Measurement of the Radiative Muon Decay as a Test of the V − A
Structure of the Weak Interactions
Emmanuel Munyangabe
Kigali, Rwanda
B.S., National University of Rwanda, 2006
A Dissertation presented to the Graduate Faculty
of the University of Virginia
in Candidacy for the Degree of
Doctor of Philosophy
Department of Physics
University of Virginia
June, 2012
Abstract
Measurements of the radiative muon decay can be used to test the validity of the
V − A form of weak interactions. All the Michel parameters can be extracted from
the analysis of the ordinary muon decay µ+ → e+ νe ν¯µ , with the exception of the
η̄ parameter which is measured by analyzing the radiative decay µ+ → e+ νe ν¯µ γ.
This analysis is based on more than 5.1 × 105 radiative muon decays recorded by
the PIBETA experiment at the Paul Scherrer Institute (PSI), Switzerland in 2004.
Based on these events, the experimental branching fraction was measured to be B =
[4.365 ± 0.009 (stat.) ± 0.042 (syst.)] × 10−3 . The η̄ parameter was extracted using
the least squares method and the experimental value was found to be η̄ = 0.006 ±
0.017 (stat.) ± 0.018 (syst.). This result is to be compared to the V − A Standard
Model value η̄SM = 0. Our experimental result of η̄ gives an upper limit of: η̄ ≤
0.028 (68.3 % confidence), a fourfold improvement in precision over the existing world
average.
Contents
1 Introduction
1.1 The Standard Model . . . . . . . . . . . . . . .
1.1.1 Weak Interactions . . . . . . . . . . . . .
1.1.2 V-A Theory . . . . . . . . . . . . . . . .
1.2 Muon decay . . . . . . . . . . . . . . . . . . . .
1.2.1 Michel Decay: µ+ → e+ νe ν µ (γ) . . . . .
1.2.2 Radiative Michel Decay: µ+ → e+ νe ν µ γ
1.2.3 Motivation . . . . . . . . . . . . . . . . .
2 PIBETA Detector
2.1 Introduction . . . . . . . . .
2.2 Experimental area . . . . .
2.3 Beam-defining elements . . .
2.3.1 Passive Collimator .
2.3.2 Active beam-defining
2.4 Detector Tracking System .
2.5 Calorimeter . . . . . . . . .
2.6 Cosmic muon veto detectors
2.7 PIBETA Triggering System
2.7.1 Introduction . . . . .
2.7.2 Random Triggers . .
2.7.3 Beam Triggers . . . .
2.7.4 Calorimeter Triggers
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elements
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3 Event Reconstruction
3.1 Introduction . . . . . . . . . . . .
3.2 Data Analysis Software . . . . . .
3.2.1 Tracking System algorithm
3.2.2 Clump Algorithm . . . . .
3.3 Particle Identification . . . . . . .
3.4 Monte Carlo Simulation . . . . .
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1
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ii
3.5
Data Calibration . . . . . . . . . . . . . . . .
3.5.1 Introduction . . . . . . . . . . . . . . .
3.5.2 Energy Calibration (ADC Calibration)
3.5.3 Time Calibration (TDC Calibration) .
4 Data Analysis
4.1 Branching Fraction . . . . . . . . . .
4.1.1 Introduction . . . . . . . . . .
4.1.2 Muon decay time distribution
4.1.3 Non-Radiative Muon Decay .
4.1.4 Radiative Michel decay . . . .
4.1.5 Background Signal . . . . . .
4.1.6 Kinematic cuts . . . . . . . .
4.1.7 Systematic errors . . . . . . .
4.1.8 Results (branching fraction) .
4.2 Extraction of η̄ and ρ Parameters . .
4.2.1 Systematic error . . . . . . .
4.3 Conclusions . . . . . . . . . . . . . .
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51
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98
A The functions fi (x, y, θ)
100
B The Least Squares Method
103
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
Elementary particles and
Beta decay . . . . . . . .
f1 (x, y, θ ≥ 45◦ ) . . . . .
f1 (x, y, θ ≥ 90◦ ) . . . . .
f1 (x, y, θ ≥ 135◦ ) . . . .
f1 (x, y, θ ≥ 157◦ ) . . . .
(f2 /f1 )(x, y, θ ≥ 45◦ ) . .
(f2 /f1 )(x, y, θ ≥ 90◦ ) . .
(f2 /f1 )(x, y, θ ≥ 135◦ ) . .
(f2 /f1 )(x, y, θ ≥ 157◦ ) . .
(f3 /f1 )(x, y, θ ≥ 45◦ ) . .
(f3 /f1 )(x, y, θ ≥ 90◦ ) . .
(f3 /f1 )(x, y, θ ≥ 135◦ ) . .
(f3 /f1 )(x, y, θ ≥ 157◦ ) . .
gauge bosons
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4
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2.1
2.2
2.3
Detector cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decay chain signal . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The pion stop signal . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
32
37
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
The primary vertex x0 , y0 , z0 . . . . . . . . . . . . . . . . . . . . . . .
CsI-MWPC angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Positron (Michel) θ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Positron Michel) φ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Positron θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Photon φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Photon θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Positron φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Positron energy in CsI calorimeter for 1-arm low trigger events. . . .
Positron energy in plastic veto (PV) for 1-arm low trigger events. . .
Walk-correction effect on the energy and time of PV. The straight
horizontal line indicate the removal of energy-time dependence in the
observed TDC values. . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
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iii
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55
iv
3.12 Walk-correction effect on the energy and time of CsI calorimeter. The
straight horizontal line indicate the removal of energy-time dependence
in the observed TDC values in the energy range of 10-53 MeV. . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
4.24
4.25
4.26
Time coincidence of positron and photon in the CsI calorimeter. . . .
The time distribution of Michel events . . . . . . . . . . . . . . . . .
The time distribution of radiative Michel events. This is measured as
+
the time difference ( 21 (teCsI + tγCsI ) − tDeg ). . . . . . . . . . . . . . . . .
The time ratio before selection . . . . . . . . . . . . . . . . . . . . . .
The time ratio after selection . . . . . . . . . . . . . . . . . . . . . .
The positron energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
The photon energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The opening angle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The positron energy cut stability . . . . . . . . . . . . . . . . . . . .
The photon energy cut stability . . . . . . . . . . . . . . . . . . . . .
The signal window cut stability . . . . . . . . . . . . . . . . . . . . .
The opening angle cut stability . . . . . . . . . . . . . . . . . . . . .
The χ2 as a function of ρ from analysis of non-radiative Michel decays
(f2 /f1 )(x, y, 150◦ < θ ≤ 160◦ ) . . . . . . . . . . . . . . . . . . . . . .
(f2 /f1 )(x, y, 160◦ < θ ≤ 170◦ ) . . . . . . . . . . . . . . . . . . . . . .
(f2 /f1 )(x, y, 170◦ < θ ≤ 180◦ ) . . . . . . . . . . . . . . . . . . . . . .
(f2 /f1 )(x, y, 157◦ < θ ≤ 180◦ ) . . . . . . . . . . . . . . . . . . . . . .
The χ2 as a function of photon energy scale and η̄ for 170◦ < θ ≤ 180◦
bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The χ2 as a function of η̄ for 170◦ < θ ≤ 180◦ bin, with Cγ = 0.994. .
The χ2 as a function of photon energy scale and η̄ for 160◦ < θ ≤ 170◦
bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The χ2 as a function of η̄ for 160◦ < θ ≤ 170◦ bin, with Cγ = 0.994. .
The χ2 as a function of photon energy scale and η̄ for 160◦ < θ ≤ 180◦
bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The χ2 as a function of η̄ for 160◦ < θ ≤ 180◦ bin, with Cγ = 0.994. .
The χ2 as a function of photon energy scale and η̄ for 150◦ < θ ≤ 160◦
bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The χ2 as a function of η̄ for 150◦ < θ ≤ 160◦ bin, with Cγ = 0.997. .
The χ2 as a function of photon energy scale for 160◦ < θ ≤ 180◦ bin,
with η̄ = 0.006.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
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93
List of Tables
1.1
1.2
1.3
Range and lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weak isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Muon decay modes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
12
2.1
Trigger pre-scaling factors . . . . . . . . . . . . . . . . . . . . . . . .
39
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
Pibeta time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michel cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Event generator level cuts. . . . . . . . . . . . . . . . . . . . . . . .
Event reconstruction cuts . . . . . . . . . . . . . . . . . . . . . . .
Systematic errors (largest accessible phase space . . . . . . . . . . .
Experimental branching fraction . . . . . . . . . . . . . . . . . . . .
Values of η̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Systematic errors of the region 0.5 < x ≤ 0.75, 0.28 < y ≤ 0.48 . . .
Systematic errors of the region 0.5 < x ≤ 0.75 and 0.5 < y ≤ 0.75 .
Systematic errors of the region 0.28 < x ≤ 0.48 and 0.5 < y ≤ 0.75 .
Systematic errors of the region 0.28 < x ≤ 0.48 and 0.28 < y ≤ 0.48
Statistics for bin 170◦ < θ ≤ 180◦ . . . . . . . . . . . . . . . . . . .
Statistics for bin 160◦ < θ ≤ 170◦ . . . . . . . . . . . . . . . . . . .
Statistics for bin 160◦ < θ ≤ 180◦ . . . . . . . . . . . . . . . . . . .
Statistics for bin 150◦ < θ ≤ 160◦ . . . . . . . . . . . . . . . . . . .
Systematic errors η̄ parameter . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
Introduction
1.1
The Standard Model
The Standard Model of particle physics [1] is a theory that describes the behavior of
subatomic particles. This behavior is manifested by the interactions between these
particles. So far, there are four known fundamental interactions; electromagnetic,
strong, weak and gravity. The Standard Model theory was developed throughout the
mid to late 20th century, the current formulation was finalized in the mid-1970s upon
experimental confirmation of the existence of quarks and the intermediate bosons (W
and Z). Because of its success in explaining a wide variety of experimental results,
the theory is sometimes regarded as a theory of almost everything.
Yet, the theory fails to incorporate the physics of general relativity, such as gravitation and dark energy or dark matter particles that are deduced from cosmological
1
Chapter 1: Introduction
2
evidence. Nevertheless, the Standard Model theory is an important tool to explain
particle physics.
The Standard Model includes 12 elementary particles of spin-1/2 known as fermions
and their corresponding anti-particles. The fermions of the Standard Model are classified according to how they interact. Some interact strongly, weakly, and electromagnetically (quarks), while others never interact via strong interactions (leptons).
There are six quarks (up, down, charm, strange, top, and bottom) and six leptons
(electron, electron neutrino, muon, muon neutrino, tau, tau neutrino), grouped into
three generations as shown in Fig 1.1.
The main difference between quarks and leptons is that quarks carry strong color
charge, therefore, only quarks can interact via strong interaction. Quarks bound to
one another form color-neutral hadrons. Hadrons are of two types: mesons (composed of a valence quark and anti-quark) and baryons (composed of three valence
quarks). Quarks can also interact with other fermions via electromagnetic and weak
interaction, because they carry electric charge and weak isospin, respectively.
Leptons and their corresponding neutrinos do not cary color charge and hence do
not interact strongly. Furthermore, the three neutrino flavors and their corresponding
anti-neutrinos cannot interact electromagnetically as they do not carry electric charge
and hence their interaction is only through the weak force. However, charged leptons
interact both electromagnetically and weakly.
Each member of a generation has greater mass than the corresponding particles
Chapter 1: Introduction
3
of lower generations. The first generation of charged particles (up, down quarks
and electron) being the lightest particles, are kinematically forbiden to decay into
other particles. This is the reason why all ordinary (baryonic) matter is made of such
particles. Specifically, all atoms consist of electrons orbiting atomic nuclei, ultimately
constituted of up and down quarks. Second and third generation charged particles,
on the other hand, decay with very short half-lives, and are observed only in very
high-energy collisions. Neutrinos of all generations also do not decay, and even though
they pervade the whole universe, they are hard to detect because they rarely interact
with baryonic matter. The Figure 1.1 shows the elementary particles and the bosons
that mediate the interactions.
1.1.1
Weak Interactions
This work focuses on testing the V − A structure of weak interactions [2], therefore
I will give an overview of weak interactions and give a brief theory about V − A
structure of weak interactions.
Firstly, weak interactions affect all known fermions; quarks and leptons interact
weakly by the exchange (emission or absorption) of W ± and Z 0 bosons. There are two
types of weak interactions: charged and neutral interactions. W ± bosons participate
in the charged interactions while Z 0 bosons participate in the neutral interactions.
The best known example of this weak force is the neutron beta decay, where an
electron, proton and a neutrino are emitted as shown in Figure 1.2
Chapter 1: Introduction
4
Figure 1.1: Elementary particles and the gauge bosons that mediate
the interactions between them
The Z and W bosons masses are very heavy mW ∼ 80 GeV and mZ 0 ∼ 90 GeV;
this large mass accounts for the very short range of the weak interaction and long
lifetimes for all particles decaying via weak interactions. The range of interaction is
inversely proportional to the mass of the gauge boson that mediates the interaction,
while the life time is inversely proportional to the decay rate. The relative decay time
for strong, electromagnetic and weak interactions is shown in Table 1.1
Chapter 1: Introduction
5
The weak interaction has another unique property − namely quark flavor changing where quarks can swap their flavors (i.e, changing from one type of quark into
another). In addition, it is the only interaction that violates parity (P)-symmetry.
Weak interactions introduce a term called weak isospin (T3 ) which is a property
(quantum number) of all particles that governs how particles interact in the weak
interaction. As a comparison, weak isospin is to the weak interaction what electric
charge is to the electromagnetism, and to what color charge is to the strong interaction. Elementary particles have weak isospin values ± 21 . For example, up-type
quarks (u, c, t) have T3 = + 12 and always transform into down-type quarks (d, s, b),
which have T3 = − 12 . During this transformation, a quark never decays weakly into
a quark of the same T3 . As is the case with electric charge, these two possible values
are equal, except for sign, as shown in Table 1.2.
