only to the plastic counter. The second
comes from the reconstruction deadtime with pileups, which is the inefficiency for LYSO pulses
immediately followed by a pileup. The deadtime depends on the amplitude of the pileup, which canbe up to 30 ns in the worst case. If the LYSO pulse finding is successful, both the timing and theenergy are reconstructed, while only the timing is reconstructed otherwise.

4.3.2

Selection with LXe timeMultiple hits are usually observed with the RDC, among which only one should be selected for thephysics analysis. The selection is based only on the timing; the hit closest to the LXe time, tLXe , isselected as shown in Fig.4.18.

4.4

Reconstruction of combined kinematics

4.4.1

Reconstruction of combined kinematics

The kinematics of the gamma-ray at the target is finally reconstructed when pairs are made with thereconstructed positrons. The gamma-ray angle is reconstructed as the direction of the reconstructedconversion position in the liquid xenon seen from the reconstructed vertex position. The TOF ofthe gamma-ray from the target to the conversion point is also estimated from the distance between
them. The gamma-ray emission time is reconstructed accordlingly as tγ = tLXe − tγ−TOF . The
parameters for the combined kinematics are defined as follows,Θeγ = opening angle between positron and gamma-ray,θeγ = (π − θe ) − θγ ,φeγ = (π + φe ) − φγ ,teγ = tγ − te .

(4.18)(4.19)(4.20)(4.21)

Chapter 4. Event reconstruction and selection

58

RDC time distribution
Before selection

30000
After selection

25000
20000
15000
10000
5000
0
−40

−20

0

20
40
tRDC− tLXe [ns]Figure 4.18: The RDC detection timing distribution before and after the selection with the offset
from the LXe time. The offset peak position is intentionally introduced to maximize the RMD
tagging efficiency in the asymmetric hit time distribution.

4.4.2

Fiducial volume of the detector

The fiducial volume of the LXe detector is defined such that reconstructed gamma-rays are accepted
when the following conditions are satisfied.
|uγ | < 23.9 cm|vγ | < 67.9 cm
0 < wγ < 42 cm.

(4.22)On the other hand, the positron acceptance is defined on an angle basis; reconstructed positrons are
accepted when the opposite of the positron emission direction points to the LXe fiducial volume.

4.4.3

Pair selection

Multiple positrons are reconstructed in general, and the combined kinematics is reconstructed for
all the possible positron and gamma pairs. However, only a single pair must be finally selected so as
not to bias the µ → eγ search, which demands that the data samples be independent of each other.
Therefore, single-pair selection is introduced at the end of the selection. Here, the selection criteria
are optimized to keep the signal efficiency as high as possible as well as to avoid possible biases in
the physics analysis.
The single pair selection is applied in the following steps;
1. Pre-selection of pairs with a wide window just to reduce the number of candidates
2. Selection of one pair from the pre-selected pairs
In the first step, cuts are applied on the angle, the positron momentum, and the time difference.
• |teγ | < 3 ns
• 45 MeV < Ee < 56 MeV• cos Θeγ < −0.95

4.4. Reconstruction of combined kinematics

59

After these cuts, 0.24 % of events have multiple candidate pairs. In the second step, the pair thathas the largest opening angle (Θeγ ) is selected, which keeps the signal inefficiency as small asO(0.01 %).

60

Chapter 4. Event reconstruction and selection

Chapter 5
Calibration and performance evaluation
5.1

DRS calibration

5.1.1

Time calibration

The sampling interval between the DRS capacitor cells is calibrated beforehand by inputting sine
wave signals. This calibration is applied to the waveform data used in the offline reconstruction. In
addition, to correct event-by-event fluctuation of the time synchronization between different DRS
chips, the 80 MHz master clock from the DCB (see Sec.2.6.1) is input to a dedicated DRS channel.
The clock waveform is fitted with a sine function to evaluate the leading-edge point (Fig.5.1) closest
to −590 ns, which corresponds to the trigger latency. The difference in the fitted zero-crossing timein different DRS chips is then used for the time alignment.

5.1.2

Voltage calibration

The voltage of each DRS cell is calibrated by measuring the response for different input DC voltages
provided internally by the DRS module. Though this calibration is applied when the waveform datais written out, some additional calibrations are necessary offline. Fig.5.2 shows a comparison of
the summed MPPC waveforms of the LXe detector before and after the offline calibration, where
the noise situation significantly improves. Data samples for the offline voltage calibration arecollected in the absence of the muon beam with a random-timing trigger. Four different sources are
Chip 120

0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.65
−0.64

−0.62

−0.6

−0.6

−−0.58
0.55

−0.56

−0.5
[sec]

×10−6

Figure 5.1: Display of clock analysis, which is used to align the DRS time offset. The red line is
the fitted waveform and the green marker indicates the fitted leading-edge point of the sine wave.
The green marker has 20 mV offset because of the voltage offset in the sine wave.
61

Chapter 5. Calibration and performance evaluation

Voltage[V]622w/ Noise Reduction

1

w/o Noise Reduction

0
−1
−2
−3
−4
−600

−400

−200
Time[ns]

Figure 5.2: Effect of offline DRS voltage calibration for the summed waveform of the LXe detector
channels [48]. The noisy black waveform before the calibration is corrected to get the red waveform.
The noisy regions in the edges of the time window (caused by fluctuations of edge effects) are notincluded in the waveform analysis and thus are not harmful.

responsible for the remaining offline voltage miscalibration, and these are all calibrated with thisrandom-trigger data.
The first calibration is further corrections of residual voltage offset of each DRS cell, whichcauses low-frequency noise. This comes from the incompleteness of the online voltage calibration.
This is calibrated by averaging the voltage at each cell with the calibration samples.
The second calibration is a subtraction of high-frequency electronics noise, which is harmful tothe time resolution. This noise comes coherently with the 80 MHz master clock signal of Fig.5.1 andcan be reduced by subtracting typical noise templates. The templates are obtained by accumulating
the waveform with a proper time alignment.
The third calibration is a correction of temperature-dependent noise, which originates from aleakage current of the DRS cells. Due to the accumulation of leakage current during the voltage
hold time, this results in a slope in the recorded voltage. Furthermore, this slope value is also
dependent on the temperature of the DRS chips. This is corrected with samples collected at
different temperatures.
The final calibration is a correction of noise correlated with the first DRS cell of the waveform,
whose origin has not been identified. This noise is found to have a large impact on the LXe energy
reconstruction with the resolution contribution being O(MeV). This is calibrated by accumulating
the waveform with a grouping on the first DRS cell index.

5.2. LXe sensor alignment and position resolution

y

63

15
10
5
0
−5
−10
−15

−10

0

10
x

Figure 5.3: (Left) The surface of the MPPCs is scanned with the laser scanner [48]. (Right) An
example of the measured height (perpendicular to the surface) distribution. The MPPC package
structure (hosting four chips) is visible.

5.2

LXe sensor alignment and position resolution

5.2.1

MPPC alignment

Optical scan at room temperature
During the MPPCs installation into the detector, the surface of the MPPCs was scanned by a threedimensional laser scanner (FARO Edge ScanArm [97]) as shown in Fig.5.3. The MPPC positions
are reconstructed by fitting the scanned points with a model describing the package structure of fourchips. The quality of the fitting was enough for 426 MPPCs, whereas the other low-quality data
points (those far from the reach of the scanner’s support arm) come from diffused reflections of thelaser. To have positions of unfitted MPPCs, the well-fitted ones are interpolated taking into accountthe PCB structure.

Alignment at liquid xenon temperature with gamma-rays
As the MPPC positions are changed by the thermal contraction, the optical scan at room temperature
is complemented by a measurement at the liquid xenon temperature. In this measurement, collimated120 keV gamma-rays from 57 Co were injected into the liquid xenon from the outside of the detector
cryostat. The shallow gamma-ray conversion within 3 mm is desirable for the alignment; only afew MPPCs located directly below the conversion point are illuminated with the scintillation light.
The gamma-rays were collimated to have a spot size of 1.5 × 40 mm, whose direction was changed
by a moving stage to cover the whole detector area (Fig.5.4). The distribution of the measured
gamma-ray rate as a function of z in Fig.5.4 is fitted with a flat top and Gaussian tails at both ends.
Here, the MPPC position is estimated to be at the center of the fit function with a 100 µm precision.
In addition to the main alignment measurement, validation measurements were also performed
using 16 lead strips (1 × 1 × 25 mm) attached to the entrance face of the LXe detector (Fig.5.5). In
the distribution of the gamma-ray conversion position, 16 defects are expected just behind the strips,
which is the principle of this validation measurement. The consistencies between the reconstructed
position of the lead strips and the optical survey were σz = 0.43 mm and σφ = 0.68 mrad.

Laser Emitter
φ motion

Rotation Stage

Rate [Hz]Chapter 5. Calibration and performance evaluation

64

14
12
10

Bubble

Beam

8
6
4

Collimator

2z motion0Linear Stage

80

90

100

110

120
z [mm]

EventRate[Hz]Figure 5.4: (Left) Gamma-ray source with a collimator used for the MPPC alignment [48]. (Right)
A position scan of the event rate at an MPPC.

25
20
15
10
5
0

−10

0

10

20
30
40
Beam Position[mm]Figure 5.5: (Left) Lead strips used in the validation measurement [48]. (Right) A position scan ofthe event rate around a lead strip. The decreased rate around 5 mm corresponds to the lead position.

5.2. LXe sensor alignment and position resolution

65

TargetGamma-ray
Event Distribution

collimator

PMT

support structure

COBRA magnet

y

MPPC

x
LXe detectorzφFigure 5.6: (Left) Setup for position resolution measurement with the 17.6 MeV gamma-rays [48].
(Right) Picture of the lead collimator used in the position resolution evaluation measurements.
Combination of two alignment methods
The two methods, the optical scan and the collimated gamma-rays, are finally combined to have afull three-dimensional alignment. The optical scan can give a three-dimensional alignment, which
is likely to change due to the thermal contraction. On the other hand, the collimated gamma-ray
measurements were performed at the liquid xenon temperature but can provide information only inthe uv plane. A combination of the above two methods complements each other accordingly.
The position at the liquid xenon temperature is assumed to be related to the position at the room
temperature as,~xfit = (1 − a)REular · ~xroom + ~xoffset ,(5.1)
where a is the scaling parameter for the thermal contraction, REular is the Eular rotational matrix,~xroom is the measured position in the optical scan at the room temperature, and ~xoffset is an additional
offset term. The parameters of the coordinate transformations are estimated in a fitting to minimize
the residual in the uv plane between the ~xfit and the estimation from the collimated gamma-rays.
The overall alignment uncertainties are ∼ 600 µm in the z coordinate and ∼ 0.7 mrad in the
φ coordinate [48]. The z uncertainty is dominated by the reproducibility of the measurementswith collimated gamma-rays, which are performed twice. The φ uncertainty is dominated by theslight inconsistency between the measurements with collimated gamma-rays and the cross-checkmeasurements with the lead strips.

5.2.2

LXe position resolution evaluation

The position resolution of the LXe detector was measured with the 17.6 MeV gamma-rays whichwere collimated by a 25 mm thick lead collimator in front of the LXe inner face (Fig.5.6). The
collimator has several 5 mm wide slits in 50 mm intervals. The reconstructed position distribution
with this setup shows a peak corresponding to the slit position, which is smeared by the position
resolution and the position spreading of 17.6 MeV gamma-ray origins (Fig.5.7). Here, the vertex
size of the 17.6 MeV gamma-rays is as small as 0.6 mm, which is small enough compared to the
expected resolution. The resolution can thus be obtained from the smearing evaluation in theobserved distribution. The collimated gamma-ray data was taken with two different slit rotations,
one for the u resolution evaluation and the other for that of v.

Chapter 5. Calibration and performance evaluation

66

v[cm]

0

vrec 2.0 <w< 5.0 cm, Data

−5

40

−10
−15

Events

45

100
Data

90

35

80

MC(Smeared)

30

70

MC Truth

25

60

20

50
40

−20

15
30

10

−25
−30
−15

−10

−5

0

5

10
15
u[cm]

20

5

10

0

0
−24

−23

−22

−21

−20

−19 −18
vrec[cm]

Figure 5.7: Observed position distribution in a σv measurement [48]. The observed distribution in
the one-dimensional plot is fitted with a smeared MC distribution to evaluate σv .
In the resolution measurements, the observed position distribution was fitted with a simulated
truth distribution smeared by the resolution parameter. Fig.5.8 shows the estimated resolution at
17.6 MeV as a function of the conversion depth, w. The observed resolutions are worse than the
expected ones in particular for deep events, whose cause is not understood.
The resolution for 52.8 MeV gamma-rays is expected to be different from that for 17.6 MeV
gamma-rays. The resolution at 52.8 MeV was estimated with detector simulations, which is thencorrected according to the observed resolution discrepancy between the data and the simulation for17.6 MeV. The results are shown in Fig.5.9.

5.3

LXe energy calibration and resolution

5.3.1

Sensor gain and excess charge factor

PMT calibration
The absolute PMT gain value is calibrated with the dedicated LED data, where the Poisson fluctuation of the number of photoelectrons plays a key role. When the mean of the number ofphotoelectrons is given as N̄phe , the mean of the integrated charge can be expressed as
Q̄ = G · N̄phe ,(5.2)

where Q̄ is the average ofp On the other hand, the Poisson fluctuation has
pthe charge and G is the gain.
a standard deviation of N̄phe , which gives a G · N̄phe contribution to the standard deviation of
the charge. When the contribution from the electronics noise is included, the full standard deviationof the charge thus follows
2
σQ= G2 · N̄phe + noise term

= G · Q̄ + noise term,(5.3)

where σQ is the standard deviation of the charge distribution in fixed N̄phe data.
2
As the noise term in Eq.(5.3) is independent of N̄phe , the gain can be calibrated from a σQvs

5.3. LXe energy calibration and resolution

67v Resolution

0.8

σv[cm]

σu[cm]

u Resolution

0.7

0.8
0.7

0.6

0.6

0.5

0.5

0.4

0.4
0.3

0.3

MEG II Data

MEG II Data

0.2

0.2

MEG II MCMEG II MC

0.1
0
0

0.1

MEG MC

2

4

6

8

10
12
w[cm]

MEG MC

0
0

2

4

6

8

10
12
w[cm]

σ [cm]

Figure 5.8: The estimated σu and σv values at 17.6 MeV as a function of w [48]. The measured andsimulated resolutions for the MEG II detector are compared with the MEG resolution. The MEG II
detector improved the position resolution especially in the shallow region thanks to the improved
granularity of the photosensors on the inner face.

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3

σu

0.2

σv

0.1
0
0

5

σw
10

15
w [cm]

Figure 5.9: Estimated position resolution as a function of w for signal gamma-rays [48].

Chapter 5. Calibration and performance evaluation

PMT 4092

×10−3

PMT

Variance[1018e2]

68

0.7

35

0.6

30

0.5

25

0.4

20

0.3

15
10

0.2

Raw

0.1

5

Fit, Gain=767697.6

×10
1000
Gain

3

0.1

0.2

0
700

0.3
Charge[109e]

Weighted Sum of N

phe(a.u.)

2
Figure 5.10: σQ
vs Q̄ plot for a PMT [48].

29.3 ×10

6

30

29.2

25

29.1

20

29

15

28.9

10

28.8

5

28.7

04/11

0

07/11

750

800

850

900

950

Figure 5.11: PMT gain distribution at the beginning of 2021 run [48].

Time

Figure 5.12: Time variation of PMT gain in
beam-on condition [48].

PMT Response (normalized)

0
0

1.1
LED Charge

1.05
Gain

1
0.95
0.9
0.85

31/08/21 30/09/21 30/10/21 29/11/21 29/12/21
Time

Figure 5.13: PMT gain decrease during thebeam period [48].

Q̄ plot scanned over various N̄phe value. Fig.5.10 shows such an example, where the measurement
points (each corresponds to one N̄phe value) are fitted with a linear line and the slope parametercorresponds to the gain. To have a high uniformity of the PMT gain, it was adjusted at the beginningof the 2021 run (Fig.5.11) by tuning the PMTs’ high voltage.
Once the absolute gain value was estimated at a point in the middle of the 2021 run, the time
variation was then followed with constant LED intensity data, which was collected also in the muon
beam. The absolute gain value at a fiducial point was scaled to each point during the run according tothe observed charge. The gain difference between the beam-off and beam-on conditions (Sec.2.4.2)
can be corrected this way. It is also advantageous because the time variation can be followed in ashort time scale (Fig.5.12).
The gain decrease explained in Sec.2.4.2 can be found in Fig.5.13, where two different gain
calibration methods are compared in dedicated beam-off calibration dataset. The overall decreaseduring the whole period is around 10 %. A small discrepancy between the two methods can also beseen in the gain decrease speed, which suggests a systematic uncertainty in the absolute gain value.
This is suspected to come from the quality of Eq.(5.3); the LED light may not be completely stableor the gain itself may have a non-negligible contribution from the fluctuation in the multiplication.
Still, the time variation is well followed by the method with constant LED light, and the above

Events

5.3. LXe energy calibration and resolution

69

80
70
60
50
40
30
20
10
0
−0.1

0

0.1

0.2

0.3
0.4
9
Charge [10 e]

Figure 5.14: LED charge distribution on a MPPC [48].
systematics are not expected to cause energy miscalibration. This is because the absolute gain
uncertainty is finally absorbed in the later steps as long as the gain calibration is consistently
applied.
MPPC calibration
The absolute gain value and the excess charge factor of the MPPCs are calibrated with LED
data where the expected number of photoelectrons is adjusted to ∼ 1. By resolving the single
photoelectron peaks, the gain is estimated from the differences in the peak positions. Fig.5.14
shows the distribution of the MPPC charge on such data, which is fitted by a sum of two Gaussianfunctions with the mean difference corresponding to the gain. The excess charge factor, FECF , canbe estimated by the Poisson photoelectron statistics [96]. The expectation value for the number ofthe incident photoelectrons, N̄phe , can be estimated from


Nzero
,
(5.4)
N̄phe = − log
Nallwhere Nzero /Nall is the fraction of the zero photoelectron events in the dataset. The correlated noise
enhances the observed number of photoelectrons from the incident one as
FECF =

N̄observed
,
N̄phe

(5.5)

where N̄observed is the average of the observed number of photoelectrons in Fig.5.14.
Though the MPPC charge is reconstructed in integration over 150 ns, such a long integration
gives a large electronics noise contribution to the charge distribution and the single photoelectron
peaks cannot be resolved. Therefore, the gain calibration was based on analyses with shorter
integration ranges, tint . The results from shorter integration ranges were then extrapolated to theparameters with tint = 150 ns astint − t0
G(tint ) = G × 1 − exp −
,
(5.6)
τfail

Chapter 5. Calibration and performance evaluation

1.8

λ

1.6

Fit

9

2

Gain [10 e]

λ

70

1.4

0.14

Peak

0.12

Fit

0.1

1.2
0.08

1

0.06

0.8
0.6

0.04

0.4
0.02

0.2
0
0

50

100

0
0

150
tint [ns]

50

100

150
tint [ns]1800 ×10

3

ECF

Gain

Figure 5.15: Integration range dependence of (left) the excess charge factor and (right) gain [48].
3.5
A
B

1700
3

C
D

1600
2.5

1500
1400

2

1300

A

1.5

B

1200

C
D

1100
0

2000

4000

6000
Serial Number

1
0

2000

4000

6000
Serial Number

Figure 5.16: Distribution of the (left) gain and (right) the excess charge factor of MPPCs [48].
where G, t0 , and τfail were fitted to the shorter tint results (Fig.5.15). The distribution of the calibrated
gain and the excess charge factors as a function of the serial number are shown in Fig.5.16, where
production lot dependences can be seen.
The time variation of the MPPC gain was also followed by the constant LED light data. Fig.5.17
shows the time variation of the charge in constant LED data for beam-on and beam-off periods.
The overall decrease during the beam time was 4 %, which came from the PDE decrease due to theradiation damage (see Sec.2.4.2). The instability observed in early November came from a ∼ 1 K
temperature fluctuation of the liquid xenon, which increased the MPPC breakdown voltage and thus
decreased the gain. The discrepancy between the beam-off and the beam-on data likely came fromthe voltage drop by the current from the constantly illuminated MPPCs. This is a behavior discussed
in [44], and the observed difference agrees with the expectation.

5.3.2

QE and PDE of sensors and liquid xenon light yield

The MPPC PDE and the PMT QE are estimated with alpha-ray events from 241 Am sources. The
number of photoelectrons on each sensor is estimated in simulation studies making use of the
known position of the α-sources, the known energy deposit of the alpha-rays, and an assumption
on the QE,PDE . This simulation-based expectation is then compared with the observed number of

QLED (normalized)

5.3. LXe energy calibration and resolution

71

1.02
w/ Beam

1.01
1w/o Beam

0.99
0.98
0.97
0.96
0.95
0.94
31/08/21

30/09/21

30/10/21

29/11/21

29/12/21
Time

Figure 5.17: Time variation of the MPPC gain [48].
photoelectrons to estimate the PDE or QE asQE,PDE =MC assumed · N̄observed−phe,N̄simulated−phe · FLY(5.7)

where MC assumed is the assumed QE,PDE value in the simulation, N̄observed−phe is the observed
average number of photoelectrons, N̄simulated−phe is the average in the simulation, and FLY is thecorrection factor of the light yield. As this method directly observes the LXe scintillation light, it
is not sensitive to the QE,PDE dependence on the wavelength.
In the calibration data analysis, the alpha-ray events are separated from cosmic backgrounds
according to the difference in the amplitude vs charge ratio shown in Fig.5.18. This difference
comes from the energy deposit dependence of the liquid xenon scintillation process described inSec.2.4.1. The source of the incident alpha-ray is identified from the other 24 sources with the light
distribution of the PMTs, whose separation power is shown in Fig.5.19. N̄observed−phe /N̄simulated−phe
is evaluated from a plot as shown in Fig.5.20. The light yield is determined with an assumption thatthe average of the PMT QE is stably at 16 % during the data taking, which is based on experiencesin the MEG experiment.
The distribution of the observed QE value in 2021 is shown in Fig.5.21. Here, the PMTs withy > 83 cm had lower QE values than the others because the liquid was not full in the detector andthe sensors above the level were not fully sensitive to the energy deposit.
The time variation of the light yield and the PDE were also observed as Fig.5.22. The lightyield was kept within 93 ± 2 % during the data taking. The final PDE value at the end of the datataking was 6 %, above the minimum required 4 %. Though the absolute values of the PDE and QE
also have uncertainties, they are absorbed in the next calibration of the energy scale.

5.3.3

Energy scale calibration

Several calibration tools are exploited for the energy scale calibration: α-particles, 17.6 MeV
gamma-rays, 55 MeV gamma-rays, cosmic-rays, and background gamma-rays from muon decay.
The calibration with 17.6 MeV gamma-rays and 55 MeV gamma-rays are straightforward; they use

Chapter 5. Calibration and performance evaluation

72

120
100

Events

80
60

3500

Cosmic40

3000

Wire 1

20
0−
120

−

100

Alpha

80
60
40

2500

Wire 2

20
0−
120

−

100

2000

80
60
40

1500

Wire 3

20
120
0−

−

100
80

1000

60
40

Wire 4

20

500
0
0

5

10

1200−
100
80
60
40
20
0
−40

15
QA ratio

45

0

20

40
Z [cm]

0.35
0.3

40

0.25

35
30
25
20

α source

0.2

Linear Fit

0.15

15

0.1

10

Slope: 1.22

5

χ2/NDF: 1.86

0
0

10

20

30

0.05
0

40
NpheMC

Figure 5.20: N̄observed−phe vs N̄simulated−phe
plot for an MPPC [48].
0.1

0.095
0.09
0.085
0.08

−100

−50

0

50

100
y [cm]

Figure 5.21: The PMT QE values as a function of y position of the PMTs [48].
Light Yield

PDE

Wire 5−20

Figure 5.19: Identification power of alpha ray sources [48].
PDE

NpheData

Figure 5.18: Charge vs amplitude ratio for cosmicray events and α-ray events [48].

−

1
0.98
0.96
0.94

0.075

0.92

0.07
0.9

0.065
0.06

0.88

0.055
0.05

0.86
31/08/21 30/09/21 30/10/21 29/11/21 29/12/21
Time

31/08/21

30/09/21

30/10/21

Figure 5.22: Time variation of the light yield and the PDE [48].

29/11/21
Time

73

EventsEvents

5.3. LXe energy calibration and resolution

1200
1000

1000
800

800
600
600
400
400
200

200
0
10

15

20

25
E[MeV]

Figure 5.23: Reconstructed spectrum of
17.6 MeV gamma-ray data [48]. The fitted
peak position at 17.6 MeV is used for energy scale calibration. The secondary peakis the 14.8 MeV gamma-ray byproducts of
the 73 Li(p, γ)84 Be reaction.

