Development of a Muon Polarimeterfor
the J-PARC T-violation search experiment
Master’s thesis

Aine Kobayashi

35-116038
Department of Physics
Graduate School of Science, University of Tokyo
January 7th 2013

Abstract
This thesis reports the performance evaluation of a muon tube polarimeter for
the J-PARC E06 (TREK) experiment which aims to measure T violation via ameasurement of the transverse muon polarization in K + → π 0 μ+ ν decay.

iContents
1 Introduction

12 Physics motivations
2.1 Matter and antimatter asymmetry . . . . . . . . . . . . . . . . . . .
2.2 C,P,CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 T-violation search experiments . . . . . . . . . . . . . . . . . . . . .

3
3
4
4

3 J-PARC TREK+
3.1 Phenomenology of Kμ3
decay . . . . . . . . . . . . . . . . . . . . . .
3.2 Transverse Polarization PT . . . . . . . . . . . . . . . . . . . . . . .
3.3 Theoretical Predictions . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Final state interactions . . . . . . . . . . . . . . . . . . . . .
3.3.3 Multi-Higgs doublet model . . . . . . . . . . . . . . . . . . .
3.3.4 SUSY models . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 J-PARC TREK Experiment . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Experiment design . . . . . . . . . . . . . . . . . . . . . . . .

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4 Polarimeter
4.1 Muon polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Passive polarimeter for E246 . . . . . . . . . . . . . . . . . .
4.1.3 The analysis method of the E246 polarimeter . . . . . . . . .
4.1.4 Requirements for the active polarimeter . . . . . . . . . . . .
4.1.5 Expected performance of the polarimeter . . . . . . . . . . .
4.2 Plate type polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Muon Tube Polarimeter - MTP - . . . . . . . . . . . . . . . . . . . .
4.3.1 Drift tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Muon polarimeter magnet . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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iii

5 Beam Test at TRIUMF
5.1 TRIUMF M11 beam line . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Beam tuning . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 μSR measutement . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Data Analysis
6.1 Basic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Data quality . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Wire eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Drift time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Tracking algorithm . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Muon decay parameters . . . . . . . . . . . . . . . . . . . . .
6.4.2 Result of asymmetry measurement by the MTP . . . . . . . .
6.4.3 Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.4 Muon incident angle . . . . . . . . . . . . . . . . . . . . . . .
6.5 Analyzing Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 More detailed analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Conclusion

71

Appendix A Beam tuing at J-PARC K1.1BR

A–1

Appendix B Charge division

B–1

Appendix C Tracking system for the TREK detector

C–1

Appendix D Read-out system for the polarimeter beam tests

D–1

Acknowledgements

D–0

iv

List of Figures
1.1

TREK sensitivity

3.1

The intensity distribution (a), and the three components of the muonpolarizations (b) the longitudinal component PL , (c) the normalcomponent PN , and (d) the transverse component PT /Imξ. . . . . 6
Kμ3 decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Radiative corrections in the Kμ3 decay . . . . . . . . . . . . . . . . 9(a) One of the diagrams of the ﬁnal state interactions contributing
to PT em ; from one photon exchange calculated in Ref. [1] and also(b) one of the diagrams from two photon exchanges calculated inRef. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
End and side views of the TREK decector sysmtem . . . . . . . . . 14

3.2
3.3
3.4

3.5

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

4.5
4.6
4.7
4.8
4.9

Kμ3 decay and Michel μ decay, showing helicity conservation . . . .
Incident muon y distribution in the stopper measured by the C4chamber (top), and the intrinsic geometrical asymmetry for thosemuons (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between passive and active polarimeter . . . . . . . . .
Muon stopper made of parallel plates which serve also as drift chamber cell structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MTP (size: 865mm × 284mm × 330mm, 75kg) . . . . . . . . . . . .
MTP YZ-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic end view of the MTP . . . . . . . . . . . . . . . . . . . .
Time development of an avalanche in a proportional counter. [3] . .
Schematic front view of the muon ﬁeld magnet . . . . . . . . . . . .

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

Layout of M11 beam channel . . . . . . . . . . . . . . . . . . . . . .
HV curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setup for beam tuning and degrader tuning . . . . . . . . . . . . . .
TOF spectrum at 170 MeV/c . . . . . . . . . . . . . . . . . . . . . .
Timing chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical chamber signal . . . . . . . . . . . . . . . . . . . . . . . . .
Conﬁgulation of the DAQ system . . . . . . . . . . . . . . . . . . .
Gap between CP signals . . . . . . . . . . . . . . . . . . . . . . . .

4.1
4.2

4.3
4.4

v

2

15

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5.9

Degrader tuning ; Telescope (E1 and E2 counters) was placed on
MTP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Results of the degrader tuning for π . . . . . . . . . . . . . . . . . .
5.11 Results of the degrader tuning for μ . . . . . . . . . . . . . . . . . .
5.12 Muon decay curve: The parameter (1/τ ) corresponds to 2.2μs . . .
5.13 The conﬁrmation of muon stopping area in MTP. The colored points
means wire positions and all of the 256 channels worked well. . . .
5.14 Schematic view of charge division study set up . . . . . . . . . . . .
5.15 Fiber scintilator for charge division study to deﬁne the incident position with a good precision . . . . . . . . . . . . . . . . . . . . . . .
5.16 Set up for μSR measurement . . . . . . . . . . . . . . . . . . . . . .
5.17 Al sample (5 cm× 2.4 cm × 2 cm ) . . . . . . . . . . . . . . . . . . .
5.18 S3 counter timing spectrum : Muon stopping timing (the spike at ∼2500) and the oscillation pattern are seen. . . . . . . . . . . . . . .
5.19 Fitting result of the μSR oscillation spectrum . . . . . . . . . . . .
5.20 Calibrated time spectrum of the S3 signals. . . . . . . . . . . . . . .
The S2 counter time spectrum. The S2 counter was used for determing the TDC stop timing. . . . . . . . . . . . . . . . . . . . . . .
6.2 Wire eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Eﬃciency as a function of the drift distance . . . . . . . . . . . . . .
6.4 Schematic view of the wire eﬃciency of z-direction; the wire eﬃciency is expressed as (the middle of the A wire hit)/(all of the A,
B, C wires are hit) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Drift time distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Fit to the drift time distribution . . . . . . . . . . . . . . . . . . . .
6.7 X2 distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Schematic view of the deﬁnition of position for analyzing the X-T
relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 X2 (X1 , X3 ) versus t2 . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 X2 (X1 , X3 ) versus X2 (t2 ) . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Schematic view of the track reconstruction . . . . . . . . . . . . . .
6.12 Tracking result with weight for the positron . . . . . . . . . . . . .
6.13 Tracking result without weight for the positron . . . . . . . . . . . .
6.14 Another example of positron tracking . . . . . . . . . . . . . . . . .
6.15 Incident muon tracking reconstruction ﬁt . . . . . . . . . . . . . . .
6.16 Distribution of decay positrons as a function of cos θ and x . . . . .
6.17 Schematic view of the tracking angle . . . . . . . . . . . . . . . . . .
6.18 Opening angle distribution . . . . . . . . . . . . . . . . . . . . . . .
6.19 Example of bad events: Its opening angle is small, 0.387, becauseof there were two incident beam partially and the second beam misregarded as the positron, so the average of the positron hit was
mis-calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.20 The coordinate system deﬁnition . . . . . . . . . . . . . . . . . . . .

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6.1

vi

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6.21 Schematic view of the muon decay for positron in the MTP . . . . .
6.22 cos ψ by analytic calculation . . . . . . . . . . . . . . . . . . . . . .
6.23 sin ψ by analytic calculation . . . . . . . . . . . . . . . . . . . . . .
6.24 cos ψ generated by Geant4 simulation . . . . . . . . . . . . . . . . .
6.25 sin ψ generated by Geant4 simulation . . . . . . . . . . . . . . . . .
6.26 Deﬁnition of the ﬁducial volume. The area in the dashed line means
the read-out region. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.27 cos ψ distribution of the muon data . . . . . . . . . . . . . . . . . .
6.28 sin ψ distribution of the muon data . . . . . . . . . . . . . . . . . .
6.29 F orward or backward asymmetry of the muon data . . . . . . . . .
6.30 U p or down asymmetry of the muon data . . . . . . . . . . . . . . .
6.31 cos ψ distribution of the pion data . . . . . . . . . . . . . . . . . . .
6.32 sin ψ distribution of the pion data . . . . . . . . . . . . . . . . . . .
6.33 F orward or backward asymmetry of the pion data . . . . . . . . .
6.34 U p or down asymmetry of the pion data . . . . . . . . . . . . . . .
6.35 Deﬁnition of the ﬁducial volume . . . . . . . . . . . . . . . . . . . .
6.36 cos ψ distribution of the muon data with small ﬁducial volume . . .
6.37 sin ψ distribution of the muon data with small ﬁducial volume . . . .
6.38 F orward or backward asymmetry of the muon data with small ﬁducial volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.39 U p or down asymmetry of the muon data with small ﬁducial volume
6.40 cos ψ distribution of the pion data with small ﬁducial volume . . . .
6.41 sin ψ distribution of the pion data with small ﬁducial volume . . . .
6.42 F orward or backward asymmetry of the pion data with small ﬁducial volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.43 U p or down asymmetry of the pion data with small ﬁducial volume
6.44 no cut cos ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.45 no cutsin ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.46 After cut of χ2 /N DF < 20 sin ψ distribution . . . . . . . . . . . . .

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A.2 Detectors for beam measurement . . . . . . . . . . . . . . . . . . . . A–2
A.3 π/K diﬀerential Fitch-type Čerencov counter . . . . . . . . . . . . . A–3
A.4 The layout of K1.1BR beamline . . . . . . . . . . . . . . . . . . . . A–3
A.5 Results of K + /π + separation curve using narrow slit . . . . . . . . A–5
B:.2 The schematic view of charge division system. . . . . . . . . . . . . B–1
B:.3 The equivalent electric circuit of the charge division system. . . . . B–2
C.2 Schematics of the tracking system in the TREK setup (right) compared with that of the E246 experiment (lef t). . . . . . . . . . . . . C–2

vii

List of Tables
4.1
4.2

Comparison between plate type and tube type . . . . . . . . . . . . 24
Detail of MTP spec . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.1

Comparison between particles. . . . . . . . . . . . . . . . . . . . . 44

C.2 Main parameters of the charged particle tracking. . . . . . . . . . . . C–3
D.2 The main functions of ASB . . . . . . . . . . . . . . . . . . . . . . . D–2
D.3 The specs of VME discriminator board . . . . . . . . . . . . . . . . D–2ix

Chapter 1

Introduction
The Standard Model [4] explains very well the behavior of matter and interactions.
In 2012, the discovery of the Higgs boson at the Large Hadron Collider (LHC) [5]experiments completed the Standard Model predictions. However, some theoreticalconsideration point out that the Standard Model is not the ultimate theory and”New Physics” is needed.
For example, in order to explain the asymmetry of matter and antimatter in theuniverse, C and CP violation are needed. Although today’s Kobayashi-Masukawa
theory can explain CP violation in the quarks sector, it is inadequate to explain
the size of the asymmetry of matter and antimatter (baryon number asymmetry)
today. Therefore CP violation beyond the Kobayashi-Masukawa Standard Model
theory is required to explain why our universe mainly consists of baryons and not
antibaryons [6]. Also in 2005, the neutrino oscillation was observed at the KamLAND experiment [7]. It is an important key to solve the neutrino ﬂux problem.
According to relativistic local-ﬁeld theory, CPT [8] symmetry is conserved,
hence when CP violation exists, T violation also must exist by necessity. Measurement of T violation in the neutral K meson system means a measurement direct
CP violation, this is the eﬀective way for discovery of new physics beyond the Standard Model. I have joined the J-PARC E06 (TREK) experiment [9], which aims toperform a precision measurement of direct T violation by means of a measurementof the transverse polarization PT in K + → μ+ π 0 ν (Kμ3 ) decay.
PT is deﬁned as the vertical component of the polarization normal to the Kμ3
decay plane, and this value is superior in terms of following features:• very small (< 10−7 ) Standard Model prediction• very small Final State Interaction (∼ 10−5 )Therefore, measurement of a PT value as large as 10−3 ∼ 10−4 is a probe of T
violation and we could discover new physics and possibly explain the origin of the1

2

1.0 Introduction

baryon asymmetry. I describe this theory in Chapter 3.
It is necessary to conduct a very high statistical precision experiment due tosmallness of PT . The KEK-PS E246 experiment [10], which was the TREK predecessor works, obtained PT = −0.0017 ± 0.00023(stat) ± 0.0011(syst) [10] as theworlds record, and the result was consistent with no T violation. The largest uncertainty was due to the large ambiguity on the decay positron arising from muon
multiple scattering. In the TREK experiment, we will upgrade the E246 detectors
to reduce the systematic error. Fig 1.1 shows the TREK sensitivity.

Figure 1.1: TREK sensitivity

In this experiment, I have worked on the research and development of themuon spin polarimeter, which has an important role in the measurement of PT .
The TREK polarimeter, which I will describe in Chapter 4, stops muons in middle,
deﬁnes the positron emission angle by tracking both muons and decay positrons.
Since the positrons are likely to be emitted to the same direction as the muons dueto helicity conservation. Polarimeter has a role not only as a tracker but also asa stopper, so multiple scattering is unavoidable. It’s important to make a trackreconstruction algorithm to solve this problem for higher accuracy, which I madeand describe in Chapter 6.3.
We built a prototype polarimeter and made a beam test at TRIUMF [11] in OctNov 2011. We have done not only the experimental set up such as the arrangementof the polarimeter and the electronicl circuit but also TOF particle identiﬁcation,
beam tuning and degrader tuning. I will describe the beam test in Chapter 5, andthe data analysis in Chapter 6.

In addition, I joined the beam tuning at J-PARC [12] K1.1BR in June 2012,
which is mentioned in the appendix A.

Chapter 2

Physics motivations
2.1

Matter and antimatter asymmetry

In the very early universe, the symmetry of matter and antimatter was conserved,
pair production and pair annihilation always occur [13].
γ + γ  q + q̄

(2.1)

As the temperature of the universe gradually dicreased, pair production andpair annihilation slowly decreased. And there must have been processes causinga small matter-antimatter asymmetry, leading to the matter dominant univers oftoday.
The quantity diﬁned as the matter and antimatter asymmetry is expressed asa ratio of the baryon number density and its entropy density. This ratio remains
unchanged under cosmic adiabatic expansion, as long as there is no baryogenesis
or entropy production such as reheat. According to the observation value,nB= (0.67 − 0.92) × 10−10 .
s/k

(2.2)

Here, nB is the baryon number density, s is the entropy density, and k is theBoltzmann constant.