Figure 1.2: Feynman diagram for beta decay of a neutron into a proton,
electron and electron anti-neutrino, via an intermediate W gauge boson.
Chapter 1: Introduction
6
Table 1.1: Comparison of interactions in terms of energy scales, range
and lifetimes. The strong interaction is asymptotically free as quarks
are essentially free to move when very close to each other and essentially
impossible to separate as they move apart.
Interaction
Mediator
Strength
Range
Lifetime(s)
EM
photons(γ)
1/137
long
10−20
Weak
W ±, Z
10−5
short
10−8
Strong
Gluons(G)
1
short
10−23
Gravity
gravitons(g)
10−41
long
long
Table 1.2: Weak isospin for fermions. In any decay through a weak
interaction, weak isospin is conserved: the sum of the weak isospin
numbers of the particles exiting a reaction equals the sum of the weak
isospin numbers of the particles entering that reaction
Weak Isospin
First Generation
Second Generation
Third Generation
+ 12
Up
Charm
Top
− 12
Down
Strange
Bottom
− 12
Electron
Muon
Tau
+ 12
Electron neutrino
Muon Neutrino
Tau Neutrino
1.1.2
V-A Theory
Parity Violation
In physics, a parity transformation (also called parity inversion) is the flip of the sign
of spatial coordinates. The parity operator P̂ corresponds to a discrete transformation
x → −x, y → −y etc.
Chapter 1: Introduction
7
Under the parity transformation, vectors change sign while axial-vectors are un~ is an axial-vector then:
changed. For example, if ~r is a vector and L
~ =L
~
P̂ ~r = −~r and P̂ L
For a very long time, parity symmetry was considered to be conserved in all
particle interactions before experimental evidence showed that weak interaction do
not conserve parity. This was demonstrated by C.S.Wu in 1957 [3] in an experiment
on Beta decay of Cobalt-60 nuclei:
60
Co →60 Ni∗ + e− + ν̄e
(1.1)
She observed that electrons were emitted preferentially in the direction opposite to
an applied magnetic field. This was contrary to parity conservation, because if parity
was conserved, then equal number of electrons would be emitted in the directions
along and opposite to the applied magnetic field. This demonstrated that parity is
violated in the weak interaction.
V − A Structure of the Weak Interaction
The most general matrix element for the decay that embodies kinematics and spindependent interactions [4] is given by:
M=
X
γ
γ
gαβ
heα |Ôγ |(νe )ih(νµ )|Ôγ |µβ i.
(1.2)
Chapter 1: Introduction
8
There are only 5 possible combinations of two spinors and the gamma matrices that
are Lorentz invariant, called bilinear covariants:
Scalar: ΨΨ (1 component)
Pseudo-Scalar: Ψγ 5 Ψ (1 component)
Vector: Ψγ µ Ψ (4 components)
Axial-Vector: Ψγ µ γ 5 Ψ (4 components)
Tensor: Ψ(γ µ γ ν − γ ν γ µ )Ψ (6 components)
In total we get 16 elements of a general 4 × 4 matrix which correspond to the
decomposition into Lorentz invariant combinations. The most general form of the
interaction between a fermion and a boson is a linear combination of bilinear covariants. From experimental evidence, the form for weak interaction is found to be Vector
minus Vector-Axial (V − A).
The parity violation of weak interactions is cause by the helicity structure of the
weak interaction. The charged current(W ± ) weak vertex is given by:
−igw 1 µ
√
γ (1 − γ 5 ),
2 2
(1.3)
1 µ
γ (1 − γ 5 ),
2
(1.4)
and since
Chapter 1: Introduction
9
projects out left-handed chiral particle states then:
1
Ψ γ µ (1 − γ 5 )Ψ = Ψγ µ ΨL ,
2
(1.5)
Ψ = ΨR + ΨL
(1.6)
ΨR γ µ ΨL = 0,
(1.7)
1
Ψ γ µ (1 − γ 5 )Ψ = ΨL γ µ ΨL ,
2
(1.8)
and by writing
and as we know from QED [4] that
hence
which means that only the left-handed chiral components of particle spinors and righthanded chiral components of anti-particle spinors participate in charged current weak
interactions. Moreover, at very high energy (E m), the left-handed chiral components are helicity eigenstates, and this property gives the added information that in
the ultra-relativistic limit only left-handed particles and right-handed anti-particles
participate in charged current weak interactions. It is this helicity dependence of the
weak interaction that results in the parity violation.
The decays of pions provide a good demonstration of the role of helicity in the
Chapter 1: Introduction
10
weak interaction. Experimentally, we observe that:
Γ(π − → e− ν̄e )
= 1.23 × 10−4 ,
Γ(π − → µ− ν µ )
(1.9)
which is contrary to what is expected, i.e., we should expect the decay to electrons to
dominate due to increased phase space. In fact the opposite happens because electron
decay channel is helicity suppressed.
This observation can be explained as follows: the pion’s spin being zero, the spins
of muon and neutrino are opposite to each other and as the weak interaction only
couples to right-handed chiral anti-particles and since neutrinos are almost massless,
then they must be in right-hand helicity state. Momentum conservation forces the
muon to be emitted in a right-hand helicity state.
From the general right-handed helicity solution to the Dirac equation [4]:
→
→
1
|−
p|
1
|−
p|
)uR + (1 −
)uL
u ↑= PR u ↑ +PL u ↑= (1 +
2
E+m
2
E+m
(1.10)
and as the right-hand chiral and helicity states are identical in the limit E m,
this means, even though only left-handed chiral particles participate in the weak
interaction, still the contribution from right-hand helicity states is not necessarily
zero. Therefore, one should expect the matrix element to be proportional to left-
Chapter 1: Introduction
11
hand chiral component of right-hand helicity electron/muon spinor, that is
→
1
|−
p|
m
Mf i ∝ (1 −
)=
.
2
E+m
mπ + m
(1.11)
Since the electron mass is much smaller than the pion mass the decay
π − → e− ν̄e ,
(1.12)
is heavily suppressed.
1.2
Muon decay
All elementary particles interact through weak interactions, but we chose muons to
test the V − A structure of weak interaction, because muons do not interact strongly
while hadrons, which are composed of quarks, are harder to test for weak interactions
as they also interact strongly. Strong interaction effects are generally harder to calculate. Even though the electromagnetic interaction is present in the decay of muons,
the effects of this additional interaction are well known, and appropriate corrections
are applied and hence can be separated in the experimental results. The decays of
the muon, being one of the few purely leptonic weak decays, is a an important tool
for studying properties of weak interactions. Table 1.3 lists the decays and their
branching fractions.
Chapter 1: Introduction
12
Table 1.3: Muon decay modes and the corresponding branching fractions
Decay
µ+ → e+ νe νµ (γ)
µ+ → e+ νe νµ γ(Eγ > 10M eV )
µ+ → e+ νe νµ e+ e−
µ+ → e+ γ
Branching Ratio
≈ 100%
(1.4 ± 0.4)%
(3.4 ± 0.4) × 10−5
< 1.2 × 10−11
Muons decay into an electron and the corresponding neutrinos and sometimes the
decay is also accompanied by a photon, µ+ → e+ νe ν̄µ (γ). The emission of photon
is most often an emission of a soft photon where the energy of the photon is below
a certain threshold and therefore is not distinguished by the experiment from the
normal decay which has no photon emission. However, there is also a probability for
the emission of a hard photon. Hard photon emission involves photons emitted at a
large angle with respect to the positron and with energy of order of MeV or more. We
distinguish two types of decays: Michel decay (photon of any energy) and radiative
Michel decay (hard photon emission decay).
The Hamiltonian describing muon decay requires Lorentz invariance and leptonnumber conservation. The general form of this Hamiltonian has contributions from
scalar, pseudo-scalar, tensor, vector and axial-vector transformation properties. Although experimental evidence of muon decay is consistent with V − A terms, there is
still a probability of smaller contributions from scalar, pseudo-scalar or tensor transformations.
Chapter 1: Introduction
13
The matrix element for the muon decay [2] is given by:
GF X γ
M = 4√
gαβ [eα |Ôγ |(νe )][(νµ )|Ôγ |µβ ]
2 γ
(1.13)
where GF is the Fermi coupling constant while α and β denote left or right handedness
of the positron and muon respectively. The label γ denote the allowed bilinear covariγ
ants: Scalar, Vector, Pseudo-Scalar, Axial-Vector and Tensor and gαβ
are coupling
constants.
1.2.1
Michel Decay: µ+ → e+ νe ν µ (γ)
The differential decay rate for Michel decay calculated from the matrix elements given
by (1.13) is:
mµ 4 2
d2 Γ
= 3 Weµ
GF
dxd(cos(θ))
4π
q
x2 − x20 (FIS (x) + Pµ+ cos θFAS (x))(1 + P̃e+ (x, θ) · ζ̂)
(1.14)
where Weµ = (m2µ + m2e )/2mµ , x = Ee+ /Weµ and x0 = me /Weµ , mµ and me are the
masses of muon and electron respectively, while Ee+ is the energy of the positron.
The allowed range of the positron energy is x0 ≤ x ≤ 1. The variable θ is the angle
between the muon polarization P~µ and the positron momentum; ζ̂ is the unit vector in
the direction of the positron spin polarization with respect to an arbitrary direction.
P~e+ is the polarization of positron along the direction of its momentum.
Chapter 1: Introduction
14
The functions FIS and FAS are expressed as:
2
FIS (x) = x(1 − x) + ρ(4x2 − 3x − x20 ) + ηx0 (1 − x),
9
(1.15)
and
1
FAS (x) = ξ
3
q
2
x2 − x20 (1 − x + δ[4x − 3 + ( 1 − x20 − 1)]).
3
q
(1.16)
The parameters ρ, η, ξ and δ are known as Michel parameters [5].
The expression for the differential decay can be simplified in the case of no polarizations and becomes:
dΓ
mµ 4 2
= 3 Weµ
GF
dx
4π
2
x2 − x20 [x(1 − x) + ρ(4x2 − 3x − x20 )]
9
q
(1.17)
In this unpolarized decay, the decay rate is dependent on only Michel parameter
ρ. The Standard Model predicted value of ρ is :
ρSM =
3
4
(1.18)
which is consistent to experimental value from Particle Data Group [6]:
ρ = 0.7503 ± 0.0004
(1.19)
Chapter 1: Introduction
1.2.2
15
Radiative Michel Decay: µ+ → e+ νe ν µ γ
Experimentaly, the radiative muon decay is characterized by detection of a photon
accompanied by a positron. Photons being massless particles, introduces infrared
divergences and collinearity.
Infrared divergences originate from massless particles with a vanishing momentum in the small energy soft limit. Physical states like, for example, a single charged
particle, are degenerate with states made by the same particle accompanied by soft
photons. This corresponds to the impossibility of distinguishing a charged particle
from the one accompanied by soft photons due to the finite resolution of any experimental apparatus. An infrared divergence appears in QED when the energy of the
photon goes to zero as a factor of the form
Z
I=
0
1
d 0
(1.20)
where = Eγ /E is the fraction of the energy of the photon with respect to the total
available energy E for the process. On the other hand, collinearity, instead, comes
from photons having a vanishing value of the relative emission angle between photons
and positrons. Any experimental apparatus with finite angular resolution, cannot
distinguish between them. The above mentioned restrictions make it impossible to
measure total branching fraction for radiative Michel decay. Correction for the above
measurement restrictions, infrared and collinearity, is accomplished by setting a lower
Chapter 1: Introduction
16
energy cut-off on photon energy Eγ , and a lower cut-off on the opening angle between
positron and photon respectively. The spectrum of the radiative muon decay has
been calculated by several authors [7, 8]. The diffential branching fraction of radiative
muon decay after integrating over positron, photon and muon polarization is given
by:
d3 B(x, y, θ)
4
= f1 (x, y, θ) + η̄f2 (x, y, θ) + (1 − ρ)f3 (x, y, θ),
dxdy2πd(cos θ)
3
(1.21)
where
x = 2Ee+ /mµ ,
y = 2Eγ /mµ ,
cos θ = p̂e+ · p̂γ ,
and fi (i = 1, 2, 3) are polynomials in x, y and ∆ = 1−β cos θ with β = |p̂e+ |/Ee+ . The
definitions of fi are given in the Appendix A. Energy and momentum conservation
requires:
∆≥
2(x + y − 1)
.
xy
(1.22)
Energy and angular distributions of the radiative muon decay are sensitive to the
parameters ρ, and η̄. These muon decay parameters are all related to the coupling
γ
constants gαβ
[9]. This additional parameter, η̄, that is only determined from radia-
tive Michel decay, is sensitive to deviation from a V − A structure.
Chapter 1: Introduction
17
The nominal Standard Model value of η̄ is:
η̄SM = 0,
(1.23)
which is consistent with existing experimental measurements [9, 10]. Any substantial deviation from the η̄SM value, would imply deviation from a pure V − A weak
interaction. The main goal of this work is to extract experimental value of η̄ and also
to measure the branching fraction of radiative muon in the largest possible region of
phase space, keeping in mind restrictions from infrared divergence and the collinearity
angle between photon and positron. The differential branching fraction (1.21) implies
that when η̄ and ρ have Standard Model values, only f1 is involved in the calculations
of branching fraction. Figures (1.3, 1.4, 1.5 and 1.6) show f1 spectra against x, y and
various opening angles and it is clear that the f1 term takes on larger values when a
photon is emitted at small angles with respect to the positron.
The sensitive kinematic phase space regions for η̄ and ρ are obtained from the
region where |f2 /f1 | and |f3 /f1 | are at a maximum, respectively. Figures (1.7, 1.8,
1.9 and 1.10) show that the value of |f2 /f1 | increases as the opening angle increases.