0
45

50

55

60

65
E[MeV]

Figure 5.24: Reconstructed spectrum of
55 MeV gamma-rays [48].

the peak in the energy spectrum shown in Fig.5.23 and Fig.5.24. The calibration with cosmic-rays
uses the Landau distribution of the energy deposit with the peak value around 170 MeV. Here,
only cosmic-rays that cross the entrance face of the LXe detector are selected to have a uniform
pass length, which drops the low-energy peak in Fig.5.25 of short-path length cosmic-rays. The
calibration with background gamma-rays is based on a background spectrum fitting to the energy
spectrum around 52.8 MeV (Fig.5.26), which is based on a model detailed later in Eq.(6.15).
Time variation of energy scale
As introduced in Sec.4.2.5, NMPPC and NPMT are separatedly reconstructed. Here, the time
variation of the energy scale and NMPPC /NPMT value are corrected in the energy reconstruction.
Accordingly, the time variation during the physics data taking was monitored with α-particles,
17.6 MeV gamma-rays, and cosmic-rays (Fig.5.27). Similar trends were observed between different
calibration sources each for NMPPC and NPMT . They are understood to represent the time variationof the light yield of LXe caused by liquid impurities. The difference in the trend between NMPPC
and NPMT , on the other hand, is understood to arise from the difference in the distance from theα-sources to the sensors, which can give different effects of scintillation light absorption to the PDE
and QE calculations. In the energy reconstruction, this difference in the time variation is absorbedby the time-varying RMPPC parameter introduced in Sec.4.2.5
Position dependence of energy scale
The uγ , vγ , and wγ -dependence of the energy scale comes from several mechanisms: the position
dependence of the light collection efficiency and the systematics in the sensor calibration. It iscorrected by introducing Fface (u, v) in Eq.(4.14) and SE (u, v, w) in Eq.(4.17).
The face factor, Fface (u, v), was optimized to have the best resolution for 17.6 MeV γ-rays. The(u, v)-dependence was taken into account by dividing the calibration dataset into different subsets
based on the position. The optimized Fface (u, v) values in different (u, v)-segments were then fittedwith a symmetric function of u and v, and the result, shown in Fig.5.28, is used in the energyChapter 5. Calibration and performance evaluation

74

450

Raw

400

Position Selection

350

Landau Fit

300
250

Events / ( 0.4 MeV )EventsBackground γ -ray spectrum
Data
BG γ -ray component
Cosmic-ray component
Sum function
Sum function (in fit range)

104

103

200
150

102

100
50
0
0

100

200

300

Figure 5.25: Reconstructed spectrum of
cosmic-rays used for energy scale calibration [48]. The 170 MeV peak of the Landau
distribution is used to calibrate the time variation of the energy scale.

1.05
1.04
1.03

1.04
1.03
1.02

1.01

1.01

1

1

0.99

0.99
0.98

Cosmic-ray

0.97

CW-Li

0.96
0.95

0.052 0.054 0.056 0.058
Eγ (GeV)

1.05

1.02

0.98

0.05

Figure 5.26: Fitting of background γ-ray
spectrum around 52.8 MeV. The solid lineis the best-fit spectrum, which is modeled asa sum of the background gamma-ray component (cyan line) and cosmic-ray component(green line).

nMPPC[a.u.]nPMT[a.u.]

10
0.046 0.048400
500Energy[MeV]Alpha
31/08/21 30/09/21 30/10/21 29/11/21 29/12/21
Time

(a)

0.97

Cosmic-ray

0.96

CW-Li

0.95

31/08/21 30/09/21 30/10/21 29/11/21 29/12/21
Time

(b)

Figure 5.27: Time variation of measured value of (a) NMPPC and (b) NPMT for different calibration
sources. Three independent results show good agreements each other [48].

5.3. LXe energy calibration and resolution

60

1.9

v[cm]

2

1.8

40

US

1
60

0.9
0.8

40

1.7

0.7

0.6

1.5

0.4

−20

1.3

1.1

0.2
0.1

−60

1
0

−20

20
u[cm]

60

0.9
0.8

40

0.2
0.1

−60

0
0

−20

20
u[cm]

60

0.9
0.8

40

0.5
0.4

−20

0.2
0.1

−60

0
20
u[cm]

1
60

0.9
0.8

40

0.7
20

0.6
0

0.5
0.4

−20

0.3
−40

20
u[cm]

0.7
20

0.6
0

0
0

BOTTOM

1

0.7
20

0

0.3
−40

TOP

1

v[cm]

DS

−20

0.4

−20

v[cm]

−20

0.5

0

0.3
−40

1.2

−60

0.6

0.5

0

1.4

−40

0.8

20

1.6

−20

0.9

40

20

0

1
60

0.7

20

v[cm]

v[cm]

OUTER

INNER

v[cm]

75

0.6
0

0.5
0.4

−20

0.3
−40

0.2
0.1

−60
−20

0
0

20
u[cm]

0.3
−40

0.2
0.1

−60
−20

0
0

20
u[cm]

Figure 5.28: The position-dependent face factor Fface (u, v) used in the energy reconstruction [48].
reconstruction. As an overall behavior, we can see that a larger weight is used when incident γ-rays
are closer to the face.
The SE (u, v, w) factor was then calibrated with 17.6 MeV gamma-rays, background gammarays from muon decay, and 55 MeV gamma-rays. Here, the calibration with 17.6 MeV gamma-rays
has systematics due to the energy dependence; the position dependence is found to have energy
dependence both in data and MC samples. On the other hand, 55 MeV γ-rays are close to the signal
energy, but they were not measured during the physics data-taking period. Therefore, possible time
variation in the position dependence can cause systematic biases in the 55 MeV calibration. The
background spectrum fitting also has its unique difficulty because of the use of the kinematical
endpoint of the monotonically decreasing spectrum. This causes a large fit uncertainty due tothe limited statistics around 52.8 MeV, large error correlations between fit parameters, and large
systematic uncertainties from the validity of the fit modeling. Given the different uncertainties ineach method, the SE (u, v, w) was calibrated finally by combining three different methods with aweight optimization according to the uncertainties.
Energy scale uncertainty
The uncertainty of the energy scale has three contributions: the linearity of the energy response, thetime variation of the energy scale, and the position dependence. The uncertainty from the energyChapter 5. Calibration and performance evaluation76

response linearity, which originates from the small difference between 52.8 MeV and the calibration
source (55 MeV), was evaluated to be 0.1 % according to the uncertainty in an extrapolation ofa linearity plot between 55 MeV, 83 MeV, and 129 MeV. The time variation uncertainty wasevaluated to be 0.3 % from the agreement between the cosmic-ray calibration, 17.6 MeV gammaray calibration, and background gamma-ray spectrum fitting. The position dependence uncertainty
was similarly evaluated to be 0.2 %.

5.3.4

Energy resolution evaluation with π 0 → γγ data

The energy resolution was evaluated with 55 MeV gamma-rays by fitting the reconstructed spectrum(as shown in Fig.5.24) with
"

2 #
E
−
µ

γ
E
γ


exp −
(Eγ > µEγ + τ ),

2σE2 γ"(5.8)
f (Eγ ) =
#

τ
τ
/2
−
E
+
µ

γ
E
γ


(Eγ ≤ µEγ + τ ),
 exp
σE2 γ
which represents a Gauss function smoothly connected with an exponential low energy tail. As aresult, we obtained the σEγ /µEγ value of 2.0 % (1.8 %) for wγ < 2 cm (wγ > 2 cm).

5.4

LXe time calibration and resolution

The time calibration and the resolution evaluation exploit the π 0 → γγ events. Here, the fast preshower counters on the opposite side of the LXe detector are used as a reference. The pre-shower
counter gives the interaction time at the LXe as
tref = tpreshower + tγγTOF ,(5.9)

where tref is the referenced value of the tLXe in Eq.(4.13), tpreshower is the time measured at the
pre-shower counter, and tγγTOF is the TOF correction. The TOF correction here assumes that thetwo gamma-rays always originate from the center of the π 0 → γγ decay vertex though the vertex
itself has a finite spread in reality.

5.4.1

Time walk and offset

Compared with Eq.(4.12), the scintillation photon arrival time on each sensor can then be referencedastpmref i = tref + tpropi .
(5.10)
From Eq.(4.10), the walk and the offset parameters thus follow
twalk + toffset = tpulse − tref − tprop ,(5.11)

which is used in the walk and offset calibration.
When the walk parameter was calibrated, the sensors were separated into six groups: the PMTs
on the outer face, the PMTs on the lateral face, and the MPPCs from four different production lots.
This grouping was empirically determined according to the difference in the waveform shape; thetypical waveforms for these six groups are different from each other. Though the walk and the offset

29 ×1077

Lot A time walk−9

Offset between LXe and reference

1000

28
800
27

tfit − tref [ns]Time residual [sec]5.4. LXe time calibration and resolution

600

26

2500

0.4
0.3

2000

0.2
0.1

1500

0
25

400

24
200

23
22

1000

−0.1
−0.2

500

−0.3
0.02

0.04

0.06

0.08

0.1

0.12 0.14
1/ Nphe0

Figurep5.29: Distribution of tpulse −tref −tprop
vs 1/ Nphe for lot-A MPPCs. The offset
dependence on Nphe is calibrated as the walkeffect and the y-axis spread is used to evaluate
the single sensor resolution.

−0.4
40

50

60

70

80
90
Eγ [MeV]0

Figure 5.30: Comparison of tLXe − tref between the 55 MeV and 83 MeV with a notoptimized veff .

are independent of each other, the calibration is entangled due to the above grouping. Therefore, thewalk calibration and the offset calibration were iterated by updating the calibration of each other.
Fig.5.29 shows the tpulse − tref − tprop distribution as a function of Nphe for the lotA MPPCs.
Here, O(100 ps) dependence on the Nphe is found and corrected after the calibration. In addition,
the same plot after the calibration was used to evaluate the σi in Eq.(4.12), which also depends on
Nphe and the grouping. The single sensor resolution with large amplitude pulses was as good as
500 ps for both MPPCs and PMTs.

5.4.2

Effective velocity of scintillation light

The tprop parameter also needs a calibration of the effective velocity of the scintillation lightpropagation in the liquid xenon, veff . When veff parameter is changed, the tfit − tref offset becomesinconsistent between 55 MeV and 83 MeV as shown in Fig.5.30. The effective velocity parameter
was thus optimized to 8.4 cm/ns to minimize the time offset dependence on Eγ . Though this value
is different from the correct physical propagation velocity of the scintillation light (11 cm/ns groupvelocity), this choice of veff gives the best time resolution.

5.4.3

Position dependence of time offset

The fitted time in Eq.(4.12) showed a position-dependent bias and w dependence was especiallystrong as shown in Fig.5.31. This is corrected by introducing the correction term in Eq.(4.13),
which was calibrated as a sum of three one-dimensional functions of u, v, w eachFt (u, v, w) = Ftu (u) + Ftv (v) + Ftw (w).

(5.12)The Ftw (w) was the largest one among them and its value is as large as 400 ps peak-to-peak asshown in Fig.5.31.

Chapter 5. Calibration and performance evaluation

78

W dependence correction

Time offset [ns]tfit − tref [ns]W dependence of time reconstruction

5000

0.6
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10

−0.25
0

0

15

5

10

15

20

w [cm]

25

30
w [cm]

EventsFigure 5.31: Calibration of w-dependent time offset, Ftw (w). (Left) Distribution of tfit − tref vs w.
(Right) Calibrated Ftw (w) plot, which is obtained by profiling the left plot.
1200
1000
800
600
400
200
0
−0.2

−0.1

0

×10−9
0.1
0.2
t front -tback [sec]Figure 5.32: The front to back time difference of the pre-shower counter [48].

5.4.4Time resolution in π 0 → γγ events

In the reconstruction, tγγTOF measurements have large uncertainties due to the finite spread of theπ 0 → γγ vertex. Including this contribution, the tLXe − tref distribution has a standard deviation ofσLXe−preshower as
σLXe−preshower = σLXe ⊕ σpreshower ⊕ σvertex ,(5.13)
where σLXe is the LXe time resolution, σpreshower is the pre-shower counter time resolution, andσvertex is the uncertainty contribution from the vertex spread.
The contribution from the pre-shower counter was estimated to be 28.2±0.2 ps from the intrinsic
time difference between the front counter and the back counter (Fig.5.32). The contribution fromthe vertex spread was measured with a dedicated measurement shown in Fig.5.33. Here, a counter
consisting of two plastic scintillators was placed in front of the LXe detector, whose resolution wascomparable to the pre-shower counter. The observed distribution between the opposite countersshown in Fig.5.33 indicates σvertex = 68 ± 6 ps. The uncertainty here mostly comes from the factthat the measured σvertex showed dependence on the liquid level in the hydrogen target.
The distribution of the tLXe − tref is shown in Fig.5.34. The core part of σ = 98 ps accounts
for 95 % and the tail part of σ = 290 ps accounts for the other 5 %. By subtracting the contribution

5.4. LXe time calibration and resolution

Events

79

70
60
50
40
30
20
10
0
−0.4

−9

−0.2

0

×10
0.2
0.4
t ref -tps [sec]Figure 5.33: (Left) Setup to measure the vertex spread in the π 0 → γγ data-taking. (Right) Timedifference distribution between the pre-shower detector and the reference counter shown in the left
sketch [48].

55 MeV time resolution evaluation

1800

55 MeV tLXe− tref

1600
Sigma = 98 ps

1400
1200
1000
800
600
400
200
0

−0.5

0

0.5
tLXe− tref [ns]Figure 5.34: Time difference distribution of tLXe − tref in the π 0 → γγ events. The overall spreadof 98 ps includes the contributions of σpreshower ⊕ σvertex . After its subtraction, σLXe was evaluatedto be 65 ± 6 ps.

Chapter 5. Calibration and performance evaluation

80

300
✝ ✞✟✁

250 ✙☛✁✁✂✚ ✛☛✜✄✜✂✢
200✣ ✢✡✤✚ ✥✦✁☛✢✄✁✧h
EntriesMean
Std Dev

16003! 0.0601
3.704✞✠✠✝✁☎✣ ★✁✚ ✩☛✢✟✁✪✞✜✧

150 ✫✁✧✚ ✬✞✂✦✞ ★☎✁✧
✁✂✄☎✆

100

☞✌✍✎✏✎✌✑

50
0
! 20

✒✓✍✌✔✕✏✎✌✑

! 10

0

10

✡✁✂☎✁☛

20✖✡✗✘

Figure 5.35: tchannel and veff calibrations in a pTC counter [40]. The original distribution (green)is smoothed (black), which is then fitted with a trapezoid function convoluted with a Gaussian(red). The best-fit trapezoid function (blue) gives tchannel from the trapezoid center and veff fromthe trapezoid width.
from the σvertex and σpreshower , the core resolution of the tLXe was evaluated to be 65 ± 6 ps.

5.5

pTC time calibration and resolution

5.5.1

Internal time calibration

The hit reconstruction of the pTC in Eq.(4.2) requires veff and tchannel calibrations. In their calibration, the hit w distribution was modeled as a trapezoid function convoluted with a Gaussian as shownin Fig.5.35. The effective velocity was fitted so that the trapezoid length agrees with the 12 cm
counter geometry, which gave veff = 12.4 ± 0.4 ps. The end-to-end channel time offsets, tchannnel ,were calibrated to have the center of the trapezoid at zero. The tchannnel calibration precision was
1.1 mm-equivalent, and the estimated resolution of the w reconstruction is 1.1 cm according to the
convoluted Gaussian function.

5.5.2

Track-based time calibration

The counter time reconstruction in Eq.(4.1) requires the calibration of the counter time offset,tcounter . It is calibrated with positron tracks by minimizing
X X(thitij − tTOFij − tdecayi − tcounterj )2 ,(5.14)i∈Track j∈Hit

where thitij is the time measured at j-th counter in the i-th track, tTOFij + tdecayi gives the j-thcounter hit time of the i-th track, and tcounterj is the offset parameter of the j-th counter. Eq.(5.14)
is minimized with respect to all the tdecayi and tcounterj parameters, where tdecayi (tcounterj ) gives
NTrack (512) degrees of freedoms in total.
Though it has a large number of degrees of freedoms, the whole matrix that appears in the
regression is sparse because each track can leave only up to ∼15 counters. In addition, this minimization is an idealistic linear least square problem. Therefore, there exists an efficient algorithm,
and it is implemented in the MillePede package [98], which was thus used in the calibration.

5.6. Combined teγ calibration and its resolution

81

[ns]

Toffset in laser mass test

60
58
56
54
52
0

100

200

300

400

500
PositionID

Figure 5.36: The measured time offset from the optical length of the laser system [42].

5.5.3

Laser based calibration

The track-based time calibration method discussed above is not sensitive to the global offset between
the upstream and the downstream sectors. It is therefore complemented by the laser-based calibration
method. In this method, the laser pulse gives the reference time with corrections on the opticallength of the system, where the correction parameters were measured beforehand (Fig.5.36). The
uncertainty of this method was evaluated to be 27 ps overall [41].

5.5.4

Performance evaluation with even-odd method

The pTC timing resolution was evaluated by the even-odd method, which splits hits into even hits
and odd hits in Eq.(4.7) asn

1X(thit2i − tTOF2i ),teven =n i=1
n1X
todd =(thit2i+1 − tTOF2i+1 ).
n i=1(5.15)The resolution of teven − todd is then evaluated as a function of 2n. With an assumption that theerror of the counter time measurements is independent, the full resolution is expected to be the sameas the resolution of (teven − todd )/2.
Fig.5.37 shows (teven − todd )/2 resolution as a function of 2n, which is fitted with a model of√
σn−hits = σsingle / npTC . This result indicates σsingle of 111.5 ± 0.8 ps, which is worse than the∼ 100 ps result evaluated previously in 2017 [40]. This is understood to come from the accumulated
radiation damage in these years.

5.6

Combined teγ calibration and its resolution

5.6.1

Offset calibration

The offset between the position decay time and the gamma decay time was calibrated with RMD
events observed in the 45 MeV < Eγ < 48 MeV region at a precision of 3 ps.
In addition to the offset between the positron and gamma, the residual offset between theChapter 5. Calibration and performance evaluation

82

Even odd resolution [ps]Even odd analysis120

χ2 / ndf
σsingle100

4.718 / 6
111.5 ± 0.7788

80
60
40
20
0

5

10

15
Number of hits

Figure 5.37: The time resolution evaluated with the even-odd method. The fitted value gives
σsingle = 111.5 ± 0.8 ps.
upstream pTC and the downstream pTC was also observed. This was calibrated by combining the
RMD events observed in the 45 MeV < Eγ < 48 MeV region and the RMD-enhanced calibration
samples. The residual offset was corrected according to the observed residual of 32 ± 3 ps. Thissuggests that the laser-based upstream vs downstream calibration was not perfect or had some biases,
which is still not fully understood.

5.6.2

Calibration for trigger and data pre-selection

As described in Sec.2.7.2, tpTC − tLXe without the TOF correction is used in the data pre-selection
scheme. Therefore, the window for tpTC −tLXe was optimized making use of the RMD events. Here,events with |te − tγ | < 4 ns were selected and the maximum and minimum value of tpTC − tLXe
defined the window. Similarly, the online time-coincidence trigger in Sec.2.6.2 was optimized at
the beginning of the physics data taking.

5.6.3

Full evaluation of teγ resolution

The teγ resolution was evaluated by decomposing it into the pTC contribution and the others,
where the pTC contribution is different event-by-event and the other contribution is constant. Thisis because the pTC average contribution becomes larger with a smaller Ee (presented later in
Fig.6.10), which means that the average σte for RMD events is worse than that for the signal events.
The decomposed resolution is estimated in a simultaneous fitting to the RMD-subsets categorized
by the number of hits on the pTC (Fig.5.38). The RMD response in each category is modeled as√
σTC / npTC ⊕ σconst , where σTC = 112 ps is from the even-odd analysis and σconst is the estimatedparameter. A naive fitting gave σconst = 73 ± 4 ps as a result. This term, expected to be dominatedby the LXe time resolution, agrees with the π 0 → γγ evaluation of 65 ± 6 ps. By combining these5.6. Combined teγ calibration and its resolution

180
160
140
120
100

160
140
120
100

−2

−1

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t eγ (ns)

120

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−1

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Events / ( 0.1 ns )

250

−1

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t eγ (ns)23t eγ (ns)23t eγ (ns)

RMD fitting: 6 ≤ nTC ≤ 7

RMD fitting: nTC = 5

Events / ( 0.1 ns )RMD fitting: nTC = 4

Events / ( 0.1 ns )

140

40

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−3

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Events / ( 0.1 ns )RMD fitting: nTC = 2

200

Events / ( 0.1 ns )Events / ( 0.1 ns )RMD fitting: nTC = 1

83

250
200
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400
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−1

RMD fitting: 12 ≤ nTC ≤ 29

Events / ( 0.1 ns )

500

0
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−2

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Events / ( 0.1 ns )Events / ( 0.1 ns )RMD fitting: 8 ≤ nTC ≤ 9

−1

2

3
t eγ (ns)

0
−3

−2

−1

0

1

Figure 5.38: Time resolution evaluation in the peak fitting to RMD events. Event samples are
divided into the shown 9 subsets according to the number of pTC hits and a simultaneous fitting
is performed (blue lines in each subset). The data distribution indicates a resolution improvementwith a larger number of npTC .

Chapter 5. Calibration and performance evaluation

84

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

Energy scale

Energy scale parameter vs channel

0.2

0.4

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2
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1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.8
1
charge (a.u.)

Figure 5.39: Spectrum of the observed
LYSO self radiation [49]. The peak corresponds to 596 keV, which is fitted (redline) to obtain the energy scale.

2.2

20

40

60

80
ChannelFigure 5.40: Energy scale distribution of LYSO
channels in the 2021 run after a high-voltage
adjustment.

two estimations, the full estimation of the teγ resolution became
112 ps⊕ 70 ps,√npTC(5.16)

whose average over signal events is σteγ ∼ 85 ps.

5.7

RDC LYSO energy scale calibration

Though the resolution requirements for the RDC detector are not high, moderate optimization andcalibration are still necessary. In particular, the LYSO counters need energy calibration in O(10 %)precision to have a uniform response to RMD-associated low-energy positrons. The RDC energy
scale is calibrated with the intrinsic radioactivity of the LYSO crystals, which gives the total gammaray energy deposit of 596 keV. The peak in Fig.5.39 corresponds to the 596 keV activity and thefitted parameter is used to calibrate the energy scale. The high voltage of the 76 MPPCs for the
LYSO crystals is also adjusted to have a high uniformity in the energy scale (Fig.5.40). The overall
precision of the energy scale calibration is a few percent, which is smaller than required.

5.8

CDCH calibration

5.8.1

CDCH Z measurement calibration

The CDCH z measurement in the hit reconstruction was calibrated by comparing the reconstructed
track z position with the reconstructed hit position by Eq.(4.4) and Eq.(4.5). The Gdiff and twireends
were calibrated to eliminate the offset between the measured z and the track-fitted z position. In
addition, non-linearity was found between the hit and the track (Fig.5.41), which were modeledwith polynomials and calibrated by fitting the observed distribution accordingly.
Positron tracks leave most of the hits in the wire center region whereas many cosmic-ray track5.8. CDCH calibration

85

200

zTrack vs zTimeDiff

histQGlobal

150
100

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1.527

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Std Dev x

55.41

Std Dev y

43.33

1000
800

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150
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Entries2053257

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Mean y0.181

Std Dev x

69.34

Std Dev y

43.04

50

0
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600

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400

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− 100
200
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0

100
200
zChargeDiv [cm]

0

1000

600
400
200

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− 200
−200

1200

800

− 100

− 150
− 200
−200

zTrack [cm]zTrack [cm]zTrack vs zChargeDiv

−100

0

100
200
zTimeDiff [cm]0

Figure 5.41: Observed nonlinearity of the z measurement in CDCH hit reconstruction for charge
division method (left) and time difference method (right). The profile of each scatter plot is fittedwith a polynomial (magenta curve).

hits can be found also in the wire end region. The Gdiff and twireends calibration, which does not
require hits in the end region, can be calibrated both with the positron tracks and cosmic-ray tracks.
On the other hand, the non-linearity calibration used cosmic-ray events because of the necessity ofa higher hit occupancy in the wire end regions.
The z measurement resolution in the hit reconstruction was evaluated on the positron tracks asshown in Fig.5.42. Though both the time difference method and the charge division method have acore resolution of 9 cm, the charge division method has a larger tail in the measurement. Therefore,a higher weight is assigned to zT in the full hit z reconstruction as zhit = 0.3 · zQ + 0.7 · zT , giving
an overall core resolution of 7.5 cm.

5.8.2

CDCH wire alignment

Though the CDCH wire positions were measured during the assembly, residual biases were observed
between the fitted track position and the initially estimated hit position (Fig.5.43). This was correctedin an iteration of wire position shifting by the observed average of the residual. Due to the DAF
algorithm in the track reconstruction, the hits used in Fig.5.43 are biased because hits are dropped
from the track fitting when the measured position is ill-consistent with the reconstructed track
position. Furthermore, each track is not sensitive to the wire misalignments in the parallel direction
to the track, which means that the averages of the residual are biased to give even smaller values
than the actual ones. Therefore, the wire shifting by the residual average needs to be iterated until
the update converges. Fig.5.44 shows the resulting x and y position shift before and after the
track-based wire alignment. The final corrections (after the iteration) to the wire shown in Fig.5.43
were ∼ 300 µm for both x and y coordinates, which are larger than the observed average residual of∼ 100 µm in Fig.5.43 because of the biases explained above.