This phenomenon can only be explained if the three conditions of Sakharov [6]are fulﬁlled:• There must be an interaction violating the conservation of baryon number.
• There must be an interaction violating CP invariance, where C is the particleantiparticle transformation and P is the space inversion operation.
• There must be phases of the expansion without thermodynamic equilibrium.
34

2.3 Introduction

The ﬁrst condition is natural, since there was no baryon asymmetry in early universe, if all of the interactions between particles conserve the baryon number, there
should be no baryon asymmetry now. And this criterion would imply that protons are not stable (searches for such a decay have been unsuccessful, showing thatthe lifetime of the proton is longer than 1033 years). However, currently, there is
no experimental evidence of the particle interactions, where the conservation of
baryon number is broken. The second condition was ﬁrst shown to be fulﬁlled by
the discovery of the CP violation in the decay of neutral K mesons, in 1964,by
J.Christenson, J.Cronin, V.Fitch and R.Turlay. The third condition ensures thatthe reverse process cannot happen with the same probability as the violation of theconservation of baryon number process. This condition can be met in cosmologicalmodels by inﬂationary fast expansion or by a ﬁrst-order phase transition in theelectroweak interaction of the Standard Model [14].

2.2

C,P,CP violation

The CPT theorem asserts that a Lorentz-invariant ﬁeld theory is unchanged under the combined CPT operation, thus conserving CPT. According to the CPT
invariance however, the observation of the CP violation in the K 0 and B 0 systems
requires the existence of T violation. Thus, T violation has a great impact on our
understanding of nature. Before the ﬁrst observation of indirect T violation by
CPLEAR experiment in 1988 [15], the T- violation was regarded as kind of statistical ﬂuctuation due to an increase of entropy in our universe. The observationmeans that the T violation can be explained as the result of particle interactions.

2.3

T-violation search experiments

A large number of experiments to search for T violating eﬀects have been performed
such as the particle and the nuclear electric dipole moment (EDM) experiments [16].
On the other hand, we aim to measure the eﬀect in K + → π 0 μ+ νμ (Kμ3 ). Morethan 50 years ago, it was suggested by J.Sakurai [17] that the nonzero transverse
muon polarization (PT ) is a good test for T violation. The system has several
advantages such as the smallness of the ﬁnal state interactions which can mimic T
violation by inducing a T-odd eﬀect and the very small Standard Model prediction.

Chapter 3

J-PARC TREK
3.1+
Phenomenology of Kμ3
decay

Based on the V-A theory, the decay matrix element can be written as [18], [19]M=  
 GF
sin θC f+ q 2 pλK + pλπ + f− q 2 pλK − pλπ · [ūν γλ (1 − γ5 ) vμ ] .
2(3.1)

where,
GF : the Fermi constantθC : the Cabibbo angle
pK , pπ , pμ , pν : the four-momenta of the kaon, pion, muon and antineutrino
with two form factors f+ (q 2 ) and f− (q 2 ) of the momentum transfer squared to the
lepton pair, q 2 = (pK − pπ )2 . Using pK = pπ + pμ + pν , this amplitude can be
rewritten as

 


GF
sin θC f+ q 2 2pλK · ūν γλ (1 − γ5 ) vμ + ξ(q 2 ) − 1 mμ ūν (1 − γ5 )vμ .
M=
2
(3.2)
2
where the parameter ξ(q ) is deﬁned as
ξ(q 2 ) = f− (q 2 )/f+ (q 2 ).

(3.3)

The ﬁrst term of Eq. 3.2 corresponds to the vector and axial vector amplitude,
and the second term corresponds to the scalar and pseudscalar amplitude. The
parameters f− and f+ depend on q 2 asf± (q 2 ) = f± (0) 1 + λ± (q 2 /m2π ) .

(3.4)

In general, both f− and f+ can be complex. If time reversal (T) is a good symmetry,the parameter ξ is real. Any non-zero value of Imξ would imply T-violation. An
experimentally observed T-violationg muon polarization PT is directly proportional
56

3.1 J-PARC TREK

to Imξ. The currently adopted values are given below;

λ+ = 0.0284 ± 0.0027(3.5)

ξ(0) = −0.14 ± 0.05

(3.6)

λ− = 0(3.7)

Figure 3.1: The intensity distribution (a), and the three components of the muonpolarizations (b) the longitudinal component PL , (c) the normal component PN ,and (d) the transverse component PT /Imξ.

The Dalitz distribution for Kμ3 decay is given by Fig 3.1.
ρ(Eπ , Eμ ) ∝ f+2 (q 2 ) A + Bξ(q 2 ) + Cξ 2 (q 2 )

(3.8)

7

3.2 J-PARC TREKwith
A = mK (2Eμ Eν − mK Eπ ) + m2μ1 
E − Eν
4 π

1
B = m2μ Eν − Eπ
2
1 2 
m E
C =
4 μ π
Eπ = (m2K + m2π − m2μ )/(2mK ) − Eπ

(3.9)
(3.10)(3.11)(3.12)

Here, Eπ , Eμ and Eν are the energies of the pion, miuon and neutrino in the kaon
center-of-mass frame, and MK , mπ and mμ the masses of the kaon, pion and muon,
respectively. MK is the kaon mass. The Dalitz plot is shown in Fig 3.1.

3.2

Transverse Polarization PT

In the three body decay, three orthogonal components of the muon polarization
vector can be deﬁned: the longitudinal (PL ), normal (PN ) and transverse (PT ) as
the components parallel to the muon momentum pμ , normal to PL in the decay
plane, and normal to the decay plane, respectively.

Figure 3.2: Kμ3 decayWith the polarization vector σμ , they are expressed below:
PL =
PN

=

PT

=

σμ · pν|pμ |σμ · (pμ × (pπ × pμ ))|(pπ × pμ ))|σμ · (pπ × pμ )|pπ × pμ |(3.13)(3.14)(3.15)The PT changes sign under the time reversal operation, therefore making it a Todd observable. With the decay probability, the muon polarization in the kaon rest
frame can be written as(3.16)
σμ = P /|P |,8

3.2 J-PARC TREK

where P is determined as follows

P = a1 (ξ) − a2 (ξ) (mK − Eπ ) + (Eμ − mμ )(pπ · pμ )/|pμ |2− a2 (ξ)mμ pπ + mK mμ Im(ξ)(pπ × pμ ).

pμ(3.17)with
a1 (ξ) = 2m2K Eν + Re b(q 2 ) (Eπ∗ − Eπ ) ,2a2 (ξ) = m2K + 2Re b(q 2 ) mK Eμ + b(q 2 ) m2μ ,(3.18)(3.19)

 
b q2 =(3.20)

Eπ∗(3.21)

1 2
ξ(q ) − 1 , and2 2= mK + m2π − m2μ /2 (mK ) .

These three polarization components are shown in Fig 3.1. In the presence of the
predominant in-plane component of the polarizations, PL and PN , PT (Eq 3.15)
can be rewritten in terms of Imξ and a kinematical factor as
PT = Imξ ·

|pμ |
mμ

.
mK Eμ + |pμ | nμ · nν − m2μ /mK(3.22)The quantity Imξ, sensitive to T-violation, can be determinedfrom a PT measurement. The advantage of Kμ3 over Ke3 K + → π 0 e+ ν is apparent as PT is
proportional to the lepton mass. The kinematic
factor as a function of the π 0 en
ergy (2Eπ0 /mK ) and μ+ energy 2Eμ+ /mK is the contour shown in Fig 3.1 which
has an average value of ∼ 0.3 yielding a full detector acceptance relation ofPT  ∼ 0.3Imξ.

(3.23)In order to see the connection between Imξ and the eﬀective parameters of
New Physics appearing in the coeﬃcients of generic exotic interactions, interesting
results can be obtained with following eﬀective four fermion Lagrangian:GF
L = −  sin θC s̄γa (1 − γ5 ) uν̄γ α (1 − γ5 ) μ2+ GS s̄uν̄ (1 + γ5 ) μ + GP s̄γ5 uν̄ (1 + γ5 ) μ+ GV s̄γα uν̄γ α (1 − γ5 ) μ + GA s̄γα γ5 uν̄γ α (1 − γ5 ) μ + h.c.. (3.24)
Here, GS and GP are the scalar and pseudo-scalar coupling constants and GV and
GA are the exotic vector and axial-vector coupling constants, respectively. Tensor
interactions are neglected. Imξ is caused only by the interference between the SMterm and the scalar term, namely by the complex phase of GS [20] [21], which canbe written as 2
mK − m2π ImG∗S(3.25)Imξ = √
2 (ms − mu ) mμ GF sin θC
where ms and mu are the masses of the s-quark and u-quark, respectively. Thus,
PT can constrain the exotic scalar interactions.

3.3 J-PARC TREK

3.3

Theoretical Predictions

3.3.1

Standard Model

9

The very small SM contribution for PT is one of the important motivations of
this experiment. A T violating amplitude arises from the relative phases between
diagrams or complex coupling constants in a diagram. Since only a single element of
the CKM matrix, Vus , is involved for the W - exchange semi-leptonic Kμ3 decay in
the SM, no CP violation appears in ﬁrst order. As discussed in Chapter 3.1 this is a
general feature for vector and axial vector type interactions. The SM contribution
comes only from higher order eﬀects. The possible size of its contribution was
suggested qualitatively in [22] to be less than PT < 10−6 . An actual value basedon the lowest-order vertex radiative corrections to the ūγμ (1 − γ5 ) sW μ vertex(Fig 3.3) has been presented in the textbook (Bigi and Sanda [23]). This has been
estimated to be less than 10−7 . This fact constitutes the main motivation for a PT
experiment as a search for New Physics. As shown in the next section, considering
FSI eﬀects, an observation of a non-zero PT implies unambiguously the existenceof CP violation mechanisms beyond the SM.

Figure 3.3: Radiative corrections in the Kμ3 decay

3.3.2

Final state interactions

Strictly speaking, the observation of measured non-zero PT value does not mean T
violation automatically. Spurious PT F SI can be induced by ﬁnal-state-interactions(FSI). This has been a concern in PT experiments for quite some time. In the case0 decays, the FSI are mainly due to electromagnetic interactions. The FSI
of Kμ30 decay,
contributions to PT in the present case are much smaller than those in Kμ30 ) ∼ 10−3 ,which has two charged particles interacting in the ﬁnal state, PT F SI (Kμ3+
has been extensively investigated. The single-photon contribution
the FSI in Kμ3
+
F SI
to PT
in Kμ3 decay is due to the imaginary part of the two-loop diagrams. An

10

3.3 J-PARC TREK

example is shown in Fig 3.4. This was estimated more than 20 years ago to be
PT F SI ≤ 10−6 [1]. Quite recently two-photon exchange contributions to PT em havebeen studied in Ref. [2], where the transverse polarization is proportional to theimaginary parts of the diagrams shown in Fig 3.4. The value of PT em averaged overthe Dalitz plot was found to be less than 10−5 . Thus, a non-zero PT in the rangeof 10−3 ∼ 10−4 will constitute a clear signal of contribution from a New Physics.

Figure 3.4: (a) One of the diagrams of the ﬁnal state interactions contributing toPT em ; from one photon exchange calculated in Ref. [1] and also (b) one of the
diagrams from two photon exchanges calculated in Ref. [2].

3.3.3

Multi-Higgs doublet modelMulti-Higgs doublet models have been considered as the minimum and naturalextension of the SM with one Higgs doublet, and a number of papers [24], [25], [23]have applied this models to PT . In the class of models without tree-level ﬂavor
changing neutral current, new CP violating phases are introduced in the charged
Higgs mass matrix if the number of doublets is more than two. The couplings of thequarks and leptons to the Higgs boson is expressed in terms of the Lagrangian [24,
25]
2

1
√
αi u¯L V MD dR Hi+ + βi u¯R MU VdL Hi+ + γi ν¯L ME e R Hi+ + h.c.,
L = (2 2GF ) 2

i=1(3.26)
where MD , MU , ME are diagonal mass matrixes, V is the CKM matrix, and αi , βiand γi are the new complex coupling constants associated with the charged Higgs
interactions. The coeﬃcients, αi , βi and γi can have complex phases, and PT is
calculated asm2
Im (γ1 α1∗ )(3.27)Imξ = Km2H

113.3 J-PARC TREK

where αi and γi are the quark and lepton couplings to the lightest charged Higgs
boson. The E246 result [10] yielded |Im (γ1 α1∗ )| < 544 (mH /GeV )2 as the moststringent limit for this parameter. It is also constrained by the semileptonic decayof the B meson [26], B → τ νX but the result is less stringent than the PT constraint. According to the recent result on B → τ ν [27], its impact on this modelin now being checked. Other constrains this model come from the neutron EDM(dn ), b → sγ [26] and b → sl¯l [28] complementing the PT result in a diﬀerent
manner, since these channels limit Im (α1 β1∗ ). These two parameters are related as
Im (α1 β1∗ ) = − (v3 /v2 )2 Im (γ1 α1∗ ) through the ratio of the vacuum expectation values v2 and v3 . An interesting scenario assumed in [24] has v3 /v2 ∼ mtau /mt ∼ 1/80
thus making PT the most sensitive test of the three Higgs doublet model.Anotherimportant feature of this model is the prediction of PT in the radiative decay
K + → μ+ νγ (Kμνγ ). The following expression was obtained [20],PT (Kμνγ ) ∼= −0.1m2K
Im (γ1 α1∗ )m2H(3.28)

with the same sign as PT (Kμ3 ) but with diﬀerent size. E246 provided a result [10],however it lacked good statistics. Although the ﬁnal state interaction is large≈ 10−3 , also we may pursue this mode at J-PARC in the future.