Therefore the region of large angles is used to extract the η̄ parameter. Figures (1.11,
1.12, 1.13 and 1.14) show that the phase space region sensitive to ρ parameter involves
the smaller angles, since |f3 /f1 | decreases as the opening angle increases. In this
analysis the angles greater than 150◦ are of interest in the extraction of η̄ parameter.
Chapter 1: Introduction
18
y
f 1(x,y, θ > 45)
1
0.9
40
8.5
35
0.8
30
0.7
0.6
22.8425
22.5377
25
0.5
0.4
20
29.0038
33.5316
39.5568
15
0.3
10
0.2
0.1
00
31.8571
0.1
39.0735
0.2
0.3
0.4
40.86
0.5
0.6
41.5
0.7
0.8
0.9
5
1
x
0
Figure 1.3: The contours of constant f1 using events passing θ ≥ 45◦ ,
as a function of photon energy(y) and positron energy(x).
y
f 1(x,y, θ > 90)
1
0.9
14
2.86
3.7
12
0.8
0.7
0.6
10
4.78333
4.767
5.5
8
0.5
0.4
6.20423
7.26471
9.06782
6
10
0.3
4
0.2
0.1
00
8.5
0.1
11.093
0.2
0.3
0.4
13.3448
0.5
0.6
2
14.9286
0.7
0.8
0.9
1
x
0
Figure 1.4: The contours of constant f1 using events passing θ ≥ 90◦ ,
as a function of photon energy(y) and positron energy(x).
Chapter 1: Introduction
19
y
f 1(x,y, θ > 135)
1
2.5
0.9
1.41667
1.08971
0.864583
0.8
2
0.7
0.6
1.67118
1.43259
1.14375
1.025
1.5
0.5
0.4
1.80789
1.83985
1.8172
1.86328
1.98077
2.21296
2.58462
2.61389
1
0.3
0.2
0.1
00
0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
0
Figure 1.5: The contours of f1 using events passing θ ≥ 135◦ , as a
function of photon energy(y) and positron energy(x).
y
f 1(x,y, θ >157)
1
0.9
1.06696
0.749449
0.391807
0.175
1.25941
0.939538
0.545927
0.361842
1.17045
1.01033
0.734416
0.514151
1.01293
1.0625
0.881329
0.714286
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1.6: The contours of f1 using events passing θ ≥ 157◦ , as a
function of photon energy(y) and positron energy(x).
Chapter 1: Introduction
20
y
|f2/f1|: θ > 45
1
0.9
0.104547
0.156699
0.284024
0.400504
0.11826
0.153733
0.247799
0.339219
0.127478
0.136904
0.146305
0.180819
0.116267
0.118605
0.111455
0.105378
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1.7: The contours of (f2 /f1 ) using events passing θ ≥ 45◦ , as a
function of photon energy(y) and positron energy(x).
1.2.3
Motivation
The goals of this analysis are to measure the branching fraction of the decay µ+ → e+ νe ν µ γ
using the largest accessible region of phase space, and to determine the η̄ parameter.
Similar analysis was done by Brent VanDevender in 2006 [10], however during
that analysis, data calibration had not been fully optimized. Therefore, there was a
need to do an improved analysis with a better data calibration.
Additionally, this analysis revises the selection of phase space region used for the
extraction of the η̄ parameter. In this analysis, events in the region of very large
Chapter 1: Introduction
21
y
|f2/f1|: θ > 90
1
0.9
0.106588
0.157414
0.284012
0.400504
0.132817
0.168727
0.24913
0.339219
0.150603
0.163308
0.159698
0.181496
0.130884
0.134562
0.121676
0.10767
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1.8: The contours of (f2 /f1 ) using events passing θ ≥ 90◦ , as a
function of photon energy(y) and positron energy(x).
opening angle (θ > 150◦ ) were used while in the previous analysis [10], a broader
opening angle (θ > 90◦ ) cut was used. However, in both analyses, the measurement
of experimental branching applied similar region of phase space (Eγ > 10 MeV and
θ > 30◦ ).
An overview of PIBETA detector is described in Chapter 2. The details of data
calibration and event reconstruction are given in Chapter 3, while the analysis strategy and results are presented in Chapter 4.
Chapter 1: Introduction
22
y
|f2/f1|: θ > 135
1
0.9
0.119453
0.167769
0.283952
0.400729
0.19549
0.25334
0.277725
0.339706
0.239667
0.29627
0.240239
0.201652
0.197774
0.218363
0.17737
0.124986
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1.9: The contours of (f2 /f1 ) using events passing θ ≥ 135◦ , as
a function of photon energy(y) and positron energy(x).
Chapter 1: Introduction
23
y
|f2/f1|: θ > 157
1
0.9
0.138161
0.211931
0.30015
0.394601
0.246254
0.382459
0.405341
0.356935
0.303646
0.443363
0.425877
0.278642
0.267354
0.350335
0.312488
0.181712
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1.10: The contours of (f2 /f1 ) using events passing θ ≥ 157◦ , as
a function of photon energy(y) and positron energy(x).
y
|f3/f1|: θ > 45
1
0.9
0.553972
0.568058
0.568112
0.55
0.546467
0.8
0.5
0.7
0.6
0.357368
0.461514
0.539516
0.45
0.557082
0.4
0.5
0.4
0.212099
0.265512
0.486618
0.575447
0.35
0.3
0.3
0.2
0.1
00
0.349799
0.1
0.215892
0.2
0.3
0.4
0.404364
0.5
0.6
0.585217
0.7
0.8
0.9
0.25
1
x
Figure 1.11: The contours of (f3 /f1 ) using events passing θ ≥ 45◦ , as
a function of photon energy(y) and positron energy(x).
Chapter 1: Introduction
24
y
|f3/f1|: θ > 90
1
0.9
0.550011
0.567663
0.568112
0.55
0.546467
0.8
0.5
0.7
0.6
0.309114
0.43927
0.538819
0.45
0.557082
0.4
0.5
0.4
0.207446
0.226515
0.469152
0.35
0.575256
0.3
0.3
0.2
0.1
00
0.347471
0.1
0.211944
0.2
0.3
0.4
0.371733
0.5
0.6
0.584084
0.7
0.8
0.9
0.25
1
x
Figure 1.12: The contours of (f3 /f1 ) using events passing θ ≥ 90◦ , as
a function of photon energy(y) and positron energy(x).
y
|f3/f1|: θ > 135
1
0.9
0.55
0.5203
0.552591
0.566428
0.546467
0.5
0.8
0.45
0.7
0.6
0.211565
0.33255
0.507544
0.555775
0.4
0.5
0.4
0.35
0.200553
0.164465
0.385754
0.564641
0.3
0.3
0.25
0.2
0.1
00
0.324864
0.195321
0.29824
0.566164
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1.13: The contours of (f3 /f1 ) using events passing θ ≥ 135◦ , as
a function of photon energy(y) and positron energy(x).
Chapter 1: Introduction
25
y
|f3/f1|: θ > 157
1
0.9
0.488683
0.51808
0.539818
0.5
0.542157
0.8
0.45
0.7
0.6
0.173173
0.234974
0.393966
0.4
0.526689
0.35
0.5
0.4
0.188863
0.133109
0.273038
0.3
0.508976
0.3
0.25
0.2
0.1
00
0.2
0.301319
0.17703
0.231477
0.518251
0.15
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1.14: The contours of (f3 /f1 ) using events passing θ ≥ 157◦ , as
a function of photon energy(y) and positron energy(x).
Chapter 2
PIBETA Detector
2.1
Introduction
The PIBETA detector is well described in Ref [11]. In this Chapter, I will give a
brief description of different components that make up the detector and explain the
tasks performed by each component. The PIBETA detector is a large solid angle
nonmagnetic detector optimized for measurement of photons and positrons in the the
energy range of 5-150 MeV. The main sensitive components shown and labelled in
Fig. 2.1 are:
• a passive lead collimator, PC, a thin forward beam counter, BC, two cylindrical
active collimators, AC1 and AC2, and an active degrader, AD, all made of
plastic scintillators and used for the beam definition.
26
Chapter 2: PIBETA Detector
27
• active plastic scintillator target, AT, used to stop the beam particles
• two concentric cylindrical multi-wire proportional chambers, MWPC1 and MWPC2
surrounding the active target, AT, used for tracking charged particles
• a segmented fast plastic scintillator hodoscope, PV, surrounding the MWPCs
used for particle identification.
• a high-resolution segmented shower CsI calorimeter surrounding the target region and tracking detectors.
• a cosmic muon plastic scintillator veto counters, CV, around the entire detector
(not shown).
The PIBETA detector triggering system and its data acquistion system (DAQ) will
also be briefly explained.
2.2
Experimental area
The detector is situated at the Paul Scherrer Institute (PSI) in Villigen, Switzerland.
The PSI ring synchrotron accelerates protons to an energy of 590 MeV. The protons
are subsequently transported to two target stations where pions and muons are generated by collision with thick carbon targets. The generated pions and muons are
then transported toward multiple experimental areas through secondary beam-lines.
The accelerator operates at the frequency of 50.63 MHz producing 1 ns wide proton
Chapter 2: PIBETA Detector
28
pulses separated by 19.75 ns. The PIBETA experiment was set up in the PSI πE1
experimental area whose 16 m long beam line is designed to supply intense low energy
pion beams with good momentum resolution. The πE1 beam line can deliver a pion
beam with a maximum momentum of 280 MeV/c, and a Full-Width-Half-Maximum
(FWHM) momentum resolution of < 2%. The choice of a particular beam momentum
is governed by the need for good time-of-flight (TOF) separation of pions, positrons,
and muons between production target and the forward beam counter (BC).
2.3
2.3.1
Beam-defining elements
Passive Collimator
In order to reduce positron contamination, a 4 mm thick carbon degrader is inserted
in the path of the πE1 beam. The momentum-analyzed pions and positrons have
different energy losses in the carbon absorber, and are therefore spatially separated
in a horizontal plane after passing through the absorber. The lead collimator PC is
positioned immediately upstream of the fixed beamline’s exit window. Only the beam
pions pass through the collimator, while positrons stop in the collimator material,
since at this point they are separated from pions by 40 mm in the horizontal plane
by magnetic momentum analysis (bending).
The passive collimator is composed of two stacked lead brick blocks with individual
dimensions of 250 × 250 × 50 mm3 . Both pieces have a central hole with a diameter of
Chapter 2: PIBETA Detector
29
50 mm and a step bore extending the hole to 70 mm. This is the last beam defining
element before pion beam enters the active detector counter.
2.3.2
Active beam-defining elements
The forward beam counter BC, is the first active detector counter. This counter is
made of plastic scintillator material with dimensions 25 × 25 × 2 mm3 . Immediately
after BC, the pion beam passes through a quadrupole triplet magnetic, and is focused
through two active beam collimators (AC1 and AC2). These collimators are cylindrical rings made of plastic scintillators and their importance is to suppress background
signals caused by detector hits that are not associated with the pion beam.
Following the active beam collimators is an active degrader. The beam of pions
arriving in central area of the detector has a relatively high momentum of about 113.4
MeV/c which needs to be reduced before pions get to the active target where they
are subsequently stopped. The active degrader counter reduces the average kinetic
energy of the pions from 40.3 MeV to 27.6 MeV. The active degrader is made of
plastic scintillator in the shape of a truncated cone to ensure that the degrader’s
downstream projection covers the whole target area (40 mm diameter), while at the
same time particles entering parallel to the beam axis always traverse the same 30
mm scintillator thickness.
The last active beam-defining element is the active target. This target is made of
cylindrical plastic scintillator, 50 mm in length and with a 40 mm diameter. In the
Chapter 2: PIBETA Detector
Figure 2.1: A schematic cross section of the PIBETA detector showing
the main components: forward beam counter (BC), two active collimators (AC1 and AC2), active degrader (AD), active target (AT), two
multi-wire proportional chambers (MWPC1 and MWPC2), plastic veto
(PV) and CsI calorimeter
.
30
Chapter 2: PIBETA Detector
31
2004 experimental runs, two types of active target were used. In early runs (5000051017), a 9 segments target was used, while a one-piece target was used for later runs
(52000-52471). The 9 piece target was used in previous high rate runs (1999-2001).
The lower rate 2004 run dedicated to radiative decays was better served by a one
piece target with improved light collection properties. Both targets were used in the
the 2004 run in order to establish consistency with the earlier runs.
The relatively high π-stop rate, around 105 s−1 , makes the target the most complicated component of the Pibeta detector in terms of data analysis or simulation of the
its performance. The π-stop event rate in the target is roughly 100 kHz, which is very
high compared to each calorimeter cystal, which bear less than 0.5% of the π-stop
event rate. Once the pion is stopped in the target, a chain of decays follows. Nearly
every pion mostly decays into a muon and the muon comes to rest in the target and
it also decays. Figure 2.2 shows a signal produced by a stopping pion and subsequent
pion decay into a muon. In this event the muon itself travels a short distance (1.4
mm) in the target before stopping and decaying.
2.4
Detector Tracking System
The Pibeta detector tracking system is comprised of 2 multi-wire proportional chambers (MWPC1 and MWPC2) and a 20-bar plastic scintillators hodoscope (PV) [11].
The wire chambers are made of low-mass material in order to minimize photon con-
Chapter 2: PIBETA Detector
32
Figure 2.2: Example of signal produced by decaying pion and subsequent muon decay. The first peak shows the stop of pion before decaying to muon. The muon produces a shoulder on the pion stop signal
(at about 47 ns). The muon then decays into a positron (at about 80
ns) that causes the final peak as it exits the active target
.
versions. They are highly efficient and are capable of handling a high rate of up to
107 tracks/s minimum-ionizing particles (MIP). They also have good radiation hardness and are very stable operationally. These wire-chambers are cylindrical, each
having one anode wire plane along the longitudinal direction, and two cathode strip
planes in a stereoscopic geometry.