Chapter 5. Calibration and performance evaluation

86

Charge division resolution

14000
12000

Time difference resolution

20000

hQ
EntriesMean
Std Dev
χ 2 / ndf
p0
p1
p2
p3
p4
p5

2746632
0.05404
17.81
1565 / 494
9675 ± 23.9
− 0.7962 ± 0.0150
9.309 ± 0.024
4612 ± 23.7
0.9757 ± 0.0292
25.84 ± 0.06

hT
Entries
Mean

18000

15.78
4216 / 494

p1
p2

14000

10000

0.02091χ 2 / ndf
p0

16000

2746632

Std Dev

1.437e+04 ± 2.176e+01
0.2295 ± 0.0100
8.658 ± 0.015

p33533 ± 19.0

p4

− 0.3781 ± 0.0318

p5

25.76 ± 0.07

12000

8000

10000

6000

8000
6000

4000

4000
2000
0

2000
−40

−20

0

0

20
40
z difference [cm]

−40

−20

0

20
40
z difference [cm]y (global) [cm]Figure 5.42: Estiamted z position difference between the measurement in hit reconstruction andthe fitted track position for the charge division method (left) and the time difference method (right).
The Gaussian fitting to each distribution indicates a 9 cm precision from each method, which gives
7.5 cm resolution when combined with an optimized weight.

0.1

histXYResidual-Wire239

0.08

Entries

439375

Mean x− 0.01109

0.06

Mean y0.01281

Std Dev x

0.02191

0.04

Std Dev y

0.02314

6000
5000
4000

0.02
0

3000

− 0.022000− 0.04− 0.061000− 0.08
− 0.1
− 0.1

−0.05

0

0.05
0.1
x (global) [cm]0

Figure 5.43: Distribution of the difference between the measured position of the hit reconstruction
and the fitted track position on a wire. The asymmetry of the distribution is caused by a misalignmentof the wire. Two line-shaped peaked regions are produced by the typical track crossing angle on thewire (one for tracks towards the inner direction and the other for those towards the outer direction).

5.9. Positron momentum resolution and calibration

87

Y update by wire alignment

[cm]

[cm]

X update by wire alignment

0.05
0.04

0.05
0.04

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0.03

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1500
wire id

−0.05
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500

1000

1500
wire id

Figure 5.44: Wire position change before and after the wire alignment in x-coordinate (left) andy-coordinate (right). As indicated, typical misalignment is O(100 µm).

5.8.3

CDCH drift time to distance calibration

The time-to-distance relation, ζtxy (t) in Eq.(4.6), is based on Garfield++ simulation results. This
simulation-based conversion can also be used to give a drift time estimation for each hit as−1
tdrift = ζtxy
(dtrack ),(5.17)

where tdrift is the drift time estimated from the gas table, and dtrack is the estimated drift distanceaccording to the track reconstruction. The wire-to-wire time offset of the CDCH readout can thenbe aligned bytwireoffset = thit − tdrift − tpTC + tTOF ,(5.18)
which compares the pulse time with the estimated arrival time of the gas cluster on the wire.
The full calibration refinement of the ζtxy (t) itself, on the other hand, was performed by amachine learning-based approach, which was trained on the data. The machine learning model has
several inputs to include different entangled dependencies: the layer index, the reconstructed hit
time, and the amplitude of the waveform. The final resolution of the drift distance measurement is
150 µm as shown in Fig.5.45.

5.9

Positron momentum resolution and calibration

5.9.1

Energy scale and response evaluation

To estimate the energy scale and the detector resolution, the energy spectrum of the Michel positron
was fitted with the below model
Fit(Ee ) = Theory(Ee ) × Acceptance(Ee ) ⊗ Resolution(Ee ).
(5.19)The theory spectrum uses the result in [33], which includes one-loop radiative corrections. The
acceptance function, which describes the efficiency dependence on Ee , was modeled with errorChapter 5. Calibration and performance evaluation

88

Drift distance resolution
h

14000

Entries

600654Mean

8.861e − 05

Std Dev0.0198312000

χ 2 / ndf

1937 / 194

p0

1.287e+04 ± 3.994e+01

p1−2.819e −05 ± 2.704e − 05

10000

p2

0.01529 ± 0.00005

8000

p3

1225 ± 43.0

p4

0.0006491 ± 0.0001350

p5

0.03357 ± 0.00033

6000
4000
2000
0
−0.1

−0.05

0

0.05
0.1
dhit− dtrack [cm]Figure 5.45: Distribution of the difference between the measured drift distance in the hit reconstruction and that of the fitted track. The Gaussian fitting to the distribution indicates a 150 µm
precision.

Michel edge fitting

6000

4000

2000

0

40

45

50

55

[MeV]

Figure 5.46: Observed positron momentum spectrum and the fit (red line) to evaluate the energy scaleand resolution. The dashed blue line shows the estimated response to 52.8 MeV positron, giving
the resolution value of 90 keV. The gray curve is the expected spectrum without the acceptancefunction, i.e. the convoluted spectrum of the theory function with the resolution function.

5.9. Positron momentum resolution and calibration

89

Momentum uncertainty in track fit

9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0

50

100

150
200
E e uncertainty [keV]

Figure 5.47: Distribution of positron momentum uncertainty estimated in the track fitting.
functions. The resolution function was modeled with a sum of three Gauss functions. Fig.5.46
shows the fit result for the reconstructed energy spectrum. The energy scale of the positrons wascalibrated with a 0.01 % precision by correcting the offset of the response function. The momentum
resolution was evaluated to be 90 keV from its resolution parameters.
In the positron tracking, the Kalman filter estimates the uncertainty for each fitted parameter.
The estimated momentum uncertainty is shown in Fig.5.47, where the overall behavior agrees withthe estimated average resolution of 90 keV. The resolution dependence on the tracking uncertainty
is also evaluated by fitting to the Michel edge of sliced data samples, which will later be shown in
Fig.6.8. As a result, we found a clear correlation between the tracking uncertainty and the estimatedresolution in the Michel fitting, as shown later in Fig.6.9. This, however, also indicates 5 – 10 %underestimation of the resolution by the track fitting, which likely suggests that some effects are
missing when the tracking uncertainty is estimated. Though we have not yet reached any conclusion,
one suspicious cause is that the scattering inside the wire material is not correctly taken into account.
In the distribution of momentum uncertainty, we see the lower bound at ∼ 55 keV, which isunderstood to be dominated by the uncertainty due to the scattering in the material. On the other
hand, the broader range of the uncertainty distribution reflects the quality of the track fitting. The
peak around 60 keV is made by the double-turn tracks, which have abundant hits along its path in
the CDCH. The small bump around 100 keV is made by tracks that are not efficient in finding hits
in the last half-turn before entering the pTC.

5.9.2

Alignment of magnetic field

The gradient magnetic field requires a good alignment with the CDCH to have a good uniformity
in the energy scale with respect to the positron emission angle. The impact of the magnetic fieldmisalignment is illustrated in Fig.5.48, which shows θe and φe dependence of the energy scaleevaluated from the Michel edge fitting to sliced data samples. The misalignment in the positive
x direction results in a decreasing energy scale with an increasing φe . The misalignment in thepositive y direction results in an energy scale maximized at |φe | = 0. The misalignment in theChapter 5. Calibration and performance evaluation

90

φ dependence

offset [MeV]

offset [MeV]cosθ dependence

0.04

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0.02

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Original alignment
Original + X shift 1mm
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Original + Z shift 1mm−0.06

Original alignment
Original + X shift 1mm

−0.08
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Original + Z shift 1mm

−0.2

0

0.2

0.4
cosθ

−0.08
−50

0

50
φ [deg]Figure 5.48: Magnetic field alignment with the angular dependence of the energy scale. The impactof misalignment in each direction is shown as well as the observed angular dependence. The y-axis
indicates the bias on the positron momentum for 52.8 MeV tracks.
positive z direction results in an increasing energy scale with an increasing cos θe .
The magnetic field was thus aligned to minimize the observed energy scale dependence on theemission angle. As a result, the magnetic field calculated from the COBRA design was shifted by
100 µm in x, 700 µm in y, and 300 µm in z, where the estimated alignment precision in x and z
direction was 100 µm and that in y was 200 µm.

5.10

Positron resolution in double turn analysis

The vertex resolution and the angle resolution of the positron tracking were evaluated with doubleturn tracks described in Sec.4.1.9, which account for 10 % of all the tracks. The first and the second
turn tracks are separately fitted and the reconstructed position and angle are compared at the middle
point. The difference receives the contribution of the first and second turn resolution. In addition,
the first turn includes hits only in the first turn, which is less than the usual 1.5 turns included inmost of the tracks. These effects were evaluated with MC samples, in which the true MC resolution(σ MC truth ) can be evaluated as well as the MC double turn resolution (σ MC DT ).
The data resolution was finally evaluated by scaling the data double turn resolution with thiscorrection factor;
σ data = σ data DT ×

σ MC truth.
σ MC DTThe estimated resolution on the data is σye = 0.7 mm, σze = 2.0 mm, σθe = 7.2 mrad, and
σφe = 4.1 mrad.
The double turn resolution can also be evaluated on pull, which for θe is defined as δθe /σθ0 e .
Here, δθe is the error of θe measurement and σθ0 e is the nominal θe uncertainty according to the trackfitting. The resolution of the pull, denoted hereafter as sφe , sθe , sze , sye , was also evaluated with the5.11. Global detector alignment

91

double turn analysis asθefirst−turn − θesecond−turn
DT
q
sdata
=
,
θe
0 second−turn 2
0 first−turn 2)) + (σθe(σθe(5.20)

which was then corrected with the correction factors estimated with MC samples. The correction
factors, sMC truth /sMC DT , are introduced to correct the difference in the pull resolution between the
double turn analysis and the MC truth, which were observed to be ∼ 10 % in MC samples. As aresult, the estimated pull resolution is sφe = 1.3, sθe = 1.2, sze = 1.9, and sye = 1.2.
The double turn analysis can also be used to cross-check the momentum resolution evaluatedin the Michel edge fitting (Sec.5.9) though the latter is more reliable because it is based on thefull tracks (i.e. not based on tracks split into different turns). In particular, it is interesting toDTevaluate the pull resolution according to Eq.(5.20). The estimated values are sdata= 1.2 andEe
MC truth MC DT
sEe/sEe= 0.85, which gives sEe = 1.05. This agrees with the result of Michel fitting to thesliced data samples (Fig.6.9), including the suggested 5 – 10 % underestimation of the resolution in
the track fitting.

5.11

Global detector alignment

5.11.1

Target deformation and time shift of position

The target shape was measured by a CT scan before the installation, which was then combined withthe optical marker position measurement after the installation. The time variation of the overalltarget rotation and translation was also followed by the printed dot markers reconstructed in thephotograph data. Due to the absence of a photograph taken simultaneously with the optical marker
measurement, the target position history is not fully known exactly. As a compromise, the positionin the optical measurement was assumed to be the same as that in the first target photograph. The
possible difference between them will be compensated by the target hole analysis described later in
Sec.5.11.2. The position during the absence of the camera is assumed to be randomly distributed,
and a large angular uncertainty is assigned to reconstructed positrons in those periods.

5.11.2

Alignment between CDCH and target

The target hole analysis evaluates the residual misalignment of the nominal hole position, which is
obtained by combining the CT scan and target photographs. This analysis makes use of the holes
in the reconstructed position distribution on the target (Fig.5.49). The y and z positions of eachhole can be straightforwardly estimated from the dip position. The x misalignment geometrically
produces an angle-dependent bias in the y-reconstruction as δy ≈ tan φ·δx. Therefore, x-coordinate
can also be aligned as shown in Fig.5.50 from the dependence of the estimated y-position of eachhole on the φ emission angle. With the nominal position given for each of the holes, this analysis
independently estimates hole-by-hole residual misalignment. The full detail of the analysis method
and its systematics are described in Appendix.D.
The estimated precision of the alignment at each hole was composed of 100 – 200 µm statisticaluncertainty in each of x, y, z coordinate, and the systematic uncertainty of (50 µm, 100 µm, 200 µm)(see Tab.D.1 for the detail). The global translation and rotation were estimated by combining theresults on different holes, and the uncertainty was 100 µm for the translations and 6 mrad (1.4 mrad)
for the rotation around the target long (short) axis. The estimated shift, (100 µm, 800 µm, 400 µm),

Chapter 5. Calibration and performance evaluation

92

y [cm]

5

800

4
700

3
2

600

1

500

0

400

−1

300

−2
−3

200

−4

100

−5
−15

−10

−5

0

5

10

15
z [cm]

0

Figure 5.49: Reconstructed position distribution on the muon stopping target projected on the yz
plane. Six holes are visible as the dips of the distribution.

Y center [cm]tanΦ dependence

0.14

χ2 / ndf

7.658 / 5Prob0.1761

0.12 p00.06185 ± 0.00503

p1

0.02928 ± 0.00817

0.1
0.08
0.06
0.04
0.02

−1

−0.5

0

0.5

1
tanΦ

Figure 5.50: Estimated y position of a hole with positrons of different tan φ. The slope of the linear
fit (300 µm) gives an estimate of x-misalignment, and the offset of the fit (600 µm) gives an estimateof y-misalignment.

5.11. Global detector alignment

Reconstructed position of LXe
Extrapolated track position

Counts

93∆z
Entries3481
Mean−0.08793
Std Dev
4.047
Prob0.3703
Constant
45.6 ± 1.0
Mean−0.1014 ± 0.0800
Sigma
4.346 ± 0.070

60

50

40

Typical 𝜸-ray direction

30

20

10

Cosmic-ray track

0
−10

−8

−6

−4

−2

0

2

4

6

8

10

zLXe - zCDCH [cm]Figure 5.51: Illustration of alignment of
the LXe detector and CDCH with linear
cosmic-ray tracks with an indication of biases.

Figure 5.52: zLXe − ztrack distribution and its
fit with a Gauss function [58]. The fit indicatesa 1 mm misalignment between the LXe and
CDCH.

Table 5.1: Alignment summary. The estimated relative position with respect to the CDCH nominal
position is tabulated.
Sub-detector
Magnetic field (Sec.5.9.2)
Muon stopping target (Sec.5.11.2)
LXe detector (Sec.5.11.3)

δx100 ± 100 µm
80 ± 100 µm
no evaluation

δyδz

700 ± 200 µm 300 ± 100 µm
860 ± 100 µm 500 ± 100 µm
no evaluation 1000 ± 800 µm

is attributed mostly to the CDCH misalignment, because a similar result was obtained also in CDCH
vs COBRA alignment discussed in Sec.5.9.2.

5.11.3

Alignment between CDCH and LXe

The CDCH and the LXe were aligned with cosmic-ray tracks in data taken without a magnetic
field. The reconstructed linear tracks in the CDCH were extrapolated to the inner face of the
LXe (r = 64.97 cm), and the reconstructed position in the LXe was compared with that of theextrapolated track (Fig.5.51). Here, the position reconstruction in the LXe had a bias to give a
deep point (not on the inner face), which biased the reconstructed u, v position according to thetrack direction. Because of this bias and the angle distribution of the cosmic-rays, the v-coordinate
difference cannot be reliably aligned. On the other hand, the u-coordinate (or equivalently zcoordinate) can be aligned by exploiting the geometrical upstream vs downstream symmetry of thedetector; namely, the bias, which cannot be eliminated track-by-track, was expected to be cancelled
out in the overall zLXe − ztrack distribution. Fig.5.52 shows its distribution suggesting a 1 mmmisalignment between the LXe and CDCH.

5.11.4

Alignment summary

The results of detector alignment are summarized in Tab.5.1. Here, we see a good agreement
between the target position and the magnetic field position within each uncertainty. However, bothof them show large shifts in the y-axis with respect to the nominal CDCH position according to anChapter 5. Calibration and performance evaluation

94

2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0

EventsEventsEBGO with BGO self trigger

400

Subtracted

350

Scaled BG

300
250
200
150
100
50
50

100
150
EBGO[MeV]

0

0

20

40

60
80
EXEC[MeV]

Figure 5.53: The background estimation in the efficiency measurement [48].
optical survey of wire ends. The z misalignment also shows a similar trend; all three sub-detectors
are offset in the positive direction and all the shifts agree with each other within the uncertainties.
Though these are understood to be likely a real misalignment of the CDCH, they leave a ∼ 1 mm
uncertainty in the absolute CDCH position. This uncertainty on the CDCH absolute position mattersin the global CDCH vs LXe alignment because there was no direct alignment method between themin x and y directions.
Final alignment uncertainty
With all the above considerations, the alignment uncertainties are concluded as below.
• The uncertainty on relative x-shift between the CDCH and the LXe is 100 µm,• The uncertainty on the shift between the CDCH and LXe is 1 mm.

5.12

LXe efficiency evaluation

The detection efficiency of the LXe detector, defined as the fraction of the number of detected signal
events out of those coming in the fiducial volume, was evaluated with the π 0 → γγ events triggeredby logic only on the BGO detector. By selecting BGO hits estimated to be from 83 MeV gammarays, 55 MeV gamma-ray samples emitted in the LXe direction can be obtained. By counting theefficient ones among them, the efficiency of the LXe gamma-ray detection can be evaluated.
In the efficiency measurement, there were background events associated with neutron capture
events, π − p → nγ, with the final state gamma of 129 MeV (recall Panofsky ratio in Eq.(2.6)). The
contamination in the BGO-based event tagging, which was evaluated from the spectrum fitting to
the 129 MeV peak in the reconstructed BGO energy (Fig.5.53), gave an uncertainty of ∼ 1 %.
In order to correct the difference in the material budget between the physics data and π 0 → γγ
data, the measured efficiency was compared with MC samples simulating π 0 → γγ events thatleave hits in the BGO detector as shown in Fig.5.54. There was an unexplained 2.9 % discrepancy
between them, which is suspected to be from the wrong incorporation of the material budget eitherof the LXe detector components or the liquid hydrogen target components. Here, the former case

5.13. Positron efficiency evaluation95

Efficiency curve

Efficiency

0.75
0.7

0.65
0.6
MEG II

0.55
MEG II MC

0.5
44

46
48
50
52
54
Lower energy threshold [MeV]

Figure 5.54: Data vs MC comparison of the efficiency analysis result [48]. 1.8 % out of the 4.9 %discrepancy is understood to come from inactive volume not included in the simulation.
results in a reduction of the efficiency also in the µ → eγ detection while the latter has no impact
on µ → eγ. This therefore gives an additional uncertainty to the efficiency evaluation for the signal
gamma-ray detection. With all the uncertainties included, the estimated efficiency for the 52.8 MeV
gamma-rays is γ = 0.67 ± 0.02.
This, so far, considers only the efficiency associated with the gamma-ray interaction inside thematerial, which does not include the inefficiencies of the analysis. In reality, the final µ → eγ
analysis is also subject to the inefficiency of the gamma-ray selection described in Sec.4.2.6, which
is dominated by the selection of the pileup analysis. This additional contribution was evaluated to
be 93 ± 2 %, as will be detailed in Sec.6.3.1.

5.13

Positron efficiency evaluation

5.13.1

CDCH hit efficiency evaluation

The hit efficiency on a layer of CDCH was evaluated by tracking without the wires belonging to thelayer and then extrapolating the tracks to the turned-off wires. This gave candidate hit positions on
the turned-off wires with estimations of ddrift . These candidate hits were then matched, if existed,with reconstructed hits on the wires. In order not to pick up hits from pileup positrons, a consistency
cut was applied on the difference between the measured drift distance and the track-estimated drift
distance (Fig.5.55). The efficiency was measured from the fraction of consistently matched hits on
the turned-off wires in all the candidate hits.
This analysis was performed both on data and MC simulation samples, and the results are
summarized in Tab.5.2.
The data and MC simulation agreed with each other within theiruncertainty of 3 %, which is dominantly from the ambiguities in the definition of pileup rejection.
The degradation of the hit efficiency with an increased beam intensity is expected because pileupsChapter 5. Calibration and performance evaluation

96

ddrift comparison for hit candidates

1600"pileup""efficient""pileup"

0

0.1
0.2
dhit− dtrack [cm]

1400
1200
1000
800
600
400
200
0
−0.2

−0.1

Figure 5.55: Distribution of drift distance difference between the wire-crossing position of tracks
and found hits on the same wires. The peak is from hits that are really from interaction with thetested track, while a longer tail than the distribution in Fig.5.45 is also present because of pileup
hits. Hits that happen in the peak region of the distribution are counted as efficient hits, which aredefined by the cut range depicted in the plot.

Table 5.2: Results of hit efficiency evaluation of the CDCH both for MC and data.
Condition
Hit efficiency
7
Data at 3 × 10 µ/s
67 ± 3 %7
Data at 5 × 10 µ/s
64 ± 3 %7
MC simulation at 3 × 10 µ/s 70 ± 3 %5.13. Positron efficiency evaluation

97

Table 5.3: Results of positron tracking efficiency evaluation. Estimated efficiency with and without
the use of the Machine Learning method in the hit reconstruction (introduced in Sec.4.1.2) iscompared as well as the muon stopping rate dependence.
Condition
Data at 3 × 107 µ/s
Data at 5 × 107 µ/s

Tracking efficiency
67 %
60 %

Tracking efficiency(without ML hit finding in CDCH)
60 %
48 %

in the waveform analysis can make hits inefficient. This inefficiency, however, was found to be small
in this analysis.

5.13.2

Positron tracking efficiency evaluation

The positron tracking efficiency is defined by the fraction of high-quality positron reconstruction
out of all the 52.8 MeV positrons emitted in the opposite of the LXe fiducial region. It was evaluatedwith data samples triggered only on the pTC, which gives minimum-biassed samples of Michel
positrons. The efficiency was estimated by dividing the counted number of Michel positrons by the
expected number. The denominator in this calculation was evaluated as a product of the beam rate,
geometrical acceptance of the LXe detector, and the trigger efficiency for the pTC-only trigger. The
counting method for the numerator is used also in a normalization estimation method discussed
later in Sec.6.3.1, and all details about the other necessary corrections will be described there.
The estimated efficiencies are summarized in Tab.5.3. The uncertainty of the evaluation isdominated by the muon-stopping rate estimation. It can also be seen that the ML hit finding methodin the waveform analysis gives a higher efficiency; ×1.1 higher at 3 × 107 µ/s and ×1.2 higher
at 5 × 107 µ/s. We can also find an efficiency decrease with an increased beam rate. As the hit
efficiency in Tab.5.2 is comparable even at a higher beam rate, this inefficiency is mostly attributed
to algorithm inefficiency of the track finding in a high pileup environment.

98

Chapter 5. Calibration and performance evaluation

Chapter 6
Analysis
6.1

Analysis framework

6.1.1

Likelihood function

An extended unbinned maximum likelihood fit to the data samples is performed to estimate
Nsig , which is the expected number of signal events in the dataset. As nuisance parameters,
NAcc , NRMD , XTGT are also introduced in the fitting to incorporate part of systematic uncertainties
in the fitting. Here, NAcc is the expected number of accidental events (see Sec.1.2.2), NRMD is thatof RMD events (see Sec.1.2.3), and XTGT is a parameter to incorporate uncertainties in the targetalignment (see Sec.5.11).
The full likelihood function on these fit parameters is described as
L(Nsig , NRMD , NAcc , XTGT )(XTGT )2:= exp −
2
2σTGT




(NRMD − hNRMD i)2(NAcc − hNAcc i)2
× exp −
× exp −
2
2
2σRMD
2σAcc
Nobs

e−(Nsig +NRMD +NAcc ) Y
×Nsig S(~xi |XTGT , q~i ) + NRMD R(~
xi |~qi ) + NAcc A(~
xi |~qi ) ,
Nobs !
i=1(6.1a)(6.1b)(6.1c)

where Eq.(6.1c) is the usual extended likelihood function [99], Eq.(6.1a) is an external constraint
term for the detector alignment uncertainty, and Eq.(6.1b) is terms giving constraints to the number ofthe background events. These constraints are based on uncertainties from subsidiary measurements.
In the extended likelihood term, Eq.(6.1c), Nobs is the number of observed events. S(~
xi |XTGT , q~i ),
R(~
xi |~
qi ), and A(~
xi |~qi ) are probability density functions (PDF) for signal, RMD, and accidental
events, respectively. Their details will be described in Sec.6.7 (for signal), Sec.6.8 (for RMD background), and Sec.6.6 (for accidental background). The x~i is a set of fit observables that discriminate
signal from backgrounds, and q~i is a set of conditional observables, whose details will be discussed
later in Sec.6.1.2. The implementation of this analysis is based on RooFit [100, 101], which is a
dedicated toolkit for statistical data analysis and is widely used in high-energy physics experiments.
Conversion to branching ratio
As the likelihood analysis just estimates Nsig , we need to normalize it to obtain information about
the branching ratio. As has already been introduced in Eq.(1.8), we obtain the branching ratio by99

Chapter 6. Analysis

100

Table 6.1: Values used in the fitting
Condition

Used tRDC (ns)

Used ERDC (MeV)

10

−1
−1
20
ERDC

|tRDC | > 9.5 ns|tRDC | < 9.5 ns∧ Inefficient LYSO matching|tRDC | < 9.5 ns ∧ ERDC > 20 MeV|tRDC | < 9.5 ns ∧ ERDC > 20 MeV

tRDC
tRDC
tRDC

a conversion of B(µ → eγ) = Nsig /k. Here, the k parameter corresponds to the effective numberof measured muon decays. The estimation of this parameter in the 2021 data will be discussed inSec.6.3.