3.3.4

SUSY models

A number of other models also allow PT at an observable level without conﬂicting with other experimental constrains. Hence a non-observation of PT can constrain these models. Some Minimal Super-symmetric Standard Models (MSSM)
allow sizable values. One interesting case is the model discussed by G.-H.Wu and
J.N.Ng [29]. In this model the complex coupling constant between the charged
Higgs boson and strange and up-quarks is induced through squark and gluino loops.
Then, the PT value when the muon and neutrino momenta are at right angles, isgiven as


∗
∗
H+
DL
UR
Im
V
V
V
2
33
32
31
pμ (μ + At cot β) (100GeV )+PT H ≈ 3.5 × 10−3 IH +2
Eμmg
sin θCMH(3.29)
for tanβ ≈ 50 (For the meanings of various symbols see Ref. [29] except to notethat we assumed the top quark mass to be 180 GeV). If we allow large ﬂavor
mixing coupling in the squark-quark vertices, there is an allowed parameter region
for large PT . The E246 PT upper bound corresponds to MH > 140GeV . In viewof the many assumptions mode, this bound should be considered as a qualitative
estimate [29]. It is noteworthly that PT (Kμ3 ) and PT (Kμνγ ) have opposite signs inthis model.
Another interesting SUSY model is the R-parity violating model [30], in which
the interactions are described by the Lagrangian using the λijk , λijk and λijk coef-

12

3.4 J-PARC TREK

ﬁcients of the super potential as
1
λijk ν̄Lc i eLj ẽ∗Rk + ēRk νLi ẽLj − (i ↔ j)2
+ λijk [ν̄Lc i dLi d˜∗Rk + d¯Rk νLi d˜Rj + d¯Rk dLj ν̃Li− ēcRi uLj d˜∗Rk − d¯Rk eLi ũLj − d¯Rk uLj ẽLi ]L =(3.30)

by assuming λijk =0 because of the non-observation of proton decays. Here thesubscripts i(k) are the slepton family indices. There are two kinds of contributions
to PT ; one is from slepton exchange and the other is from down-type squark
exchange. They are given, respectively asImξ l =

iImξ d =

im2K
Im[λ2i2 (λi12 )∗ ]
·
4 2GF sin θc (ml˜i )2 mμ msIm[λ21k (λ22k )∗ ]m2K
·
4 2GF sin θc (md˜i )2 mμ ms(3.31)(3.32)

Thus, the parameters of Im[λ2i2 (λi12 )∗ ]/m2 and Im[λ21k (λ22k )∗ ]/m2 are constrained.
These are many experimental constraint for λijk and λijk [31], but for the relevant
four combinations of λ(λ )∗ and λ (λ )∗ the constraint from other experimental limits are not stringent enough, thus PT determines their limits as a function of theirmass scale m.

3.4

J-PARC TREK Experiment

The 40 year history of PT experiments show a rather slow improvement in the upper
limit. This is due to two reasons: the ﬁrst point is that the statistical sensitivityof an asymmetry measurement scales as 1/ N , while the single event sensitivity
in rare decay experiments scales as 1/N . The second reason is the nature of thishigh precision experiment which must be conducted and analyzed very carefully.
The understanding and reduction of systematic errors can only be achieved stepby-step. We prefer to this approach to the J-PARC experiment and to proceed in
the goal of SM+FSI signal region of 10−5 in steps.
The E246 result was essentially statistics-limited. The largest systematic error
in the error table, which was due to multiple scattering, is statistical in nature andit could have been improved as was forseen at the start of the E246 experiment.
We propose to improve the E246 result by at least a factor 20 (δPT < 2 × 10−4 ), by
improving both the statistical and the systematics uncertainties. This sensitivity
puts the experiment well into the region where new physics eﬀects can be discovered,and even a null result would set tight constraint on various theoretical models. If
warranted, further sensitivity improvement towards 10−5 will be proposed in the3.4 J-PARC TREK

13

next stage after we have been convinced of the possibility to pursue this experimentto such a high precision region. In that sense, the TREK experiment may beconsidered as a prelude to precision frontier experiments at J-PARC.
We have to upgrade the E246 detectors.
• New polarimeter system (an active polarimeter + muon polarimeter magnet)
is adopted to reduce the μ+ multiple scattering and the uncertainty of themagnetic ﬁeld in the polarimeter.
• New electronics and read-out, in particular CsI(Tl)read-out, systems will be
adopted to handle the higher event rates.
• Additional tracking GEM chambers for charged particle is adopted for improving the uncertainty in the determination of decay planes• A smaller target with ﬁner segmentation is adopted in order to improve theuncertainty in the K + stop position.
We plan to perform the TREK experiment using the K1.1BR beamline in the
Hadron Experimental Hall in J-PARC. At present we are waiting for the realizationof high power accelerator operation (currently ∼ 5kW intensity beam for slow
extraction is available), and after ﬁnishing another experiment (E36) at K1.1BR
which can be performed with lower accelerator power, we will start to perform the
TREK experiment. The preparation of all the detectors is in progress.

3.4.1

Experiment design

In order to overcome the limitations of the E246 experiment, several improvements
to the detector system must be undertaken.
The TREK experiment will be performed using a toroidal spectrometer setup in
conjunction with a stopped K + beam at the low momentum separated kaon beamline K1.1BR in the Hadrom Experimental Hall of the J-PARC 50GeV synchrotron.
The spectrometer consists of an iron-core superconducting toroidal magnet with 12
gaps. The setup is shown in Fig 3.5. The K + beam will be stopped in an active
target located at the center of the magnet. Kμ3 decays will be identiﬁed by detecting a π 0 with a CsI(Tl) barrel surrounding the active target, a muon with a charged
particle tracking system consisting of GEM chambers (C0, C1), MWPCs(C2, C3
and C4) and with particle identiﬁcation by means of time-of-ﬂight(TOF). This arrangement will enable a measurement of the decay pions ranging over all directions
covering all regions of the decay phase space for a given μ+ momentum. As described next, this scheme allows for a ”double ratio” measurement and, thus, anumber of systematic errors can be suppressed. The CsI(Tl) barrel has 12 holes toadmit charged particles into the magnet gaps. Since the barrel does not cover thefull 4π solid angle, not all events will have the two photons from π 0 decay detected.

14

3.4 J-PARC TREK

Figure 3.5: End and side views of the TREK decector sysmtem

However, one photon with relatively large energy can also be used to determine theπ 0 momentum. Muons are bent by about 90 deg, momentum-analyzed, and thenthey enter the muon polarimeter system. After passing through a spectrometerand Cu degrader, the muons are ﬁnally stopped in the active polarimeter. The
muon polarization is measured by means of the decay positron asymmetry. Whenπ 0 s are detected in the forward (FWD) or the backward (BWD) directions relative
to the beam axis, the transverse component PT lies in the azimuthal directions ineach polarimeter clockwise (cw) or counterclockwise (ccw), respectively.Thus, themagnitude of the transverse polarization PT ∼ σμ · pπ0 × pμ+ can be measured as
the azimuthal asymmetry of positron emission, namely a cw and ccw emission rate
diﬀerence.
Our aim is to perform an experiment which, in comparison to E246, will haveabout 10 times more acceptance (using active polarimeter described next chapter),about 20 times the integrated beam ﬂux, and few times higher analyzing power toachieve nearly a factor at least 20 improvements in the statistical sensitivity, i.e.,δPT (one σ limit) ∼ 10−4 .

Chapter 4

Polarimeter
4.1

Muon polarimeter

4.1.1

Principle

As already mentioned in the previous Chapter, PT is small, hence it is importantto perform a precision measurement with both high statistics and with small systematics. Therefore, the developlment of an active muon polarimeter is necessary
to observe T violation.
The most important feature of the TREK experiment is the adoption of anactive polarimeter in contrast to E246 where a passive polarimeter with a separate
system of a muon stopper and positron counters was used. A decay positron ofhigh energy (more than 30 MeV) from the Michel decay can preserve the muon spin
direction due to the helicity conservation (Fig 4.1). This is a basic mechanism of
the polarimeter. Hence, the polarimeter has to measure the emitted direction of the
decay positron precisely along the transverse axis, PT is measured as a transverse
asymmetry of the decay positron (AT ).

Figure 4.1: Kμ3 decay and Michel μ decay, showing helicity conservation

15

16

4.1.2

4.1 Polarimeter

Passive polarimeter for E246

The advantage of the passive system was its simplicity with the consequence of verysmall systematic errors associated with the analysis. The systematic cancellation
scheme when the asymmetry was summed over the 12 sectors was also based on theuse of positron counters as clockwise and counterclockwise counters at the sametime. However, this was done at the cost of positron detection acceptance andpolarization analyzing power.

4.1.3

The analysis method of the E246 polarimeter

The transverse polarization was measured as an asymmetry deﬁned asA=

Ncw − Nccw
Ncw + Nccw

(4.1)

with the clockwise (cw) and counterclockwise (ccw) counts, Ncw and Nccw , respectively. A measurement to determine the null asymmetry A0 , was also needed toensure an accurate PT . For measuring the null asymmetry, the data, which did not
require the π 0 direction (f wd direction or bwd direction), could be used. In thiscase, the data could include Kπ2 events. After conﬁrming the data quality fromthe null asymmetry measurement, the transverse polarization (PT ) measurement
could be performed. In order to take better quality PT data from Kμ3 decay, adouble ratio method was performed for two opposite kinematic situations whichcorrespond to the π 0 direction, respectively (f wd/bwd). Paying attention to justone e+ counter, it could count both Ncw and Nccw . This means that the eﬃciencyof this counter is canceled out when we sum up the 12 sector data with 12-foldazimuthal symmetry. This method played an important role in reducing the systematic errors. However, several diﬃculties still remained. When the transverse
asymmetry was measured, it was essential to consider the fact that the muon stopping distribution could aﬀect the asymmetry very strongly. In particular, a shift
along the y direction was critical. The asymmetry A0 (y) for the muons is shownin Fig 4.2.
The eﬀect of the shift on the asymmetry is δA = k×δ < y > with k = 0.015/mm
for the average shift δ < y >. Considering such a spurious eﬀect, the asymmetry
was analyzed diﬀerentially using the y-information from the C4 chamber which was
located in front of the polarimeter. The y direction of the T violating asymmetry
AT (y) was calculated as
AT (y) =
1  f wd
A (y) − Abwd (y) ,
2

(4.2)

4.1 Polarimeter

17

Figure 4.2: Incident muon y distribution in the stopper measured by the C4 chamber (top), and the intrinsic geometrical asymmetry for those muons (bottom).

18

4.1 Polarimeterwith the y-dependent f wd and bwd asymmetries
f wd
1 Ncw
Af wd (y) = [ f wd − 1],
2 Nccw

1 N bwd
Abwd (y) = [ cw
− 1].
bwd
2 Nccw(4.3)

Here, Ncw (y) and Nccw (y) are the event number distributions from C4-y. The T
violating polarization at each y point was then extracted using the relevant analyzing power function of α(y), which was determined by the polarimeter structure,and detector acceptance, as
PT (y) =AT (y).
α(y) < cos θT >(4.4)

The attenuation factor, < cos θT > determined by the average of the angles cos θπ0
is independent of y. This scheme corresponds to the case in which a number ofsliced muon stoppers were inserted and the polarization measurements were donefor each stopper. The ﬁnal PT result for one data set was obtained as the statistical
average of PT (y) asPT ≡ < PT >= PT (y)w(y)dy,= −0.0017 ± 0.0023(stat) ± 0.0011(syst),(4.5)

with the normalized weight function of w(y) ∼ 1/σP2 T (y). Thus, PT was unaﬀectedby the muon stopping distribution in the stopper.
As mentioned above, it was impossible to avoid the eﬀect of multiple scatteringand there is no way to know the decay vertex. This means that some systematicaluncertainties are still remaining. Actually, the uncertainty from muon multiple
scattering was larger than any other systematic error.Considering these points, theE246 polarimeter should be improved in order to achieve a sensitivity of δPT ∼10−4 .

4.1.4

Requirements for the active polarimeterIn comparison with the situation of E246, where the positron signals were associated with non-negligible constant background events, the new active polarimeter
determines the muon stopping position for each event, and this in turn, renders
the experiment free from the systematic error associated with the ambiguities in
the muon stopping distribution. Since the decay positron tracks are measured, thedecay vertices will be determined event-by-event.
The active polarimeter should have the following functions and advantages.

4.1 Polarimeter

19• Determination of the muon vertices event-by-event
Detection of the decay positrons in all directions by a polarimeter with alarge acceptance with nearly 4π solid angle. In E246 the positron countersolid angle was limited to about 10% on each side. The detector acceptancebecomes 10 times larger, even though the sensitivity does not scale by thisfactor. The ability to measure the positron emission provides the possibility
to use not only the f wd/bwd pion scheme but also the lef t/right pion scheme
which was not possible in E246.
• Measurement of the positron emission angle
The asymmetry changes as function of both the positron emission angle andthe energy. A weighted analysis brings about a signiﬁcant increase in theanalyzing power resulting in higher sensitivety. It is of interest to note that
this superior performance is achieved in the case of measuring the positronemission angle precisely.

The requirements for the polarimeter design are listed below.
• The polarimeter should have a large muon stopping eﬃciency in relatively
small volume. This means that the polarimeter should be made of a material
with high average density.
• On the contrary, the polarimeter should enable a clean detection of positrons
without interections such as bremsstrahlung or annihilation in ﬂight. Thismeans that the polarimeter should be made of a material with low averagedensity.
• A drift chamber is and adequate choice of the detector.
• The cell structure is essentially determined by the ﬁne structure of the stopper. Thus, the stopper has to be designed taking this point into account.
• Optimization for the azimuthal ﬁeld arrangement sets the highest priority to
the f wd/bwd pion analysis. Thus, the stopper structure must be optimized
for the cw/ccw asymmetry measurement.
• Internal cancellation mechanism against any local ineﬃciency is essential forthe polarimeter. The cw/ccw ineﬃciency cancellation scheme of the E246
experiment is maintained by using the polarimeter.

20

4.2 Polarimeter

Now, there are two models for the active polarimeter, a plate type or a tube
type. The former was studied in 2009 and I will brieﬂy describe this in the next
section. The latter is the improved version, which I will discuss as the main topic
of this thesis.

Figure 4.3: Comparison between passive and active polarimeter

4.1.5

Expected performance of the polarimeter

An active polarimeter should have the abilities to measure both incident muonsand decay positron tracks with high position resolution, typically 100μm. The
coordinate along the wire will be determined by charge division technique. The
coordinate in the plane perpendicular to the wires is determined by drift time
information. The angular determination of the decay positron track can be done
after the determination of 3-dimentional position.

4.2

Plate type polarimeter

The plate polarimeter consists of several Al muon stopper plates with gaps in
between, and drift chamber layers in these gaps (Fig 4.4).
In this design, the plates are normal to the azimuthal direction, along which
we measure the left/right positron asymmetry, the sense wires are strung in theradial direction. The cell structure of the chamber is chosen to have a rather large
aspect ratio of 2.0 after elaborate GARFIELD calculations by two independentgroups in Japan and Canada taking into account the cost which can be reduced
proportionally to the aspect ratio. Since the tracking performance is limited mainlyby the scattering in the stopper plates (Fig 4.4), there is considerable room to4.2 Polarimeter

21

Figure 4.4: Muon stopper made of parallel plates which serve also as drift chambercell structure

compromise the chamber performance. Regarding the tracking resolution a seriesof Monte Carlo simulation calculations have been performed.
One of the important things to consider was determing the selection of the
stopper material. The analyzing power, deﬁned as AT /PT , should be as large as
possible to obtain the highest sensitivity in the experiment. The requirements were,
without any initial polarization loss during stopping and without any signiﬁcantspin relaxation. Considering the decay positron interaction, the material should
also be a light element. Hence, the E246 experiment adopted a pure Al plate with
99.99% purity. However, in the TREK experiment, pure Al is not suitable since
the muon stopper material itself constitutes a component of the active polarimeter
drift chamber mechanical structure. In this case the stopper plates have to be rigid
enough as a structual component, while pure Al is too soft and easily deformed.
The most important prerequisite is good spin polarization behavior which could be
investigated with a real muon spin relaxation measurement.
By studying Al and Mg material at TRIUMF in 2007, A5052 was selected as
the frame material of the polarimeter and pure Al(JIS1100) was also selected as
the stopper material.