The plastic scintillator hodoscope PV is located in the interior of the calorimeter
Chapter 2: PIBETA Detector
33
surrounding the two concentric wire chambers (MWPCs). The PV consists of 20
independent plastic scintillator staves that form a cylinder of 598 mm long and 132
mm radius. The PV covers the whole solid angle subtended by the calorimeter as
seen from the center of target.
The main parts of PV hodoscope are: plastic scintillator staves, two light guides,
and two attached photomultiplier tubes. The dimensions of individual staves are
3.175 × 41.895 × 598 mm3 . They are designed in such a way that a particle from the
target passes through the PV before entering the calorimeter. The PV hodoscope provides efficient charged particle identification. It is reliable at differentiating charged
particles (cosmic muons, positrons and protons) from photon events. Furthermore, it
provides precise timing information for charged particles.
2.5
Calorimeter
The segmented PIBETA calorimeter [11] consists 240 Cesium-Iodide (CsI) crystals,
220 of these crystals are truncated hexagonal and pentagonal pyramids covering the
total solid angle of 0.77 × 4πsr, while the remaining 20 crystals cover two detector
openings for the beam entry and target readouts and act as electromagnetic shower
leakage vetoes. The inner radius of the calorimeter is 26 cm, and its active depth is
22 cm, corresponding to 12 CsI radiation lengths (X0 = 1.85 cm).
The calorimeter is the most important part of the detector. The charged particles,
Chapter 2: PIBETA Detector
34
namely positrons, electrons and protons deposit their full energies in the calorimeter
while the photons that come from radiative decay are well confined in the calorimeter.
It is designed in a such way that it can handle a high event rate and it also has
a good energy and time resolution to minimize the background signals. Another
very important function of the calorimeter is that it gives the basis for trigger logic
described in Section 2.7.4.
2.6
Cosmic muon veto detectors
The detector is shielded from background radiation in the experimental hall as well
as from the cosmic ray background. The active parts of the detector are doubly
protected. A lead house enclosure provides inner passive shielding. It is in turn lined
on the outside with active cosmic muon veto counters.
2.7
2.7.1
PIBETA Triggering System
Introduction
The triggering system accepts events that satisfy certain predetermined conditions.
These conditions are based on energy thresholds and temporal coincidences. The
triggers used in this analysis are random triggers, beam triggers, and calorimeter
triggers.
Chapter 2: PIBETA Detector
2.7.2
35
Random Triggers
The random triggers are made using a plastic scintillator with dimensions of 190 ×
20 × 8 mm3 . That counter is placed above the electronic racks, parallel to the area
floor, about 3 m away from the main detector. The scintillator is shielded from
the experimental radiation by a 50 mm thick lead brick wall as well as a 500 mm
thick concrete wall. It is designed in such a way that events being triggered by
this trigger are random backgrounds events which are unrelated to events from the
primary detector. The main sources of these random background signals are cosmic
muons and electronic noise in the active detector. The information from these random
triggers is used to determine the ADC pedestals in the energy spectra. For example,
the average deposited energy in each CsI crystals due to random events is 0.15 MeV.
2.7.3
Beam Triggers
The beam triggers are defined by a coincidence between the beam counter BC, the
active degrader AD, the active target AT, and the RF accelerator signal. The signals
from the beam line elements BC, AD and AT are discriminated to produce 10 ns wide
logic pulses. These pulses from beam elements along with the accelerator RF make
up a pion stop signal πS, which is discriminated to be 10 ns wide:
πS = BC • AD • AT • RF.
(2.1)
Chapter 2: PIBETA Detector
36
This coincidence is only possible if the pion that was produced at a primary target by
a proton beam which is in coincidence with the RF, traverses the beam line elements
and stops in the active target AT. The πS signal initiates another signal called pion
gate, πG, which is 180 ns long and designed to start 50 ns before πS. The events
that occur during the time when the πG is open are the only recorded events. To
suppress prompt background signals, events occuring within a few nanoseconds of the
pion stop are rejected. The main source of the prompt signals are pion absorption,
elastic and inelastic pion scattering and single-charge-exchange interactions between
pions and the stopping target material:
π+ + A → π0 + B
π + + A → p + B etc.
(2.2)
These πS and πG are the basis of PIBETA trigger system. Other triggers are coincidences formed of calorimeter logic signals and beam signals. Figure 2.3 shows the
typical timing of beam trigger components.
2.7.4
Calorimeter Triggers
The CsI calorimeter is composed of 240 crystals. These crystals are grouped into
60 clusters, composed of 9 crystals each. These clusters are also grouped into 10
superclusters whereby 6 adjacent and overlapping clusters form one supercluster.
The simplest calorimeter trigger is the one-arm CsI trigger made in two versions:
Chapter 2: PIBETA Detector
37
Figure 2.3: The pion stop signal is defined as a four-fold coincidence
of beam counter (BC), active degrader (AD), active target (AT) and a
19.75 ns RF cyclotron signal. The snapshot shows signals from top to
bottom; BC, AD, AT and RF
Low-Threshold (5 MeV) and High-Threshold (53 MeV). It requires the firing of at
least one supercluster:
CS = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9,
(2.3)
Chapter 2: PIBETA Detector
38
where the numbers 0-9 represent the indices of the above mentioned superclusters.
The supercluster fires whenever energy deposited in one of the constituent clusters
exceeds the threshold.
The next more complicated type of calorimeter trigger is a two-arm trigger. The
two-arm trigger requires firing of two non-neighboring super-clusters, and is defined
as:
CSS = 0 ◦ (2 + 3 + 5 + 6 + 9) + 1 ◦ (3 + 4 + 5 + 6 + 7) + 2 ◦ (0 + 4 + 6 + 7 + 8) + 3 ◦
(0 + 1 + 7 + 8 + 9) + 4 ◦ (1 + 2 + 5 + 8 + 9) + 5 ◦ (0 + 1 + 4 + 7 + 8) + 6 ◦ (0 + 1 + 2 +
8 + 9) + 7 ◦ (1 + 2 + 3 + 5 + 9) + 8 ◦ (2 + 3 + 4 + 5 + 6) + 9 ◦ (0 + 3 + 4 + 6 + 7).
As the two-arm trigger requires firing of two superclusters, it follows that there
are three possible versions of it:
HH
• high thresholds on both superclusters: CSS
LL
• low thresholds on both superclusters: CSS
HL
• one high, one low thresholds: CSS
The above calorimeter triggers play a role in the event reconstruction for the PIBETA
detector. For example, non-radiative Michel decay, µ+ → e+ νe ν µ requires one-arm
LT trigger (CSL ) while the radiative Michel decay, µ+ → e+ νe ν µ γ requires the firing
L
of two-arm LT trigger (CSS
).
There are different decay types recorded by the PIBETA detector. Some are
common and some are rare. In order to optimize event statistics, trigger prescaling is
Chapter 2: PIBETA Detector
39
Table 2.1: The prescaling factors for different PIBETA triggers. Notice
higher prescaling factors on low-threshold triggers to suppress common
muon decays which would otherwise saturate the data acquistion system and prevent recording of events from rare pion decays
Trigger
CSL
CSH
LL
CSS
HH
CSS
LH
CSS
Prescale Factor
512
1
16
1
1
applied during data taking and it ensures that rare events like π + → π 0 e+ νe are always
recorded, while common events like Michel decay are prescaled appropriately. During
data analysis, the prescaling factors are applied in the number of events counting to
determine the actual number of events that occured based on the trigger prescaling
factor. The corresponding trigger pre-scaling factors for various PIBETA triggers are
shown in Table 2.1.
Chapter 3
Event Reconstruction
3.1
Introduction
This chapter describes the PIBETA data analysis software, energy and time calibrations, and the detector simulation. The data analysis software reconstructs different
decay modes. The software consists of different parts, with each part corresponding
to a component of the PIBETA detector system described in the previous chapter. In
order to optimize resolution of energy and time measurements, detailed and careful
calibrations were performed and I will highlight the procedure that was used. The
GEANT package simulation is described in detail in Ref [12]. This chapter presents
a summary of how we simulated the decay modes of interest.
40
Chapter 3: Event Reconstruction
3.2
41
Data Analysis Software
The software that is used for data analysis is written in the C language and is divided
into smaller routines which describe how particles are reconstructed in the individual
parts of the detector. For example, there are routines for Multi-Wire Proportional
Chamber (MWPC), a routine for Plastic Hodoscope (PV), CsI crystals etc.
The analysis software is built in such a way that a particle originating from the
active target AT can be well reconstructed by following its path through different parts
of the detector. The PIBETA detector identifies particles by using the tracking sytem
plus energy and timing information extracted from different sub-detectors. Charged
particles will leave information in the MWPC’s, and deposit energy in the PV and
CsI calorimeter. Neutral particles (photons) leave no signature in the MWPC’s, but
are identified in the calorimeter.
3.2.1
Tracking System algorithm
The tracking algorithm identifies charged tracks that register hits in the wire chambers. The algorithm considers hits in two wire chambers separated by an azimuthal
angle of less than 30 degrees. The line through the hits is projected back to the target
and forward to the struck calorimeter clump. The angle between this line and the
calorimeter clump is required to be less than 13 degrees, and the energy deposited
in CsI to be greater than 5 MeV. The algorithm takes the point (x0 , y0 , z0 ) shown in
Chapter 3: Event Reconstruction
42
Fig. 3.1 as the decay vertex. This point is calculated by using the information from
wire chamber tracks. The point (x0 , y0 , z0 ) is taken as the point on the track closest
to the z-axis. Fig. 3.2 shows the distribution of angles between the intersection of the
wire-chamber tracks with the calorimeter face (x1 , y1 , z1 ), and the angular coordinates
of the corresponding calorimeter clump.
A comparison between simulation and measurement was performed to check if
tracks were well reconstructed. Cosine of polar angle θ and azimuthal angle of a
positron from the Michel decay are shown in Figures (3.3 and 3.4), respectively. Similarly, for radiative Michel decay, the matching between simulation and measurement
are shown in Figures (3.5–3.8). Note that in this matching process for radiative decay,
simulated background is also shown in the histograms. Photons being neutral leave
no hits in the MWPCs or PV so they are assumed to come from the center of the
stopping distribution (x0 , y0 , z0 ). The direction cosines and calorimeter intersection
point (x1 , y1 , z1 ) are based on angles obtained from clump algorithm, explained in the
next section.
3.2.2
Clump Algorithm
A clump is defined as a centrally hit crystal plus 5-to-6 neighboring crystals. The
central crystal is the one with the highest energy deposition. If a track intersects a
crystal centrally, the particle deposits more than 90% of its energy in that crystal.
For the algorithm to accurately reconstruct the energy of an incident positron, it
Chapter 3: Event Reconstruction
43
× 10
Arbitrary Units
3
MC
8000
DATA
7000
6000
5000
4000
3000
2000
1000
0
-25
-20
-15
-10
-5
0
5
10
15
20
25
X0[mm]
× 10
Arbitrary Units
3
MC
8000
DATA
7000
6000
5000
4000
3000
2000
1000
0
-25
-20
-15
-10
-5
0
5
10
15
20
25
Y0[mm]
× 10
Arbitrary Units
3
Data
4500
MC
4000
3500
3000
2500
2000
1500
1000
500
0
-20
-10
0
10
20
30
40
Z0 [mm]
Figure 3.1: The primary vertex x0 , y0 , z0
Distribution of coordnates values of the primary vertex x0 , y0 , z0 obtained from the
wire-chambers tracks by backtracking to the point of closest approach to the z-axis.
needs to take into account the shower energy that leaks into the neighboring crystals.
The algorithm finds the centrally hit crystal and subsequently sums the energy of this
Chapter 3: Event Reconstruction
44
3
Number of Events
×10
4000
MC
3500
DATA
3000
2500
2000
1500
1000
500
0
0
2
4
6
8
10
12
14
αCsI-MWPC(Degrees)
Figure 3.2: The angular separation between a wire-chamber track and
the coordinates of the corresponding calorimeter clump
crystal and the energies in the neighboring crystals that are hit within |∆t| < 14 ns
of the central crystal. The algorithm also makes sure that two neighboring crystals
cannot be the centers of distinct clumps. The time of the clump is calculated as the
energy-weighted average of the all clump members, while clump angular coordinates
(θ, φ) are obtained using energy-weighted centers of the crystals in a clump:
θCsI
Pn
wi θi
= Pi=1
,
n
i=1 wi
(3.1)
φCsI
Pn
wi φi
= Pi=1
,
n
i=1 wi
(3.2)
and similarly,
Chapter 3: Event Reconstruction
Number of Events
× 10
45
3
DATA
4000
MC
3500
3000
2500
2000
1500
1000
500
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Cos θe
Figure 3.3: Cosine of polar angle θ of a positron track with respect to
the beam axis passing 1-arm trigger (Michel events).
where θi and φi are polar and azimuthal angles of the ith crystal and n is the number
of crystals in the clump. The weight wi is defined as:
Ei
wi = a0 + ln Pn
i=1
Ei
.
(3.3)
The simulation of calorimeter energy resolution gave as the best value of parameter
a0 = 5.5.
3.3
Particle Identification
Particle identification is needed to distinguish protons from positrons as both may
seem to be of the same type if no particle identification criteria are applied. The
separation of the positrons from protons is done through the relative amount of energy
Chapter 3: Event Reconstruction
46
×10
Number of Events
3
3000
2500
2000
DATA
1500
MC
1000
500
0
50
100
150
200
250
300
350
Positron Trk φ
Figure 3.4: Azimuthal angle φ of a positron track passing 1-arm trigger
(Michel events).