6.1.2

Observables of PDFs

Fit observables
The fit observables, x~i , consist of the following parameters.
1. Ee ; positron momentum
2. Eγ ; gamma energy
3. teγ ; time difference between positron and gamma
4. npTC ; the number of positron hits on the pTC
5. θeγ = (π − θe ) − θγ ; difference in θ
6. φeγ = (π + φe ) − φγ ; difference in φ
7. tRDC ; timing of the selected RDC hit
8. ERDC ; energy of the selected RDC hit
Handling of RDC parameters
In the likelihood analysis, the RDC parameter ranges are defined as −10 ns < tRDC < 10 ns
and −1 MeV < ERDC < 20 MeV. When no RDC hit is found within the timing range of|tRDC | < 9.5 ns, the event uses tRDC = 10 ns, ERDC = −1 MeV, which represents inefficient
events. In the other events, the reconstructed time is filled in tRDC . Among them, ERDC = −1 MeV
is used if LYSO matching is inefficient, ERDC = 20 MeV is used when the reconstructed energy
is larger than 20 MeV, and the reconstructed energy is filled otherwise. Summarized in Tab.6.1 isthe value used in the fitting in each condition. Note here that the RDC energy reconstruction is
implemented so that the reconstructed ERDC value is always numerically positive.
Handling of npTC parameter
When the reconstructed npTC is larger than 16, nhit = 16 is used instead and the overall timing
resolution of the pTC measurement is also calculated accordingly. This is because such a large
number of hits is from rare tail events, which may occasionally cause a large bias in the fitting.
√
Another reason is that the σte resolution improvement as σsingle / npTC saturates there.

6.1. Analysis framework

101

Conditional observables for per-event PDFs
The conditional observables, q~i , are introduced to build the PDFs considering event-by-event differences of detector response (per-event PDFs). They consist of the following parameters
1. vγ and φe ; the gamma conversion point in XEC local v coordinate, or equivalently the positron
φ angle.
2. wγ ; the gamma conversion point in XEC local w coordinate
3. ~σe ; the tracking uncertainties estimated by the Kalman filter on positron angle, position, andmomentum.

6.1.3

Toy MC simulation

Once we have a full set of PDFs for the fitting, it can also be used to simulate experiments bygenerating events. This simulation method, often called toy MC, is a powerful tool to study thestatistical behavior of the experiment. Note that this simulation is distinguished from the full
simulation described in Sec.2.8. The toy MC simulates experiments based on an already estimateddetector response, which is incorporated in the PDFs. The full simulation, on the other hand, aims
to understand the detector response itself.
As we build per-event PDFs, the toy MC event generation depends on the conditional observables,
which themselves must be generated with correct weights. This is achieved by directly samplingthem from observed events in data, which ensures that all the complications (such as correlations) of
the conditional observables are correctly taken into account. Each toy MC experiment is generated
by making a set of events taking into account the Poisson fluctuation according to Nsig , NAcc , and
NRMD .

6.1.4

Frequentists-based statistical method

This analysis gives a result based on the frequentists approach, where the confidence interval on
Nsig is constructed based on the Feldman-Cousins approach [102]. The test statistic is built fromthe profile likelihood ratio, λ(Nsig ), which is given by

 
ˆ
ˆ
ˆ

L Nsig , N̂Acc (Nsig ), N̂RMD (Nsig ), X̂TGT (Nsig )





(when N̂sig > 0)
L N̂sig , N̂Acc , N̂RMD , X̂TGT


(6.2)
λ(Nsig ) =ˆ (N ), N̂
ˆ
ˆ

L
N̂
(N
),
X̂
(N
)
N
,
sig
Acc
sig
RMD
sig
TGT
sig





(when N̂sig < 0).
ˆ (0), N̂ˆˆL 0, N̂(0), X̂
(0)
Acc

RMD

TGT

Here, the single hat variables — N̂sig , N̂Acc , N̂RMD , and X̂TGT — are the values that maximize theˆ (N ), N̂ˆˆlikelihood, and the double hat variables — N̂
Acc
sig
RMD (Nsig ), and X̂TGT (Nsig ) — are thevalues that maximize the likelihood with a fixed value of Nsig . Eq.(6.2) is split into two lines because
the best-fit value in the denominator is searched for within the physical region, namely Nsig ≥ 0.
This test statistic offers a good property for a statistical test; a larger − log λ(Nsig ) suggests that thetested value of Nsig is less likely the case.
We then introduce CL(Nsig ), which is the probability of − log λ(Nsig ) being smaller thanthe observed − log λ(Nsig ) value when the tested value of Nsig is correct 1. This probability is

1Note here that (by definition) 1 − CL(Nsig ) is identical to the p-value of the hypothesis that the true value is Nsig .

Chapter 6. Analysis

102

directly calculated from an ensemble of toy MCs simulated with the given Nsig . Note that theasymptotic formula of − log λ(Nsig ) distribution presented in [103], which is widely used in manyother experiments, is not used in this analysis because this formula is known not to be precise in alimited statistics regime.
The confidence interval is set by requiring CL(Nsig ) to exceed the defined threshold; CL(Nsig ) >0.9 is required in this analysis to set a 90 % confidence interval. This interval of Nsig is finallytranslated into the interval of B(µ → eγ) with a multiplication of 1/k.

6.1.5

Theory independence of the analysis

The theory uncertainty on the signal polarization appears in θe distribution. In this analysis, however,θe is not included as an observable — neither as a fit observable nor as a conditional observable —,which means any θe dependencies are integrated out from all the PDFs. In particular, the signal
PDF is formalized as
Z
S(~x|~q) = S(~x|~q, θe )S(θe )dθe .
(6.3)
Here, thanks to the upstream vs downstream symmetry of the detector and the anti-symmetry of theθe -dependence in Eq.(1.4), the S(~x|~q) is polarization independent; namely
Z
Z
S(~x|θe )SR (θe )dθe = S(~x|θe )SL (θe )dθe ,(6.4)
where SR(L) (θe ) is the θe distribution in the full right (left) polarization case. Thus, any models
with an arbitrary |AR |/|AL | ratio can be consistently tested with a single PDF obtained as Eq.(6.3).

6.1.6

Incorporation method of systematic uncertainties

The systematic uncertainties are included in two methods. The first one randomly samples uncertain
parameters during the toy MC generation (analogous to [104, 105]), which broadens the distributionof − log λ. As this distribution is used in the CL evaluation, the broader distribution results ina reduction of CL value at each Nsig giving an enlarged confidence interval. The other method
includes uncertain parameters as fitted nuisance parameters in the likelihood functions; the XTGT
parameter in Eq.(6.1), for example. As stated in Wilk’s theorem [106], it is expected not to changethe − log λ distribution calculated by the toy MC production. Instead, when it is adapted to data,
the profiling reduces − log λ at non-best Nsig points. Therefore the CL value compared at the same
test
best
|Nsig
− Nsig| gets smaller than the case without profiling, which broadens the confidence interval.
As a general behavior, the profiling method is more robust in particular when the signal strengthis finite. Its drawback is a large increase in the CPU cost, and it is a serious problem, especially inthe fitting with unbinned PDFs or per-event PDFs; in our case, the fitting takes ×4 or more if one
nuisance parameter is added to incorporate PDF uncertainties. Therefore, only important parameters
can be profiled in the analysis. Such parameters are roughly identified by the relative uncertainty
level with respect to the resolution parameters used in the signal PDFs. Among the uncertainties, thelargest relative uncertainty is on the signal φeγ PDF due to the large contributions from alignment
uncertainty. The introduced nuisance parameter, XTGT is an approximate combination of target x
misalignment and LXe misalignment in (x, y) plane. The other uncertainties are incorporated in
the first method, namely the numerical sampling of uncertainties in the toy MC generation.
The profiled XTGT parameter changes the center of the φeγ PDF as
S(φeγ |XTGT , θeγ , Ee · · · ) = G(φeγ , Φ(XTGT ) + F(θeγ , Ee · · · ), Σ(θeγ , Ee · · · )),(6.5)

E γ [MeV]

6.2. Analysis window

103

60

90

58

80
70

56

60
54

50

52

40

50

30

48

20
10

46
−3

−2

−1

0

1

2

3
t eγ [ns]0Figure 6.1: Distribution of accessible events before the unblinding on (teγ , Eγ ) parameter space.
48 MeV < Eγ < 58 MeV and |teγ | < 1 ns region is null because of the blinding.
where G represents a Gaussian function and Φ is a function that geometrically transforms XTGT into
event by event change in φeγ value. The other parameters appearing in the equation, F(θeγ , Ee · · · )and Σ(θeγ , Ee · · · ), will be fully given in Sec.6.7.4.

6.2

Analysis window

After applying the selections described in Sec.4.4.3, data samples in the analysis window defined
below are finally used in the likelihood analysis;• |teγ | < 0.5 ns• 52.2 MeV < Ee < 53.5 MeV• 48 MeV < Eγ < 58 MeV• |θeγ | < 40 mrad
• |φeγ | < 40 mrad
In order to avoid psychological human biases, events that satisfy both 48 MeV < Eγ < 58 MeV
and |teγ | < 1 ns were blinded until the final fit in the analysis region as can be seen in Fig.6.1, which
is a projection of events onto Eγ vs teγ plane. In addition to the main analysis window, energy
sideband (45 < Eγ < 48 MeV) and timing sideband (1 < |teγ | < 3 ns) regions are used to examine
background events.

6.3

Normalization

6.3.1

Michel positron counting method

One method to evaluate the normalization factor, k, is based on a Michel positron counting, where
the number of positrons with Ee > 50 MeV is counted in the dataset taken by the pre-scaled

Chapter 6. Analysis

104

Table 6.2: Parameters used in the k-estimation with the Michel positron counting method.
B eν ν̄
eγ
trg
ν̄
eν
trg
eγ eν ν̄e /eP eν ν̄
γ
Aeγ
γ
eγ
sel

0.101
0.77 – 0.79
∼ 0.9 depending on beam rate (Eq.(6.7))
1.09
6
6
2 × 10 – 7 × 10 depending on periods
0.67 (detection efficiency) × 0.93 (analysis efficiency)
0.97
0.93

self-trigger on the pTC. The parameter is estimated as
kMichel =eγN eν ν̄ trg eγ
P eν ν̄eγe···· γ · Aeγγ · sel .
ν̄
eν ν̄
eγ
B eν ν̄ eν

P
trg
e

(6.6)

Here, N eν ν̄ is the counted number of Michel positrons, which amounts to 114739 in total. B eν ν̄ =0.101 is the branching ratio of decay with Ee > 50 MeV. P factors are the pre-scaling factor of thephysics trigger and pTC self-trigger. The pre-scaling for the physics trigger, which ideally should
be 1, was not 1 at the beginning of the data taking as was presented in Fig.3.5. The pre-scaling
factor for the pTC self-trigger ranged between P eν ν̄ = 2 × 106 – 7 × 106 , also depending on the
period.  and A factors, of which detail is discussed in the next paragraphs, are introduced to correct
efficiencies and detector fiducial volume. The numbers are summarized also in Tab.6.2.
Trigger efficiency correction
The trigger inefficiency of the pTC self-trigger was caused by the deadtime of the trigger logicν̄(τdead = 50 ns), which causes a paralysis of the trigger functionality. Its correction factor, eνtrg , isthus evaluated from the measured rate of fired trigger (rfired ) by solving
rfired = rtrue exp(−τdead · rtrue )rfiredrfired= eν ν̄ exp −τdead · eν ν̄ ,
trg
trg

(6.7)

where rtrue = rfired /trg is the true rate of the events that meet the trigger logic. The estimatedcorrection factor from this inefficiency is ∼ 1.1 depending on the muon stopping rate.
The trigger efficiency of the physics data, eγtrg , was estimated by comparing the teγ , Eγ , Θeγdistribution with that in data taken with a looser trigger threshold; namely wide timing logic, low
Eγ threshold logic, and wide-angle logic. As a result, eγtrg was estimated to be 0.77 – 0.79 dependingon the data-taking period, which reflects the difference in the trigger threshold optimization.
Positron efficiency correctioneν ν̄The positron efficiency correction, eγ
was evaluated from the energy spectrum fitting of thee /eMichel positrons, which was introduced in Eq.(5.19). Here, the fitting was performed on the pTC
self-trigger data because the efficiency dependence on Ee is different between the timing sideband
data (i.e. the data collected by the physics trigger) and the pTC self-trigger data. This differenceis understood to be from the direction match logic of the physics trigger. The estimated correctioneν ν̄factor is eγ= 1.09.
e /e6.3. Normalization

105

Gamma-ray acceptance

Positron acceptancez
Figure 6.2: Difference in the definition of acceptance between positron and gamma-ray. The redregion in the triangular lines corresponds to the positron acceptance and the blue rectangular region
corresponds to the gamma-ray acceptance. The gamma-ray acceptance is smaller than the positron
acceptance, which is taken into account in Aeγγ .
Gamma efficiency correction
One contribution to the gamma efficiency is from the shower leakage or the absorption of gammarays in the material in front of the detector. This was evaluated to be 67 ± 2 % with BGO-single
dataset taken during the π 0 → γγ calibration as presented in Sec.5.12.
The other contribution to the gamma efficiency is from the reconstruction efficiency with thepileup analysis. This was evaluated with signal MC samples and corrected according to a data vsMC comparison. The data and MC were compared with the background gamma-rays; i.e. the datain the timing sideband is compared with the MC simulation for the background gamma-rays. As
a complementary method to a direct comparison of the behavior for background gamma-rays, we
also studied the pileup analysis results for gamma-rays tagged by the RDC, which are supposed tobe less contaminated by the AIF background. The estimated value of the reconstruction efficiency
is 93 ± 2 %, and the uncertainty is dominated by the data vs MC correction.
Acceptance correction
The acceptance correction factor, Aeγγ , corrects the definition difference in the acceptance region
between that in the Michel counting and that for the µ → eγ signal (Fig.6.2). In the Michel
counting, all the positrons are accepted if the opposite of their direction at the target points tothe acceptance region defined at the inner face of the LXe detector. In the µ → eγ signal events,
however, the gamma-rays can convert in a deeper region, which can be outside of the acceptance even
if the positron direction is accepted. The |θγ | dependence of the acceptance correction factor was
evaluated as shown in Fig.6.3, which is then weighted by the signal positron efficiency dependence
on |θe |. Thig gave an overall correction factor of Aeγ
γ = 97 %.
Selection efficiency correction
The selection efficiency is introduced to correct inefficiencies that are relevant for the likelihood
analysis but cannot be included in the Michel counting. The estimated efficiency is 93 % with two

Chapter 6. Analysis

Acceptance

106

1
0.8
0.6
0.4
0.2
0

−0.4

−0.2

0

0.2

0.4
− cosθγ

Figure 6.3: Gamma-ray acceptance correction factor as a function of θγ emission angle evaluatedwith MC samples. The acceptance gets smaller at large cos θγ because gamma-rays are accepted
only if they are converted in the shallow region.
major effects taken into account;
1. Positron missing-turn tracks that are dropped in the pair timing selection• This gives 4 % inefficiency.
2. Tails in angular and momentum measurements that are dropped in angle or momentum cuts
according to the analysis window explained in Sec.6.2.
• 0.6 % inefficiency from Ee tail,• 2 % from θeγ tail,• 0.4 % from φeγ tail.
Estimated kMichel value and uncertainty
The uncertainty of the Michel counting method mostly comes from the selection efficiency estimations. The uncertainty of γ is 3 %; 2 % from the evaluation with the π 0 → γγ data and 2 %about the reconstruction efficiency evaluation. The sel uncertainty is also 3 %, which also includes
the uncertainty about the efficiency of the likelihood fitting itself (i.e. PDF efficiency). This isincluded because there may exist missing effects in the PDF uncertainty evaluation, which will
be later discussed in Sec.6.10. The other contributions are small, giving up to 1 % uncertainties
each. The total uncertainty about kMichel was thus evaluated to be ∼ 5 %. With all these taken into
account, the k-estimation from the Michel positron counting method iskMichel = (2.55 ± 0.13) × 1012 .

6.3.2

(6.8)

RMD normalization

In the RMD normalization, the number of RMD events in the energy sideband, estimated by a time
fitting, is scaled as
eγ
eγ
eγ
eγ
N eν ν̄γ trgγ
(6.9)
kRMD = eν ν̄γ · eν ν̄γ · eνeν̄γ · eν ν̄γ · eνselν̄γ ,
B
trg
e
γ
sel

6.3. Normalization

107

Table 6.3: The results of k-estimation with RMD events with analyses performed in different angle
ranges.
Θeγ range [deg]164 – 168
168 – 172
172 – 176
176 – 180

Estimated kRMD ± (stat)(3.05 ± 0.36) × 1012
(2.73 ± 0.17) × 1012
(3.60 ± 0.24) × 1012
(3.47 ± 0.28) × 1012

where the definitions of parameters are similar to those in Eq.(6.6). In this analysis, events in
Ee > 45 MeV, 45 MeV < Eγ < 48 MeV, and 164° < Θeγ < 180° were used. The angle range wasfurther divided into four sub-ranges as tabulated in Tab.6.3 and kRMD was estimated independently
in each sub-range.
As the RMD samples used in this analysis were triggered on the main trigger for µ → eγ,
the difference in the kinematics gives rise to the difference in the trigger efficiency between theeν ν̄γµ → eγ events and the RMD events, eγtrg /trg . It consists of additional inefficiencies from direction
match and the Eγ threshold logics, which make RMD event collection less efficient. The direction
match effect is evaluated in an analysis of Θeγ -dependent efficiency with accidental events. The
Eγ -threshold effect is evaluated in the background spectrum fitting with a model described later ineν ν̄γ), the acceptance function
Eq.(6.15). In the estimation of positron efficiency correction (eγe /eeν ν̄γ) estimation,from the Michel fit result is again used. In the gamma-ray efficiency difference (eγγ /γthe counting of events with the 45 MeV < Eγ < 48 MeV cut on the reconstructed energy has bothinefficiencies (Eγtrue inside the range but Eγreco outside the range) and overestimations (vise versa)
due to the finite resolution. The correction factor for these effects is evaluated by convoluting the
RMD spectrum with the expected energy response, which is the same as the Eγ PDF for the µ → eγeν ν̄γevents (see Sec.6.7.1). In the selection efficiency difference (eγsel /sel ), the missing turn positron
tracks are irrelevant because they are inefficient in both of µ → eγ and RMD events. Therefore,
we considered only the inefficiencies due to the kinematical tails, which are smaller in the N eν ν̄γestimation because of the wider cut ranges and the continuous spectrum.
The result of kRMD estimation in each angle sub-range is presented in Tab.6.3, and the combined
result is shown in Fig.6.4. The uncertainty presented in the table includes only the statisticaluncertainty in N eν ν̄γ estimation. The slight disagreements between the results in different angle
sub-ranges can be due to systematic effects in the evaluation of angle cut effects though it is not fully
concluded. However, such systematic effects, if exist, likely yield an anti-correlation between theerrors in different angle subranges. Therefore, its contribution to the uncertainty of the combined
result is estimated as < 3 %, which is smaller than the observed ∼ 10 % disagreement between
different angle sub-ranges. The systematic uncertainty that is common in all the four angle subeν ν̄γranges was estimated to be 10 % in total. The dominant contribution is from eγ, with theγ /γuncertainty evaluated to be close to 10 % because of its high impact on the kRMD estimation.
This evaluation comes from a comparison of the results with different (but within the calibration
uncertainties) assumptions on the shape of the response function; a Eγ spectrum based on that in
π 0 → γγ events fully calibrated with the π 0 → γγ dataset gave a ∼ 10 % higher kRMD , whereas aspectrum that strongly considers disagreements between different calibration sources gave a ∼ 10 %smaller result. The systematics of N eν ν̄γ estimation was evaluated to be 3 %. This is an outcomeof the uncertainty in the time response function for RMD events, which was used to fit the data ineν ν̄γeγeν ν̄γeν ν̄γthe energy sideband. The other correction factors — eγ, eγ— weretrg /trg , and sel /sele /eevaluated to have ∼ 2 % uncertainties respectively. With all the above discussions considered, theChapter 6. Analysis

108

k RMD [× 1012]

RMD normalization

3.8

χ2 / ndf

3.6

Prob
p0

0.01261
3.103 ± 0.117

165

170

10.84 / 3

3.4
3.2
3
2.8
2.6
175
180
Angle range [deg]Figure 6.4: The combined estimation of kRMD in a χ2 fitting to the results in different angle subranges.
final estimation of the RMD normalization is as follows,
kRMD = (3.1 ± 0.11 (stat) ± 0.3 (syst)) × 1012 .

6.3.3(6.10)

Combined result and uncertainty

From Eq.(6.8) and Eq.(6.10), the discrepancy between kMichel and kRMD is ∼ 1.7σ. The combined
normalization is also given as
kcombined = (2.64 ± 0.12) × 1012 .

6.4

Background estimation

6.4.1

Number of accidental background(6.11)The number of accidental backgrounds was directly counted in the timing sideband region; 1 ns <|teγ | < 3 ns. Here, the angle cut was widened to |θeγ | < 100 mrad and |φeγ | < 100 mrad to have anenriched control sample, which improves the statical uncertainty at a cost of introducing a systematic
uncertainty from the angle scaling according to the different cut range. With the widened sideband
region, the uncertainty of NAcc was ±3.5, which is better than ±4 statistical uncertainty that wewould have with the nominal angle range. The final estimated number of accidental background
events were
NAcc = 68 ± 3.5.
(6.12)

6.4.2

Number of RMD backgroundThe number of RMD background events was estimated by extrapolating the RMD normalization.
As was the case in the RMD normalization, the extrapolation again had a large uncertainty (10 %)6.5. Period-dependent event weight109

due to the Eγ energy response uncertainty. In addition, the ∼ 10 % disagreement between different
angle subranges in Tab.6.3 is fully relevant in the NRMD estimation. Note that the impact of thisdisagreement in the RMD normalization was concluded to be smaller because it is likely from anticorrelated systematics in different angle sub-ranges (see Sec.6.3.2). This suppression of systematics
thanks to the anti-correlation, however, is not the case here because NRMD is estimated in the narrowangle-range, namely |φeγ | < 40 mrad and |θeγ | < 40 mrad. Therefore, ∼ 15 % uncertainty was
assigned to NRMD giving the estimation as
NRMD = 1.2 ± 0.2.

6.5

(6.13)

Period-dependent event weight

In the 2021 data-taking, the beam intensity was changed several times, ranging between 2 × 107 –5 × 107 µ/s. As the accidental background rate gets larger with larger beam intensity, the signal-tobackground ratio is dependent on the period. Therefore, the whole dataset is divided into different
periods according to the beam intensity, and period-dependent weights are included in the fitting.
Hereafter, we will use the following period index; id = 0 for 3 × 107 µ/s, id = 1 for 2 × 107 µ/s,
id = 2 for 4 × 107 µ/s, and id = 3 for 5 × 107 µ/s.
The weight for the accidental background events is evaluated by counting the accidental eventsin the timing sideband for each period. The weight for the signal events is obtained from theMichel positron counting result in each period (Sec.6.3.1). Since the RMD rate is expected to be
proportional to the normalization factor, the weight assigned for the RMD events is identical to thatfor the signal.

6.6

Accidental background modeling

The accidental background PDF is decomposed as
A(Eγ , Ee , teγ , θeγ , φeγ |~q)
= A1 (Eγ |vγ , wγ )×
A2 (Ee |σe )×
A3 (teγ , npTC |Ee , wγ )×
A4 (θeγ )×
A5 (φeγ |vγ )×
A6 (tRDC , ERDC |Eγ ).

6.6.1(6.14)

Accidental gamma-ray energy PDF: A1 (Eγ |vγ , wγ )The accidental Eγ PDF is modeled as
A1 (Eγ |vγ , wγ ) = (SpectrumMC ⊗ Gadditional ) + r · Spectrumcosmic × Acceptancetrigger , (6.15)
where SpectrumMC is a spectrum of reconstructed energy in background simulations, Gadditional isa Gaussian term to additionally smear it, Spectrumcosmic is a cosmic-ray spectrum, r is a factor toaccount for the cosmic-ray fraction, and Acceptancetrigger is an acceptance function to represent
the trigger Eγ threshold window. The parameters in this model were estimated by fitting to thebackground spectrum (see also Fig.5.26).

Chapter 6. Analysisv (cm)vγ (cm)

110

0

60
2

3

4

5

6

7

40

0

60
2 3

4

5

6 7

8

9

1011 12131415 16

17

1819 20212223 24

25

40

20

20

0

0
8

9

−20

−20
10

11

12

13

−40

−40

−60

−60

1

0

10

20

30

(a) Accidental Eγ PDF

40wγ (cm)

0

10

1
20

30

40
w (cm)(b) Signal Eγ PDF

Figure 6.5: Eγ PDF segmentation according to the conditional observables, v, w.