After a charge division study, at the Fuji Test Beam Line (FTBL) at KEK
in 2008, we adopted SUS304 for the sense wires. The details of this experimentis described in Ref. [32], and charge division method is mentioned in Appendix
B. The result was that 2 mm in the position resolution σ is good enough for our22

4.3 Polarimeterrequirement, and the sense wire which has 1.9 KΩ intrinsic resistance can fulﬁll therequirements.

In the positron track reconstruction analysis, there were diﬃculty caused bya rectangle cell structure. In a case of the rectangle, the shape of the equi-drift
time line is very complicated. Hence, it is diﬃcult to determine the drift velocity
precisely. As a result, the calculation of the drift velocity have to rely on thesimnulation if we pursue higher position resolution. Another possible issue is due toits geometry. Misalignment if the plates with respect to the average muon incident
direction may result in a situation where muons enters in the plates dorminantly
from one side, leading to an asymmetry of positron eﬃciency in the transverse
direction. The beamtest studying for this plate model was performed in Autumn
2009, and the evaluation was summarised in Ref. [33].

4.3

Muon Tube Polarimeter - MTP -The other choice of TREK polarimeter is tube type polarimeter, MTP. I describe
the beamtest to evaluate the performance of this model in Chapter 5, and describe
analysis in Chapter 6.
The MTP is composed of a stack of drift tubes. Like the plate type polarimeter
the MTP can track not only incident particles but also the decaying particles.
Hence the MTP can deﬁne exactly emission angle for each event, and so is freefrom the systematic error associated with the ambiguities in the muon stopping
distribution that were present in E246.
Table 4.2 shows the detail of the MTP. The tube thickness was decided basedon a Geant4 simulation, for stop muon middle of the MTP.The equi-drift time
plane is completely circular, hence it is easier to reconstruct tracks compared to
the rectangular cells.Broken wires do not aﬀected to the other wires. Comparedwith the plate type, the MTP has fewer wires, so it’s easy to read-out, data-taking,analysing, and manufacturing. I have summarized the comparison between twomodels in table 4.1.
Tracking methods are• YZ-plane : equi-drift time plane (use of tracking)• X-axis : charge division

23

4.3 Polarimeter

Figure 4.5: MTP (size: 865mm × 284mm × 330mm, 75kg)

Figure 4.6: MTP YZ-plane24

4.3 Polarimeter

Figure 4.7: Schematic end view of the MTP

Table 4.1: Comparison between plate type and tube type
Plate type
Tube type
number of sense wires∼500 channel 371 channel
number of potential wires
∼500
none
distance between sense wires
16mm
14.2mmequi-drift time line
complicatedcircledrift analysis for small angle
good
good
drift analysis for large ange
not good
good
tracking for small angle
good
good
tracking for large angle
not good
good
manufacturing
easy
not very easy
maintenaice
not good
good
Table 4.2: Detail of MTP spec
spec size
284mm × 865mm × 330mm
gas
Ar : Ethane = 50 : 50
sence wire
SUS304 20μm
number of channels
371
read-out AMP
ASB card
material density / full volume
24%

25

4.3 Polarimeter

4.3.1

Drift tube

A drift tube detects charged particles by means of gaseous ionization and its detection.Passing through gas, charged particles ionize the gas molucules, atoms andloss energy due to Coulomb interactions. The average diﬀerential energy loss (lossper unit length) is expressed by the Bethe-Bloch Formula in the framework of relativistic quantum mechanics, and can be written follows (in the electrostatic unit
system):
Z ρ
dE
=K
−
dX
A β2



2mc2 β 2 EM
2πN z 2 e4
2
ln 2
,
K
=
−
2β
I (1 − β 2 )
mc2

(4.6)

N : the Avogadro numberm, e : electron mass and charge
Z, A, and ρ : atomic number and mass, and the density of the medium
I : eﬀective ionization potential
z : the charge
β : the velocity in units of the speed of light c of the projectile
The rest energy of the electron is mc2 = 0.511 MeV, so K = 0.154 MeV ·The maximum energy transfer allowed in eachinterection is EM , and simple two-body relativistic kinematics gives the following.

cm2 /g for unit charge projectiles.

EM =2mc2 β 2
1 − β2(4.7)

Gas detector produces the signal pulse arising from the energy loss of the incident
particle in the gas after avalanche ampliﬁcation. By these signal, the position of
the trajectory and the energy deposit can be determined.
Two types of gas, base gas and quench gas are usually ﬁlled in a drift chamber.
The former play a role as gas ampliﬁcation to make the gain larger, and the latterdoes adjustment of ampliﬁcation in order to defend electrical discharge, since it haslarge photon absorption cross-section. We used Argon for the base gas and Ethane
for the quench gas mixture 50:50 for the the MTP operation.

There are 371 thin sense wire (SUS304, 20μm) stretched along the x axis of
the MTP. When a strong electric ﬁeld is applied around the electrodes it causes
avalanche multiplication, boosting the signal amplitudes by several orders of magnitude.The electric ﬁeld in one tube is at its maximum at the surface of the anodewire, i.e. sense wire ,and rapid decreases as r−1 towards the tube inner surface
which play the role of the cathode. In order to produce high values of the ﬁeld
close to the anode, a thin wire is used.

26

4.4 PolarimeterIn the region where the charges are produced by the primary interaction processes, a single primary electron proceeds towards the anode, on the contrary, thepositive ion proceeds towards the cathode. The electric ﬁeld near the wire is strong
enough to start multiplication as the electrons come close to the anode. A drop-likeavalanche developes surrounding the wire. Due to lateral diﬀusion and the small
radius of the anode, the avalanche surroundings the wire as shown Fig4.8, electrons
are collected and positive ions begin to drift towards cathode.

Figure 4.8: Time development of an avalanche in a proportional counter. [3]4.4

Muon polarimeter magnet

A uniform muon polarimeter magnet is essential for the TREK experiment, whereas
a passive ﬁeld was used by guiding and trimming the main ﬁeld of the superconducting magnet in the E246 experiment. The unavoidable consequence was that
there was a non-uniform strength distribution and a curved ﬂux distribution at
the stopper. A uniform ﬁeld parallel to the PT component provides the maximum
analyzing power. Some requirements of the muon polarimeter magnet, size, ﬁeld
strength and uniformity, are as follows.
To accommodate the polarimeter with range stacks on both sides the parallel
gap of the dipole magnet must be about 30 cm. The area is determined to produce
a uniform ﬁeld distribution in the polarimeter region. From the point view of
1. spin relaxation suppression,
2. stray ﬁeld decoupling,a strong ﬁeld is preferable. However, the ﬁeld is limited by the interference withthe toroidal magnet, in particular with its SC coils. Point 2 is regarded as the4.5 Polarimeter

27

determining factor; assuming 0.3 Gauss of an unwanted component in the shielded
magnet gap, a ﬁeld strength of at least 300 Gauss is necessary to obtain a ﬁeldalignment of 10−3 . The ﬁeld asymmetry across the median plane is important but
a non-uniformity of 10−2 in strength as well as in vector distribution is tolerable in
the positron energy analysis.
A one sector prototype of the muon polarimeter magnet has been tested in
the North Hall of KEK. After the symmetry measurement using the opposite coilcurrent method of the pole faces and coil without a return yoke, the magnet is now
equipped with the low-weight yokes and a return yoke as shown in Fig4.9, and anormal ﬁeld distribution with a uniformity good enough for the polarimeter has
been achieved in the ﬁducial volume. This magnet can be applied to the TREK

Figure 4.9: Schematic front view of the muon ﬁeld magnet

experiment. It would be, however, necessary to compare the distribution with the
3D TOSCA calculation with the ﬁnal realistic geometry. An improved shim system
which realizes a larger ﬁeld-ﬂat volume might be possible. We don’t know if theperpendicular Bx in the ﬁducial volume is small enough to suppress the systematic
error due to δz rotation, although its detailed estimate depends on the actual muon
stopping distribution in the polarimeter and this is not yet known.

4.5

Systematic errors

In the TREK experiment, two sources of systematic errors will dominate. While
one source is the misalignment of the detector elements, in particular of the muon

28

4.5 Polarimeterpolarimeter, the other source will be given by the background contamination of+− dif events).
muons from the decay-in-ﬂight of Kπ2 pions (Kπ2
• The eﬀects of polarimeter misalignments, in particular the ﬁeld rotation δz ,are not more relevant to the PT determination. If necessary, they can becalibrated using data. Monte Calro simulation studies assuming considerably large misalignments for the rotation parameters r , z , δr , and δz showed
the associated systematic error to be smaller than 10−4 in the discrepancy
between the ﬁt value and the input value.
• The inﬂuence of decay phase space distortion parameterized by the decay
plane angular parameters θr and θz should be corrected. The error associatedwith these corrections is essentially a statistical one and is estimated to be
far less than 10−4 for both θr and θz . The validity of the correction method
can be checked by introducing an artiﬁcial asymmetry in, for example, thekaon stopping distribution in the target to produce signiﬁcant θr and θz .
+• The error due to Kπ2− dif background contamination can be suppressed bymeans of the new upgrated tracking system (Appendix C) down to less than
5 × 10−5 .

• There is a new potential source of error which was not present in E246, namely
the error coming from the active polarimeter analysis. The eﬀects of Ee+ andθe+ ambiguities have to be suppressed to the level smaller than 10−4 .
• The largest systematic error in E246, which was the ambiguity of muon stopping point due to scattering, does not exists in the TREK polarimeter anymore.
Other potential sources such as the misalignments of the tracking elements are
regarded as rather harmless since correction based on the alignment calibration canbe done accurately enough.

Chapter 5

Beam Test at TRIUMF
In order to investigate the detailed performance of the Muon Tube Polarimeter,
we conducted a beam test experiment at TRIUMF, one of the world’s leading subatomic physics laboratories, in Vancouver in Oct-Nov 2011. The main purpose wasto see whether the MTP was suited for TREK experiment. The test program include basic evaluation of the drift tube, performance of muon and positron tracking,and performance as a muon polarimeter.

5.1

TRIUMF M11 beam line

TRIUMF has a 500 MeV cyclotron which produces the primary proton beams.
Fig.5.1 shows the layout of the M11 beam channel. The primary proton beam
strikes the meson production target T1, which is typically 1 cm thick graphite.
The momentum of the channel is determined largely by the magnetic ﬁeld of thetwo bending magnets B1 and B2. Focusing is achieved by using the 6 quadrupole
magnets Q1 to Q6. There are also sextupole magnets available for a better focus,
but these are not currently being used, since the object, and hence the image, is
ill-deﬁned in the absence of the septum magnet. The two quadrupoles Q1 and Q2
provide a double-focus at the midplane (focused in both horizontal and verticalplanes), such that particles of the same momentum emanating beam the object
point at various angles, will all intersect at the same point at this focus.
The midplane of the channel is a plane of momentum focus, such that particles
of diﬀerent momenta will come to a focus at diﬀerent horizontal locations along
this focal plane. The momentum dispersion is 18 mm/%Δp/p.
With the horizontal slits open 90 mm full aperture, the beam channel admits agive value momentum bite. Also located at the midplane are a set of vertical slits
which are considered to be fully open at 30 mm aperture when the septum was inoperation, but which may need a larger opening to increase the ﬂux.
The total length of the M11 channel is 15.299 m, from the production target to the29

30

5.1 Beam Test at TRIUMF

Figure 5.1: Layout of M11 beam channel

5.2 Beam Test at TRIUMF

31

nominal ﬁnal focus. The momentum range is from 60 MeV/c to 450 MeV/c.
Pions and muons travel down the M11 channel where they could be momentum
selected. Particles with a momentum of 170 MeV/c were used for our beamtest.

5.2

Preparation

5.2.1

Preparation

We used an Ar/Ethane 50/50 ﬂammable gas mixture for the MTP operation. The
gas ratio is mainly associated with the drift velocity of the avalanche ions, so keeping
a constant gas ratio was important. Before shipping from KEK to TRIUMF, we
purged the MTP with pure Argon gas, which is not ﬂammable, into the polarimeter.
Prior to the instllation of the MTP in the M11 experimental area, we pumped out
the pure Argon gas and ﬁlled the MTP with the Ar/Ethane mixture gas.
After ﬁlling the gas, piping delivered the gas mixture from outside the area,
and we could measure the gas inﬂow and outﬂow rates in order to monitor any gas
leakage.
For a stable operation of MTP, it is important to set the proper high voltage.
According to the HV curve (Fig 5.2), I used +1980 mV for the MTP, and thesignal threshold was set at -30 mV by an analog signal check. The electric current
of MTP was stable during the beam test, at ∼=1.8 μA.

HV_Curve Threshold103.0mV
10000
9000

Counts/min

8000
7000
6000
5000
4000
3000
1700 1750 1800 1850 1900 1950 2000 2050 2100
HV[V]Figure 5.2: HV curve

32

5.2 Beam Test at TRIUMF

5.2.2

Beam tuningFig 5.3 shows the experimental set up. We tuned the beamline to extract andFigure 5.3: Setup for beam tuning and degrader tuning

separate both muons and pions with a ﬁxed magnet current. We achieved particle
identiﬁcation by Time-of-ﬂight (TOF) relative to the CP timing (Capacitive-Prove,
proton bunch signal). This is a method of mass spectrometry by time measurement
using the mass-to-charge. In beamline, particles have the same momentum as anyother particle.
When the particle beta factor, β = v/c, where v is a particle’s speed, it can bepresented using particle’s momentum p, its energy E, and rest mass m as follows:
β=

pc
pc
=
E
p2 c2 + m2 c4(5.1)

Therefore, the time t spent when traveling a distance d is
d
d
=
t=
βc



p 2 c 2 + m2 c 4
pc2

(5.2)

The distance of the M11 beamline is roughly 13 m.
Using TOF measurement, we tuned the beamline using S0, S2 and the CP
timing.Here, M11 beamline produced proton bunches separated 43.37 ns. The
CP on the 1A proton beamline sends out a pulse to each of the experimental
areas whenever a proton bunch passes. We used this pulse for timing the arrivalof the particles for TOF. The stop timing of TOF was determined by the CP
signal. We found out that the beam momentum was best at 170 MeV/c, the pion
contamination was larger at 180 MeV/c. We selected the particle timings for thepion gate 8 ns delay, for the muon gate 24 ns delay and for the fastest three mixed
gate (muon, pion and positron) 38 ns delay. Table 5.1 shows the comparison ofdelay times.