× 10
6
2400
2200
2000
1800
1600
Signal MC
1400
1200
Background MC
1000
DATA(measured)
800
MC(signal) + MC(background)
600
400
200
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Cos θpositron
Figure 3.5: Cosine of polar angle θ of a positron track passing 2-arm
trigger (radiative Michel events).
Chapter 3: Event Reconstruction
47
×10
6
2500
2000
Signal MC
1500
Background MC
DATA(measured)
MC(signal) + MC(background)
1000
500
0
50
100
150
200
250
300
350
Photon φ
Figure 3.6: Azimuthal angle φ of a photon passing 2-arm low trigger
(radiative Michel events).
× 10
6
2500
2000
Signal MC
1500
Background MC
DATA(measured)
1000
MC(signal) + MC(background)
500
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Cos θphoton
Figure 3.7: Cosine of polar angle θ of a photon passing 2-arm low trigger
(radiative Michel events).
Chapter 3: Event Reconstruction
48
×10
6
2200
2000
1800
1600
1400
Signal MC
Background MC
1200
DATA(measured)
1000
MC(signal) + MC(background)
800
600
400
200
0
50
100
150
200
250
300
350
Positron φ
Figure 3.8: Azimuthal angle φ of a positron track passing 2-arm trigger
(radiative Michel events).
that each particle deposits in the PV. The positrons, being minimum-ionizing particles
deposit smaller amount of energy in the PV, while protons being non-relativistic,
deposit greater amounts of energy in PV. Photons deposit very little (if any) energy
in the PV, charged particle identification is performed through a comparison between
how much energy a particle deposited in the PV and the total energy deposited in
both PV and CsI calorimeter. Two functions of energy deposited in PV (EPV ) and
energy in CsI (ECsI ) (fγ and fe ) are applied as boundaries for separating positrons,
protons and photons. If energy in PV is less than fγ then the particle is identified as
photon. Positrons and protons are separated by a condition of whether a particles’
energy in PV (EP V ) lies between fγ and fe or if its EP V is above fe . The functions
Chapter 3: Event Reconstruction
49
fγ and fe are defined as follows:
fγ = 0.2 exp(−0.007(EPV + ECsI ))
(3.4)
fe = 2.3 exp(−0.007(EPV + ECsI ))
(3.5)
and
Particle identification is such that:
3.4
EPV < fγ ⇒ photon,
(3.6)
fγ ≤ EPV < fe ⇒ positron,
(3.7)
EPV ≥ fe ⇒ proton.
(3.8)
Monte Carlo Simulation
The simulation of the PIBETA detector is done using standard GEANT3 Monte
Carlo software package [12] where both passive and active parts of the detector are
defined. The simulation takes into account all physics processes taking place in different detector systems. For example, the electromagnetic showers in the CsI as well
as electromagnetic interactions in the active target and the tracking system are well
described.
Chapter 3: Event Reconstruction
50
The Monte Carlo simulation begins with defining all subdetectors; beam line active
detectors (upstream beam counter, active degrader and target); two cylindrical MWPCs, the 20 cylindrical plastic hodoscope staves and the 240-crystal CsI calorimeter.
The user can modify the simulation code according to a number of different criteria,
e.g., the run year (due to different active targets), beam properties, or reaction/decay
final states. The user can select the final state of any particular decay channel with
its relative probability defined by the reaction cross-sections and decay branching
fraction. Each reaction is characterised by its own kinematics, and the event is generated by using specific event generators such as Michel or radiative Michel decays.
The simulation of energy deposition is done by simulating ADC values, and the time
is simulated through TDC simulations. The simulation of ADC values and TDC hits
takes into account individual detector photoelectron statististics, axial and transverse
light collection nonuniformities, ADC pedestals, and electronics noise. Once the event
has been generated, the simulation of the particle’s passage through the detector material is performed in such a way that the event is reconstructed using the same event
reconstruction algorithm as for the measured data.
Chapter 3: Event Reconstruction
3.5
51
Data Calibration
3.5.1
Introduction
Good measurement resolution is critical for a successful data analysis. In the PIBETA
experiment, energy and time distributions of the detected particles are used to identify
different decay channels. Therefore, we performed a careful calibration of energy and
time for the main parts of the PIBETA detector (PV and CsI calorimeter).
The entire process of calibration involved calibrating energy and time for both PV
and CsI calorimeter modules. The PV is composed of 20 individual staves and there is
a variation of energy and time recorded in each one. Similary, the CsI has 240 crystals,
each one having its own energy and time resolution parameters. This disparity is a
source of degraded resolution for the measured energy and time of any decay mode
compared to an ideal detector. Therefore, to increase our resolution we performed
calibration by aligning each subdetector response to one common reference point.
In addition to calibrating the PV and CsI detectors, the calibration was performed
in such a way that run dependence on over all energy and time distributions was
removed.
3.5.2
Energy Calibration (ADC Calibration)
The energy calibration for both CsI and PV was done by requiring the energy peak of
each segment of PV or CsI to match the peak value obtained from the MC simulation.
Chapter 3: Event Reconstruction
52
Number of Events
×103
DATA
3000
MC
2500
2000
1500
1000
500
0
0
10
20
30
40
50
60
Positron CsI Energy[MeV]
Figure 3.9: Positron energy in CsI calorimeter for 1-arm low trigger
events.
To obtain these values, we used gaussian plus exponential functions to fit energy
distributions; subsequently the fit values for each segment were applied to obtain the
multiplying factor which is the ratio between the reference peak and the different
peaks from each segment. These multiplying factors were then applied in the offline
data analysis for each PV stave and CsI crystals. The energy distribution of the decay
π + → µ+ → e+ νe ν¯µ was used as a reference for both PV and CsI energy calibration.
The energy distribution peak reference values that were obtained from simulation for
perpendicular PV energy and CsI energy are 0.5588 MeV and 63.29 MeV respectively.
After energy calibration, the CsI energy fractional resolution ∆E/E (∆E defined as
Full-Width-Maximum-Height), decreased from 8.0% to 4.5%.
Chapter 3: Event Reconstruction
53
×10
Number of Events
6
DATA
12
MC
10
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2 1.4 1.6 1.8
2
Positron Energy in PV[MeV]
Figure 3.10: Positron energy in plastic veto (PV) for 1-arm low trigger
events.
3.5.3
Time Calibration (TDC Calibration)
The PV timing calibration for channels in the PV was done by using prompt trigger
events and finding the time difference between each PV stave and the time zero (pionstop time). A similar method was aplied to the CsI time calibration where we had
to find the time difference between the pion-stop time and each subsequent hit in the
CsI counter. Events passing the prompt trigger were the ones used in time calibration
and the measured time offsets were applied in the offline data analysis. Consequently,
these offsets were applied to all other PIBETA decay modes including Michel decays.
This is done by subtracting the offsets from all raw TDC values in the offline data
analysis.
Chapter 3: Event Reconstruction
54
The ADC readouts in the PV and CsI have different amplitudes. The smaller
amplitude signal takes a longer time to rise to the fixed discriminator threshold than
a larger signal. Consequently, there is energy dependence of observed TDC values with
higher energy signals registering earlier time while lower energy signals registering at
later time. For a complete time calibration we need to perform a time walk correction
in order to correct for this energy-time dependence. The correction was performed
using Michel (µ+ → e+ νe ν¯µ ) events by fitting each PV stave and each CsI channel
T DC (time) vs ADC (energy) distribution. The fitting function is of the functional
form
CT DC = T DC0 + a.(ADC − b)c
(3.9)
where T DC0 , a, b, and c are free parameters of the fit, ADC is the calibrated ADC
value proportional to the deposited energy. The resulting corrected value, CT DC
is the final TDC value that is applied in the offline analysis. Figures 3.11 and 3.12,
show the energy-time distribution before and after the walk correction procedure
is performed. Notice that after the walk-correction there is no more energy-time
dependence. This is shown in Fig 3.12 by a straight horizontal line in the energy
range of 10-53 MeV.
Time
Chapter 3: Event Reconstruction
55
2
After Walk-Correction
1
Before Calibration
0
-1
-2
-3
-4
-5
-6
0
0.5
1
1.5
2
2.5
PV Energy[MeV]
Figure 3.11: Walk-correction effect on the energy and time of PV. The
straight horizontal line indicate the removal of energy-time dependence
in the observed TDC values.
CsI time - PV time](ns)
CsI Walk Correction validation
Entries
Mean
Mean y
RMS
RMS y
5
4
p2
8408818
30.63
-2.555
10.95
1.766
3
2
1
0
-1
-2
-3
-4
-5
0
10
20
30
40
50
60
CsI Energy [MeV]
Figure 3.12: Walk-correction effect on the energy and time of CsI
calorimeter. The straight horizontal line indicate the removal of energytime dependence in the observed TDC values in the energy range of
10-53 MeV.
Chapter 4
Data Analysis
4.1
4.1.1
Branching Fraction
Introduction
As described in Chapter 1, the primary objectives of this analysis are to determine the
experimental branching fraction of the radiative muon decay in the largest accessible
phase space, and to extract the Michel parameter, η̄. This chapter explains how
the branching fraction was measured and how the Michel parameters were extracted
using both measurement and simulated data.
Although the design of the PIBETA detector is not optimized for a precise measurement of the ρ parameter compared to the dedicated experiments [13], ρ parameter
was extracted for the purposes of a consistency check of our data and analysis. Ex-
56
Chapter 4: Data Analysis
57
traction of the ρ parameter was performed by minimizing the chi-square difference
between the observed and simulated positron energy spectra of the non-radiative
muon decay.
The experimental branching fraction was determined by the following expression:
Br(µ+ → e+ νe ν¯µ γ)Eγ >10 MeV ,θ > 30◦ =
Nµ+ →e+ νe ν¯µ γ Aµ+ →e+ νe ν¯µ (γ)
·
,
Nµ+ →e+ νe ν¯µ (γ) Aµ+ →e+ νe ν¯µ γ
(4.1)
where Nµ+ →e+ νe ν¯µ γ and Nµ+ →e+ νe ν¯µ (γ) are number of observed events for radiative and
non-radiative muon decays, respectively, Aµ+ →e+ νe ν¯µ (γ) and Aµ+ →e+ νe ν¯µ γ are the MC
calculated acceptances for non-radiative and radiative muon decays, respectively. The
number of events for both non-radiative and radiative muon decays are determined
from measured data, while the detector acceptance for each decay channel is obtained
from detailed Monte Carlo simulations.
The Michel parameter η̄ is extracted by minimizing the chi-square difference between the experimental branching fraction and theoretical (expected) branching fraction. Theoretical branching fraction is calculated using Monte Carlo integration of
the following equation:
BR
Theo
Z
x2
Z
y2
Z
cos θ2
(η̄, ρ) = 2π
x1
y1
cos θ1
4
dxdyd(cos θ) f1 + η̄ · f2 + (1 − ρ) · f3 . (4.2)
3
Chapter 4: Data Analysis
4.1.2
58
Muon decay time distribution
The decay chain that leads to a Michel decay or radiative Michel decay proceeds
through the sequence of a pion stopping in the Active Target and subsequently decaying into a muon. The muon from the decay π + → µ+ νµ will travel about 1-2 mm
before coming to rest in the target. The target radius (2 cm) is very large compared
to muon travel distance and the number of decaying muons is essentially equal to the
number of the stopped pions.
The parameters that are needed to describe the muon decay time [10] are: accelerator period (19.75 ns), width of pion gate, pion and muon lifetimes and the rate at
which the pions are stopped in the target.
The PIBETA experiment time scales were designed based on the lifetime of the
charged pion. For example, from Table 4.1, we see that the width of pion gate is
wide enough for five pion lifetimes to allow the decay of almost all the stopped pions.
The situation becomes different for the muons as their lifetime is very large, allowing
muons to accumulate in the target; the decay of these piled-up muons can occur in a
later pion gate with which they are not causally connected.
The rest of this section will describe in detail the time structure of the muon
decays. The decay chain that leads to the muon decay following the stop of pion in
the target proceeds as follows:
π + → µ+ νµ ,
(4.3)
Chapter 4: Data Analysis
59
Table 4.1: Time scales involved in the PIBETA detector. The rate of
stopping pions is around 105 /sec.
Time scale
pion lifetime (τπ )
cyclotron period (Trf )
pion gate width
muon lifetime (τµ )
Pion Stop Period (1/rπ )
value (ns)
26.02
19.75
180
2197.03
10000
followed by
µ+ → e+ νe ν̄µ (γ).
(4.4)
The probability distributions over time for these decays are given by:
fπ (t) =
exp(−t/τπ )
,
τπ
(4.5)
fµ (t) =
exp(−t/τµ )
,
τµ
(4.6)
and
hence the probability for sequential decay is given by;
Z
fπ→µ→e (t) = fs (t) =
0
t
(e−t/τµ − e−t/τπ )
,
fu (t − t )fπ (t )dt =
τµ − τπ
0
0
0
(4.7)
where fs (t) is the probability per unit time that the decay chain π + → µ+ νµ , µ+ →
e+ νe ν̄µ (γ) will happen at time t after the pion stops in the target and this probability
Chapter 4: Data Analysis
60
is normalized as
Z
∞
fs (t)dt = 1.
(4.8)
0
As noted above, muons accumulate in the target and they can subsequently decay
in a later pion gate with which they are not causally connected. Since the rate of
stopping pions is comparable to the muon decay rate, we have to consider pileup
effects in our study of muon time structure. The probability of having a pion in a
particular beam pulse is
p = rπ Trf .
(4.9)
Hence the rate of pileup muon decays is calculated as the sum over all beam pulses
of the causal decay rate given by (4.7), weighted with the probability (4.9):
fpu (t) =
∞
X
pfs (t + nTrf ).