The background MC simulation has AIF, RMD, and decay in flight (DIF) gamma-rays, which
are mixed according to the muon beam rate. In the calculation of the fraction of each source, we use
the branching ratio of 0.579 % for Eγ > 20 MeV RMD and from the stopping efficiency at the targetof 83 % for DIF. As the reconstructed spectrum on the MC samples is used, the effect of shower
leakage, which can cause a low-energy tail, is already included. Therefore, an additional Gaussian
smearing is intended to account for the resolution difference between the data and the MC.
The cosmic-ray component is introduced in the background spectrum. It gives a non-negligible
contribution in the high Eγ region, where the gamma-ray contribution is vanishing. The spectrum
was directly estimated from the data taken with the beam-off condition, which was fitted with apolynomial. Its contribution can be seen in Fig.5.26.
The trigger acceptance function is represented as

1
Acceptance =
4



 


Eγ − µacc−Eγ + µacc−high√√1 + erf
1 + erf,
2σacc
2σacc−high(6.16)

where erf(x) is the error functions, µacc ∼ 44 MeV corresponds to low energy trigger threshold,and µacc−high ∼ 100 MeV corresponds the high energy trigger threshold. The shape of the trigger
window is based on an evaluation with Eγ -single trigger data.
The PDF dependence on the conditional observables, v and w, are included by dividing thefiducial volume into 14 segments as shown in Fig.6.5(a); the top and bottom edges are separately
treated because of the worse resolution than in the central regions and the increased contaminationof cosmic-ray events. As the PDF dependence on the beam intensity is found, it is also included byevaluating the PDFs period-dependently.
As the accidental energy PDF was evaluated directly with the data, its uncertainty evaluation isstraightforward. The only contribution to the PDF uncertainty is the fitting uncertainty due to thestatistical limitation of the data samples.

6.6. Accidental background modeling

6.6.2

111

Accidental positron momentum PDF: A2 (Ee |σe )The accidental positron energy PDF is based on the Michel spectrum fitting with the below
parametrization,

A2 (Ee |σe ) = (Michel) × (Acceptance) ⊗ Resolution.
(6.17)
Here, the per-event PDF is based on a categorization according to the covariance (~σe ) of the fit
uncertainties estimated by the Kalman filter.
The acceptance function introduces inefficiencies in both low and high momentum sides with aparametrization as


 


1
Ee − µacc−Ee + µacc−high√√Acceptance =1 + erf
1 + erf.
(6.18)
42σacc
2σacc−high
The small acceptance on the low-energy side arises from the geometrical coverage of the positron
spectrometer, where positrons can be reconstructed only when they have a large radius that canleave hits on the CDCH and pTC. The introduction of the acceptance function in the high Ee side
is motivated by a comparison of the Michel MC spectrum of the Ee -truth with the true theoretical
spectrum (Fig.6.6). Here, we can see that the high Ee inefficiency is especially large for tracks
that have small tracking uncertainties. This behavior is understood to be from the detector material
distribution, which affects the tracking uncertainty via scattering.
A sum of three Gaussian is used to describe the resolution function, which is motivated byobserved distributions of Eereco − Eetrue in detector simulation (Fig.6.7). Because of the kinematical
upper limit of Eetrue < 52.8 MeV, Eereco > 53.5 MeV data samples cannot be explained within a 2σ
discrepancy by the core (100 keV resolution) or the first tail (300 keV resolution). Therefore, theintroduction of the long-tail component (σ & 1 MeV) is especially important when Ee > 53.5 MeV
is included in the range of background fitting.
Fig.6.8 shows distributions of background data samples and their fit in each Kalman-covariancebased category. The best-fit values for some representing parameters are also shown in Fig.6.9,
where higher resolution and smaller fractions of long-tail components can be seen for tracks withsmaller tracking uncertainties. These parameter dependences were studied also with MC samples
to validate this method, and we found that the data and the MC agree within the fit uncertainties.

6.6.3

Accidental time PDF: A3 (teγ , npTC |Ee , wγ )The npTC distribution for the accidental background was evaluated from the timing sideband
samples. With the observed correlation shown in Fig.6.10, the npTC parameter is conditioned by the
positron momentum. The accidental teγ distribution was not perfectly flat due to a time walk effect inthe timing-coincident logic of the trigger, which was parametrized with a linear function (Fig.6.11).
The slope parameter was correlated only with wγ , and it was confirmed to be independent (or havenegligible dependence even if correlations exist) of the other fit or conditional observables. Hence,
the PDF is further decomposed as A3 (teγ , npTC |Ee , wγ ) = A3a (teγ |wγ ) × A3b (npTC |Ee ).

6.6.4

Accidental angle PDF: A4 (θeγ ) and A5 (φeγ |vγ )The accidental angle PDFs were directly evaluated from the timing sideband data. We found weakcorrelations between θeγ , φeγ , and Ee , which is understood to come from the direction-matching logicof the trigger. These, however, are weak enough that we can approximate them to be independentof each other as long as we study the PDFs within a narrow window. Its correlation with Eγ , on

Chapter 6. Analysis

112

1.2
1
0.8
0.6
0.4
0.2
44

46

48

50

65 keV < E e uncertainty < 75 keV
1.2
1
0.8
0.6

0.2

1
0.8
0.6

72342049.75
2.15
157.7 / 131
0.05608
1.044 ± 0.000
47.94 ± 0.00
2.632 ± 0.003
67.67 ± 27.46
1.61 ± 0.93

0.4
0.2
0

44

46

48

50

52
E e [MeV]

44

46

48

50

75 keV < E e uncertainty < 85 keV

0.8
0.6

EntriesMean
Std Dev
χ 2 / ndf
Prob
p0
p1
p2
p3
p4

1
0.8
0.6

21974250.01
2.064
135.2 / 131
0.38281.144 ± 0.010
49.16 ± 0.04
2.971 ± 0.035
89.54 ± 0.16
0.7662 ± 0.0000

0.4
0.2
0

44

46

48

50

52
E e [MeV]

142421049.49
2.192
140.6 / 131
0.2684
1.166 ± 0.022
47.17 ± 0.05
2.217 ± 0.047
55.54 ± 0.30
2.502 ± 0.391

0.4
0.2
0

52
E e [MeV]

EntriesMean
Std Dev
χ 2 / ndf
Prob
p0
p1
p2
p3
p4

1

95 keV < E e uncertainty < 105 keV
Acceptance normalized at 52.8 MeV

Acceptance normalized at 52.8 MeV

85 keV < E e uncertainty < 95 keV
EntriesMean
Std Dev
χ 2 / ndf
Prob
p0
p1
p2
p3
p4

116389549.4
2.23
104 / 131
0.9603
1.2 ± 0.0
46.99 ± 0.05
2.174 ± 0.051
55.46 ± 0.29
2.531 ± 0.385

0.4

0

52
E e [MeV]

EntriesMean
Std Dev
χ 2 / ndf
Prob
p0
p1
p2
p3
p4

Acceptance normalized at 52.8 MeV

1.4

19536948.91
2.374
123.1 / 131
0.67571.476 ± 0.057
46.05 ± 0.09
2.133 ± 0.172
53.87 ± 0.09
2.264 ± 0.337

44

46

48

50

52
E e [MeV]

105 keV < E e uncertainty < 120 keV
Acceptance normalized at 52.8 MeV

EntriesMean
Std Dev
χ 2 / ndf
Prob
p0
p1
p2
p3
p4

Acceptance normalized at 52.8 MeV

Acceptance normalized at 52.8 MeV

50 keV < E e uncertainty < 65 keV

EntriesMean
Std Dev
χ 2 / ndf
Prob
p0
p1
p2
p3
p4

1
0.8
0.6

11675350.17
2.129
125 / 131
0.6303
16.54 ± 2.09
62.69 ± 0.69
6.863 ± 0.161
54.24 ± 0.17
1.704 ± 0.144

0.4
0.2
0

44

46

48

50

52
E e [MeV]

Acceptance normalized at 52.8 MeV

120 keV < E e uncertainty < 190 keV
EntriesMean
Std Dev
χ 2 / ndf
Prob
p0
p1
p2
p3
p4

1
0.8
0.6

78653
49.82
2.175
106.4 / 131
0.94392.583 ± 0.868
51.13 ± 1.07
3.623 ± 0.343
53.4 ± 1.0
3.893 ± 0.530

0.4
0.2
0

44

46

48

50

52
E e [MeV]

Figure 6.6: Positron acceptance function in Michel MC samples for different σEe value estimatedin the track fitting. The top-left plot shows the acceptance curve for tracks estimated to be σEe <65 keV, for example.

Events / (20 keV)

6.6. Accidental background modeling

113

Simulated E e response

104

No kinematical cut

With kinematical cuts

103
102

10

1
−3

−2

−1

0

1
2
3
E reco − E truth [MeV]

Figure 6.7: Positron response function in detector simulation. The violet histogram is a distribution
without any kinematical cuts and is fitted with triple Gaussian (black line). The hatched blue
histogram is a distribution after applying angle and momentum cuts and fitted with double Gaussian(red line). The long tail component reduction, visible around δEe = ±0.6 MeV, originates fromangle cut, which affects the momentum spectrum due to the δEe vs δφe correlation.
the other hand, was not observed, which suggests that we can enlarge the Eγ range to have larger
statistics to study the accidental distribution. Therefore, the angle PDFs were studied in the rangeof 52.2 < Ee < 53.5 MeV, 42 < Eγ < 58 MeV, |θeγ | < 200 mrad, and |φeγ | < 200 mrad.
Both the θeγ and φeγ PDFs are parametrized with polynomials up to fourth order (Fig.6.12).
The φeγ PDF was evaluated after slicing the full acceptance range into five v ranges. The observed
strong dependence on vγ is due to the edge effect of the acceptance cut defined in Eq.(4.22). The
overall non-flat distribution is understood to be an efficiency curve that comes from the directionmatching algorithm of the trigger. The PDF uncertainty is dominated by the fitting uncertainty ofthe polynomial modeling due to the statistical limitation of the data samples.

6.6.5

Accidental RDC PDF: A6 (tRDC , ERDC |Eγ )The accidental RDC PDF was estimated with the samples in the timing sideband region. Here,the PDF is conditioned by Eγ because its dependence is induced by the difference in the Eγ
spectrum between the RMD-originated background gamma-rays and AIF-originated ones. In
the PDF extraction, the dataset was divided into Eγ -based subsets with the thresholds at Eγ =48, 49.5, 51, 53, 58 MeV.
The PDF was parametrized in a histogram-based format, which, in the unbinned likelihood
fitting, means the use of a step function. The parameter width in the step function was flexibly
changed according to the tradeoff between the available statistics and the importance of having
detailed information within each step width. The RDC PDF projected on tRDC -axis and ERDC -axis
are shown in Fig.6.13.
The RDC PDFs are strongly period-dependent for two reasons. The first one is from the hit
rate of accidental positrons, which is mostly determined by the muon stopping rate on the target.

Chapter 6. Analysis

114

3

10

102

104

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10

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45

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Michel spectrum: 0.105000 < EUncert < 0.120000

Events / ( 0.2 MeV )

104

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Michel spectrum: 0.095000 < EUncert < 0.105000

Events / ( 0.2 MeV )Events / ( 0.2 MeV )

3

10

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10

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Michel spectrum: 0.085000 < EUncert < 0.095000

104

104

102

Ee (MeV)

1045

Michel spectrum: 0.075000 < EUncert < 0.085000

Events / ( 0.2 MeV )Michel spectrum: 0.065000 < EUncert < 0.075000

Events / ( 0.2 MeV )Events / ( 0.2 MeV )Michel spectrum: 0.050000 < EUncert < 0.065000

104

104

3

10

102

50

55

45
Ee (MeV)50

55
Ee (MeV)

Events / ( 0.2 MeV )Michel spectrum: 0.120000 < EUncert < 0.190000

3

10

102

10
45

50

55
Ee (MeV)Figure 6.8: Evaluation of accidental positron momentum PDF, with the same σEe -based categorization as that in Fig.6.6. The increased trend in the fraction of the long-tail resolution component(σ & 1 MeV) can be seen with a larger σEe value.

115

Resolution of core component

1

Core resolution [MeV]Fraction of each component

6.6. Accidental background modeling

0.16

0.9
0.8

0.14

0.7
0.6

0.12

Long tail fraction

0.5Core fraction

0.4

Tail fraction

0.1

0.3
0.2

0.08

0.1
0

(a)

0.06

0.06 0.08 0.1 0.12 0.14 0.16 0.18
E e tracking uncertainty [MeV]Core to tail fraction parameter in the response function.

(b)

0.06 0.08 0.1 0.12 0.14 0.16 0.18
E e tracking uncertainty [MeV]

Ee resolution parameter for the core partof the response function.

Figure 6.9: Parameters for the Ee response as a function of the tracking Ee uncertainty parameter(σEe ), which is estimated by the Michel spectrum fitting.

npTC

n pTC vs E e with reconstructed data

20
1400

18
16

1200

14

1000

12
800

10
8

600

6

400

4

200

2
46

48

50

52

54
E e [MeV]

0

Figure 6.10: Dependence of the npTC parameter on Ee , observed for background positrons.

Chapter 6. Analysis

116

Events / ( 0.1 ns )Accidental teγ

5000
4000
3000
2000
1000
0
−3

−2

−1

0

1

2

3
t eγ (ns)Figure 6.11: Time distribution for the accidental background events and its fit with a linear function(blue line). No events are found in the |teγ | < 1 ns region due to the blinding.

Events / ( 0.01 rad )Events / ( 0.01 rad )

500
400
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200

200
180
160
140
120

40
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Events / ( 0.01 rad )Accidental φeγ (-20.00<v<20.00)

Accidental θeγ
600

−0.1

0

0.1

(a) Accidental θeγ PDF.

0.2
θeγ (rad)

0
−0.2

(b)

−0.1

0

0.1

0.2
φeγ (rad)Accidental φeγ PDF for|v| < 20 cm region.

0
−0.2

(c)

−0.1

0

0.1

0.2
φeγ (rad)Accidental φeγ PDF forv < −55 cm region.

Figure 6.12: Evaluation of angle PDFs for accidental events. (a): The accidental θeγ PDF is notconditioned by other parameters; the uγ dependence is integrated out, and the other dependencies
are weak enough and approximated to be independent. (b): φeγ distribution for the central region
in the acceptance range of v. (c): That in the edge region. The difference in (b) and (c) indicates astrong dependence of φeγ distribution on v, which is included in the conditional PDF, A5 (φeγ |vγ ).

6.7. Signal modeling117

RDC PDF projection to t RDC

1

RDC PDF projection to E RDC

1
BG PDF projection

BG PDF projection

10−1

Signal PDF projection

10−1

Signal PDF projection

10−2

10−2

10−3
−10 −8 −6 −4 −2

0

2

4

×10−9
6 8 10t RDC [s]10−3−0.005

0

0.005

0.01

0.015

0.02 0.025
E RDC [GeV]

Figure 6.13: RDC PDF projected on tRDC axis (left) and ERDC axis (right) for 49.5 < Eγ <51.5 MeV events. The rightmost (leftmost) bin in the tRDC (ERDC ) projection corresponds toevents with no hit detection in the RDC.
This makes a difference in the probability of having events in the off-peak region of the timing
distribution. The other period dependence comes from the fact that the RDC was not installed at
the beginning of the beam time for safety reasons.
The uncertainty of the PDF comes from the used statistics in the PDF evaluation. The bin-by-bin
uncertainties are assigned according to the Poisson fluctuation.

6.7

Signal modeling

The full signal PDF is decomposed as follows,
S(Eγ , Ee , teγ , θeγ , φeγ , tRDC , ERDC , npTC |~q)
= S1 (Eγ |vγ , wγ )×
S2 (Ee |~σe )×
S3 (teγ , npTC |Eγ , Ee )×
S4 (θeγ |Ee , vγ , wγ , ~σe , XTGT )×
S5 (φeγ |Ee , θeγ , vγ , wγ , ~σe , φe , XTGT )×
S6 (tRDC , ERDC ).

6.7.1(6.19)

Signal gamma-ray energy PDF: S1 (Eγ |vγ , wγ )The signal PDF modeling is based on the function introduced in Eq.(5.8); the sum of two suchfunctions is used for the PDF. Dependences on the conditional observables (vγ and wγ ) were
included by dividing the fiducial volume into 26 segments shown in Fig.6.5(b). Though thesegmentation must be identical to that for the accidental in principle, the signal PDF segmentation
has a larger number of segments. This, however, is based on a data-driven reasonable approximationthat the accidental spectrum does not show strong correlations with v and w, and hence does notcause biases in the fitting.
As the PDF is not conditioned by uγ , the uγ dependence is integrated out as discussed inChapter 6. Analysis

118

hPeakScale_5

hCoreSigma_5

hCoreTransition_5

180

hPeakScale_5

400

hCoreSigma_5

160

Entries

1000

350

Entries

1000

Mean

0.9992Mean

0.01948

Std Dev0.0005468

140
Std Dev300

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120

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0.05280.05275
0.05270.00110.001080.00106
0.00104
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0.00098

0.05255

0.000960.00110.00115

Sigma (GeV)

×10
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20.5

20.45

1000

Mean

2 0.009288

Std Dev0.0006117

0Transition

0.001120.05260.00105

0.05

0.001140.05265

0.001

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0.053

0.03

Entries4020
020.03 20.025 20.02 20.015 20.01 20.005

Sigma (%)
Sigma (GeV)

Energy (GeV)

Energy scale

hCoreTransition_5

23

20.4

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23

×10
20.4
20.42
20.44
20.46
20.48
20.5
20.52
20.54
20.56
20.58

0.0526

0.05270.05280.05290.053

Energy (GeV)

Figure 6.14: Parameters of signal Eγ parameters in a segment. The top three plots are the
distributions of energy scale, sigma, and transition. The bottom three are scatter plots between two
parameters out of three.
Sec.6.1.5. Still, we may have energy-scale non-uniformities in u sub-segments of each v, w-based
segment if its calibration is not correct. This energy-scale misalignment broadens the shape ofthe spectrum when u dependence is integrated out. Such an effect should be taken into accountin the PDF according to the uncertainties in the u-dependent non-uniformity calibration. Thiswas evaluated by making an ensemble of u-integrated PDFs, in which each sampled PDF models
different u-dependent energy scale miscalibration.
The full procedure of making a signal PDF in the ensemble is
1. extract the spectra in different u regions obtained in the π 0 → γγ data,
2. randomize the energy scale according to its uncertainty in each segment,
3. simulate (MC) the θγ distribution of signal events including efficiencies,
4. integrate uγ -segmented spectra with a weight according to the simulated θγ distribution.
In segments that do not have enough π 0 → γγ statistics in the first step (see Fig.3.6), the spectra inneighboring segments are interpolated.
By iterating the above procedure, an ensemble of 1000 PDFs was generated in each of 26
segments divided by v, w. The distribution of the PDF parameters in this ensemble is shown in
Fig.6.14. Here, the mean of each distribution corresponds to the best estimate of PDF parametersconsidering the energy scale calibration uncertainty effects in u integration. On the other hand, thecovariance of each distribution gives the uncertainty of PDF parameters.
Uncertainty of signal gamma-ray energy PDF
The uncertainty of the Eγ scale has two contributions, a global uncertainty that is common among
all the segments and segment-by-segment uncertainty. The global Eγ scale uncertainty comes fromthe uncertainty of energy scale parameters calculated in background Eγ spectrum fitting (0.2 %), a
disagreement with the energy scale calculated from π 0 → γγ (0.2 %), and statistical uncertainty in
π 0 → γγ (0.1 %). The overall global Eγ scale uncertainty is 0.3 % in total. The segment-by-segment

6.7. Signal modeling

119

uncertainty comes from different trends of energy scale among different calibration sources, which
is 0.23 % on average.
In addition to the global energy scale, uncertainties of the PDF shapes are included. The
distribution in Fig.6.14, which spans the PDF parameter uncertainties, is used to quantify the
impacts.

6.7.2

Signal positron momentum PDF: S2 (Ee |σe )The signal positron momentum PDF has the tracking momentum uncertainty as a conditional
observable. The Michel fit technique for the background spectrum presents a good agreement withresponse studies based on Eereco − Eetrue in MC samples. Therefore, both the energy scale and theresolution parameters are based on the best-fit value of the background spectrum fitting.
In the signal PDF, however, the angle cuts of |φeγ | < 40 mrad and |φeγ | < 40 mrad must
be taken into consideration. These cuts reduce the long tail component of the response function
because δφe and δEe are correlated as will be discussed later in Sec.6.7.4. This mechanism does
not apply to the backgrounds because such an angle cut is not associated with the real δφe whencombined with accidentally coincident gamma-rays. Therefore, the resolution function estimatedin the Michel fit is further corrected to obtain the signal PDF.
The effect of the angle cuts was evaluated with the detector simulations. It was found there thatthe signal PDF can be parametrized as a sum of two Gaussian functions (the hatched distribution
and its fit in Fig.6.7). Here, the differences before and after the angle cuts are found only inthe fraction parameters and no systematic effect was found in the resolution. Therefore, only thecorrections on the core-to-tail fraction were applied when the response function from the Michel fit
was transformed into the signal PDF. The fraction of the core component is ∼ 80 %, which is 5 –10 % larger than that for the background response function.
Uncertainty of signal Ee PDF
The PDF uncertainty is based on the fit uncertainty of the resolution function parameters in theMichel fitting, which was summed with additional uncertainties from the corrections based on MC
samples. The MC-based corrections were adopted to the core to tail fraction of the double Gaussian
response function, which yields a 3 % additional uncertainty to the parameter.

6.7.3

Signal time PDF: S3 (teγ , npTC |Eγ , Ee )The npTC distribution for the signal was evaluated from MC samples, which was parametrizedindependently of the reconstructed Ee because no correlation was found among them. Therefore,
the signal PDF is further decomposed as S3 (teγ , npTC |Eγ , Ee ) = S3a (teγ |npTC , Eγ , Ee )×S3b (npTC ).
The teγ PDF, S3a (teγ |npTC , Eγ , Ee ), uses a sum of two Gaussian functions, which considersσ. The full time response is thus parametrized
event-by-event positron time resolution of σte = √single
npTC
as σteγ = σte ⊕ σconst , where σconst part is decomposed into two Gaussian functions. The σsingle
parameter, the averaged single counter resolution of the pTC, was evaluated to be 112 ps by theeven-odd analysis (Sec.5.5.4). The σconst response term was then evaluated to be σconst = 70 ±3 ps (220 ± 30 ps) for core (tail), which accounts for 94 ± 3 % (the remaining 6 %) of the full
response. This was evaluated by fitting to the RMD events in the energy sideband (Sec.5.6.3) with
fit parameters constrained by the resolution in the π 0 → γγ dataset (Sec.5.4.4). Here, the constraints
from the π 0 → γγ resolution exploit the fact that the σconst is dominated by the LXe resolution, andthe contribution from the positron TOF measurement is small (∼ 10 ps). This additional constraint

Chapter 6. Analysis

TOFreco − TOFtruth [ns]120t e γ vs E e correlation

0.03

300

0.02

250

0.01

200

0

150

−0.01

100

−0.02

50

−0.03

−0.4

−0.2

0

0.2
0.4
E reco − E truth [MeV]0

Figure 6.15: Correlation between positron TOF reconstruction error and Ee reconstruction error
evaluated with signal MC events.
reduces the uncertainty of the time response estimation.
There is a correlation between teγ and Ee because Ee reconstruction error results in TOF
measurement error. This small effect of 18.9 ± 0.5 ps/MeV is included in the signal PDF. The
correlation between teγ and Eγ is also included as a resolution dependence on Eγ , which wasevaluated with the π 0 → γγ dataset.
Uncertainty of signal timing PDF
The uncertainty of the npTC PDF comes from the statistical uncertainty of the used data samples,
which is thus evaluated according to the Poisson statistics. The uncertainty of the σsingle is dominatedby the systematic uncertainty of the even-odd analysis, which gives 5 % uncertainty to its scale. As
the σconst evaluation uses σsingle as an input, their uncertainties are correlated, which was evaluatedby profiling σsingle parameter in the RMD fitting. This correlation changes the σconst value by −1 ps
when σsingle is overestimated by 5 %. The uncertainty in the global time offset and the shape of teγ
resolution function come from the fit uncertainty in the RMD peak fitting. Here, the uncertainty inthe shape is suppressed also by the π 0 → γγ data because of its use as a fit constraint.