5.2 Beam Test at TRIUMF

33

Figure 5.4: TOF spectrum at 170 MeV/c

5.2.3

Experimental setup

Fig 5.3, Fig 5.5 and Fig 5.6, Fig 5.7 show the conﬁguration of the beam test. The
data acquisition system consisted of 16 ASB cards, 16 VME-discriminator boards,
8 CAEN QDC modules and 4 CAEN TDC modules. These ampliﬁers and modules
are explained in detail in Appendix D.
The trigger system consisted of several trigger counters. The trigger signal
was selected by a triple coincidence which consisted of S0, S2 and the delayed
TOF signals (S0×S2×delayed TOF). The trigger timing was determined by the S2
counter, which was placed just before the MTP. The delay module located before
the triple coincidence unit (S0×S2×delayed TOF) could select the kind of particle
according to the length of the delay cable. The TDC common stop was the sametiming as the end of the trigger gate.
We scanned the MTP with the beam to see there were no broken wires, all of
the 256 reading channels worked well. It was important to pay attention to checkif there was wrong cabling or channel connection.

For TDC calibration, I checked the gap between the CP signals. Here, thefrequency for the TRIUMF cyclotron was 23.06 MHz so the time separation of thebeam pulses was 43.37 ns. The gap was 445 channels, so I got 1 channel = 97.46 ps(Fig 5.8).
Next, we made the beam particles stop in the middle of the MTP by adjusting34

5.2 Beam Test at TRIUMF

Figure 5.5: Timing chart

Figure 5.6: Typical chamber signal

5.2 Beam Test at TRIUMF

Figure 5.7: Conﬁgulation of the DAQ system

Figure 5.8: Gap between CP signals

35

36

5.2 Beam Test at TRIUMF

the thickness of the Al degrader plates.

Figure 5.9: Degrader tuning ; Telescope (E1 and E2 counters) was placed on MTP• beam tuning: to focusing beam on the trigger counter
The S3 counter was used as a veto counter since we required the muon beam
to stop in the middle of the MTP for our experiment. Here, we arranged
E1 and E2 counters as a telescope pointing to the center of the MTP.The
telescope was used in the range curve measurement.
• degrader tuning: to control the muon or the pion beam stopping position
In the beam test, the entire volume of the MTP should be tested to determine
whether or not the MTP detection eﬃciency depended on position. Hence,
we varied the stopping position of the incident particles. In order to control
the stopping position, some 0.5 and 1.0 inch size Al plates were used as thedegrader. The Al degrader was located in front of the S2 counter.
In this measurement, a telescope (the set of E1 and E2 counters) was used
and although it was just counting decay positrons, the number of counts
correlated with the decay position. The telescope should be placed in thetransverse direction relative to the other counters, so we placed it on top of
the MTP.The number of telescope counts ( E1 × E2 × TOF ) was normalizedusing the number of S0 and S2 coincidence (S0 × S1) counts.
When muon or pion stopped in the middle of the MTP, positrons should beemitted, so the thickness which enable the maximum telescope count rate issuitable. The results were the following;37

5.2 Beam Test at TRIUMF
For π + : 1.5 inch Al (Fig 5.10)
For μ+ : 3.25 inch Al (Fig 5.11)
Degrader thickness for pi

E1xE2xTOF/S0xS1 (x100)

2

Counts/(S0xS2xTOF)

S0xS2xTOF/S0xS1

1.5

S0xS2xS3barxTOF/S0xS2xTOF

1

0.5

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Degrader thickness (inch)

Figure 5.10: Results of the degrader tuning for π

Degrader thickness for mu
E1xE2xTOF/S0xS1 (x1000)

Counts/(S0xS2xTOF)

12

10

S0xS2xTOF/S0xS1

8

S0xS2xS3barxTOF/S0xS2xTOF

6

4

2

0
0

1

2

3

4

5

6

Degrader thickness (inch)

Figure 5.11: Results of the degrader tuning for μ

We found that the telescope timing presents that the decay positrons comes fromthe incident muons as expected(Fig 5.12).
Fig 5.13 shows the incident muon stopping point in the MTP. The last hit of38

5.2 Beam Test at TRIUMF(TDC[103][0]-TDC[261][0]+6313)*0.09746 {TDC[103][0]>10}
Entries224356

Mean

1431

RMS

854.2
7.934 ± 0.011

Constant

Counts/bin

Slope -0.0004422 ± 0.0000070

103

500

1000

1500
2000
Time[ns]

2500

3000

Figure 5.12: Muon decay curve: The parameter (1/τ ) corresponds to 2.2μs

the muons position are plotted in this Figure. We can see that the beam particles
stopped at the center of the MTP.
TDC stop position dist.

tbottom

20
18000

18

16000

Y [wire position]

16

14000

14

12000

12

10000

10
8

8000

6

6000

4

4000

2

2000

0
0

5

10

15
20
25
Z [wire position]

30

35

0

Figure 5.13: The conﬁrmation of muon stopping area in MTP. The colored points
means wire positions and all of the 256 channels worked well.

During beamtest, diﬃcult things were below:• The cabling was confusing as already mentioned.

5.4 Beam Test at TRIUMF

39• The connection between the cables and the modules were not very good, so
we had to check the signal size by amps. This should be improved.
• The MTP signal was noisy without an Al foil shield. The cover of the mounting face for the cables and amps should more protected if there were no
shield.

5.3

Measurements

We collected the data of diﬀerent kinds as listed below.
• Normal run (using the muon trigger, high statistics μ+ → e+ data collection)• MTP rotated run (rotated forward and backward)• Charge division run
Fig 5.14 shows the schematic view of charge division study set up and Fig 5.15
is the ﬁber scintilator to deﬁne the incident position of the muons with goodprecision.Reading both ends of wires, and moving x-direction.

Figure 5.14: Schematic view of charge division study set up• Pion run
Degreder was set for the pion, and the pion trigger was used.
• μSR run
For the determination of the beam polarization. I describe in the next section.

5.4

μSR measutement

In order to determine the muon beam polarization of M11, we measured μSR, on
the same place after MTP beam test. Since the muons with the momentum of
170 MeV/c generated upstream of the M11 beamline was considerd to be the cloud
muons. The polarization of cloud muons was not very well known.

40

5.4 Beam Test at TRIUMF

Figure 5.15: Fiber scintilator for charge division study to deﬁne the incident position with a good precision

The muon is a spin -1/2 particle, its mass is 106 MeV/c2 , and its lifetime is
2.2 μsec. Its magnetic moment interacts with mgnetic ﬁeld.We use μSR measurement, Muon-Spin-Rotation or Muon-Spin-Relaxation, to know the polarization of
the muon beam.
The muon’s magnetic moment μμ interacts with the local magnetic ﬁeld Bloc
inside a solid. When a muon enters a material and stops there by ionization energy
loss, the muonspin precess about the local magnetic ﬁeld. When the muon decay
via weak interaction, it emits a positron:
μ+ → e+ + ve + v¯μ(5.3)

High energy positrons tend to be emitted in the direction of the muon spin. In
order to detect the muon spin polarization P (t), we can place counters which detect
positron, typically, in the back and the front of the sample, and the asymmetry of
their number of counts will tell us the direction of the spin. The number of hits at
diﬀerent times is plotted for each detector. When muons always feel same direction
and same intensity of the magnetic ﬁeld, the ensemble of the positron timing showsa cosine curve. Superimposed on the exponential decay of the muon is an oscillation
showing extra counts as the muon’s spin sweeps past the detector. If the ﬁeld is
non-uniform, the frequency of the muon spin is variess and the polarization to be
gradually relaxed.
Fig 5.16. shows our set up for μSR measurement. We placed a magnet with
366 gauss instead of the MTP, and the Al sample (Fig 5.17), which was made ofsame material as the MTP, was placed at the center of the magnet. Using the same

5.4 Beam Test at TRIUMF

41

experimental electronic circuit for the MTP beamtest, we changed the counters andtheir positions around the magnet. We used four counters, named S2, S3, E1 andE2, and I used the S3 counter for this analysis since its precession pattern was themost clear. S3 counter was located at the down stream of the Al sample along thebeam direction.

Figure 5.16: Set up for μSR measurement

Fig 5.18 shows the μSR precession pattern. The ﬁtting function is following.
t(1 + A cos(ωt + φ)) + B.G.
τμt
(1 + p2 cos(p3 t + p4 )) + p5
≡ p0 exp −p1

f (t) = N0 exp −

(5.4)
(5.5)

Here, N0 is the nomalization coeﬃcient, t is the time,τμ is the muon lifetime, A isthe amplitude corresponding to the asymmetry we want to determine, and ω is theangular velocity, φ is the initial phase, B.G. is the background.
According to the ﬁtting result, asymmetry was 5.694 ± 0.005 [%].
Considering time start origin, the initial spin direction could be known by theμSR spectrum and ﬁtting result. The counters counted positrons, which were
emitted by the muon and had the same direction of the muon spin by helicity
conservation, so substitute the time when muons incident from ﬁtting result Eq 5.5,
we could determine the initial phase.

42

5.4 Beam Test at TRIUMF

Figure 5.17: Al sample (5 cm× 2.4 cm × 2 cm )

MSR
MSR

2000

Entries
139573
Mean 2.386e+04
RMS 2.125e+04

1800
1600

Counts

1400
1200
1000
800
600
400
200
0
0

10000

20000

30000

40000 50000
time 1ch = 97ns

60000

70000

80000

90000

Figure 5.18: S3 counter timing spectrum : Muon stopping timing (the spike at ∼2500) and the oscillation pattern are seen.

43

5.4 Beam Test at TRIUMF
MSR
MSR
600

Entries
139573Mean
1.846e+04 ± 35.98
RMS
1.124e+04 ± 25.44
χ2 / ndf
502.1 / 414
p0550.2 ± 3.9
p12.242e+04 ± 6.272e+02
p2
0.05694 ± 0.00505
p3
0.003018 ± 0.000008
p4
1.796 ± 0.158
p5
17.62 ± 5.34

500Counts

400

300

200

100
5000

10000

15000

20000
25000
30000
time 1ch=97ns

35000

40000

Figure 5.19: Fitting result of the μSR oscillation spectrum

Using the TDC value of muon stopping time T0 = 2548.3, calibrating muon
stopping time for time zero, the relation between time t ns and T is
T = T0 + βt, β = 1/0.09746.

(5.6)

Therefore f (t) is expressed asT0
p1t= a0 exp −a1f (t) = p0 exp −

exp −

β
t (1 + p2 cos(p3 βt + p3 T0 + p4 )) + p5
p1

(1 + a2 cos(a3 t + a4 ) + a5 )(5.7)

where a0 = p0 exp(−T0 /p1 ) = 490.9 ± 3.8. a1 = p1 /β = 2185.1 ± 61.1 (ns) ismuon decay constant, which is consitent with the PDG value (2.19698(μs). a2 =p2 = 0.05694 ± 0.00505 is the oscillation amplitude (5.69%). a5 = p5 . Fig 5.20
shows the time calibrated ﬁt function (Eq 5.7).
a3 = p0 β = 0.03097 ± 0.00008 is the muon spin angular frequency ω = 2πf .
Therefore the frequency was f = 4.929 ± 0.013(MHz), which is consistent with the
expected value 4.96 MHz, since the muon gyromagnetic ratio is 153.3 MHz/T andthe magnetic ﬁeld was 0.0366 T.
Those above-mentioned parameters were obtained from the measurement andﬁtting, now I wanted to know a4 , which is initial muon spin phase φ0 . a4 =T0 p3 + p4 = 3.204 + 2π ± 0.1783, and 3.204 rad = 183 degree. S3 counter wasplaced on the beam direction. Hence, it could be concluded that the muon spin
was polarized backward, and forward muons.
In conclusion, the beam polarization we used was about 17% (= amplitude 5.69
× 3) (Chapter 6.4.1) and backward polarized.

44

5.4 Beam Test at TRIUMF

490.9*exp(-x/2185.1)*(1+0.05694*cos(0.03097*x+9.487))+17.62

500
450
400

counts

350
300
250
200
150
100
50
0

500

1000

1500

2000
2500
time[ns]

3000

3500

4000

4500

Figure 5.20: Calibrated time spectrum of the S3 signals.

Table 5.1: Comparison between particlesπ+μ+e+Lifetime
26.033(5) ns 2.19703(4) μs> 4.3×1023 y
2
Mass (MeV/c ) 139.56995(35) 105.8389(34) 0.51099907(15)Charge (e)
+1
+1
+1
1
1
Spin ()022Delay (ns)82438

Chapter 6

Data Analysis
6.1

Basic Analysis

6.1.1

Data quality

In order to evaluate the performance of the data, the time spectrum of the S2
counter was checked ﬁrst. The S2 counter was used for determing the TDC stop
timing. As shown in Fig 6.1, the time resolution of the S2 counter was obtained as
σ = 1.276 ns. This resolution was good enough to evaluate MTP signals because
the typical sense wire time resolution scale, which was determined by the drift
velocity of the avalanche ions and the cell structure, was much larger than the S2
counter time resolution scale.
Time_reference
Entries

60000

50000

209831

Mean

11.25

RMS

3.192

Counts/ns

Constant6.106e+04 ± 2.078e+02

40000

Mean

11.15 ± 0.00

Sigma

1.276 ± 0.005

30000

20000

10000

0
-30

-20

-10

0

10
Time[ns]

20

30

40

50

Figure 6.1: The S2 counter time spectrum. The S2 counter was used for determing
the TDC stop timing.

45

46

6.1 Analysis
Wire Efficiency
100

Efficiency[%]

99.5

99

98.5

98

97.5
5

10

15

20
25
Wire position

30

35

Figure 6.2: Wire eﬃciency

6.1.2

Wire eﬃciency• Wire eﬃciency
It is important to have high detection eﬃciency as a tracker, which is usuallyrequired to have more than 99%. The tube eﬃciency was studied in thefollowing way.
– The three wires parallel to the incident beam were selected for the wholevolume of MTP.
– I examined whether the center of wire was tagged or not when both thesurrounding wire were hit.
The wire eﬃciency of the chamber was evaluated using the muon beam datain our measurement, and is deﬁned as follows and shown in Fig 6.2. Here theeﬃciency is shown as a function of z coordinate of the tube.
The decrease of the wire eﬃciency was due to the material density since thispolarimeter plays a role as not only the detector but also the stopper. This
polarimeter was made of material in order to stop the incident particles in thevolume, this is a point of diﬀerence from other trackers. In general, a tracker
should be made of light material in order to prevent secondary emission or
particle scattering. Larger density like the MTP causes some bad tracks withlarge multiple scattering because it was diﬃcult to prevent these eﬀects. The
apparent position dependence of the wire eﬃciency could be seen in Fig 6.2.
• Dependence of eﬃciency on drift distance47

6.1 Analysis
Efficiency

100

Efficiency[%]

95
90
85
80
75

0

1

2

3

4

Drift distance[mm]56Figure 6.3: Eﬃciency as a function of the drift distance

The number of primary ionization, therefore the amount of collected charge
on the wire, is proportional to the track length inside the tube, which varies
as a function of the distance of the track from the wire.
I checked the uniformity of eﬃciency by means of computing the X-T relation,
which I will explain in the next chapter, per unit mm. The deﬁnition of theeﬃciency here is:– The three wires parallel to the incident beam were selected for the wholevolume of the chamber.
– I checked each X2 (X1 , X3 ) whether the center wire was hit or not (Fig 6.4).