(4.10)
n=1
This series can be simplified by summing it as follows:
∞
X
p
(e−(t+nTrf )/τµ − e−(t+nTrf )/τπ ),
pfs (t + nTrf ) =
τ
−
τ
µ
π
n=1
n=1
∞
X
(4.11)
where after further summation of the series (4.11)
fpu (t) =
p
e−t/τµ
e−t/τπ
( T /τµ
− T /τπ
).
τµ − τπ e rf − 1 e rf − 1
(4.12)
Chapter 4: Data Analysis
61
By ignoring the pion term (second term), which is a lot smaller compared to the
first term and evaluating this periodic function at t = 0 we obtain
fpu (0) ≈ rπ .
(4.13)
This means that the rate of muon pile-up in the target is approximately equal to the
rate at which pions are stopped in the target. In summary, the probability distribution
for all times is given by:
f (t) =




fpu (t) if
t<0
(4.14)


 fpu (t) + fs (t) if t ≥ 0.
Figure 4.2 shows the actual measured time structure of the decay µ+ → e+ νe ν̄µ ,
plotted as relative time between degrader (t = 0) and the time of CsI calorimeter
showers, while Figure 4.3 is the decay time structure for radiative muon decay which
+
is taken as time difference between the average of CsI time of positron (teCsI ) and
photon (tγCsI ) and the degrader time (tDeg ), expressed as:
1 e+
(tCsI + tγCsI ) − tDeg .
2
(4.15)
The time spectrum of muon decays is an important factor in branching fraction
measurement, because the number of events for non-radiative Michel decay is obtained
from the time distibution (Figure 4.2) after time window selection (explained below)
Chapter 4: Data Analysis
62
CsI Time Coincidence
Entries/0.5 ns
×103
1200
1000
N0. of Events = 518813 ± 946
σ = 0.71 ns
800
600
400
200
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
∆t = [t + - tγ ] (ns)
e
Figure 4.1: Time coincidence of positron and photon in the CsI
calorimeter.
is applied. The number of events from the radiative decays is obtained from the fit
of the histogram of time coincidence of positron and photon reconstructed as per
Equation (4.16) and is shown in Figure 4.1
+
∆t = teCsI − tγCsI .
(4.16)
The time window is selected by dividing radiative time distribution, Figure 4.3 to
non-radiative distribution, Figure 4.2 and taking a region where the ratio can be
fitted by a straight horizontal line. As it can be seen from Figure 4.4, care is needed
Chapter 4: Data Analysis
Number of events
× 10
63
3
1400
1200
1000
800
600
400
200
0
-50
0
50
100
150
tposi - tdegr [ns]
Figure 4.2: The decay time distribution of Michel events. The time is
the difference between CsI time (tCsI ) and degrader time (tDeg ).
especially at the edges of the pion gate and also at the prompt veto where there is a
discrepancy between the time structure of non-radiative and radiative decay modes.
In the measurement of branching fraction, we normalize the number of radiative
muon decays to the number of non-radiative muon decays. It is therefore necessary
to select a time window such that there is no uncertainty in counting events due to
time structure difference between the two decays.
From the fit of Figure 4.5, only events in the time window of
t ∈ (−37, −6) ns ∪ (13, 144) ns,
were selected in the determination of experimental branching fraction.
(4.17)
64
Number of Events
Chapter 4: Data Analysis
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
-50
0
50
100
0.5*(t
posi
150
+ tphot ) - t
degr
Rel. Time Ratio (Rad./Non-rad)
Figure 4.3: The time distribution of radiative Michel events. This is
+
measured as the time difference ( 21 (teCsI + tγCsI ) − tDeg ).
2
1.5
1
0.5
0
-0.5
-50
0
50
100
150
tCsI - tdeg[ns]
Figure 4.4: The ratio of radiative decay time to non-radiative muon
decay time before selecting the time window.
Rel. Time Ratio (Rad/Non-Rad.)
Chapter 4: Data Analysis
65
2
1.5
1
0.5
0
-0.5
-50
0
50
100
150
tCsI - tdeg[ns]
Figure 4.5: The ratio of radiative decay time to the non-radiative muon
decay time after selecting only the range where the plot can be fitted
by a straight horizontal line.
4.1.3
Non-Radiative Muon Decay
Event Reconstruction
The reconstruction of non-radiative muon decay begins by taking any charged track
that passes the one-arm low CsI trigger requirement. Furthermore, the track is required to have at least one hit in each wire-chamber (MWPC1 and MWPC2) and
also at least one hit in the PV. The candidate events need to satisfy further requirements including positron identification selections, requiring the energy measured to
be above the CsI low-threshold (5 MeV) and also to be in the range of kinematically
allowed energies of a positron coming from the decay of a muon. The background
Chapter 4: Data Analysis
66
Table 4.2: Michel cuts
Positron Energy(CsI) gen. level
Positron Energy(CsI) reconstructed
Plastic Veto(PV) Energy
CsI Veto Energy
Prompt Cut
5 MeV < ECsI ≤ Emax = (m2µ + m2e )/2mµ
12 MeV < ECsI ≤ Emax = (m2µ + m2e )/2mµ
EPV > 0 MeV
Veto
ECsI
< 5 MeV
EPV ≤ −0.02 ∗ ECsI + 2.8
events from the decay π + → e+ νe are of the order of 10−4 of the signal because
Brπ+ →e+ νe
. 10−4 .
Brµ+ →e+ νe ν̄µ
(4.18)
Even though the monoenergetic positron from the decay π + → e+ νe has a peak well
above the endpoint of the energy distribution of a positron from the decay µ+ →
e+ νe ν¯µ , there are still some events from π + → e+ νe that are detected well below the
peak due to response function of the calorimeter. Any track identified as a positron
that fires the one-arm trigger with energy 10 < EeCsI
+ < 52.83 MeV was taken as a
candidate for non-radiative muon decay. Additional kinematic selection applied was
to reject events with more than 5 MeV of energy in the calorimeter veto crystals (221240), which are situated in the calorimeter perimeter of the beam openings. This cut
on energy deposited in the veto crystals helps us to eliminate the events that have
energy leaking from boundaries of the calorimeter. Table 4.2 summarizes the applied
kinematic cuts in the selection for non-radiative muon decay.
Chapter 4: Data Analysis
4.1.4
67
Radiative Michel decay
At the tree level, the radiative decay of the muon involves no complications from
the strong interaction couplings or structure. This fact favors radiative muon decay
in tests of the V − A structure of the weak interaction. The differential branching
fraction of muon radiative decay was calculated by several authors, for example [8, 7].
From Equation (1.21), we see that the branching fraction is a function of the kinetic
energy of the positron, photon energy and the opening angle between the positron
and the photon.
Event Reconstruction
In the PIBETA analysis software, the radiative muon decay event is selected as follows: (1) an event that passes either low threshold trigger (one-arm trigger or twoarm trigger) condition is kept, (2) the software requires a minimum of at least two
reconstructed electromagnetic showers. For better track reconstruction, the software
requires (3) at least one hit in each subdetector of the tracking system (MWPC1,
MWPC2 and PV). Once the above three conditions are met, further requirements
are introduced to have a well defined radiative muon decay event.
If one particle is identified as a positron and another as a photon, then a calorimeter time difference between the two particles is calculated. Virtually all coincident
positron and photon pairs in low threshold triggers will have a signature of radiative
muon decay as shown in Figure 4.1. The event reconstruction also requires a rejec-
Chapter 4: Data Analysis
68
tion of an event that contains a hard photon emmited at a small angle relative to the
positron. This kind of hard photon events are typically due to external bremstraulung
and are one of the sources of background signals described in Section 4.1.5 below.
The event reconstruction algorithm calculates the angle between the positron and
photon. This is done as follows:
cos θ =
V~1 · V~2
1
= 2 (v1x · v2x + v1y · v2y + v1z · v2z ),
|V1 ||V2 |
R
(4.19)
where v1x , v1y , v1z , v2x , v2y , v2z are the particles’ trajectory intercept coordinates
with the inner surface of the calorimeter and R is the inner radius of the calorimeter
(26 cm). Lastly, the angle between the calorimeter clump and the trajectory of the
positron measured by the wire chamber is calculated. Once all above conditions
are satisfied and the angles have been calculated, the analyzer software writes the
information in an ASCII file which is converted into ROOT file for further offline
analysis.
4.1.5
Background Signal
In the offline analysis, we need to study the source of background events that potentially contaminate the radiative muon decay process. There are two types of
background events [14, 10] . The first source is the combinatoric backgrounds events
which come from random photons and positrons that are coincident in the calorime-
Chapter 4: Data Analysis
69
ter. As shown in Figure 4.1, they form a flat time distribution before and after the
peak of ∆t = te+ − tγ . These are removed by taking only events falling inside the
window
|∆t| ≤ 5 ns,
(4.20)
and subtracting those in the side time window
5 ns ≤ |∆t| ≤ 10 ns.
(4.21)
These two windows are of equal width in the time coincidence distribution. Assuming
that this combinatoric background is uniformly distributed in the whole spectrum of
∆t, the subtraction process gives an accurate estimate of the number of radiative
events that are coincident in the calorimeter.
Another source of background comes from events that are non-radiative but pass
the criteria of reconstructing radiative events. In order to obtain the most accurate
value for the branching fraction, it is very important to determine the fraction of
background signals masquerading as radiative muon decays. The source of these misidentified events can be explained by the way showers develop in an electromagnetic
shower calorimeter, such as the PIBETA detector.
Shower development in the calorimeter can give rise to background signals from
non-radiative Michel decay. For example, a positron in the calorimeter can lead to a
false radiative decay if a photon from its shower manages to travel in the calorimeter
Chapter 4: Data Analysis
70
far enough to produce a shower in another clump. The clumping algorithm mentioned
earlier will identify this new shower as a distinct clump and hence take it as radiative
decay. Due to the fact that these two showers will be in close temporal coincidence,
the event reconstruction procedure described above will not be able to distinguish this
background signal from a radiative event. This background source takes into account
all random photons that are generated inside the calorimeter, and as in ordinary
bremsstrahlung, they are most likely emitted at small angles with respect to the
original particle. Therefore, by applying a cut-off angle, most of these background
signals are cut out. Figure 4.8 shows the applied cut where only events that register
an opening angle above 45 degrees were accepted.
Even by applying the opening angle requirement, there are still some background
signals that occur when a secondary shower is emitted at a very large angle with
respect to the positron. The source of these background is the external bremstraulung
interactions. The effect of this background is estimated in simulation by determining
how many events are misidentified as radiative muon decays. The correction of the
order of 2.45% was then applied to the number of radiative muon decay events by
taking into account the MC estimation of this background. Figures 4.6, 4.7 and 4.8
show the simulated background, measured and simulated radiative distributions for
opening angle, positron energy and photon energy respectively.
Chapter 4: Data Analysis
71
Table 4.3: Event generator level cuts.
Positron Energy
Photon Energy
Opening Angle (θ)
Kinematic Cut
4.1.6
0.0005 ≤ x < 1
0.189 ≤ y < 1
30◦ ≤ θ < 180◦
∆ ≥ 2(x+y−1)
xy
Kinematic cuts
One goal of this work is to measure the branching fraction over the largest accessible
phase space of positron energy, photon energy and the corresponding opening angle.
It is necessary to apply a lower energy cut off on photon energy to avoid the infrared divergences described in Section 1.2.2. The numbers of events for both Michel
and radiative Michel decay modes were determined using the procedure described
in Section 4.1.2. The experimental acceptance accounts for those events that were
generated but were lost due to kinematic plus geometrical cuts (i.e., energy and
opening angle cuts). The acceptance is calculated as number of simulated events that
were generated and passed reconstruction cuts divided by the total number of events
that passed generator level cuts
A=
N recon
.
N gen
(4.22)
Table 4.3 and Table 4.4 show the event generator level cuts and event reconstruction
cuts, respectively, that were applied in the branching fraction calculation.
Chapter 4: Data Analysis
72
Table 4.4: Event reconstruction cuts
Positron Energy(CsI)
Photon Energy(CsI)
Opening Angle (θ)
Kinematic Cut
Plastic Veto(PV) Energy
CsI Veto Energy
Prompt Cut (MeV)
0.227 > x ≤ 1
0.3789 > y ≤ 1
45◦ < θ ≤ 180◦
∆ ≥ 2(x + y − 1)/xy
EPV > 0 MeV
ECsIVeto < 5 M eV
EPV < −0.02 · ECsI + 2.8 MeV
40000
Signal MC
Misid Events
35000
Signal MC + Misid. events
DATA(measured)
30000
25000
20000
15000
10000
5000
10
15
20
25
30
35
40
45
50
55
Positron Energy [MeV]
Figure 4.6: Radiative muon positron energy distribution in the
calorimeter (CsI). The histogram shows distributions for the measured
energy, simulated energy (signal MC), simulated background (background MC) plus the sum of signal MC and background MC.
.
Chapter 4: Data Analysis
73
60000
Signal MC
Misid Events
50000
Signal MC + Misid. events
DATA(measured)
40000
30000
20000
10000
0
15
20
25
30
35
40
45
50
55
Photon Energy [MeV]
Figure 4.7: Photon energy distribution in the calorimeter (CsI). The
histogram shows distributions for the measured energy, simulated energy (signal MC), simulated background (background MC) plus the
sum of signal MC and background MC.
.
Chapter 4: Data Analysis
× 10
74
3
Signal MC
100
Misid Events
80
Signal MC + Misid. events
DATA(measured)
60
40
20
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Cos θ
Figure 4.8: The cosine of opening angle between positron and photon.
The histogram shows distributions for the measured cosine of opening
angle, simulated cosine of opening angle (signal MC), simulated background (background MC) plus the sum of signal MC and background
MC.
Chapter 4: Data Analysis
4.1.7
75
Systematic errors
A systematic uncertainty is a possible unknown variation in a measurement, or in
a quantity derived from a set of measurement, that does not randomly vary from
data point to data point. It is any error that is not statistical. It represents a
constant (not random) but unknown uncertainty whose size is independent of the
number of events. In the process of measuring the branching fraction, main sources
of systematic errors were identified to be energy calibration, subtraction of random
combinatoric background signals, selection of kinematic cuts, time window selection,
and the misidentified events.