6.7.4

Signal angle PDF: S4 and S5

The signal angle PDF is parametrized based on the combination of the positron angle and vertexing
resolutions and the gamma position resolution. The gamma position resolution, as discussed inSec.5.2.2, depends on wγ , which is included as the event-by-event gamma position resolution. The
positron resolution is parametrized by multiplying the pull sigma with the event-by-event tracking
uncertainty, which is based on the evaluation with the double-turn analysis presented in Sec.5.10.
Correlations are also considered in the signal PDF as is evident in Eq.(6.19); the θeγ PDF is
conditioned also by Ee , and the φeγ PDF is conditioned by Ee and θeγ . The φe dependence of the
PDF represents the dependence of φeγ vs Ee correlation on φe , which will be discussed later in thissection.
To describe the angle resolution and correlation, we introduce the following notations,6.7. Signal modeling

121

𝜃𝑗
𝜎𝑗

𝜎𝑗,inner𝑝𝑖𝑗 𝜎𝑖 = 𝜌𝑖𝑗 𝜎𝑗𝜃𝑖𝜎𝑖Figure 6.16: Parametrization of correlation (Original figure from Figure 40.5 of [2], modified toinclude only necessary parameters and to have a consistent notation).
δ: error; δφe represents the difference between the reconstructed and the true value,µ describes center of PDF; µφeγ represents the center of the φeγ PDF,
σ: true resolution,
σ 0 : nominal reconstruction uncertainty,
s: resolution of pull, namely s × σ 0 = σ,pij : slope parameter to describe correlation between variables (see Fig.6.16),
ρ: correlation coefficient between variables, ρ = pij · σi /σj ,p0 : correlation slope parameter between pull, namely p0ij = pij · σi0 /σj0 ,σinner and sinner : inner pull resolution after correlation correction (see Fig.6.16), which followpσy,inner = σy2 − p2xy σx2 ,psinner = s · 1 − ρ2 .
Handling of δφe vs δEe correlation
An error of Ee is equivalent to an error in the estimated track radius, which changes the backpropagated trajectory between the CDCH and the target. This geometrically results in an error of
φe as illustrated in Fig.6.17. Note here that the correlation parameter, pEe φe , also depends on φebecause of the geometrical mechanism.
The φe dependent correlation is parametrized as
p0Ee φe = qcφ − kφ tan φe.

(6.20)

σφ2 e (0) + (kφe tan φe )2The denominator parametrization is motivated by the fact that the track fitting uncertainty (σφ0 e )is evaluated including the effect of σE0 e and this correlation, namely the σφ2 e (0) is introduced toChapter 6. Analysis

122

𝑦
𝛿𝑦𝑒

𝛿𝜙𝑒

𝑥

Figure 6.17: Mechanism of correlation of δφe vs δEe and δye vs δEe . An error of δEe is equivalent
to an error of track radius estimation, which geometrically results in φe and ye error during the track
back-propagation to the target.
represent the φe uncertainty at φe = 0. As the true positron momentum is known for signal events,
the φe shift is evaluated straightforwardly by multiplying p0Ee φe with δEe = Ee − 52.8 MeV. As δφeshifts the φeγ accordingly, the center of the φeγ PDF is dependent on Ee .
After subtracting this correlation, the inner pull resolution of φe becomesv
uu 1 − (c2φ − 2cφ kφ tan φe )/σφ2 e (0)sφe ,inner = sφe u.
(6.21)
2tkφ
1 + σφ (0) tan φeeThis is one of the contributions to the φeγ resolution, which will be summed up with the othercontributions discussed hereafter.
Handling of δye vs δEe correlation
The δye vs δEe correlation also arises from the geometrical mechanism in Fig.6.17. Here, pEe ye
is used instead of p0Ee ye because the former was found to better describe the observed correlationin data. The pEe ye value was evaluated to be 280 cm/GeV from the double-turn analysis. This isused to shift the center of φeγ PDF according to the measured Ee by geometrically converting δye
to δφeγ . The inner ye resolution after the correlation correction is also added to the φeγ resolution.
Handling of δze vs δθe correlation
The ze and θe measurements are both relevant to the θeγ measurement. The correlation between
these two parameters enhances the error of θeγ measurement because δθe induces δze in a way thatδθe and δθγ have the same sign contribution to δθeγ (Fig.6.18). When the positron resolution, δzevs δθe correlation, and uγ resolution are all included, the full θeγ resolution followsqσθeγ = s2θe σ 0 2θe + Cz2 s2ze σ 0 2ze + 2Cz p0θe ze s2θe σθ0 e σz0 e + Cu2 σu2γ ,(6.22)
where Cz (Cu ) is a factor that converts δze (δuγ ) into δθeγ . This conversion factor is simply
calculated considering the geometry of the detector. The full derivation of this formula is based on6.7. Signal modeling123

z-axis𝜹𝜽𝒆Reconstructed vertex
𝜹𝒛𝒆True vertex𝜹𝜽𝜸𝛾 conversion point

Figure 6.18: An illustration of the effect of δze vs δθe correlation in the combined error of δθeγ .
Positive δθe likely produces positive δze , which geometrically results in an enhancement of δθeγ .
(The drawing ignores the target slant angle for simplicity.)
a tedious algebra described in Appendix.B.
The value of the p0θe ze parameter was evaluated to be 1.1 from signal MC samples. Here, thedouble-turn analysis method cannot be used because it is insensitive to θe vs ze correlation. This isbecause the sign of the correlation is opposite between the first turn track (forward propagated to
the virtual target plane) and the second turn track (backward propagated to the same plane), whichcancels out the correlation effects in the double-turn analysis. The sign difference between twoturns can be intuitively understood with a replacement of δθγ → −δθe (first−turn) in Fig.6.18.
Handling of correlations between δφe , δze , and δθe
The δφe vs δze and δφe vs δθe are both correlated mainly because of the propagation length change
due to the slanted target geometry (Fig.6.19). The double-turn analysis method is found to besensitive only to δφe vs δze correlation among them, which is again because of the correlation
sign difference between the first and second turns. Therefore, δφe vs δze correlation was evaluatedfirst with the double-turn method. The φeγ vs θeγ correlation was then evaluated with signal MC
samples after correcting the δφe vs δze correlation. As they were both found to depend on φe , theirdependence was modeled with quadratic functions.
Note here that the δze and δθe are also correlated. This means that the δφe vs δze correlation
receives a contribution of a combined effect of the δφe vs δθe correlation and δze vs δθe correlation.
Therefore, this additional contribution has to be carefully considered, whose full detail is given in
Appendix.B.
Handling of δφeγ vs δθeγ correlation
The correlation of δφe with δθe and δze results in φeγ vs θeγ correlation. Both of these, however,receive the contribution from the LXe position resolution, namely σuγ in σθeγ and σvγ in σφeγ .
Therefore, a further transformation from the positron correlations into φeγ vs θeγ is necessary. Thistransformation, as shown around Eq.(B.23), is given aspθeγ φeγ =s2ze σz0 e σφ0 e 0s2θe σθ0 e σφ0 e 0
0
0
(q
+
q
·
p
)
+
C
·
q z e φe .
zz e φeθe ze
θ e φeσθ2eγ
σθ2eγ(6.23)

Chapter 6. Analysis

124

𝑧
𝛿𝜃𝑒

𝑦

15∘

𝑥

𝛿𝑥

𝛿𝜙𝑒

𝑥

Figure 6.19: An illustration of the mechanism for δφe vs δze and δφe vs δθe correlations. The δze ,which can be also from δθe via ze vs θe correlation, results in δφe because δze on the 15° slanted
target plane changes the track propagation length.
The δφeγ vs δθeγ correction in the δφeγ inner resolution is specifically shown in Eq.(B.24). Aftercorrecting both the δφeγ vs δθeγ and δφeγ vs δEe correlations, the inner resolution of φeγ is finallygiven as2
σφ2 eγ ,inner = s2φe,inner−Ee σ 0 φe − p2θeγ φeγ σθ2eγ + Cv2 σv2 + Cy2 s2ye ,inner σy02e .
(6.24)
The complicated correlation handling discussed above was cross-checked by comparing the φeγ
vs θeγ signal correlation between toy MC and full detector simulation. Fig.6.20 compares the µφeγ
parameter as a function of θeγ , where we can see a good agreement between them.
Uncertainties of signal angle PDF
The uncertainty of the PDF comes both from the detector alignment and the correlation and pullparameters. A wrong choice of the correlation and pull parameters changes only the shape of the
PDFs. The alignment, on the other hand, results in a shift of the peak positions of the PDFs.
The uncertainties on the correlation and pull parameters are different depending on their estimation methods. Some of the correlation parameters and all the pull parameters were estimated on datawith the double-turn analysis, and 5 % uncertainties are expected. Here, the double-turn analysis
on data showed that the agreement between the data and the MC is ∼ 10 % level. Therefore, 10 %uncertainties are assigned for those estimated only from MC samples.
As discussed in Sec.5.11, the uncertainty of the alignment between the LXe detector and CDCH
is 1 mm in all x, y, and z coordinates. This misalignment changes the reconstructed θγ and φγ .
In addition, the target alignment uncertainty in the x coordinate also affects the φe reconstruction
because of the change in the track propagation (recall Fig.6.19). The target alignment uncertainty
is 100 µm for the translation and 6 mrad for the rotation around the target long axis.
During the period without a camera photograph, there were several insertion and extraction

Center of φ eγ [rad]6.8. RMD background modeling

125

Simulated φ e γ vs θ e γ correlation

0.02
0.01
0

−0.01
Toy MC samples

−0.02
−0.04

Full detector simulation

−0.02

0

0.02

0.04
θ eγ [rad]Figure 6.20: Simulated correlation between φeγ error vs θeγ error, which is compared between toy
MC samples and full detector simulation. The center (peak position) of φeγ PDF at each θeγ slice
is shown.
operations (see Fig.3.7) and the true target position likely changed in each operation. The angle
PDF in the period is likely an overlay of multiple Gaussians that have different peak positions, andthis can be represented by increasing the resolution parameter of the PDFs. Its contribution to theoverall resolution, however, is negligible and has little impact on the likelihood analysis.

6.7.5

Signal RDC PDF: S6 (tRDC , ERDC )The signal RDC PDF was estimated with the samples in the timing sideband making use of theoff-peak region in Fig.4.18. The tRDC vs ERDC correlation is fully considered here because they
are not independent even in the signal PDF. This is because the single-hit selection makes tRDC
correlated with the pileup situation2 and thus correlated with ERDC .

6.8

RMD background modeling

The RMD background PDF is decomposed as
R(Eγ , Ee , teγ , θeγ , φeγ |~q)
= R1 (Eγ , Ee , φeγ , θeγ |vγ , wγ )×
R2 (teγ , npTC |Eγ , Ee )×
S6 (tRDC , ERDC ).

(6.25)

Here, the RDC PDF is shared with the signal because they are identical; the detected RDC hit isaccidental and not associated with the detected gamma-ray in both signal and RMD events.

2If tRDC = −5 ns for example, it means there is no hit during 10 ns after that hit.

Chapter 6. Analysis

126

6.8.1

Incorporation of RMD kinematics in R1

The differential branching ratio of RMD events is a function of the angle, the positron energy, andthe gamma-ray energy. Therefore, the PDF building starts from its theoretical formula given in [28],which is shown in Appendix.A. Here, the opening angle parameter of the formula is transformed intothe (θeγ , φeγ ) parameter space. This is further modified to include the detector responses (efficiency
and resolution) in the following way:
• The differential branching ratio is multiplied by the Ee acceptance (efficiency) curve presented
in Eq.(6.18). It is then convoluted with the Ee resolution function used in Eq.(6.17), which
includes the long tail component. Note here that the |θeγ | < 40 mrad and |φeγ | < 40 mrad
cuts have little effect on the response function of the RMD positrons because the original
RMD angle spectrum is wider than the cut range.
• The two dimensional angle spectrum of (θeγ , φeγ ) is convoluted with the angle resolution. It
is then multiplied by the angular efficiency estimated from the non-flatness of the backgroundangle distribution.
• The Eγ spectrum is convoluted with the signal PDF, which is then multiplied by the Eγ
acceptance curve estimated in the background spectrum fitting.

6.8.2

RMD timing PDF: R2

The RMD timing PDF is further decomposed into
R2 (teγ , npTC |Eγ , Ee ) = R2a (teγ |npTC , Eγ ) × A3b (npTC |Ee ).
Here, the same npTC distribution is used as that for the accidental backgrounds. R2a (teγ |npTC , Eγ )σ
. The only differenceis similar to the signal PDF, with the resolution parametrized as σconst ⊕ √single
npTC
is that the Ee dependence is removed, as is evident in the R2a (teγ |npTC , Eγ ) formalism. This isbecause the RMD positron has a continuous spectrum, and Ee is therefore negligibly correlatedwith the TOF error.

6.9

Properties of PDFs

Overall PDF shape
The projected distribution of the observables in a toy MC with Nsig = 2500, NRMD = 0, andNAcc = 2500 and its fit are shown in Fig.6.21 and Fig.6.22. Here, the true number of the generated
signal events is 2574, and that for the accidental events is 2469, and the best-fit values (fitted without
external constraints) are consistent within the uncertainties.
In the “φ after correlation correction” plot, the φeγ value is corrected for each data pointconsidering the correlations in the signal PDFs. Here, the measured φeγ is shifted to cancel the
expected signal φeγ center, which is calculated with the measured Ee and θeγ . As we cannot know
the true event type (signal, accidental, or RMD) for the data, this correction is applied also tobackground events. Here, events are dropped from the plot if the corrected value is outside the φeγ
window. The φ-corrected background PDFs are calculated by shifting the original PDF according
to the event-by-event correction and their overall average is drawn as the fit line. This φ-corrected
plot thus shows the inner φeγ resolution for the signal PDF.
In the 2021 data taking, the RDC was not installed in the first half of the whole period. Therefore,

6.9. Properties of PDFs

127

Gamma energy

600
500
400
300
200

Events / ( 0.0004 GeV )Positron momentum

Events / ( 5.2e-05 GeV )Events / ( 0.04 ns )Time difference

600
500
400
300

450
400
350
300
250
200
150

200

100
100

100

0

0

−0.4

−0.2

0

0.2

0.4
t e γ (ns)50
0.0525

400
350
300
250
200

0.05

0.052

0.054

0.056
0.058
E γ (GeV)

φ after correlation correction
Events / ( 0.0032 rad )

450

450
400
350
300
250

500
400
300

200

150

150

100

100

50

50

0
−0.04

−0.02

0

0.02

0
−0.04

0.04
θ e γ (rad)

200
100
−0.02

Run period weight

0

0.02

0
−0.04

0.04
φ e γ (rad)

−0.02

1200
1000

Events / ( 0.001 GeV )

1400

4500
4000
3500
3000
2500

800

0

0.02
0.04
Corrected φ e γ (rad)RDC detected energy

RDC detected time

Events / ( 0.5 ns )Events / ( 1 )

0
0.048

0.0535E e (GeV)

φ angleEvents / ( 0.0032 rad )Events / ( 0.0032 rad )θ angle

0.053

4500
4000
3500
3000
2500

2000

2000

1500

1500

1000

1000

600
400
200
0

500
0

1

2

3
Period id

0
−10

500
−5

0

5

10
t RDC (ns)

0

0

0.02

0.04
0.06
E RDC (GeV)

Figure 6.21: One-dimensional projection of toy MC distribution (black markers) and its fit (bluesolid line) on each parameter space drawn in a linear scale. The toy MC samples were generated
with Nsig = 2500 and NAcc = 2500, and the blue fit line is a sum of the accidental PDF (magenta
dashed line) and the signal PDF (green solid line) accordingly. The best-fit value and its uncertainty
was Nsig = 2575 ± 51, NAcc = 2468 ± 52, and NRMD = 0 ± 2, which is consistent within theuncertainties.

Chapter 6. Analysis

128

1

Events / ( 0.0004 GeV )

102

10

102

10

−0.2

0

0.2

0.4
t e γ (ns)0.05250.053

0.0535E e (GeV)

0.048

10

1

102

10

1

0

0.02

−0.04

0.04
θ e γ (rad)

0.05

−0.02

0

0.02

10

−0.04

0.04
φ e γ (rad)

−0.02

0

0.02
0.04
Corrected φ e γ (rad)RDC detected energy

Events / ( 0.001 GeV )Events / ( 0.5 ns )

102

0.056
0.058
E γ (GeV)102

RDC detected time

10

0.054

1

Run period weight
3

0.052

φ after correlation correction
Events / ( 0.0032 rad )

102

−0.02

10

φ angleEvents / ( 0.0032 rad )Events / ( 0.0032 rad )θ angle

−0.04

102

1

1
−0.4

Events / ( 1 )Gamma energy

Positron momentum

Events / ( 5.2e-05 GeV )Events / ( 0.04 ns )Time difference

103

102

103

102

10
10

10

1
0

1

2

3
Period id

−10

−5

0

5

10
t RDC (ns)0Figure 6.22: The same distribution as Fig.6.21 in a log scale.

0.02

0.04
0.06
E RDC (GeV)6.10. Summary of systematic uncertainties

129

the efficiency of background tagging became lower for this reason. Note that this inefficiency shallnot be the case in the future dataset.
Event-by-event resolution parameters for signal PDFs
The distribution of important resolution parameters of the PDFs, which are different event-by-event,are shown in Fig.6.23. The positron resolution parameters — σEe , σye , σze , σθe , and σφe — areconditioned by the uncertainty of the track fitting. Fig.6.23 shows the raw positron resolution before
correlation corrections, and the visible σφe dependence on φe comes from the stronger δφe vs δEecorrelation at larger tan φe . The gamma resolution parameters — σEγ , σuγ , σvγ , and σwγ — areconditioned by vγ and wγ . In addition to the strong resolution dependence on wγ , as can be seenin the plots, the σEγ also depends on vγ , which results in the different resolution values even in thesame wγ ranges. The full angle resolutions — the inner σφeγ and the σθeγ — shown in Fig.6.23
are the values with all the correlations taken into account. Here, it can be found that σθeγ is larger
than the simple quadrature sum of σθe , σze , and σuγ because δze vs δθe correlation enhances the
combined error of δθeγ (recall Fig.6.18). The inner σφeγ , on the other hand, is smaller than thequadrature sum of each contribution thanks to the correlation corrections.
Visualization of signal vs background separation power
The power of the analysis to separate signal events from accidental events can be evaluated in terms(part)of Rsig , which is defined event by event as the ratio of signal likelihood divided by the backgroundlikelihood; namely



S(~
x
)


 Rsig = log10 B(~x)
(6.26)
Si (xi )
part

Rsig
= log10,Bi (xi )part
where Rsig is the full ratio of the full PDF and Rsig
is its parameter by parameter decomposition
with xi indicating one of Eγ , Ee , θeγ , φeγ or teγ . The background PDF, B(~x), is a sum of theaccidental PDF and the RMD PDF, namely

B(~x) = r · A(~x) + (1 − r) · R(~x),(6.27)(part)

where r ∼ 0.02 is based on the evaluation in Sec.6.4.
The Rsig distributions for signal
part
and accidental events are compared in Fig.6.24. The shown Rsig
distributions correspond to thepartprojections of Fig.6.25, which are scatter plots of Rsig vs the kinematical observables. Note herethat each plot must be projected with the correct weight according to the distributions of signal andbackground kinematics.

6.10

Summary of systematic uncertainties

The overall impact of the uncertainties on the signal PDFs is shown in Fig.6.26 with green (yellow)
bands indicating 1σ (2σ) of the PDF uncertainties, and major uncertain parameters are tabulated
in Tab.6.4. As is visible in the plots, the angle PDFs and the gamma energy PDF have large
uncertainties. The dominant cause of the angle PDF (gamma energy PDF) uncertainties is thedetector alignment uncertainty (the energy scale calibration uncertainty).

Chapter 6. Analysis

130

Ee resolution

θe γ resolution
180

2.5

160

γ

σE [%]

Eγ resolution vs w

3

2500

140
2000

2

1500

1.5

1000

1

500

120

100

150

200

0
0

250 300
σE [keV]

400

80

300

60

200

40

0.5
50

500

100

20
0
0

600

5

10

15

20

e

φe γ inner resolution

25
30
w [cm]

100

0

0
0

5

10

15

20

25
30
σθ [mrad]eγ

θe resolution

te γ resolution
3500

1400

3000

1200

400

2500

1000

300

2000

800

1500

600

1000

400

500

200

500

200
100
0
0

5

10

15

0
0

20
25
30
σφ inner [mrad]

50

100

150

200

φe resolution

e

5

10

15

20

eγ

eγ

σφ [mrad]00

250 300
σt [ps]25
30
σθ [mrad]ez e resolution

ye resolution

30
500

450

25

400
400

350

20

300
300

250

15

200

200

10

150

5

100

0

0
0

100
50

0

50
φe [deg]

0.5

2

0
0

2.5
3
σy [mm]

600

9
500

8
7

400

6
5

300

4
200

3
2

100

1

600

9
500

8
7

400

6
5

300

15

20

25
30
w[cm]

3

4

0

600

14
12

500

10

400

8

300

200

3

200
4

2
0
0

5
6
σz e [mm]

6

100

2

0

0
0

1
10

2

wγ resolution

10

4

5

1

v γ resolution
σvγ [mm]σu γ [mm]1.5euγ resolution
10

0
0

1

σwγ [mm]

−50

5

10

15

20

25
30
w[cm]

100
5

10

15

20

25
30
w[cm]

0

Figure 6.23: Distribution of the main resolution parameters used for signal PDFs. The first fivepanels shows the full resolution of Ee , Eγ , θeγ , φeγ , and teγ . The shown θeγ and φeγ resolution
values are those after correlation correction. The next four panels (the final three panels) show thepositron (gamma) resolution parameters relevant to angle PDFs.

6.10. Summary of systematic uncertainties

part

part

Rsig (Ee)
35000
30000

131

18000

Signal

16000

part

Rsig (Eγ )70000

Signal

50000
12000

20000

10000

40000

15000

8000

30000

6000
10000

20000

4000

5000

10000

2000
−2

−1

0

1 part 2
R sig (Ee )

0
−3

−2

part

30000

−1

0

0
−3

1 part 2
R sig (Eγ )

−2

−1

part

Rsig (φeγ )35000

Accidental

14000

25000

0
−3

Signal

60000

Accidental

Accidental

Rsig (θeγ )45000

Signal40000

Accidental

35000

25000

Accidental

7000

Signal

6000

Accidental

5000

30000

20000

25000

4000

15000

20000

3000

15000

10000

1part
2
R sig (θe γ )Overall Rsig

Rsig (teγ )Signal

0

2000

10000
5000
0
−3

1000

5000
−2

−1

0

1part
2
R sig (φe γ )

0
−3

−2

−1

0

1 part 2
R sig (t e γ )

0

−4

−2

0

2

4
R sig

part
Figure 6.24: Distribution of overall Rsig and its parameter-by-parameter decomposition of Rsig
(part)
for signal events (black) and accidental events (red). The probability of finding Rsig at zero isby definition the same between signals and accidentals when contributions from RMD background
PDF are approximately ignored.

Table 6.4: Summary of uncertain parameters
Parameter
Target alignment
LXe global shift
NormalizationEγ energy scale
Ee energy scaleteγ center

Uncertainty
100 µm
1 mm
5%
0.3 %
6 keV
4 ps

Chapter 6. Analysis

132

part

part

1

1

0

−1

−1

−1

−2

−2

−2

−3

−3

−3

−4

−4

−4

53

−5
48

53.5
Ee [MeV]

50

52

Rsig (φeγ )

1

−1

−1

−2

−2

−3

−3

−4

−4
0
0.02
0.04
Corrected φe γ [rad]

−5
−0.04

−0.02

0

0.02
0.04
θe γ [rad]

1
0

−0.02

56
58
Eγ [MeV]

Rsig (teγ )

2

0

−5
−0.04

54

part

part

2

R sig (t e γ )

part

part

1

0

52.5

Rsig (θeγ )

2

0

−5

R sig (φe γ )partRsig (Eγ )2

R sig (θe γ )part

R sig (Eγ )part

R sig (Ee )part

Rsig (Ee)

2

−5

− 0.4

− 0.2

0

0.2

0.4
te γ [ns]partFigure 6.25: Rsig
as a function of kinematical observables. The data points are produced fromsamples mixed both with signal and background samples. Projections of these plots on the y-axis,
when weighted correctly with the kinematical distribution, give the distribution shown in Fig.6.24

6.10. Summary of systematic uncertainties

φ

eγ

133

θ e γ PDF

PDF

t e γ PDF

80
5

40
70
35

4

60
30
50

25

40

3

20

30

15

20

10

10

5

2
1

0

−0.04

−0.02

0

0.02

−0.04

0.04
φ e γ [rad]

−0.02

0

0.02

0.04
θ e γ [rad]0− 0.4 − 0.2

0

0.2

0.4
t e γ [ns]E γ PDFE e PDF

250
5000
200

4000
3000

150

2000

100

1000

50

0.05250.053

0.0535E e [GeV]

0.048 0.05 0.052 0.054 0.056 0.058
E γ [GeV]

Figure 6.26: Impact of systematic uncertainties on the PDF shape. The solid black lines are thenominal shape of the PDFs, and the green and yellow bands are 1σ (2σ) of the PDF uncertainties.

Chapter 6. Analysis

134

Confidence interval in full MC fitting

Best-fit value of Nsig

Upper limit of Nsig

35
30
25
20

30
25
20
15

15
10

10

5

5
0
0

5

10

15

20

25
Nsig

0
0

5

10

15
20
Lower limit of Nsig

Figure 6.27: Nsig best-fit distribution (left) and its confidence interval distribution (right) for
Nsig = 10 pseudo-experiments that mix full signal detector simulation with background toy MC
up−limit
low−limit
samples. The red box in the right figure corresponds to the region with Nsig
< 10 < Nsig.
Each of the plots shows a good agreement with the generated pseudo experiments; the best-fit value
up−limit
low−limitof Nsig is consistent with 10, and 90 % experiments satisfy Nsig
< 10 < Nsig
as requested
by definition.