Figure 6.4: Schematic view of the wire eﬃciency of z-direction; the wire eﬃciency
is expressed as (the middle of the A wire hit)/(all of the A, B, C wires are hit)48

6.2 Analysis
Fig 6.3 shows the result of the radial dependence of eﬃciency. It was goodenough to detect ions from the whole volume of tube, since tube radius was
5.6 mm.

6.2

Drift time

In order to tracking precisely, it is important to deﬁne the time starting origin
and to know the drift velocity of the electrons to draw drift circle and ﬁt a track.
The drift time distribution shows the expected shape for tube type drift chamber(Fig 6.5).
It was made by choosing 1300 ch = 124.15 ns as the time origin which was themuon beam timing for a wire.
Drift_time_distribution
Drift_time_distribution

Entries
Mean
RMS

3500

14770450.68
28.88

Counts/5ns

3000
2500
2000
1500
1000
500
0

-20

0

20

40

60

80

100

120

140

time[ns]

Figure 6.5: Drift time distribution

Assuming that the muons irradiated the tube uniformly, the drift distance isalso uniform. I determined the following three parameters by means of ﬁtting thedrift time distribution.
• the time starting origin• the relation between drift distance and drift time (X-T)• the drift velocity
The slope of the drift time distribution is expressed as a linear function, withthe rise time and fall time expressed as Fermi-Dirac function. The ﬁtting function
is:

49

6.2 Analysis

f (τ ) = p0 + 
1 + exp

p1 ((τ − p3 ) + p2 )
 


1 + exp (τ −pp36)−p53− τ −pp4(6.1)

p0 : backgroundp1 : slope of the drift time distribution
p2 : intercept
p3 : time stating point
p4 : width of the rising time
p5 : max of the drift distance
p6 : width of the fall timeFig 6.6 shows the ﬁtting result.
Drift_time_distribution

Drift_time_distribution

3500

Entries

147704

Mean

50.68

RMS
χ2 / ndf

3000[counts/5ns]

p0
p1

2500
2000
1500

28.88
210.4 / 57
4.061 ± 2.822-15.29 ± 0.38

p23730 ± 20.9

p3

5.442 ± 0.079

p4

2.852 ± 0.046

p5

99.45 ± 0.14

p6

3.761 ± 0.095

1000
500
0

-20

0

20

40

60

80

100

120

140

time[ns]

Figure 6.6: Fit to the drift time distribution
I obtained the X-T relation by an integration of the linear part of the distribution. The boundary condition was the position of the wire and tube surface as
determined by the parameters p3 and p5.
x(t) = (−1.50 × 10−4 )t2 + (7.33 × 10−2 )t, t ≡ τ − p3

(6.2)

By diﬀerentiating this equation, the drift velocity could be ditermined.
v(t)

=
=

d
x(t)
dt
(−3.00 × 10−4 )t + (7.33 × 10−2 )The average drift velocity was:(tube radius 5.8 mm)/(maximum drift time 110 ns)=50 μm/ns.

(6.3)

50

6.2 Analysis

Acutually the velocity was not constant, you can see from drift time distribution,
since there was gradient of the electric ﬁeld from wire, but the average drift velocity
was equal to the standard of a typical drift chamber ﬁlled with same mixed gases.
The assumed uniformity of the incident beam positions was conﬁrmed by thedrift distance distribution, which was made using the X-T relation. In addition, thedrift distance distribution is consistent with the tube diameter, the approximationof the drift velocity was adequate.
X2
Entries
Mean
RMS

X2
147704
2.874
1.707

2500

[count/um]

2000

1500

1000

500

0

-2

-1

0

1

2

3
4
X2[mm]

5

6

7

8

Figure 6.7: X2 distribution
Furthermore, to conﬁrm this result, I checked by another X-T relation method.
For adjacent three wires, I deﬁne a the wire hit timings as t1 , t2 and t3 and theirdrift distance as X1 , X2 and X3 , respectively as shown in Fig 6.8.

Figure 6.8: Schematic view of the deﬁnition of position for analyzing the X-T
relation
Here, I derived X2 by two ways:• Using equation 6.2 derived by ﬁtting drift time distribution, X2 (t2 ) is obtained directly from t0 .

51

6.2 Analysis• Alternatively X2 can be obtained from X1 (t1 ) and X3 (t3 ) using Eq. 6.4,
assuming all the hits belong to a common straight track.
X2 = (X1 + X3 ) /2

(6.4)

By comparing the relation between X2 (X1 , X3 ) and X2 (t2 ), the consistensy of
the ﬁtting result is conﬁrmed.
x-t relation
x-t relation
Entries112317
Mean x37.66
Mean y
2.636
RMS x
25.02
RMS y
1.272

180

X2=(X1+X3)/2[mm]

6

160

5

140
120

4

100

3

80

2

60
40

1

20

0

0

20

40

60
t2[ns]

80

0100

Figure 6.9: X2 (X1 , X3 ) versus t2

X_X2

X_X2

Entries
Mean xMean yRMS x
RMS y112317
2.442
2.626
1.524
1.281

X2=(X1+X3)/2[mm]

6

300

5

250

4
200

3
150

2

100

1

50

0
-1
-1

0

1

2
3
4
X2(t2)[mm]

5

6

0

Figure 6.10: X2 (X1 , X3 ) versus X2 (t2 )52

6.3 Analysis

6.3

Tracking

6.3.1

Event selection

It was necessary to extract clean tracks with no-bias before the main analysis.
The track reconstruction analysis and the asymmetry analysis require ﬁne tracks
extracted after some selections.
First of all, I classiﬁed the TDC hit timing in order to remove bad events.
The incident particles, whose TDC is 0-2500 channels (1 channel = 97.5 ps), werecategorized as ”BEAM ”, and there no positron contamination due to the 500 ns
delayed trigger gate. After ”BEAM ”, the rest of the TDC data were categorized
into 10 categories as 2500-10000 channels, ...,80000-90000 channels, and 90000channels. There were some bad events which should be discarded, for example
having some diﬀerent timings which belong to more than 3 categories, respectively.
I selected a true track which belong to both ”BEAM ” and the one of the othergroups.
Next, I classiﬁed by geometry in order to eliminate some noise hits and torescue ineﬃcient hits. The incident track should be formed as a straight line, noise
events which did not form a line were discarded. Since the decay positron has a low
momentum, the positron tended to generate secondary particles, thus, a positron
could make two lines. In this case, it was diﬃcult to extract the true positron track,hence these events were also discarded.
I deﬁned the last hit of the muon line as a vertex. Those events which have overtwo vertices should also be discarded since they could include some hits caused bysecondary particles or event pile up.

6.3.2

Tracking algorithm

Using the X-T relation, drift circle of the point of hit wire(y, z) can be drawn. Theline touching the drift circles can be estimated by the method of least squares. For
this, following parameters were deﬁned:|ay + bz + c|
a2 + b2
Ek = Q0 (t − t0 )2 + Q1 (t − t0 )
p0
σ = √
+ p1
t − t0

Ok =

(6.5)
(6.6)
(6.7)

where k is the identiﬁcations number of the wire. Ok is the distance between
the sense wire and the ﬁtting line. Ek is the radius of the equi-drift-time line, andσk is the ﬂuctuation of the equi-drift-time line.

53

6.3 Analysis

Figure 6.11: Schematic view of the track reconstruction

Here, Ek and σk have already been determined by the X-T relation. σk included
a ﬂuctuation of the incident angle, so the equi-drift-time line could be regard asa full circle. Also, in order to consider multiple scattering, I used the positron in
number order Nk = 1, 2, 3.... for a weight of chi-square. Therefore, chi-square wasdeﬁned using these parameter asχ 2 = Σk(Ok − Ek )2
Nk σ 2(6.8)

Multiple scattering increases in proportion to subduplicate of the radiation lengthof the matter, the weight represent these things. The ﬁtting of the decay positron
was performed based on a principle of minimum χ2 , using ROOT TMinuit class.
The MINUIT package acts on a multiparameter function to which one must give thegeneric name FCN, and oﬀers the user a choice of several minimization algorithms.
The MIGRAD algorithm is in general the best minimizer for nearly all functions.
There was no information for determing the positron time origin t0 because we
could not use a counter for measuring it. The positrons decay in the middle regionof the MTP and t0 is diﬀerent for each event. There were four free parameters
corresponding to a, b, c, and t0 , more than 5 hits in a cell unit were required forthe analysis. Thus, the ﬁducial volume of the polarimeter including the vertex cell
was chosen as a small region.
The eﬀects of using ﬁtting weights or not are shown in Fig.6.12 and Fig.6.13.
Without weights, the ﬁtting line was pulled by the end of the hitting wire and thiswas not preferable because it means tracking was aﬀected by multiple scattering.
For the asymmetry analysis, we want to know the positron emitted angle uninﬂuenced by multiple scattering, so this method works well.

54

6.3 Analysis

Tracking Map
160
140

Y[mm]

120
100
80
60
40
20
0

50

100

150
Z[mm]

200

250

Figure 6.12: Tracking result with weight for the positron

Tracking Map
160
140

Y[mm]

120
100
80
60
40
20
0

50

100

150
Z[mm]

200

250

Figure 6.13: Tracking result without weight for the positron55

6.4 Analysis
Tracking Map

220

Y[mm]

200

180

160

140

0

20

40

60

80

100
120
Z[mm]

140

160

180

200

Figure 6.14: Another example of positron tracking

Tracking Map

140

Y[mm]

120
100
80
60
0

20

40

60

80 100 120 140 160 180 200
Z[mm]Figure 6.15: Incident muon tracking reconstruction ﬁtUsing same ﬁtting algorithm, I also reconstructed the incident muon tracking.
The tracking reconstruction eﬃciency was over 96% for the muons, for positrons
the tracking eﬃciency was about 68% since it was diﬃcult to ﬁt the positron parallelto the wire for 2-dimension tracking. For 3-dimensional tracking, charge division
method is required (Appendix B).

56

6.4

6.4 Analysis

Asymmetry

Measureing the polarization by the MTP is an essential element of the TREK
experiment, so the performance of the MTP must be known in advance. We useda polarized muon beam in our beam test, hence the MTP must measure the samelongitudinal polarization and have null asymmetry in the transverse direction. As
already mentioned in Section 5.4, the M11 muon beam is polarized in the forward
direction, but it had not been clearly known since it had never been measured.
Hence, the result of our μSR measurement was key to know the beam polarization.

6.4.1

Muon decay parameters

In the muon decay mode μ → eν ν̄, neglecting radiative corrections and the massof the neutrinos and e± , the diﬀerential decay probability can be written asd2 Γdx d cos θ2ρ
(4x − 3) + 3ηx0 (1 − x)/x
3
2δ
(6.9)
± Pμ · ξ · cos θ[1 − x + (4x − 3)]}.
3Here, θ is the angle between the electron momentum and the muon spin, andx ≡ 2Ee /mμ . Pμ = |Pμ | is the degree of muon polarization and x ≡ Ee /max(Ee ) is
the reduced energy, where the maximum e± energy, max(Ee ) = mμ /2 = 52.83M eV .
The Michel parameters are bilinear combinations of the weak interaction couplingγ. For the Standard Model coupling, the interaction corresponds to one
constants gμ
V being unity and all others being zero, so ρ = ξδ = 3/4, ξ =single amplitude gLL
1, η = 0 and the diﬀerential decay rate is∼ x2 · {3(1 − x) +G2F m5μd2 Γ=[3 − 2x ± Pμ cos θ(2x − 1)]x2 ].
(6.10)
dx d cos θ
192π 3
The antisymmetric part in cos θ represents the asymmetry, which depends on thee± energy.
d2 Γ
dx d cos θ∼ F (x)[1 + α(x) cos θ](6.11)

2x − 1.
3 − 2x(6.12)

α(x) = ±Pμ

The normalized symmetric part of the e± energy spectrum has the form
F (x) = 2x2 (3 − 2x)(6.13)The decay distribution are shown Fig6.16 as a function of cos θ and x.
When the polarization of the μ± is 100%, the theoretical asymmetry averaged
over all the possible e± energies from x = 0 to x = 1 is given by, 1
1(6.14)α =α(x)F (x)dx = ± .
3
0

57

6.4 Analysis
x*x*((3-2*x)-1*y*(1-2*x))

d_gamma/dx/d_theta

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

cos0.2 0_th-0.2-0.4
eta -0.6

-0.8

-1 0

x(E

0.8
0.6 0.7

0.9

1

me n tu mo ta l moT)xa/E _ m

0.5
0.3 0.4
0.1 0.2

Figure 6.16: Distribution of decay positrons as a function of cos θ and x

6.4.2

Result of asymmetry measurement by the MTP

While a track is reconstructed by ﬁtting as a straight line, I should pay attention
to the fact that there were two possible direction for a line, the true direction andits opposite direction. I measured positron hit direction from the vertex. I will
describe the selecting procedure.
When a line is expressed as az + by + c = 0, its directional vector is


b
a
,
u1 = −
a2 + b2 a2 + b2oru2 = 

b
a2 + b2, −aa2 + b2(6.15)

.

(6.16)The direction of average hit position with respect to the vertex can be expressed
as;


N
N
x
y
i
i
i
− zvertex , i− yvertex .
(6.17)
p = (Z, Y ) =N
N
Here, (zi , yi ) is the positron hit wire position, N is the number of the positron hits,(zvertex , yvertex ) is the vertex position. The true angle can be selected requiring thatthe inner product u1 · p or u2 · p is positive and close to 1. And its x component±√ 2b 2 means cosine. The events with cosine > 0 means positron emitted in thea +bforward, and cosine < 0 means positron emitted in the backward direction. By
counting those, asymmetry can be calculated.

58

6.4 Analysis

Figure 6.17: Schematic view of the tracking angle

Although I extracted clean tracks with no-bias before the main analysis, there
remained a few bad events. When the selection was done ideally and the positron
hit direction and line directional vector are the same, the opening angle of those·p(i = 1 or 2) is expected to be close to 1. In fact, as seen
vector cos ψ = |uuii||p|Fig 6.18, there were event its cos ψ close to 0, and example of those bad events,
which could not be conducted tracking reconstruction, were shown in Fig 6.19.
In order to avoid using these bad events, I regarded events with an opening
angle over 0.5 as good for the asymmetry calculation. In addition, although I
selected the ﬁducial volume of the polarimeter including the vertex cell for a small
region, I examined the systematics due to the size or position of the selection.
The asymmetry can be obtained as,
Af /b =

Nf wd − Nbwd
Nf wd + Nbwd

(6.18)

where a number of events with positrons Nf wd and Nbwd correspond to the z direction (f orward) counts and the -z direction (backward) counts, respectively.
where Nf wd and Nbwd correspond to the z direction (f orward) and the -z direction(backward), respectively.