The energy calibration for the photons cannot rely entirely on the calibration
that was perfomed using the positron calorimeter response and therefore a small
correction to the photon energy scale factor was introduced. Section 4.2.1 explains
how this factor was obtained.
We also estimated the systematic errror originating from the subtraction of random coincidences (flat temporal background) in Figure 4.1. This was done by varying
the width of signal events under the gaussian peak and the corresponding width of
flat background. After allowing a variation of one standard deviation (0.71 ns), the
difference between the value of branching fractions (before and after width change)
was taken as a systematic error.
The systematic errors originating from selection of kinematic cuts (energy and
76
-3
Br(10 )
Chapter 4: Data Analysis
4.42
4.41
4.4
4.39
4.38
4.37
12
12.5
13
13.5
14
14.5
15
15.5
16
Ee+ [MeV]
Figure 4.9: The variation of the branching fraction due to different
positron energy threshold selection cuts
opening angle thresholds) were estimated by selecting different cuts around the nominal cuts showed in the Tables (4.4 and 4.2). The variation on branching fraction from
this procedure of varying selection cuts were taken as systematics. Figures 4.11 and
4.12 show how the branching varies when a different threshold energy cut is applied
is the event selection.
All of the estimated systematic errors are shown in Table 4.5.
4.1.8
Results (branching fraction)
The experimental branching fraction was determined using the kinematic cuts as described in Section 4.1. The numbers of events for both radiative and non-radiative
decays were obtained using the procedure described in Section 4.1.2 and their respec-
77
-3
Br(10 )
Chapter 4: Data Analysis
4.51
4.5
4.49
4.48
4.47
4.46
4.45
18
20
22
24
26
28
Eγ [MeV]
Br(10-3)
Figure 4.10: The variation of the branching fraction due to different
photon energy threshold selection cuts
4.375
4.37
4.365
4.36
4.355
4.35
4.345
2
2.5
3
3.5
4
4.5
5
∆ t Cut [ns]
Figure 4.11: The variation of the branching fraction due to different
signal window selection
78
-3
Br(10 )
Chapter 4: Data Analysis
4.38
4.37
4.36
4.35
4.34
42
44
46
48
50
52
54
Opening angle Cut [Deg]
Figure 4.12: The variation of the branching fraction due to different
opening angles
Table 4.5: Systematic errors associated with the measured experimental
branching fraction in the region of phase space spanned by (0.0005 <
x ≤ 1 , 0.189 < y ≤ 1, 30◦ < θ ≤ 180◦ )
Quantity
Photon energy calibration
Background subtraction
Positron energy threshold
Positron energy calibration
Photon energy threshold
Cosine opening angle
Time window selection
Misid. events
Total Syst. Error
Syst. Error(%)
0.73
0.14
0.26
0.18
0.41
0.29
0.12
0.03
0.96
Chapter 4: Data Analysis
79
Table 4.6: The experimental results for the branching fraction of radiative muon decay in the region of phase space defined by cuts in
Tables 4.3, 4.4 and 4.2
.
Quantity
Nµobs
+ →e+ ν ν¯ γ (real plus misid. events)
e µ
misid.
Nµ+ →e+ νe ν¯µ γ misidentified events
Nµobs
+ →e+ ν ν¯ Michel events
e µ
Aµ+ →e+ νe ν¯µ γ (Accept. radiative Michel)
Aµ+ →e+ νe ν¯µ (Accept. Michel)
B Exp (misid. events excl.)
Value
518813 ± 946
12738 ± 113
37094400 ± 6090
0.0693 ± 0.0001
0.7102 ± 0.00032
(4.365 ± 0.009(stat.) ± 0.042(syst.)) × 10−3
tive acceptance were calculated using Monte Carlo events. The statistical errors on
number of events were estimated using standard Poisson distribution statistics [15].
The uncertainty on detector acceptance follows the binomial distribution [15] and is
calculated as follows:
σA =
p
A(1 − A)/N ,
(4.23)
where A is the calculated acceptance and N is the total number of generated events.
The theoretical branching fraction calculated using Monte Carlo integration in the
region of phase space given by Table 4.3 with the Standard Model values of η̄ and ρ
is:
B Theo = (4.342 ± 0.005) × 10−3 ,
(4.24)
Chapter 4: Data Analysis
80
which is close to our measured experimental value:
B Exp = [4.365 ± 0.009 (stat.) ± 0.042 (syst.)] × 10−3 .
4.2
(4.25)
Extraction of η̄ and ρ Parameters
In the Standard Model, η̄ ≡ 0 and ρ ≡ 0.75 and therefore from Equation (1.21), only
function f1 is involved in the calculation of the branching fraction. These values of
Michel parameters are consistent with V − A structure of weak interactions. To test
the validity of V − A structure of weak interaction, we extract experimental values
of both η̄ and ρ parameters. Any statistically significant deviation from Standard
Model values would indicate that weak interaction is not V − A pure and there could
be contributions from scalar, pseudo-scalar or tensor transformations.
The parameter ρ was extracted using non-radiative muon decay events, µ+ →
eνe ν̄µ . From Equation (1.17), we see that the decay rate is dependent on ρ parameter.
The simulated positron energy spectrum is dependent on the ρ parameter through
the weight Equation (1.17) which is assigned to every simulated event. The Method
of least squares was used as a criterium and the experimental value of ρ parameter
was taken as the value that minimizes the χ2 difference between experimental and
Chapter 4: Data Analysis
81
simulated positron CsI energy distributions:
2
χ (ρ) =
N
X
(yi − λi (ρ))2
i=1
σi2
,
(4.26)
where yi is the number of entries in bin i (measured), σi2 is the variance of the
number of entries in bin i , λi = E[yi ] (simulated). The number of entries predicted
in bin i , and the function that is fitted on measured positron energy histogram is
λi (ρ), obtained from simulated positron energy distribution. Figure 4.13 shows the
statistical χ2 as a function of ρ parameter. We obtained:
ρ = 0.7581 ± 0.0045,
(4.27)
which is consistent with the current PDG value:
ρ = 0.7503 ± 0.0004 (PDG).
(4.28)
Since our experiment was not optimized for a measurement of ρ, in the rest of this
work ρ will be set to its SM value. The parameter η̄ is deduced from the radiative
muon decay sample µ+ → e+ νe ν¯µ γ. From Equation (1.21), we see that η̄ will only
contribute to the branching fraction calculation if its value is different from the Standard Model value (η̄SM = 0). From Figures 4.14, 4.15 and 4.16, the region of phase
space most sensitive to η̄ was determined to be large opening angle region, for mod-
Chapter 4: Data Analysis
82
χ2
χ2 vs ρ
180
160
ρ = 0.7581 ± 0.0045
140
120
100
80
60
40
20
0
0.72
0.74
0.76
0.78
0.8
0.82
ρ
Figure 4.13: The χ2 as a function of ρ from analysis of non-radiative
Michel decays
erate values of x and y. We chose events in the region of 150 ◦ ≤ θ ≤ 180 ◦ and also in
the region defined by the following kinematic cuts on positron and photon energies:
0.25 < x < 0.85 and 0.189 < y < 0.8. The least squares statistical method [15]
was applied in the determination of experimental value of the η̄ parameter. The
η̄ value was taken as the one that minimizes the χ2 difference between theoretical
and experimental branching fraction calculated inside the above mentioned region of
phase space. The region of phase space for calculation of experimental branching
fraction was split into different parts to verify the sensitivity of η̄ paramater in different regions. The region was split into bins of 150 ◦ ≤ θ ≤ 160 ◦ , 160 ◦ ≤ θ ≤ 170 ◦ ,
Chapter 4: Data Analysis
83
°
°
|f2/f1|: θ [150 ,160 ]
y
1
0.9
0.125024
0.159389
0.248811
0.4
0.43907
0.8
0.35
0.7
0.6
0.212907
0.282154
0.253333
0.293119
0.3
0.5
0.4
0.25
0.263077
0.337378
0.251229
0.158329
0.3
0.2
0.2
0.1
0
0
0.215762
0.238272
0.171783
0.15
0.10838
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
x
Figure 4.14: (f2 /f1 )(x, y, 150◦ < θ ≤ 160◦ )
.
170 ◦ ≤ θ ≤ 180 ◦ and 160 ◦ ≤ θ ≤ 180 ◦ . As expected, the greatest sensitivity to η̄
is for opening angles greater than 160 ◦ . The result from the bin 150 ◦ ≤ θ ≤ 160 ◦
shows that sensitivity to the η̄ parameter diminishes for lower angles.
When χ2 difference between theoretical and experimental branching fraction is
calculated, the ρ parameter value was fixed at 0.75 while η̄ is varied around its Standard Model value. The value of η̄ that minimizes χ2 as explained below is taken as
our experimental value.
The value of theoretical branching fraction (B theo ) was calculated by using Monte
Carlo integration:
Chapter 4: Data Analysis
84
°
°
y
|f2/f1|: θ[160 ,170 ]
1
0.9
0.137664
0.208531
0.279084
0.367726
0.4
0.8
0.7
0.6
0.35
0.247036
0.383426
0.404634
0.328612
0.3
0.5
0.4
0.305369
0.447383
0.428395
0.257539
0.270073
0.347863
0.303267
0.160863
0.25
0.3
0.2
0.2
0.1
0
0
0.15
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
x
Figure 4.15: (f2 /f1 )(x, y, 160◦ < θ ≤ 170◦ )
.
B theo = F (x, y, θ)dV,
(4.29)
4
F (x, y, θ) = f1 (x, y, θ) + η̄f2 (x, y, θ) + (1 − ρ)f3 (x, y, θ),
3
(4.30)
dV = dxdyd(cos θ).
(4.31)
where
and
Chapter 4: Data Analysis
85
°
°
y
|f2/f1|: θ[170 ,180 ]
1
0.9
0.55
0.14713
0.261525
0.383471
0.419825
0.5
0.8
0.45
0.7
0.6
0.267073
0.451008
0.540158
0.461801
0.4
0.5
0.4
0.35
0.326315
0.510713
0.5609
0.3
0.43747
0.3
0.25
0.2
0.1
0
0
0.298534
0.448825
0.459294
0.2
0.324561
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
x
0.15
Figure 4.16: (f2 /f1 )(x, y, 170◦ < θ ≤ 180◦ )
.
F (x, y, θ) is a function that needs to be calculated in a region of phase space spanned
by dV.
Briefly, for a function of one variable, the Monte Carlo integration method works
like this:
(i) Pick N randomly distributed points x1 , x2 , x3 , ......, xN in the interval [a, b].
(ii) Determine the average value of the function
N
1 X
hf i =
f (xi ),
N i=1
(4.32)
Chapter 4: Data Analysis
86
y
|f2/f1|: θ > 157
1
0.9
0.138161
0.211931
0.30015
0.394601
0.246254
0.382459
0.405341
0.356935
0.303646
0.443363
0.425877
0.278642
0.267354
0.350335
0.312488
0.181712
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 4.17: (f2 /f1 )(x, y, 157◦ < θ ≤ 180◦ )
.
(iii) Compute the approximation to the integral
Z
b
f (x)dx ≈ (b − a) ∗ hf i,
(4.33)
a
(iv) An estimate for the error is
r
Error ≈ (b − a)
hfi2 − hf 2 i
,
N
(4.34)
Chapter 4: Data Analysis
87
where
N
1 X 2
hf i =
f (xi ).
N i=1
2
(4.35)
Finally the χ2 difference is calculated as
2
χ (η̄) =
N
X
(B Exp (x, y, θ, η̄) − B Theo (x, y, θ, η̄))2
i
i=1
σi2 (B Exp ) + σi2 (B Theo )
(4.36)
where BiExp is experimental branching fraction, B Theo (η̄) is a calculated theoretical
branching fraction and σi (B Exp ) is a statistical error for each branching fraction,
BiExp .
4.2.1
Systematic error
The systematic errors associated with each branching fraction used in the calculation
of the χ2 were estimated in a similar way as described in Section 4.1.7. The systematic
error from photon energy calibration was estimated after finding the energy scale
factor and apply its error as a systematic (Figure 4.26). As the measured experimental
branching fraction depends on the applied photon energy scale, this energy scale factor
in turn influences the value of η̄.
The minimization required the variation of both energy scale factor and η̄ and
calculating the corresponding χ2 (Equation 4.36). The minimization was performed
for each of the four chosen bins of opening angle. The analysis showed that adding
Chapter 4: Data Analysis
88
Table 4.7: The values of η̄ obtained from different regions of phase
space as showed in Tables 4.12, 4.13 and 4.15.
Angle range
170◦ − 180◦
160◦ − 170◦
160◦ − 180◦
150◦ − 160◦
η̄ value
0.012 ± 0.017
0.002 ± 0.017
0.006 ± 0.017
0.042 ± 0.031
events with θ < 160◦ significantly dilutes the sensitivity to η̄ paramter. Figures (4.18,
4.20, 4.22 and 4.24) show the χ2 as a function of photon energy scale and the η̄
parameter. The η̄ parameter values were extracted after fixing the photon energy
scale factor at a value that gives the minimum χ2 . For large angles (160◦ < θ ≤ 180◦ )
the factor was found to be:
Cγ = 0.994 ± 0.003,
(4.37)
while for the bin of 150◦ < θ ≤ 160◦ the factor obtained was:
Cγ = 0.997 ± 0.003.
(4.38)
The extracted η̄ parameter values are shown in Figures (4.19, 4.21, 4.23 and 4.25).
Due to higher event statistics the results from the bin of 160◦ < θ ≤ 180◦ were taken
as our best results.