6.11

Reliability check of analysis with full detector simulation

Let us now discuss the reliability of the likelihood analysis. As presented in Sec.6.6, the accidental
background PDF is based on the direct evaluation with data samples, and we do not expect alarge mis-modeling. For the RMD background PDF, we have a larger uncertainty in the modeling,which however is not harmful in the fitting because the RMD events are not prominent in theanalysis region. With regards to the signal PDF, on the other hand, the fitting is sensitive to signal
mis-modeling. Furthermore, the signal PDF may have complicated correlations that are missed inSec.6.7. Therefore, it is desirable to have a careful reliability check for the signal modeling.
The reliability of the signal modeling was checked by applying the likelihood analyses to signal
samples in full detector simulation. The fit samples were generated by mixing signal full MC
samples with background toy MC samples. Here, Nsig = 10, NAcc = 68, and NRMD = 1.2
expectations were used when toy MCs were generated. There is a large discrepancy in the σEγ
between the data and simulation; the data gives 2 % resolution while the simulation gives 1 %resolution. To compensate for this difference, additional 1.7 % smearing of Eγ was adopted when
the toy MCs were generated. A thousand such samples were generated and fitted in this study.
The distribution of Nsig best-fit value is presented in the left plot of Fig.6.27, where its average
agrees with the true value of Nsig = 10. The confidence interval distribution is also presented inup−limit
low−limit
the right figure of Fig.6.27, where 90 % of all the samples satisfies Nsig
< 10 < Nsig
, asrequested by definition.
(part)
Fig.6.28 shows a comparison between the toy MC and full detector simulation in terms of Rsig
defined in Eq.(6.26), where we can see good overall agreements. In the parameter by parameter
part
part
decomposition, small differences can be seen in Rsig
(Eγ ) and Rsig
(θeγ ) distributions. These canbe understood within the performance differences between the data and the detector simulation.

6.11. Reliability check of analysis with full detector simulation

part

part

Rsig (Ee)part

Rsig (θeγ )Rsig (Eγ )

500

1400

1800

Toy MC from signal PDF

Toy MC from signal PDF

Toy MC from signal PDF

1200

400
Full detector simulation

1600
Full detector simulation

1200

300

800

Full detector simulation

1400

1000

1000

600

800

200

600

400

400

100
200
0
−3

135

200
−2

−1

0

1 part 2
R sig (Ee )

0
−3

−2

part

−1

0

1 part 2
R sig (Eγ )

0
−3

−2

−1

part

Rsig (φeγ )

0

1part
2
R sig (θe γ )Overall Rsig

Rsig (teγ )200

Toy MC from signal PDF

1200

Toy MC from signal PDF

Full detector simulation

1000

Full detector simulation

Toy MC from signal PDF

180

1000

160

Full detector simulation

140

800
800
600

120
100

600

80
400

400

200

200

0
−3

0
−3

60
40
20

−2

−1

0

1part
2
R sig (φe γ )

−2

−1

0

1 part 2
R sig (t e γ )

0

−4

−2

0

2

4
R sig

Figure 6.28: Comparison of fitting to signal MC samples from full detector simulation and toy MC.

136

Chapter 6. Analysis

Chapter 7
Result and Discussion
7.1

Sensitivity

The sensitivity is defined as the median of 90 % upper limits for toy MCs produced with Nsig = 0.
The 90 % upper limit distribution of the toy MC samples is shown in Fig.7.1. The median valueof the limit is B(µ → eγ) < 8.4 × 10−13 (Nsig < 2.21) without the systematics and B(µ → eγ) <8.8 × 10−13 (Nsig < 2.31) with all the systematics. The 3-sigma discovery power is achieved
when Nsig > 6.5 or B(µ → eγ) > 2.4 × 10−12 , which is defined as the value that gives > 90 %of experiments p-value < 0.001 for the Nsig = 0 hypothesis. The breakdown of systematics
uncertainties, which account for ∼ 4 % sensitivity degradation, is shown in Tab.7.1.

7.1.1

Sensitivity cross-check in timing sideband data fitting

The timing sideband regions were divided into four sub-ranges; (−3 ns, −2 ns), (−2 ns, −1 ns),
(1 ns, 2 ns), and (2 ns, 3 ns). The corresponding upper limits on these four sidebands are overlaidin Fig.7.1. In the fourth sideband, which set an interval of 0.41 < Nsig < 6.95, the best-fit value
is Nsig = 2.65 and the p-value for the Nsig = 0 hypothesis is 4.4 %. It is understood just to be
an upward fluctuation of accidental background events. The top 10 highly ranked events, with theorder defined by Rsig in Eq.(6.26), are shown in Tab.7.2. The best-fit value of Nsig = 2.65 in the
fourth sideband can be understood to be from the top three events therein.
Table 7.1: Breakdown of impact of systematic uncertainties on limits
Parameter

Impact on sensitivity

φeγ uncertainty
Eγ uncertainty
θeγ uncertainty
Normalization uncertainty
teγ uncertainty
Ee uncertainty
RDC uncertainty
137

1.1 %
0.9 %
0.7 %
0.6 %
0.1 %
0.1 %
< 0.1 %

Chapter 7. Result and Discussion

138

100
80
60
40
20
0
0

1

×10−122
3
4
90% upper limit on Br(µ→ eγ )Figure 7.1: Distribution of 90 % upper limit in toy MCs including systematic uncertainties. The
median value is 8.8 × 10−13 . The four arrows indicate the upper limits in the sideband data fitting.

Table 7.2: List of highly ranked events in the timing sideband
Rank

Run

Event

1
2
3
4
5
6
7
8
9
10

404797402358398657406340402700404598393087403861391657405095

1970
1389
329
1678
2001
1129
1188
912
739
848

teγ
[ns]
2.374
2.491
2.593
-1.629-2.656-1.3271.318-2.525-1.643
1.519

Ee
Eγ
φeγ
[MeV] [MeV] [mrad]
52.789 52.407 18.372
52.779 49.240 3.942
52.697 51.048 -5.082
52.469 52.305 4.106
52.706 52.154 -11.193
52.821 51.237 -19.580
52.971 50.940 -18.616
52.776 48.117 19.285
52.906 49.266 -0.745
52.792 49.998 -22.754

θeγ[mrad]-11.755
8.631
3.582-5.679-3.821
16.9618.304-2.190-18.50915.431

tRDC
[ns]
10
10
10
-2.786
10
10
10
10
10
10

ERDC
[MeV]
-1
-1
-1
19.017
-1
-1
-1
-1
-1
-1

7.2. Result

CL

139

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0

×10−12
0.5

1

1.5

2

2.5
Br(µ→ eγ )Figure 7.2: CL curve with the MEG II 2021 dataset. The curve crosses CL = 0.9 at B(µ → eγ) =7.5 × 10−13 , giving a 90 % upper limit accordingly.

7.2

ResultThere were 66 events in the analysis region, and we did not find any excess of signal events among
them with the best-fit value being Nsig = −0.0003. Fig.7.2 shows the CL value (defined inSec.6.1.4) as a function of B(µ → eγ). We set a limit of B(µ → eγ) < 7.5 × 10−13 (Nsig < 1.98)
at 90 % confidence level. Fig.7.3 and Fig.7.4 show the observed event distribution. A scatter plotof Rsig vs observed date is also shown in Fig.7.5. The highest ranked 10 events, shown as the filled
red circles in Fig.7.5, are also listed in Tab.7.3. The above results were cross-checked with different
analysis methods as detailed in Appendix.C, and they agree well with each other.

7.2.1

Combined result with MEG I final analysis

This analysis was combined with MEG I final result [1], which made use of the likelihood curves
shown in Fig.7.6. Here, the combined best fit is Nsig = 0 with the physical constraint. As thefull-frequentist simulation is not practically possible for the combined analysis, the threshold on
the profile likelihood ratio was empirically determined from the full-frequentist simulations for the
MEG final analysis and the MEG II 2021 analysis. The threshold on the negative log-likelihood
value — the − log λ introduced in Eq.(6.2) — ranges between 1.2 – 1.7, depending on the tested
Br(µ → eγ) value and the difference between MEG and MEG II. Here, we found that the thresholdof the negative log-likelihood can become large when NBG is small and Nsig < 5. This time, weused the full MEG statistics, and we had a large NBG . Therefore, we adopted a threshold of 1.6
for 90 % confidence interval, which was the largest possible threshold for the MEG final analysis.
Here, note that this threshold value is larger than the 1.35 threshold value of the asymptotic method[103]. As a result, the combination of the MEG final data and the MEG II 2021 data gave a limit ofB(µ → eγ) < 3.1 × 10−13 .
The sensitivity for the combined analysis was calculated by combining MEG-only pseudo
experiments with MEG II-only pseudo experiments. The upper limit distribution is shown in
Fig.7.7, where the median sensitivity is B(µ → eγ) < 4.3 × 10−13 . The combined data upper limitChapter 7. Result and Discussion

58

t eγ [ns]E γ [MeV]

140

57

1
0.8

56

0.6

55

0.4

54

0.2 2

53

0
1

−0.2

1

52

−0.4

51

−0.6

50
2

−0.8

49
48
51.5

52

52.5

53

53.5
54
E e [MeV]−1−1−0.9998 −0.9996 −0.9994 −0.9992 −0.999
cosΘ eγ

Figure 7.3: Distribution of observed events in the (Ee , Eγ ) and (cos Θeγ , teγ ) planes. In the (Ee , Eγ )plot, cos Θeγ < −0.9995 and |teγ | < 200 ps cuts are applied with 97 % signal efficiency each. In
the (cos Θeγ , teγ ) plot, 49 < Eγ < 55 MeV and 52.5 < Ee < 53.2 MeV cuts are applied with 93 %and 97 % efficiency respectively. The green lines indicate 1σ, 1.64σ, and 2σ contours of the signal
PDF. The highest ranked five events are drawn as stars with the ranking indices beside them. The
third to fifth rank events are not found in the left plot because of the timing or angle cut.

Table 7.3: List of highly ranked events
Rank

Run

Event

1
2
3
4
5
6
7
8
9
10

401563402458403059405442401603405131402692391935401611402737

1286
22
2406
9
2718
1189
2734
1276
2589
797

teγ
Ee
Eγ
φeγ
[ns]
[MeV] [MeV] [mrad]-0.108 52.974 51.952 -2.235
0.137 52.695 49.514 3.376-0.287 52.738 52.013 -1.397-0.040 52.772 49.721 9.852-0.099 52.766 49.186 -22.790
0.243 53.153 48.518 -1.468-0.107 52.528 51.747 -21.823-0.195 52.521 52.490 20.031
0.211 52.744 48.773 -0.080-0.423 52.594 48.601 2.509

θeγ[mrad]-27.6591.311-13.269-30.290
23.95613.60332.029-37.719-13.500
13.404

tRDC
[ns]
10
10
10
10
10
10
10
10
-1.367
10

ERDC
[MeV]
-1
-1
-1
-1
-1
-1
-1
-1
4.435
-1

7.2. Result

141

Gamma-ray energy

108

Events / (0.5 MeV)

Positron momentum
Events / (50 keV)

Events / (50 ps)

Time difference
12
10
8

6

10

1

6
10−1

4
4
2
0

2
−0.4

−0.2

0

0.2

10−2

0

0.4
t eγ [ns]

52.5

53.5
E e [MeV]48

φ angle

10
8

50

52

54

56
58
E γ [MeV]

RDC detected energy
Events / (1 MeV)

Events / (5 mrad)

Events / (5 mrad)θ angle

53

12
10
8

6

102

10

1

6
10−1

4
4
2

0

0
−40

20
40
θ eγ [mrad]

−20

0

20
40
φ eγ [mrad]

0

Run period weight

RDC detected time

10

Events / (0.5)

102

−20

Events / (bin)Events / (0.5 ns)

0
−40

10−2

2

30
25
20

5

10

15
20
E RDC [MeV]R sig value
10

1

1
15
10−1
10−2
−10

10−1

10
10−2

5
−5

0

5

10
t RDC [ns]

0

1

2

3
Period id

−10

−5

0

5
Rsig

Figure 7.4: Projected distribution of observed events on the parameter space of eight fit observables
and Rsig parameter (black markers), and the expected shape of the distribution according to thebest-fit value (blue solid line). The magenta dashed (red dash-dotted) line represents the componentof the accidental background (RMD background) in the fitted PDF. The green hatched region shows
the signal PDF with Nsig = 8, which is four times larger than the upper limit.

Chapter 7. Result and Discussion

Rsig

142

5
0

−5
−10
−15
−20

30/Sep07/Oct14/Oct21/Oct28/Oct04/Nov

11/Nov

18/Novdate− logλ p

Figure 7.5: The Rsig value and the date of the observed 66 events in the analysis region. The filled
red circles are the highest-ranked 10 events and the black open circles are the other 56 events. The
number of observed events before October 15th (between October 15th – October 28th) is small
because of the inefficiency in the DAQ as shown in Fig.3.5 (because of the small beam intensity aspresented in Tab.3.1).

12
10
8

Combined
MEG II 2021
MEG 2009-2013

6
4
2
0
0

×10−120.5

11.5
Branching ratio

Figure 7.6: The negative log-likelihood ratio for the MEG I final data (magenta dashed line),
MEG II 2021 data (blue dash-dotted line), and their combination (black solid line) as a functionof B(µ → eγ). The offset of the log-likelihood value is defined so that − log λ = 0 at the best-fit
value with the Nsig = 0 constraint.

7.3. Discussion

143

Sensitivity comparison

MEG II toy MC + MEG final data combined

70
60
MEG II 2021 sensitivity

50
Combined sensitivity

40
30

MEG II only toy MC limit−12

MEG I final sensitivity

3 ×10
2.5
2
1.5

20

1

10

0.5

0
0

0.5

1

×10−121.5
2
2.5
3
Upper limit on Br(µ→ eγ )Figure 7.7: Distribution of 90 % upper limit
in toy MCs simulating a combined analysisof the MEG final data and the first MEG II
data. The median value is 4.3 × 10−13 .

0

0.2

0.4

0.6

×10−120.8
1Combined limit

Figure 7.8: Distribution of 90 % upper limit
when the final MEG data is combined withtoy MCs simulating the first MEG II data.
The x-axis shows the combined upper limit
and the y-axis shows the limits of the MEG II
toy MCs.

of 3.1 × 10−13 is equivalent to the bottom 30 % of the combined toy MC samples. This can beunderstood from the fact that both the MEG final data and MEG II data are downward fluctuated;
4.2 × 10−13 of the MEG final data is equivalent to the bottom 36 %, and 7.5 × 10−13 is equivalent
to the bottom 38 %. As a case study, Fig.7.8 shows the upper limit distribution for combinations of
MEG II-only pseudo experiments and the MEG final data. Here, the set of upper limits on data,
7.5 × 10−13 for MEG II and 3.1 × 10−13 for the combined, is found to be consistent.

7.3

Discussion

The sensitivity of this analysis (8.8 × 10−13 ) did not exceed the full sensitivity of the previous
experiment (5.3 × 10−13 ), and the upper limit (7.5 × 10−13 ) was not as stringent as the previous
one (4.2 × 10−13 ). This, however, is just because of the limited time in the data-taking of thefirst MEG II data. The author stresses that this result, as will be discussed in the next sections,demonstrates the feasibility of sensitivity improvement by an order of magnitude when we consider
the planned MEG II data taking in the next few years.
This analysis was purely limited by the statistics of the dataset and it kept almost backgroundfree conditions. On the other hand, the previous search was limited by the number of background
events. This makes an important difference in future projections. When the sensitivity is limitedby the statistics, continuous data-taking will improve proportionally to the DAQ time. In the caselimited by backgrounds, on the other hand, it becomes useless because the sensitivity improves
only by the square root of the DAQ time. Furthermore, the effective DAQ time for the data in thisanalysis is only four weeks when we scale the DAQ inefficiencies unique to the first commissioning(see Fig.3.1 and Fig.3.5 in particular). This is much smaller than the five years of data-taking in
the previous search. Therefore, it is a wonderful result that the sensitivity of this analysis closely
approached that of the previous one despite all these limitations.
The improvement of this experiment comes from several reasons. The first improvement is in theefficiency of the positron reconstruction, which is ×2 higher than that of the previous experiment.

Chapter 7. Result and Discussion

144

In addition, the interruption of the data taking was reduced thanks to the stable operation of thedetectors. This improvement gives a much higher number of effectively measured muon decays perunit time. A significant reduction in the number of backgrounds was also achieved by resolution
improvements. In particular, the positron momentum resolution improvement was significant;
380 keV → 90 keV. In addition, we succeeded in reducing the impact of systematic uncertainties
on the sensitivity, 13 % → 4 %.
At the moment with limited statistics from the MEG II, it is also useful to combine this analysis
with the previous experiment. This gave the sensitivity of 4.3 × 10−13 and the most stringent upper
limit of 3.1 × 10−13 . Note that this sensitivity record will be broken shortly (within a year fromnow) when the MEG II 2022 data analysis is completed, as will be presented in the next section.

7.4

Prospect

This thesis is based on the 2021 data, which was taken for seven weeks in total giving 4.8 weeks
of efficiently used livetime. At the time of this thesis, we have already taken additional two yearsof data. In 2022 (2023), we took the physics data for 18 (22) weeks in total from July (May) to
November, which gave 12.8 (15.9) weeks of efficiently used livetime. Though the fraction of the
livetime was small at the beginning of 2022 data taking because of some inefficiencies (in calibration
scheme, DAQ computer, etc.), it finally improved to 81 % in late 2022.
In the future, we plan to continue the experiment till 2026. Here, however, we have uncertainties
in the beamtime assignment because of the beamline sharing with other experiments, in particular the
Mu3e experiment. Based on the achievements in the DAQ scheme and the beamtime uncertainties,
our expectation of the total MEG II livetime is 60 – 80 weeks. Hereafter, we employ this expectation
as the baseline scenario of future data taking.
In the future sensitivity calculation, we fully use the knowledge from the 2021 data analysis. The
set of PDFs used in the 2021 data analysis is reused in the toy MC simulation and the fitting. The
number of effectively measured muon decays is extrapolated according to the result in Sec.6.3.1. If
we can assume that the trigger efficiency will be improved to 95 %, the increase of the normalization
factor per unit livetime would be
dkMichjel= 1.85 × 106
dtlive(at 4 × 107 µ/s beam intensity).

(7.1)

The number of background events is also based on the observation in the 2021 dataset as
Nacc= 2.8 × 10−11k(at 4 × 107 µ/s beam intensity).

(7.2)

Note here that the value on the right-hand side behaves proportionally to the beam rate, Nkacc ∝ Rµ .
According to the inputs above, we found that the sensitivity is maximized at 4 × 107 µ/s beam
intensity within 60 – 80 weeks of DAQ livetime. Therefore, we adopted 4 × 107 µ/s beam rate since
the start of the 2023 data taking. The future sensitivity was also estimated with this optimized beam
rate, and the result is shown in Fig.7.9. It indicates that a 6 × 10−14 sensitivity can be achieved
within 60 weeks of livetime.
Possible performance improvement
As well as the continuous data taking, further improvements in the detector performance are
expected. Hardware improvements are expected in the positron spectrometer and the RDC, and7.4. Prospect

Br(µ→ eγ )

145

10−12
2021 result

MEG sensitivity (90% C.L.)
MEG limit (90% C.L.)

10−13
90% UL sensitivity± 1σ of sensitivity± 2σ of sensitivity

10−14
0

3σ discovery at 90% power

20

40

60
80
DAQ livetime [weeks]

Figure 7.9: Expected sensitivity in the future MEG II data taking as a function of the accumulated
DAQ livetime. The black solid line with circular markers indicates the sensitivity (median) andthe green (yellow) band indicates a statistical fluctuation at one (two) sigma level. The sensitivity
is calculated at the point locations. The blue line with square markers shows a 3 sigma discovery
potential at 90 % power. The red cross marker at the top-left indicates the observed upper limit inthis thesis for MEG II only analysis, namely 7.5 × 10−13 .

146Chapter 7. Result and Discussion

reconstruction and calibration improvements are expected both for positron and gamma-ray. In therest of this section, we will discuss them in detail.
The positron spectrometer hardware is planned to be replaced with new ones in the 2023 – 2024
shutdown period. The CDCH, which is defective now due to the wire breaking problem mentioned
in Sec.1.4, will be replaced with a new chamber with one more outermost layer than the present
one. With a complete set of wires and the increased detection layer, we can expect a higher tracking
efficiency and a better resolution. Given this opportunity to re-install the detector, the pTC will also
be updated by replacing some scintillation tiles with new ones. This replacement targets counters ofseverely radiation-damaged counters, which thus have bad timing resolution. The reported timing
resolution in Sec.5.5.4 got worse by 10 % since the first operation in 2017 [42]. Therefore, thisreplacement is expected to improve the positron timing resolution accordingly.
In the current data taking, the RDC is only installed in the downstream end (Fig.2.26). This isbecause of the technological difficulty in achieving a high rate capability and a low material budget,which is because of the operation in the muon beam. An encouraging candidate is a Resistive Plate
Chamber based on a novel technology with Diamond-Like Carbon electrodes. In Ref.[55, 56],
both the material budget and the rate capability are demonstrated in a measurement at the MEG II
beamline. The detector scalability is the remaining challenge, and technological R&D works arein progress to solve it. This will be installed, if everything goes smoothly, in 2025 and will further
reduce the background gamma-rays from the radiative decay.
The pileup analysis in the gamma-ray energy reconstruction can be improved by fully exploiting
waveform information recorded in the trigger hardware, which is currently not referred to in thereconstruction. This waveform corresponds to the summed waveform of the LXe readout channels
and is used to implement the Eγ threshold trigger. This is recorded in a wider time window thanthat recorded by DRS, and thus, pileup pulses in the baseline region can be found more reliably.
The use of this additional information would help to search pileups with a more robust algorithm
and thus reduce the background in the Eγ spectrum.
The time calibration for the gamma-ray measurement, presented in Sec.5.4, is found not toperfectly model the sensor-by-sensor measured time. This imperfect modeling causes a correlation
between the bias in the measured time and the scintillation light propagation distance. Though thiscan naively be considered to be a wrong calibration of the scintillation light velocity, we concluded
that a simple modeling with a constant velocity cannot explain all the observed effects. This canbe mitigated by introducing a more elaborate calibration method, including the machine learning
approach, which the author expects to improve the σLXe down to 50 ps in the most optimistic case.
The positron track reconstruction efficiency is currently limited by the track finding algorithm
as indicated in Sec.5.13. Therefore, its improvement would be helpful to have larger statistics of thedataset. One possible direction of improvement is the introduction of a global pattern recognition,
which can find tracks from a set of distant hits, unlike the currently employed local-forwarding
pattern recognition. Though a huge amount of studies will be necessary here, it would be aninteresting work also in view of tracking detectors in other experiments.
If all these improvements work effectively in addition to the increased data amount, the final
sensitivity presented in Fig.7.9 can be improved by up to 20 – 30 %. After all, the experimentwill certainly achieve an order of magnitude sensitivity improvement than what we have currently
achieved.

Chapter 8
Conclusion
The MEG II experiment started data taking in 2021 to search for the lepton flavor violating muon
decay, µ → eγ. In this work, we searched for µ → eγ by observing 2.64 × 1012 muon decays,
which were collected in effectively four weeks. We built a maximum likelihood analysis and thesearch sensitivity was 8.8 × 10−13 . No signal excess was observed in the analysis and we set an
upper limit ofB(µ → eγ) < 7.5 × 10−13
at 90 % confidence level.
The sensitivity of this analysis was limited by the statistics of the dataset due to the limited
DAQ time in 2021. However, we approached the sensitivity of the previous experiment only ineffectively four weeks of data taking. This demonstrates the feasibility of the target sensitivity of
the MEG II experiment, which is an order of magnitude higher than the previous experiment. A
sensitivity calculation, which is based on the achieved performance presented in this thesis, showsthat 6 × 10−14 sensitivity can be reached within 60 weeks of data taking.
In order to achieve the highest sensitivity ever, this analysis was combined with the previousµ → eγ search [1]. The combined sensitivity was 4.3 × 10−13 , and we set an upper limit ofB(µ → eγ) < 3.1 × 10−13
at 90 % confidence level. This gives the most stringent limit for the µ → eγ branching ratio.

147

148

Chapter 8. Conclusion

Appendix A
Radiative Muon Decay
A.1

Full energy distribution in radiative decay

The differential branching ratio in muon radiative decay, drawn in Fig.A.1a, can be expressed as[28, 107]dB(µ+ → e+ ν ν̄γ) =dy
α
βdx dΩe dΩγ
3
64π
y


p
~
p
~
γ
e
G(x, y, d) − P~µ ·H(x, y, d) ,× F (x, y, d) − β P~µ ·|p~e ||p~γ |(A.1)

where dΩe,(γ) is the solid angle of electron (gamma), P~µ is the polarization vector of the muon,p~e,(γ) is the momentum of electron (gamma), β = |p~e |/Ee , x = 2Ee /mµ , y = 2Eγ /mµ , andd = 1 − β |p~p~ee|·|·p~p~γγ | . The x and y ranges are given from the four-body kinematics as√(0 < y ≤ 1 − r)√
(1 − r < y ≤ 1 − r),√
2 r <x<1+rr(1 − y) +≤x≤1+r1−y(A.2)

where r = (me /mµ )2 . The explicit form of the functions F, G, H are given as
F =F (0) + rF (1) + r2 F (2)G =G(0) + rG(1) + r2 G(2)
H =H

(0)

+ rH

(1)

2

+r H

(2)

(A.3)
.