59

6.4 Analysis

opening_angle
opening_angle
Entries 492835

5Counts/bin

10

Mean

0.897

RMS

0.2016

4

10

3

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cos_open

Figure 6.18: Opening angle distribution

Tracking Map
140

130

Y[mm]

120

110

100

90

0

20

40

60

80

100

120

140

160

180

200

220

240

X[mm]

Figure 6.19: Example of bad events: Its opening angle is small, 0.387, because of
there were two incident beam partially and the second beam mis-regarded as the
positron, so the average of the positron hit was mis-calculated.

60

6.4 Analysis

6.4.3

Systematics

It was simple if the MTP has no systematics. In fact, when I checked carefully, there
seems to exists diﬃcult systematics, then I studied how is the details. In conclusion,
with a ﬁnite size of a detector, a MTP, the problem seems unavoidanble and theunderstanding is necessary. I describe and discuss the ﬁndings in this section.

Fig 6.20 and Fig 6.21 shows the deﬁnition of emittance angles in three dimension
and two dimension, respectively.

Figure 6.20: The coordinate system deﬁnition

The deﬁnition of the coordinate can be written as,⎞⎞ ⎛sin θ cos φx⎝ y ⎠ = ⎝ sin θ sin φ ⎠
cos θ
z
⎛

(6.19)

61

6.4 Analysis

Figure 6.21: Schematic view of the muon decay for positron in the MTP

cos ψ = 

z

=

cos θ

,

(6.20)=,z2 + y2cos2 θ + (1 − cos2 θ) sin2 φtan ψ.
θ = arctan
sin φ(6.21)sin ψ = z2 + y2ycos2 θ + (1 − cos2 θ) sin2 φsin θ sin φ(6.22)In the two dimentional tracking, we can see the projected angle ψ.
The number of events emitted f orward or backward equivalent to the integralof cos ψ > 0 or cos ψ < 0, respectively. Also, the number of events emitted up or
down equivalent to the integral of sin ψ > 0 or sin ψ < 0, respectively.
Using Eq 6.11 and Eq 6.20, without polarization, the cos ψ distribution is expected to as Fig 6.22 and Fig 6.23.
Using the data analysis, in fact, these events whose positrons were emitted
parallel to tube (x-axis) were diﬃcult to two-dimensional trackng, these shape ofcos ψ distribution and sin ψ distriburion should be deformed. Fig 6.24 and Fig 6.25
shows simulation by Geant4.
Data
Although the asymmetry measured by the MTP seems to be obtained simply,
there exists systematics depending on the incident particle stopping position of the62

6.4 Analysis

cos_distribution

cos_distribution

350

350
cos_distribution

cos_distribution

Entries 10000

300

200

200

Counts250

Counts

250

150

150

100

100

50

50

-1

Entries 10000

300

-0.8

-0.6

-0.4

-0.2

0
0.2
cos_psi

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0
0.2
cos_psi

0.4

0.6

0.8

1

Figure 6.22: cos ψ by analytic calculation Figure 6.23: sin ψ by analytic calculation

cos distribution

sin distribution
cos distribution

600

500

500

400

400

300

200

100

100

-0.8

-0.6

-0.4

-0.2

0

cos_psi

0.2

0.4

0.6

0.8

1

Entries

12829

0.8

1

300

200

0
-1

sin distribution

12829

CountsCounts

Entries

600

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

sin_psiFigure 6.24: cos ψ generated by Geant4 Figure 6.25: sin ψ generated by Geant4
simulation
simulation

63

6.4 Analysis

ﬁducial volume. Fig 6.26 shows the deﬁnition of the ﬁducial volume. I devided
ﬁducial muddle of MTP area into four ﬁducial volume, a to d.

Figure 6.26: Deﬁnition of the ﬁducial volume. The area in the dashed line means
the read-out region.

• μ+ → e+ νμ νe data
According to Fig 6.27 and Fig 6.28, the distributions has some structure peakcorresponding to 0 degree, 30 degree, and 60 degree. I think this comes fromtwo dimension tracking and the ﬁlling structure of tube in the MTP.
fiducial all
40000

cos_distribution

fiducial a

30000

35000

cos_distribution

fiducial b

35000

fiducial all

fiducial c

Entries 492835

fiducial dMean -0.01678
RMS

30000

sin_distribution

fiducial afiducial b

sin_distribution

fiducial c

Entries 492835

fiducial dMean -0.02615

25000

0.7356RMS

0.6759CountsCounts

25000

20000

20000

15000

15000
10000

10000

5000

5000
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0
-1

-0.8

-0.6

cos_psiFigure 6.27:
muon datacos ψ distribution of the Figure 6.28:
muon data

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sin_psisin ψ distribution of the

The measured asymmetry is not uniform depends on the ﬁducial volume(Fig 6.29, Fig 6.30). In case of the forward or backward asymmetry, it isdiﬀerent the forward area (c and d) and the backward area (a and b) inFig 6.29. Also, in case of the up or down asymmetry, it is diﬀerent with the64

6.4 Analysis
upward area (b and d) and the downward area (a and c) in Fig 6.30. Thisshould be comes from the MTP has a ﬁnite size.
It is not understood why the asymmetry depends on the opening angle cut.
The opening angle cosine nearly one means the ﬁtting direction and the average of the positron hitting direction are the almost same.
Asymmetryabcd

Asymmetryfiducial a

0.15

abcd

0.15

fiducial a

0.1

fiducial b

fiducial b
0.1

fiducial cfiducial d

0.05

fiducial c

0.05

Asymmetry ud

Asymmetry fb

fiducial d

0
-0.05
-0.1
-0.15

0
-0.05
-0.1
-0.15

-0.2

-0.2

-0.25

-0.25

-0.3
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.3
0

0.9

0.1

0.2

0.3

opening angle cut

0.4

0.5

0.6

0.7

0.8

0.9

opening angle cut

Figure 6.29: F orward or backward asym- Figure 6.30: U p or down asymmetry of
metry of the muon data
the muon data
• π + → μ+ νμ , μ+ → e+ ν ν̄ data
Since the spin of pion is 0, the emission angle of muon from decays of stopped
pion is expected to be isotropic, therefore no asymmetry of positrons are
expected. However asymmetries are observed also in the pion data. (Fig 6.31,
Fig 6.32 and Fig 6.33, Fig 6.34).
cos_distribution
30000

25000

sin_distribution

fiducial all

cos_distribution

20000

fiducial a

Entries 280335

18000

fiducial all
fiducial a

Mean -0.04638

16000

fiducial b

RMS

14000

fiducial cfiducial dfiducial b
fiducial cfiducial d

20000

0.7519sin_distribution
Entries 280335
Mean -0.02789
RMS

0.6562

CountsCounts

12000

15000

10000
8000

10000

6000
4000

5000

2000
0
-1

-0.8

-0.6

-0.4

-0.2

0

cos_psi

0.2

0.4

0.6

0.8

1

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sin_psiFigure 6.31: cos ψ distribution of the pion Figure 6.32: sin ψ distribution of the pion
data
data
The behavior of the muon and the pion data are similar, so there must be
common systematic eﬀects with the MTP.

65

6.4 Analysis
Asymmetry

abcd

0.15

0.1

fiducial b
fiducial c

0.1

fiducial d

0
-0.05
-0.1
-0.15

-0.1
-0.15
-0.2
-0.25

0.2

fiducial d

0

-0.2

0.1

fiducial c

-0.05

-0.25
-0.3
0

fiducial b

0.05

Asymmetry ud

Asymmetry fb

fiducial afiducial a

0.05

Asymmetry

abcd

0.15

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opening angle cut

0.9

-0.3

0

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0.4

0.5

0.6

0.7

0.8

0.9

opening angle cut

Figure 6.33: F orward or backward asym- Figure 6.34: U p or down asymmetry of
metry of the pion data
the pion data
More smaller ﬁducial volume
I deﬁned the more smaller ﬁducial volume. Fig 6.35 shows the new deﬁnition of
the ﬁducial volume. I also devided ﬁducial muddle of MTP area into four ﬁducial
volume, a to d.

Figure 6.35: Deﬁnition of the ﬁducial volume• μ+ → e+ νμ νe data
There seems the same position dependency(Fig 6.36, Fig 6.37 and Fig 6.38,
Fig 6.39).
• π + → μ+ νμ , μ+ → e+ ν ν̄ data
There also seems the asymmetry (Fig 6.40, Fig 6.41 and Fig 6.42, Fig 6.43).
The behavior of the muon and the pion data are also similar. The diﬀerenceof both of their f wd/bwd asymmetry are small these results, the up/dwn

66

6.4 Analysis
cos_distribution
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Entries 153709

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2000

2000

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0

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sin_psiFigure 6.36: cos ψ distribution of the Figure 6.37: sin ψ distribution of themuon data with small ﬁducial volume
muon data with small ﬁducial volume
Asymmetryabcd
fiducial a

0.15

0.1

fiducial b

0.1

Asymmetry ud

Asymmetry fb

0

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Asymmetry

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Figure 6.38: F orward or backward asym- Figure 6.39: U p or down asymmetry of
metry of the muon data with small ﬁdu- the muon data with small ﬁducial volume
cial volume
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1000
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sin_psiFigure 6.40: cos ψ distribution of the pion Figure 6.41: sin ψ distribution of the pion
data with small ﬁducial volumedata with small ﬁducial volume

67

6.5 Analysis
Asymmetry

abcd
0.15

0.15

fiducial b

0.1

fiducial c

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fiducial afiducial b
fiducial cfiducial dfiducial d

0.05

0

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0

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opening angle cut

0.9
0

0.1

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

opening angle cut

Figure 6.42: F orward or backward asymFigure 6.43: U p or down asymmetry of
metry of the pion data with small ﬁducial
the pion data with small ﬁducial volume
volumeasymmetry are not so improved. The length of the vertical direction is shorter
than front-back direction, so this might be have some relation.

6.4.4

Muon incident angle

I checked the incident beam angle distributions with muons. Although we set thebeam incident parallel to z-axis, some muons yield to incide obliquely. Fig 6.44
and Fig 6.45 shows the results of the tracking reconstruction cos ψ distribution
and sin ψ distribution which contains some bad ﬁtting and the results of ﬁtting
χ2 /N DF value distribution, respectively (here, NDF ( = number of degree of
freedum) is ﬁtting paramerer(4) - 1. The time starting point t0 of muons could beknown.). There exists large χ2 /N DF value, hence I used the only good, χ2 /N DF <20 events. Then I got expecting shape of the muon incident angle distribution,
Fig 6.46. The RMS of distribution of 130 mrad, which was adequate the eﬀectof both the multiple scattering in the degrader (∼94 mrad) that is 94mrad for
170 MeV/c muons and the beam expansion (∼ 33 mrad).
And the structure of the three peaks may come from left right ambiguity of
tracking reconstruction.

6.5

Analyzing Power

When results of asymmetry from both the μSR measurement and the MTP measurement are obtained, the analyzing power, which is characteristic value of the
MTP itself, can be obtained.
The analyzing power of the MTP is deﬁned as
α = Af /b /Pμ .

(6.23)

68

6.5 Analysis

cos_distribution

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Entries 15401
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100
50

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Figure 6.44: no cut cos ψ

-0.8

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Figure 6.45: no cutsin ψ

sin_distribution
sin_distribution

4500

Entries107353Mean

0.006932

RMS

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4000
3500
3000
2500
2000
1500
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Counts/bin

6000

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Counts/bin

8000

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0.8

1

6.6 Analysis

69

where, Af /b is the f orward or backward asymmetry, and Pμ is the beam polarization.

6.6

More detailed analysis

A number of systematic eﬀects have been observed in the data analysis. For better
understanding and further research, the followings should be done:• Detailed Monte Calro simulation
By comparing the true track and the reconstruction track, we can estimate
the systematic errors.
• 3-dimensional tracking
I reconstructed 2-dimensional tracking algorithm, but in fact, muons decayin 3-dimension space hence the x-direction dependence should be known in
ﬁnal. With the charge division method, I refer precisely in Appendix B, theposition of the hit in the x-direction can be known by comparing the charge
quantity of both sides of the readout. The combination of 2-dimensional
tracking and charge division can provide mean 3-dimensional tracking.
• Eﬃciency estimation by track reconstruction• Improvement of tracking reconstruction
In addition, improvement of ASB cards design is desirable. During the beamtest,
there was a strange peak in the lower channels of the ADC spectrum due to the
problem with the ASB cards. Fortunately, we had no bad eﬀect by changing the
ADC threshold. However, for charge division study, the correlation between bothsides of readout was important, so we couldn’t use the lower channels.

Chapter 7

Conclusion
The TREK aims to measure the transverse polarization PT in Kμ3 decay withaccuracy δPT < 10−4 , using upgrated E246 detector. There are two choice for the
TREK polarimeter, a plate type or a tube type (MTP).
We had beam test in order to evaluate the performance of the MTP. We tuned
beam condition and determined the degrader thickness. The beam polarization weused had not been known, so we measured by means of μSR. We had a result thatthe beam polarization of M11 muon beam at 170 MeV had about 17% backward
polarization.
I studied the basical MTP performance. The wire eﬃciency was good enough(>99%), and the eﬃciency depended on drift distance was also high enough. I
estimated the drift velocity by ﬁtting the drift time distribution. I cross checked
the estimation by the X-T relation. Using the drift velocity, I made tracking
reconstruction algorithm considering the multiple scattering.
It is desirable that there exists few systematics in the MTP. However, I foundsome systematics in the MTP, so further examination is needed.

71

Appendix A
Beam tuing at J-PARC K1.1BR
The J-PARC (Japan Proton Accelerator Research Complex) consists of a series
of world-class proton accelerators and experimental facilities that make use of thehigh-intensity proton beams.
J-PARC K1.1BR at the Hadron hall is one of the beam channels of the J-PARC
Hadron Experimental Hall and the TREK is planned to in this area. In the TREK
experiment, we will use a separated K + beam with 0.8 GeV/c momentum from
K1.1BR. The K1.1BR beamline with a total length of 21.5 m was made as a branchbeamline of K1.1, namely the K1.1BR uses the upstream magnets (from D1 to M S)
of K1.1. The design intensity proton beam (50 GeV, 15 μA, 750 kW), guiding Tltarget, using secoundary particle for various atomic or particle experiments. The
momentum is about 740 MeV/c to 800 MeV/c, and expected beam intensity is
about 3 × 104 K + /spill when ﬁrst beam line intensity is 2kW.
We are plannning the following two precision measurements using the stopped
K + beam at K1.1BR.
• E36: Measurement of Γ(K + → e+ ν)/Γ(K + → μ+ ν) and search for heavy
sterile neutrino using the TREK detector system• E06(TREK): Measurement of T-violating Transverse Muon Polarization in
K + → π 0 μ+ ν decays
The beam optics of K1.1BR[ , ] was designed by J.Doornbos of the TREK
collaboration. In the design of the beamline, it is essential to increase the K/πratio higher than 2 and the kaon yield as much as possible in order to satisfy the
TREK requirement.

The detector elements and the beam optics for the K1.1BR beamline(Fig A.4)
are the following.
• Proton target (T 1) and ﬁrst bending magnet (D1)
• Quadrupole doublet Q1 − Q2 and intermediate vertical focus (IF Y )A–1

Figure A.2: Detectors for beam measurement

A–2

Figure A.3: π/K diﬀerential Fitch-type Čerencov counter

Figure A.4: The layout of K1.1BR beamline

A–3• Quadrupole doublet Q3 − Q4 and the electrostatic separator (ESS)• Quadrupole doublet Q5 − Q6 and mass slit (M S)• The ﬁnal bending magnet (D3)
• Quadrupole doublet Q7 − Q8 and horizontal focus (HF OC)• Sextupole magnets (SX1 and SX2) and octupole magnet(01)We conducted beam tuning from May 2012 to June 2012, and re-tuned the
K + beam for good condition. We determined that the condition of the beam andmagnets were recovered from autumn 2010, when TREK had beam tuning before
the big earthquake of March 2011. In addition, we measured excelent K + beam
acceptance which is large enough for E06 and E36 under the expect K/π ratio withwide slit. The detectors used for beam measurement were the following:• Hodoscope : beam proﬁle measurement• Fitch Cěrencov : K/π identiﬁcation (Fig A.3)
• Beam deﬁning counter : trigger• TOF1 and TOF2 : K/π identiﬁcation• Gas čerencov : positron identiﬁcation (veto for e+ )A–4

Figure A.5: Results of K + /π + separation curve using narrow slit

A–5

Appendix B
Charge division
Following is the brieﬂy description of the mechanism of the charge division method.
As shown in Fig B.1, the distance between particle hit position and the wire ends aredeﬁned as Z1 and Z2 , and the resistances corresponding to the distances are deﬁned
as R1 and R2 , respectively. In addition to these wire resistances we have to consider

Figure B:.2: The schematic view of charge division system.

the input impedance of ampliﬁes, which are deﬁned as r1 and r2 , respectively. In
this case, the equivalent circuit can be expressed as shown in Fig B.2.
The collected charge on the virtual capacitance is discharged through the resistance. The experimentally time-integrated T1 and I2 currents are observed as
the charges Q1 and Q2 , respectively. Using the capacitance C and voltage V, the
B–1

Figure B:.3: The equivalent electric circuit of the charge division system.

charge Q can be written as,Q=

V = I1 (R1 + r1 ) = I2 (R2 + r2 ),dV,(−I)dt = CV, −I = −(I1 + I2 ) = C
dt

(B.2)
(B.3)

thus,
C

V
V
1
1
dV
V,
=−
−
=−
+
dt
R1 + r1 R2 + r2
R1 + r1 R2 + r2
dV
(R1 + r1 )(R2 + r2 ) dV
= −r
V =−
C
R1 + R2 + r1 + r2 dt
dt

(B.4)

V can be obtained as
V (t) = V (0)exp(−t/τ )

(B.5)

and
Q
1
exp(−t/τ )R1 + r1 CQ
1exp(−t/τ ).
I2 =R2 + r2 C
I1 =

(B.6)

The charges of both ends wire are obtained by time-integration of I1 and I2 as
R2 + r2
1Qτ=
Q1 = (−I1 )dt =Q,
R1 + r1 C
R1 + R2 + r1 + r2

R1 + r1
Q
1
τ=
Q.
(B.7)
Q2 = (−I2 )dt =R2 + r2 C
R1 + R2 + r1 + r2
B–2

From the relation of

R1
Z1
=
,
Z2
R2

(B.8)

Q1 /Q2 can be described by using Q = Q1 + Q2 , R = R1 + R2 , and Z = Z1 + Z2 as,
Z2 /Z + r2 /R
Q1
R2 + r2
R2 /R + r2 /R
=
.
=
=
Q2
R1 + R2 + r1 + r2
1 + (r1 + r2 )/R
1 + (r1 + r2 )/R(B.9)

Therefore, the particle hit positions can be determined by the output charges fromboth wire ends as,
Z1
r1 + r2 Q1 r2
− ,
= 1+
Z2
R
Q
R
r1 + r2 Q2 r1
Z1
Z2
= 1+
− ,
=1−
Z2
Z
R
Q
R
r1 + r2 Q1 − Q2 r1 − r2
Z1 − Z2
= 1+
−
.
Z
R
Q
R

(B.10)In the coordinates where x is the distance of the particle hit position from thecenter (x = 0 is center), x is written as,
x
=
Z

1+

r1 + r2R

Q1 − Q2 r1 − r2
−
,
2Q
2R

where Z1 − Z2 = 2x.

B–3(B.11)

Appendix C
Tracking system for the TREK
detector
With the upgraded of the tracking system of TREK detector, the error from back+− dif will be improved to meet the requirement of < 10−4 for the total
groud Kπ2
systematic error in PT . These performance goals will be achieved both by reducing
the material budget along the track and by rearranging existing and adding new
tracking elements in replacement of the previous C1 chamber. The momentum
uncertainty of 3.6MeV/c in E246 can be reduced by at least a factor of 10
1. by employing a 6-cm- instead of 9.3-cm-wide target with a segmentation of
3.0×3.0mm2 ﬁbers instead of 5×5mm2
2. by replacing the air volume in the magnet between C2 and C3 and before C2chambers with helium bags
3. by increasing the distance between the C3 and C4 elements to 30cm from15cm.
+
− dif events we need to build a
For suﬃcient identiﬁcation and suppression of Kπ2cylindrical tracking chamber (”C0”) with a radius of 10cm and a spatial resolutionof <0.1mm. The new C0 chamber will replace the previous cylindrical C1 chamberof the E246 setup. In order to increase tracking redundancy we propose to add newplanar element (again named ”C1”) with <0.1mm resolution to cover each of the
12 gaps at the outer surface of the CsI(Tl) calorimeter. By adding these additional
elements to the tracks which do not have a kink along their path. In combination
with the higher segmentation of the ﬁber target this will be suﬃcient to suppress++− dif /Kμ3
ratio below 10−3 , rendering a spurious PT < 5 × 10−5 .
the Kπ2The planned modiﬁcations are in summery:

1. Thinner target with higher segmentation.
2. Helium gas bags in the magnet between C2 and C3, and before C2.
3. Increase in the distance between C3 and C4 to 30cm from 15 cm.
C–1

4. Addition of new tracking elements: ”C0” and ”C1” chambers based on GEM
technology.
Fig7 and tableC.2 show comparisons of the tracking system in E246 and the
TREK experiment. The GEM technology on which both C0 and C1 will be basedpresents a new generation of positron-sensitive counters that are reasonably cheap,
radiation hard, and well suited to be operated in high-rate environments.

Figure C.2: Schematics of the tracking system in the TREK setup (right) comparedwith that of the E246 experiment (lef t).

C–2

Table C.2: Main parameters of the charged particle tracking.
Item
E246
TREK
High-resolution elements
C2, C3, and C4
5×5mm + rings
3×3mm
Target ﬁber
MWDC
Cylindrical GEM
C0 chamber
Plannar GEM chamber
C1 chamber
MWPC
MWPC (not charged)
C2 chamber
MWPC
MWPC (not charged)
C3 chamber
MWPC
MWPC (not charged)
C4 chamber
15cm
30cm
C3-C4 distanceAir
He gas bag
Magnet gap
Total material thickness ∼ 7 × 10−3 X0 6.6 × 10−3 X0

C–3

Appendix D
Read-out system for thepolarimeter beam tests
The main data acquisition stream for the MTP follows below. First of alll, theMTP raw signals are input into an ASB card. The ASB card serves as Amplifer(A),shaper(S), and buﬀer(B). The ASB output signal passes through a ﬂat cable and is
input to a VME-discriminator board which requires about ± 5V for its operation.
The VME-discriminator board works as an ordinary discriminator, and has threekinds of outputs which are an analog signal and two logic signals, a LVDS and a
NIM signal.The analog signal and the LVDS signal are required for the CAEN ADC
and TDC moduls. On the other hand, the NIM signal is used in order to tag the
MTP signals. The trigger logic includes the NIM output signals. After the VMEdiscriminator board, the LVDS signals are input into the CAEN TDC modules.
The TDC modules are used to take both MTP data and all scintillation counters.
In terms of impedance matching for the ADC modules, a De-coupler card was
also required for our measurement. The De-coupler cards are connected directly
to the CAEN ADC modules for the adoption of the charge division method. The
important parts of the data acquisition system are summarized in next sections.

16ch Amp. Shaper Buﬀer card
Special ASB IC (JRK-ASB(N)-1) chips were developed by the KEK electronics
shop and the REPIC Co. The standard Amp. Shaper Discriminator (ASD) card
is unsuited for thecharge division method. Basically, an analog input signal froman anode wire is integrated for all 16 channels on one board with a 25 ns time
constant. In addition, the output signal pulse height can be changed using the gain
controller.The ASB has a small input impedance, thus, it is possible to adopt it to
the charge division method. The main functions are summarized in Table D.2.

D–1

Table D.2: The main functions of ASB
ASB ICJRK-ASB(N)-001 × 4 pcs
Analog input
negative charge input / 16 ch input
Maximum quantity of charge 1.5 pC with minimum gain
Shaping time constant
about 25 ns
Test input
negative NIM 50 Ω impedance
Analog output- 3.75 V ∼ - 1.25 V
Vth control
 + 2.5 V
Digital output
ECL output
Consumption of electricity+ 5 V 500 mA / - 5 V 300 mA

VME discriminator board MODEL RPV-240
The VME discriminator boards have not only analog outputs but also LVDS outputs. The former signals are used for the ADC and the latter ones are used for the
TDC. The discriminator modules have a function of adjusting the threshold level,
it is necessary to provide ia DC signal level (≤ 5 V) from an external DC power
supply. In addition, this modules produces a NIM signal, which is generated as an
OR output even if only one of the 16 channels has a hit. The main speciﬁcations
are summarized in Table D.3.

Table D.3: The specs of VME discriminator board
INPUT
16 CH Analog INPUT
Vth INPUT
Maximum INPUT 5 V
Analog OUTPUT 16 CH Analog OUTPUT
Digital OUTPUT
16 CH Digital OUTPUT (LVDS 110 Ω)
OR OUTPUT
16 CH OR OUTPUT (Negative NIM 50 Ω impedance)
POWER SUPPLY + 5 V / 1.1 A

De-coupler card
The De-coupler card works as an attenuator. In order to take ADC data, the output
signals must have 3 times lower pulse height than the signal input by the VME
discriminator when it exceeds 1V. Actually, the quantity of the integrated charge
is limited to the CAEN ADC’s dynamic range. So as to achieve a higher position
resolution along the wire direction, it is necessary to gain a lot of charge before thepulse shaping by the ASB card. At the same time, the quantity of charge has to be
reduced to be able to take ADC data. Furthermore, the De-coupler card also works
as a 200 ns delay module. As the trigger timing is always delayed by more than 100
ns compared to the time reference, delaying the analog signals is also necessary for
D–2

ADC data taking. As mentioned above, the De-coupler card plays an importantrole for the analog signals.

CAEN QDC V792 / CAEN TDC V1190
The CAEN ADC V792 and CAEN TDC V1190 modules were used for the dataacquisition. An ADC modules has 32 channels and provides the time data for the
charge division method, and a TDC module has 64 channels and provides data for
drift time measurement.In case a wire is hit two or more times on one event, a
multi-hit TDC modules is required so that one channel can take more than 2 data
hits for each event. This means that both the muon and the positron can pass near
the same wire. This requirement is met by using the CAEN TDC V1190. The ADC
is not multi-hit, thus we could take just the positron ADC data which was essential
to measure the positron along the wire using the charge division method. Insteada FADC will be used for the MTP data acquisition in the TREK experiment, so
it will be possible to collect both ADC and TDC data for both the muons and thepositrons.

D–3

Acknowledgements
I would like to express my sincere thankfulness to my supervisor, Prof. Tatsuo
Kawamoto for continuously encouraging and supported the whole of my study. He
gave so many helpful advice, and a lot of discussions with him including daily
conversations greatly attracted me to particle physics so much. I would like toexpress my gratitude to Prof. J. Imazato, who gave me so many chances to learn a
lot about physics through this exciting experiments. I have acquired an attitude toa physicist from him. I would like to thanks Prof. S. Asai, who allowed me to join
this experiment. I deeply appreciate Dr. T. Namba for his valuable suggestions andsupport. I am extremely grateful for Prof. M. Hasinoﬀ for carefully reading thisthesis, and also for teaching me to how to tune the beam at TRIUMF. His vivid
personality always get rid of my fatigue during the hard beam time. I would liketo express my gratitude to Dr. Y. Igarashi, who helped me when I started launch
operation of MTP and always worked with me in KEK, TRIUMF and J-PARC.
I learn a lot from him about hardware and software technique. I am thankful tohis kindness and his sense of humor in the working. I would like to express mygratitude to Dr. H. Yamazaki, who helped me with advice when I face problems,and I amazed a his wealth of knowledge about physics. I would like to express mygratitude to Dr. S. Shimizu, who suggested me the strategy of the analysis, and I
learned the physicist mind from his passion for physics experiments. I would like toexpress my gratitude to Dr. K. Horie, who also supported my study, and I admire
her clarify statement in the meeting. I would like to express my gratitude to Dr.
A. Toyoda, whose one of work of beam test for the development of the aerogelčerencov counter I had join, and I learned serious attitude as a physicist. I would
like to thank to Dr. M. Uchida for his advice operating the MTP. I appreciate thecontribution all the TREK collaborators to perform the beam test and support mystudy. I would like to express my gratitude to predecessor Mr. K. Yoshihara, whosuggested me to the importance of learn a lot about detectors and experiments
and showed me to this amazing experience. He always give me valuable advice
before I entered the graduate school, and helped when I stumbled on a diﬃculty.
I would like to thank all the ICEPP people, staﬀ including CERN in our group,
secretaries, elderly students and younger students. Especially, the support from mycolleagues, M. Moinaga, R. Kurosaki, K. Owada, T. Inada, W. Choi, I. Otani, T.
Yamada, N. Higashi, M. Tani, K. Arima helped me to live the enjoyable studying
life. I would like to thank the people and students in the KEK also supported meto enjoy studying, and encouraged me to write this thesis. I would like to thankthe people I met at the TRIUMF and J-PARC also encouraged me to studying.
Finally, I deeply appreciate all of my important people, friends and my family forcontinuous support and encouragement.

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