Each function component in the above equation is given as
8
F (0) (x, y, d) = {y 2 (3 − 2y) + 6xy(1 − y) + 2x2 (3 − 4y) − 4x3 }d+ 8{−xy(3 − y − y 2 ) − x2 (3 − y − 4y 2 ) + 2x3 (1 + 2y)}+ 2d{x2 y(6 − 5y − 2y 2 ) − 2x3 y(4 + 3y)} + 2d2 x3 y 2 (2 + y)
F

(1)

(A.4)



32
y(3 − 2y)
8(x, y, d) = 2 −− (3 − 4y) + 2x + {y(6 − 5y) − 2x(4 + y) + 6x2 }
d
x
d
(A.5)
2
2
2
+ 8{x(4 − 3y + y ) − 3x (1 + y)} + 6dx y(2 + y)149

Appendix A. Radiative Muon Decay

Differential Branching Ratio

150

1

0.8

0.6

10

10

-1

-2

0.4

0.2
10

-3

0
0
0.1

ph 0.2ot 0.3
on 0.4
en 0.5
er 0.6gy(y 0.7
) 0.8
0.9
1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y (x)

erg
on en
ositr

10

-4

0.1

0.2

0.3

0.4

0.5

(a)

0.6

0.7

0.8

0.9

1

y (photon energy)p(b)Figure A.1: Differential branching ratio of radiative muon decay as a function of positron andgamma-ray energy [28]. (a): Two-dimensional energy spectrum of positron energy and gammaray energy. (b) The one-dimensional energy spectrum of gamma-ray energy with positron energy
integrated out.

F

(2)

32
(x, y, d) = 2
d




48y
(4 − 3y)
−3 +
x
d

(A.6)

8
G(0) (x, y, d) = {xy(1 − 2y) + 2x2 (1 − 3y) − 4x3 }d+ 4{−x2 (2 − 3y − 4y 2 ) + 2x3 (2 + 3y)} − 4dx3 y(2 + y)

(A.7)

32
8
G(1) (x, y, d) = 2 (−1 + 2y + 2x) + (−xy + 6x2 ) − 12x2 (2 + y)
d
d

(A.8)

G(2) (x, y, d) = −

96
d2

8
H (0) (x, y, d) = {y 2 (1 − 2y) + xy(1 − 4y) − 2x2 y}d+ 4{2xy 2 (1 + y) − x2 y(1 − 4y) + 2x3 y}+ 2d{x2 y 2 (1 − 2y) − 4x3 y 2 } + 2d2 x3 y 3
H

(1)



y(1 − 2y)
−
+ 2y
x
8
+ {y(2 − 5y) − xy} + 4xy(2y − 3x) + 6dx2 y 2d32(x, y, d) = 2
d

H (2) (x, y, d) = −96y 48y
+
d2 x
d

(A.9)

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

A.2. Background gamma-ray spectrum for accidental events

A.2

151

Background gamma-ray spectrum for accidental events

Integrating Eq.(A.1) over the electron energy and angle variables, we can calculate the gamma
energy spectrum in the accidental background as
dB(µ+ → e+ ν ν̄γ)
1
= [J+ (y)(1 + Pµ cos θγ ) + J− (y)(1 − Pµ cos θγ )] ,dyd(cos θγ )ywhere me /mµ is neglected. Here, J± are

 

α
1 − y 171−y
J+ (y) =(1 − y) 3 ln−
+ −3 ln+ 7 (1 − y)
6π
r
2
r



1 − y 132−
(1 − y)+ 2 lnr
3
J− (y) =α(1 − y)2
6π


 

1 − y 31
1 − y 293 ln−
+ −4 ln+(1 − y)
r
4
r
3



1 − y 55
+ 2 ln−(1 − y)2 .
r
12(A.13)(A.14)(A.15)

Fig.A.1b shows the accidental gamma energy spectrum calculated by Eq.(A.13). The rough scalingof the number of accidental background gamma-rays from radiative decay can be calculated byintegrating Eq.(A.13) over y within the resolution,Z
Z 1
dB(µ+ → e+ ν ν̄γ)  α ≈(∆y)2 [ln(∆y) + 7.33].
dy d(cos θγ )dyd(cos θγ )
2π
1−∆y

152

Appendix A. Radiative Muon Decay

Appendix B
Transformation of positron correlation toθeγ vs φeγ correlation
The notation below follows that defined in Sec.6.7.4.

B.1

Correlation and covariance in a simple model

B.1.1

Relation between y vs x covariance, y inner, and px y

Let us consider a two-parameter model that follows
y = pxy × x + yinner .

(B.1)

where x and yinner are independent. In this model, the covariance follows
cov(x, y) = pxy · σx2 .

(B.2)

In an ideally distributed Gaussian model, the slope parameter can thus be transformed by dividing
the covariance by the σx2 . Therefore, we can also derive the following identity,
pyx =

σx2
pxy .
σy2

(B.3)

In addition, the y resolution follows2
σy2 = σy,inner+ p2xy σx2 ,(B.4)

which indicates the relation between the full resolution and the inner resolution.

B.1.2 y vs x correlation when x is a composite of x1 and x2
Let us next consider a model of x, y with x that can be decoupled into two independent measurements
as x = x1 +x2 , and y is correlated only with x1 and independent of x2 . Here, the covariance between
x, y is identical to that of x1 , y,cov(x, y) = cov(x1 , y).
(B.5)
When they are ideally distributed, the following are satisfied,cov(x, y) = pxy · σx2= px1 y · σx21
153

(B.6)

154

Appendix B. Transformation of positron correlation to θeγ vs φeγ correlation

Given σx2 = σx21 + σx22 , one can obtain pxy as
σx21
pxy = 2
· p x1 y .
σx1 + σx22

(B.7)

The y inner in y vs x (σy,inner−xy ) then becomes different from that in y vs x1 (σy,inner−x1 y ),2
σy,inner−xy= σy2 − p2xy σx22= σy,inner−x+1yσx22 2 2p σ .
σx2 x1 y x1

(B.8)

The enlarged inner resolution here can be qualitatively understood that the additional x2 contribution
in the y vs x plot prevents the y vs x1 correlation from getting perfectly resolved, which thus broadens
the y inner distribution.

B.2

Correlation and covariance in δφeγ vs δθeγ

We then build the full correlation of δφeγ vs δθeγ . Given the positron correlation parameters, they
can be decomposed into the following equations,δθeγ = δθe + Cz δze + Cu δuγ
δφeγ = δφe,inner−Ee + Cv δvγ
δφe,inner−Ee = δφe,inner−Ee θe ze + pθe φe δθe + pze φe δzeδze = pθe ze δθe + δze,inner(δθe = pze θe δze + δθe,inner ),(B.9)(B.10)(B.11)(B.12)(B.13)

where δφe,inner−Ee is the inner of φe after correcting only Ee correlation and δφe,inner−Ee θe ze is theinner of φe after correcting all Ee , θe , ze correlations. When we describe per-event PDF parameterization, σφe,inner−Ee = sφe,inner−Ee × σφ0 e is used hereafter. Cz , Cu , and Cv are parameters thattranslate Ze , uγ , vγ resolutions into their contributions to θeγ or φeγ resolutions. Though ye contribution, which is in our best knowledge independent of θe , φe , ze , is dropped in the above equation
for simplicity, our analysis implementation includes its contribution too.
The resolution of θeγ , which is the σ of the square of Eq.(B.9) considering ze vs θe correlation,
can be written asσθ2eγ = σθ2e + Cz2 σz2e + 2Cz pθe ze σθ2e + Cu2 σu2γ22

= s2θe σ 0 θe + Cz2 s2ze σ 0 ze + 2Cz p0θe ze s2θe σθ0 e σz0 e + Cu2 σu2γ ,(B.14)

where the below substitutions are used
σz0 e
σθ0 e

(B.15)σθe = sθe σθ0 e .

(B.16)

pθe ze = p0θe ze

B.2. Correlation and covariance in δφeγ vs δθeγ

155

Here, note that σze is the full σ including the p0θe ze correlation, not the inner after correcting it 1.
The covariance between θeγ and φeγ follows
cov(δθeγ , δφeγ ) = σθ2e (pθe φe + pze φe · pθe ze ) + Cz (σz2e pze φe + σθ2e pθe φe · pθe ze )= σθ2e (pθe φe + pze φe · pθe ze ) + Cz σz2e (pze φe + pθe φe · pze θe ),(B.17)

where a substitution like Eq.(B.3) is used. This will later be used to obtain pθeγ φeγ =cov(δθeγ , δφeγ )/σθ2eγ and the inner φeγ resolution.
The positron correlation parameters observed in data or MC studies are not necessarily pze φe orpθe φe in the above equations because we study “pze φe ” with (θe , ze , φe ) parameter space projected
onto (ze , φe ) or (θe , φe ) plane. Let us assume that we evaluate the correlation in the following
procedure;
1. φe vs ze correlation first, and introduce qze φe parameter as the correlation parameter when
projected on the (ze , φe ) plane,
2. After correcting qze φe and making a projection on the (ze , φe ) plane, we then evaluate qθe φeparameter.
By substituting Eq.(B.13) into Eq.(B.11), we obtain
δφe,inner−Ee = δφe,inner−Ee θe ze + (pθe φe pze θe + pze φe )δze + pθe φe δθe,inner .

(B.18)

Here, δφe,inner−Ee θe ze and δθe,inner are independent of δze , therefore,q z e φe = p z e φe + p θ e φe · p z e θ e .

(B.19)The qθe φe parameter can then be evaluated from
δφe,inner−Ee − qze φe δze = δφe,inner−Ee θe ze + pθe φe (1 − pze θe pθe ze )δθe − pθe φe · pze θe δze,inner , (B.20)
from which we obtainqθe φe = pθe φe (1 − pze θe pθe ze ).

(B.21)

From Eq.(B.17), Eq.(B.21), and Eq.(B.19), we then derivecov(δθeγ , δφeγ ) = σθ2e (qθe φe + qze φe · pθe ze ) + Cz σz2e qze φe .

(B.22)

The correlation parameter between φeγ vs θeγ then followspθeγ φeγ =cov(δθeγ , δφeγ )
σθ2eγ

=

s2θe σ 0 2θes2ze σ 0 2ze
(q
+
q
·
p
q z e φe
)
+
C
·
θe φe
z e φe
θe ze
z
σθ2eγ
σθ2eγ

=

s2θe σθ0 e σφ0 e 0s2ze σz0 e σφ0 e 0
0
0
(q
+
q
·
p
)
+
C
·
q z e φe .
z
θ e φez e φe
θe ze
σθ2eγ
σθ2eγ

1If one tries to parametrize it on the inner σze ,inner base, the following replacement must be adopted,
σθ2e + Cz2 σz2e + 2Cz pθe ze σθ2e=(1 + Cz pθe ze )2 σθ2e + Cz2 (σz2e − p2θe ze σθ2e )=(1 + Cz pθe ze )2 σθ2e + Cz2 σz2e ,inner ,which corresponds to the σ of δθe + Cz pθe ze δθe + Cz δze,inner .

(B.23)

156

Appendix B. Transformation of positron correlation to θeγ vs φeγ correlation

Finally, the inner φeγ resolution after correcting φeγ vs θeγ , σφeγ ,inner , follows
2

σφeγ ,inner = s2φe,inner−Ee × σ 0 φe + Cv2 σv2 − p2θeγ φeγ σθ2eγ .

(B.24)

When the above correlations are corrected in a different order, Eq.(B.23) would be changed.
There are two such possibilities,
1. φe vs θe is corrected first and then correcting φe vs ze .
2. Both φe vs θe and φe vs ze are corrected without correcting the other correlation.
In the first case, the full correlation followspθeγ φeγ =s2θe σθ0 e σφ0 e 0s2ze σz0 e σφ0 e 0q+C·(qze φe + qθ0 e φe · p0ze θe )z
θ e φeσθ2eγ
σθ2eγ

s2θ σθ0 σφ0σz0 σφ0= e 2e e qθ0 e φe + Cz · e2 e (s2ze qz0 e φe + s2θe qθ0 e φe · p0θe ze ).
σθeγ
σθeγ

(B.25)In the second case, the full correlation followspθeγ φeγ =s2ze σz0 e σφ0 e 0s2θe σθ0 e σφ0 e 0+C·qq z e φe .
z
θ e φeσθ2eγ
σθ2eγ(B.26)

Appendix C
Analysis cross check
C.1

Cross check analysis with different modeling

A cross-check analysis is constructed in the same way as Eq.(6.1), which however uses PDFs
parametrized by a different set of observables. The PDFs for this cross-check analysis use four fit
observables;
1. Ee ,
2. Eγ ,
3. teγ ,4. Θeγ .
In particular, note that this analysis uses only the opening angle, Θeγ , instead of the set of (θeγ , φeγ )as described in Sec.6.1.2. This analysis uses constant PDFs, which are commonly parametrized for
all the events, except that one conditional observable, wγ , is employed to take into account the Eγ
resolution dependence. This analysis considers the correlation only between Ee and Θeγ , and all
the other correlations described in Sec.6.7.4 are integrated out to obtain the constant PDFs.
The consistency between the two analyses is validated with shared toy MC samples. Here,the toy MCs are shared by generating events with the main analysis framework, which are thenconverted with a (θeγ , φeγ ) → Θeγ transformation. Fig.C.1 shows a scatter plot that compares theupper limits for toy MCs generated with the null signal hypothesis, where we can see a strong
correlation between them. Some of the samples with a large disagreement are investigated, andthey are concluded to arise from the difference in the angle parametrization. All the investigated toy
MC samples with disagreement are found to contain events with θeγ and φeγ having the same sign.
They are not identified as signal events in the main analysis because it considers the anti-correlation
between θeγ and φeγ (Fig.6.20). They, however, can be candidate signal events because the opening
angle parameter is not sensitive to this anti-correlation. Shared toy MC samples are generated alsowith finite signal hypothesis, in which the best-fit value of Nsig is confirmed to be consistent.
The sensitivity of this analysis is 30 % worse than that presented in Sec.7.1, which is understood
to arise from the broader response function of the constant PDFs. The obtained upper limits fordata are consistent between the two analyses, as presented in Fig.C.1 for both the analysis region(black star) and sideband regions (blue circular markers).

C.2

Cross check with counting analysis

As a cross-check of the main likelihood analysis, a counting analysis is also performed with a cut on
Rsig . The confidence interval is set according to the table presented in Ref.[102]. Fig.C.2 shows the157

Appendix C. Analysis cross check

Constant PDF UL × 1012

158

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.5

1

1.5
2
Per-event PDF UL × 1012

Cut analysis sensitivity

Signal efficiency

Figure C.1: Comparison of upper limits on B(µ → eγ) between the two analyses for shared toy MC
samples (red small dots), data samples in the sideband regions (blue circles), data samples in theanalysis region (black star) [57].

1
0.8
0.6
0.4

2.2

×10−12

2
1.8
1.6
1.4
1.2

0.2
1
0
0

1

2

3
4
5
NBG after Rsig cut

0

1

2

3
4
5
NBG after Rsig cut

Figure C.2: Plots used for cut optimization. Left: Relation between signal efficiency and the numberof background events obtained by threshold scan on Rsig obtained from cumulative of Fig.6.24. The
signal efficiency is normalized to the efficiency after the selections for the likelihood analysis. Right:
Estimated sensitivity as a function of NBG .

C.2. Cross check with counting analysis

159

Table C.1: Comparison of confidence intervals with different methods.
Data sample

Likelihood analysis intervalCounting analysis interval

Analysis region

B(µ → eγ) < 7.5 × 10−13

B(µ → eγ) < 8.8 × 10−13

Sideband−3 ns < teγ < −2 ns−2 ns < teγ < −1 ns
1 ns < teγ < 2 ns
2 ns < teγ < 3 ns

< 7.1 × 10−13< 8.0 × 10−13< 6.9 × 10−13
1.6 × 10−13 – 2.6 × 10−12< 8.8 × 10−13< 8.8 × 10−13< 8.8 × 10−13
2.8 × 10−13 – 3.1 × 10−12

sensitivity as a function of the cut threshold, and we found that the optimal threshold is Rsig > 1.2,
which gives NBG = 0.5 and 0.84 relative signal efficiency. Here, the relative efficiency is definedby normalizing that in the fit region of the likelihood analysis (Fig.6.2). The sensitivity at the
optimized point is ×1.4 larger than that for the likelihood analysis.
In the analysis region, the observed number of events is 0, giving a confidence interval ofB(µ → eγ) < 8.8 × 10−13 . In the sidebands, the observed number of events is 0 for the first
three sidebands and 3 for the fourth sideband. In the fourth sideband, the p-value of the Nsig = 0
hypothesis is 1.4 %. The intervals in the sideband are summarized in Tab.C.1 and compared withthose from the likelihood analysis. As is evident in this table, the results of the counting analysis
and those of the likelihood analysis agree with each other.

160

Appendix C. Analysis cross checkAppendix D
Full detail of the target hole analysis
D.1

Method and systematics of alignment

The hole analysis firstly estimates the position distribution that we would have without the target
holes, interpolating the position distribution in the neighboring region around each hole. This, theexpected total number of positrons without the target holes, is then subtracted by the reconstructed
position distribution. This gives expected y and z distributions of the positrons associated withthe hole (Fig.D.1), and the peak positions are used to give y and z estimates of the holes. The x
coordinate is evaluated from the correlation between the estimated hole y position and tan φe of thepositron emission angle (Fig.5.50), which is produced by an additional x misalignment between thenominal and the true hole position.
In studies with MC samples, the y and z estimation from Fig.D.1 is found to be biased by
the non-uniform beam profile, and the peak position estimation gets closer to the beam center by
100 µm. This effect is also observed in data when the hole-to-hole distance estimation is comparedwith that from the CT scan. To correct this effect, the original histogram can be scaled according
to the observed distribution around the holes (Fig.D.2). This correction, however, overestimates
the bias1 and the estimated peak position gets further from the beam center. This overestimation,
also evaluated with MC Michel samples, results in 100 – 200 µm biases. The hole positions arethus estimated by combining the results from two methods with the corrections learned from MC
samples. The remaining discrepancies between them are a source of systematic uncertainties.
In addition to the above biases, there is another source of biases unique to data, which comesfrom the non-Gaussian asymmetric beam profile. As can be seen in Fig.D.2, a non-zero baseline —not existing in MC samples — can be found in the position distribution associated with the holes.
Biases on the peak position due to this baseline structure are evaluated by fitting different models tothe distribution; simple Gaussian function, Gaussian function with a constant offset, and Gaussianfunction with a linear offset function. The differences between the fit models were found to be aslarge as 400 µm in z coordinate and 200 µm in y coordinate.
The final y and z estimation combines the results from different methods with different systematics as discussed above, and 100 – 200 µm, half of the differences between them, are included asa systematic uncertainty. The x estimate of the holes does not show strong method dependence
because the biases in the y coordinate only produce y-shift in the Fig.5.50 plot with little impact
on the slope. Therefore, the systematics in x coordinate are ∼ 50 µm, smaller than the statistical
1Probably because this correction does not correctly weight the correction according to the circular shape of the
holes

161

Appendix D. Full detail of the target hole analysis

162

Expected− observed

Estimated Z distribution

400
300
200
100
0
−100
−4

−2

0

2

4
Z [cm]

Figure D.1: Estimated ze distribution of the number of lost positrons at a target hole.

Expected− observed

Estimated Z distribution after beam profile correction

700
600
500
400
300
200
100
0
−100
−200
−4

−2

0

2

4
Z [cm]

Figure D.2: Estimated ze distribution of the number of lost positrons at a target hole after correcting
for the beam rate non-uniformity.

D.2. Results of target hole analysis

163

Table D.1: Hole analysis results
Hole position in survey
in MEG coordinate [cm]Estimated difference from photograph + CT-scan(δx, δy, δz) each with ±stat ± syst [µm](2.3601, −0.313624, −8.53115)(1.30243, −0.319128, −4.69109)(−0.0312347, 0.97017, 0.138845)(−0.0414534, −1.62367, 0.14167)(−1.32782, −0.314651, 4.97927)(200 ± 180 ± 50, 550 ± 110 ± 70, 250 ± 160 ± 120)
(280 ± 80 ± 50, 620 ± 50 ± 70, 100 ± 100 ± 150)
(200 ± 80 ± 50, 900 ± 50 ± 100, 680 ± 90 ± 130)(−50 ± 150 ± 50, 740 ± 100 ± 100, 560 ± 180 ± 130)(−240 ± 180 ± 50, 760 ± 100 ± 70, 800 ± 180 ± 200)

uncertainties.

D.2

Results of target hole analysis

The additional misalignment estimated from the hole analysis is tabulated in Tab.D.1. Fitted
to the above results, the target position and rotation — parametrized as additional corrections to
the photographic estimations — are estimated with 100 µm uncertainties in the translations and6 mrad (1.4 mrad) uncertainty in the rotation around the target long (short) axis. The overall
offsets in the best fit, 100 µm in x, 700 µm in y and 400 µm in z, are understood to be global
offset shifts of the CDCH, which were also found in CDCH vs COBRA alignment studies. The
hole-to-hole discrepancies from the best fit — especially large for z estimation — are not fully
understood. Nevertheless, the most suspicious causes are underestimations of the uncertainties or
mis-evaluation of the bias corrections according to the observations in Fig.D.1 and Fig.D.2. On
the other hand, the additional rotation was small and can be explained by the precision of the target
survey.

164

Appendix D. Full detail of the target hole analysis

Acknowledgements
First and foremost, I would like to express my deepest gratitude to my academic supervisor Prof.
Wataru Ootani for giving me an educational opportunity as a Ph.D. student. Under his mentorship,
I was able to have a smooth student life and enjoy the research activities. I am also grateful to the
spokespersons of the MEG II experiment, Prof. Toshinori Mori and Prof. Alessandro Baldini forthe opportunity to join the MEG II experiment. It was both challenging and exciting to get involved
at the start of the DAQ and complete the first analysis of the MEG II experiment. I also acknowledge
Dr. Toshiyuki Iwamoto, Dr. Angela Papa, and Dr. Malte Hildebrandt for their coordination of
the MEG II data taking. I would also like to thank Dr. Toshiyuki Iwamoto for his special support
during my stay in Switzerland.
I heartily appreciate cooperation in the analysis works. Dr. Yusuke Uchiyama gave me manysuggestions and discussions to improve the analysis as well as his contributions. He also gave me
essential support in computing and software management. Dr. Fabrizio Cei, who is in charge of thecross-check analysis, shared many important results and stimulated many meaningful discussions.
Dr. Francesco Renga was always helpful in developing the analysis with his extensive knowledge andexperience. The analysis also strongly relied on enormous contributions by Dr. Satoru Kobayashi,
Dr. Dylan Palo, and Mr. Kensuke Yamamoto. Dr. Luca Galli and Dr. Marco Francesconi werealways cooperative in the DAQ-related analyses with their expertise in the hardware. Needless tosay, I am grateful to all the other members in the positron and photon analysis groups for theircontributions and discussions to complete this work, especially Dr. William Molzon, Dr. Fedor
Ignatov, and Mr. Antoine Venturini in the positron group and Dr. Rina Onda and Ms. Ayaka
Matsushita in the photon group.
I express my special gratitude to the hardware group of the experiment, especially Dr. Toshiyuki
Iwamoto, Dr. Sei Ban, and Dr. Lukas Gerritzen for the LXe detector, Dr. Yusuke Uchiyama, Dr.
Paolo W Catttaneo, Dr. Matteo De Gerone, and Mr. Taku Yonemoto for the pTC, Dr. Marco
Chiappini for the CDCH, Dr. Angela Papa, Dr. Patrick Schwendimann, Mr. Giovanni Dal Maso,
Mr. Bastiano Vitali, and Mr. Hicham Benmansour for the beam group, and Dr. Stefan Ritt, Dr. Luca
Galli, and Dr. Marco Francesconi for the DAQ hardware. Additionally, I also sincerely appreciate
the exciting R&D works of RPC with Prof. Atsuhiko Ochi, Dr. Kei Ieki, Mr. Kensuke Yamamoto,and Mr. Masato Takahashi.
I am grateful to ICEPP secretaries and all the other laboratory members. Especially, I would liketo mention that I shared a joyful time with Dr. Shinji Ogawa, Dr. Mitsutaka Nakao, Dr. Masashi
Usami, Mr. Linghui Liu, Dr. Naoki Tsuji, Mr. Kazuki Toyoda, Mr. Kosuke Yanai, Mr. Kohei
Shimada, Mr. Ryunosuke Masuda, Mr. Fumihito Ikeda, Mr. Tatsuki Murata, Mr. Keisuke Yoshida,
Mr. Yuki Ueda, Mr. Rintaro Yokota, and Mr. Weiyuan Li.
Last but not least, I also extend my thanks to my family. Their constant love and support were
the driving forces behind my pursuit of this thesis.

165

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