Tables (4.8, 4.9, 4.10, and 4.11) show the systematic errors associated with the
Photon Energy Scale
Chapter 4: Data Analysis
89
1.002
16.325
6.817
14.14
72.92
13.93
1.73
2.11
16.96
23.14
3.154
1.469
9.722
84.74
15.893
6.302
12.957
1
0.998
0.996
0.994
0.992
0.99
0.988
-0.04
-0.02
0
0.02
0.04
χ2
Figure 4.18: The χ2 as a function of photon energy scale and η̄ for
170◦ < θ ≤ 180◦ bin.
25
η = 0.012 ± 0.017
20
χ2/NDF = 0.48
15
10
5
0
-5-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
η
Figure 4.19: The χ2 as a function of η̄ for 170◦ < θ ≤ 180◦ bin, with
Cγ = 0.994.
η
Photon Energy Scale
Chapter 4: Data Analysis
90
1.002
8.32
9.495
20.01
92.08
7.583
1.522
3.49
23.37
15.564
2.002
1.481
13.009
71.75
13.488
5.789
14.612
1
0.998
0.996
0.994
0.992
0.99
0.988
-0.04
-0.02
0
0.02
0.04
η
χ2
Figure 4.20: The χ2 as a function of photon energy scale and η̄ for
160◦ < θ ≤ 170◦ bin.
20
η = 0.002 ± 0.017
15
χ2/NDF = 0.49
10
5
0
-5
-10
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
η
Figure 4.21: The χ2 as a function of η̄ for 160◦ < θ ≤ 170◦ bin, with
Cγ = 0.994.
Photon Energy Scale
Chapter 4: Data Analysis
91
1.002
11.408
8.37
15.387
67.599
9.285
2.417
3.45
18.328
15.92
3.03
2.37
11.401
63.29
13.764
7.898
12.668
1
0.998
0.996
0.994
0.992
0.99
0.988
-0.04
-0.02
0
0.02
0.04
η
χ2
Figure 4.22: The χ2 as a function of photon energy scale and η̄ for
160◦ < θ ≤ 180◦ bin.
18
16
η = 0.006 ± 0.017
14
χ2/NDF = 0.8
12
10
8
6
4
2
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
η
Figure 4.23: The χ2 as a function of η̄ for 160◦ < θ ≤ 180◦ bin, with
Cγ = 0.994.
Photon Energy Scale
Chapter 4: Data Analysis
92
1.006
1.004
16.35
13.62
11.26
11.38
13.3
17
22.47
29.74
13.75
10.31
7.79
6.17
5.47
5.68
6.79
8.82
19.79
15.1
11.33
8.48
6.55
5.55
5.47
6.31
24.77
19.48
15.11
11.68
9.19
7.64
7.02
7.33
1.002
1
0.998
0.996
0.994
0.992
0.99
0.988
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
η
χ2
Figure 4.24: The χ2 as a function of photon energy scale and η̄ for
150◦ < θ ≤ 160◦ bin.
20
18
η = 0.042 ± 0.031
16
14
12
χ2/NDF = 1.4
10
8
6
-0.08 -0.06 -0.04 -0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
η
Figure 4.25: The χ2 as a function of η̄ for 150◦ < θ ≤ 160◦ bin, with
Cγ = 0.997.
Chapter 4: Data Analysis
12
93
Cγ = 0.994 ± 0.003
10
8
6
4
2
0.985
0.99
0.995
1
1.005
Figure 4.26: The χ2 as a function of photon energy scale for 160◦ <
θ ≤ 180◦ bin, with η̄ = 0.006..
Table 4.8: Systematic errors associated with the measured experimental
branching fraction in the the region of phase space spanned by (0.5 <
x ≤ 0.75 , 0.28 < y ≤ 0.48, and 160◦ < θ ≤ 180◦ )
Quantity
Photon energy calibration
Background subtraction
Positron energy calibration
Cosine opening angle
Time window selection
Misid. events
Total Syst. Error
Syst. Error(%)
0.75
0.15
0.24
0.32
0.14
0.04
0.87
Chapter 4: Data Analysis
94
Table 4.9: Systematic errors associated with the measured experimental
branching fraction in the the region of phase space spanned by (0.5 <
x ≤ 0.75, 0.5 < y ≤ 0.75, and 160◦ < θ ≤ 180◦ ).
Quantity
Photon energy calibration
Background subtraction
Positron energy calibration
Cosine opening angle
Time window selection
Misid. events
Total Syst. Error
Syst. Error(%)
0.72
0.18
0.25
0.28
0.12
0.03
0.84
Table 4.10: Systematic errors associated with the measured experimental branching fraction in the the region of phase space spanned by
(0.28 < x ≤ 0.48, 0.5 < y ≤ 0.75, and 160◦ < θ ≤ 180◦ ).
Quantity
Photon energy calibration
Background subtraction
Positron energy calibration
Cosine opening angle
Time window selection
Misid. events
Total Syst. Error
Syst. Error(%)
0.69
0.16
0.23
0.33
0.15
0.03
0.83
branching fractions used to extract the η̄ parameter. These errors in turn introduced
a systematic uncertainty in the measurement of the η̄ parameter. Table 4.16 shows
the details of the contribution from each source of uncertainty, with photon energy
calibration being a dominant source.
Chapter 4: Data Analysis
95
Table 4.11: Systematic errors associated with the measured experimental branching fraction in the the region of phase space spanned by
(0.28 < x ≤ 0.48, 0.28 < y ≤ 0.48, and 160◦ < θ ≤ 180◦ ).
Quantity
Photon energy calibration
Background subtraction
Positron energy calibration
Cosine opening angle
Time window selection
Misid. events
Total Syst. Error
Syst. Error(%)
0.76
0.17
0.24
0.32
0.12
0.03
0.88
Table 4.12: Event statistics for the experimental branching fractions
used in η̄ extraction in the bin of 170◦ < θ ≤ 180◦ . Calculated theoretical branching fraction in the region of phase space covered by
(0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 170◦ < θ ≤ 180◦ ) is 2.41 × 10−6 .
x Range
0.28 < x ≤ 0.48
0.28 < x ≤ 0.48
0.5 < x ≤ 0.75
0.5 < x ≤ 0.75
y Range
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
Br. Fraction (10−6 )
2.42 ± 0.04(stat.)
2.39 ± 0.05(stat.)
2.43 ± 0.04(stat.)
2.46 ± 0.04(stat.)
No. of Events
564 ± 29
581 ± 33
389 ± 36
384 ± 41
The obtained experimental result for parameter η̄ extracted using statistical error
(shown in Figure 4.23) is:
η̄ = 0.006 ± 0.017,
(4.39)
η̄ = 0.006 ± 0.017 (stat.) ± 0.018 (syst.).
(4.40)
Our final experimental result is:
Chapter 4: Data Analysis
96
Table 4.13: Event statistics for the experimental branching fractions
used in η̄ extraction in the bin of 160◦ < θ ≤ 170◦ . Calculated theoretical branching fraction in the region of phase space covered by
(0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 160◦ < θ ≤ 170◦ ) is 8.17 × 10−6 .
x Range
0.28 < x ≤ 0.48
0.28 < x ≤ 0.48
0.5 < x ≤ 0.75
0.5 < x ≤ 0.75
y Range
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
Br. Fraction (10−6 )
8.04 ± 0.01
8.17 ± 0.08
8.29 ± 0.02
8.14 ± 0.01
No. of Events
1840 ± 52
1924 ± 62
1301 ± 60
1424 ± 72
Table 4.14: Event statistics for the experimental branching fractions
used in η̄ extraction in the bin of 160◦ < θ ≤ 180◦ . Calculated theoretical branching fraction in the region of phase space covered by
(0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 160◦ < θ ≤ 180◦ ) is 1.02 × 10−5 .
x Range
0.28 < x ≤ 0.48
0.28 < x ≤ 0.48
0.5 < x ≤ 0.75
0.5 < x ≤ 0.75
y Range
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
Br. Fraction (10−5 )
1.03 ± 0.03
1.01 ± 0.03
1.06 ± 0.01
1.04 ± 0.01
No. of Events
2416 ± 62
2494 ± 72
1698 ± 66
1816 ± 81
Table 4.15: Event statistics for the experimental branching fractions
used in η̄ extraction in the bin of 150◦ < θ ≤ 160◦ . Calculated theoretical branching fraction in the region of phase space covered by
(0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 150◦ < θ ≤ 160◦ ) is 1.59 × 10−5 .
x Range
0.28 < x ≤ 0.48
0.28 < x ≤ 0.48
0.5 < x ≤ 0.75
0.5 < x ≤ 0.75
y Range
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
0.28 < y ≤ 0.48
0.5 < y ≤ 0.75
Br. Fraction (10−5 )
1.57 ± 0.05
1.58 ± 0.02
1.55 ± 0.03
1.64 ± 0.03
No. of Events
3484 ± 73
3371 ± 76
3030 ± 84
2732 ± 88
Chapter 4: Data Analysis
97
Table 4.16: Systematic errors associated with the extraction of η̄ parameter.
Quantity
Photon energy calibration
Background subtraction
Positron energy calibration
Cosine opening angle
Time window selection
Misid. events
Total Syst. Error
Syst. Error(δ η̄)
0.016
0.003
0.005
0.006
0.002
0.0003
0.018
Standard statistical method [15] of setting upper limit value for a parameter was
applied to calculate the upper limit of the η̄ parameter. In this process we used the
fact that η̄ parameter is by its definition expected to have a value between zero and
one [5]. The upper limit η̄max is thus determined by
h
i
(η̄−η̄0 )2 )
exp
−
0
2ση2¯0
i
h
C.L = R 1
(η̄−η̄0 )2
exp − 2σ2
0
R η̄max
(4.41)
η¯0
where C.L is the confidence level which can be set as 0.683, 0.95 or 0.99.
The application of Equation (4.41) on the experimental result (4.40), sets the
upper limit value of the η̄ to be:
η̄ ≤ 0.028 (68.3% confidence).
(4.42)
Chapter 4: Data Analysis
98
The combined uncertainty in this work, δ η̄ = 0.025, is about four times lower than
that in the best previous measurement [9]. Our new 68% CL upper limit is four times
lower than the previous average [6].
4.3
Conclusions
The experimental results obtained in this work can be compared to previous measurements [10] and [9]. The present experimental branching fraction is consistent with
values obtained in Ref [10] which is not surprising as similar kinematic cuts were
applied in both analyses.
The value of Michel parameter ρ obtained in this analysis is consistent with the
one obtained in [10], but the precision of η̄ values determination is significantly improved in this work. The improvement is caused by these factors: (a) a more precise
calibration of data was implemented, (b) a larger data set was used, including runs
skipped in the previous analysis, and (c) the phase space region for the η̄ fits was
restricted to the most sensitive subset. This analysis used only events that were
reconstructed in large opening angle (160◦ < θ ≤ 180◦ ) were used, in [10] the cut
on the opening angle was lowered to include all events that pass 90◦ < θ ≤ 180◦ .
From Figures (4.14, 4.15, 4.16, and 4.17), we see that the most sensitive region for η̄
extraction is in the large opening angles. In the end, the gain in sensitivity offset the
lower event statistics in the present, more restrictive region of phase spaces.
Chapter 4: Data Analysis
99
The experimental result for η̄ obtained from this analysis is still consistent with the
Standard Model value. This result, together with other experimental determinations
of Michel parameters, will be included in a global analysis in order to determine limits
to deviations from V − A structure of weak interaction.
Appendix A
The functions fi(x, y, θ)
α
(1 − RDalitz )n0V
16π 2 y
(A.1)
f2 (x, y, θ) =
α
(1 − RDalitz )(2n0S − 2n0V + n0T )
16π 2 y
(A.2)
f2 (x, y, θ) =
α
(1 − RDalitz )(2n0S + 2n0V − n0T )
2
16π y
(A.3)
f1 (x, y, θ) =
RDalitz =
ymµ
19
2α
[ln(
)− ]
3π
2me
12
(A.4)
RDalitz is the probability that the emitted photon converts into e+ e− . The other
100
Appendix A: The Functions fi (x, y, θ)
101
variables noV , noS , noT are defined as follows:
noV = 4(1 − β 2 )[2∆−2 x(x + y)(2(2 + y) − 3),
∆−1 x2 y(3 − 4(x + y)) + x3 y 2 ],
(A.5)
+∆−1 GV−1 + GVo + ∆GV1 + ∆2 GV2 ,
2n0S − 2n0V + n0T = −8xy 2 [∆−1 x(1 − β 2 + 2(1 − y − 2x) + ∆x(1 + y)],
(A.6)
2n0S + 2n0V − n0T = 4(∆−1 2[y 2 (3 − 4y) + 6xy(1 − 2y) + 2x2 (3 − 8y) − 8x3 ]
+2x[−y(3 − y − 6y 2 ) − x(3 − 5y − 10y 2 ) + 4x2 (1 + 2y)]
+∆x2 y[1.5(2 − 3y − 4y 2 ) − 2x(4 + 3y)]
(A.7)
+∆2 x3 y 2 (2 + y) − (1 − β 2 )[∆−2 2x(x + y)(3 − 4(x + y))
+∆−1 x2 y(2(4x + 5y) − 3) − 2x3 y 2 ]),
GV−1 = 8[y 2 (3 − 2y) + 6xy(1 − y) + 2x2 (3 − 4y) − 4x3 ],
(A.8)
GVo = 8[−xy(3 − y − y 2 ) − x2 (3 − y − 4y 2 ) + 2x3 (1 − 2y)],
(A.9)
GV1 = 2[x2 y(6 − 5y − 2y 2 ) − 2x3 y(4 + 3y)],
(A.10)
Appendix A: The Functions fi (x, y, θ)
GV2 = 2x3 y2(2 + y),
102
(A.11)
Appendix B
The Least Squares Method
This Appendix will outline the method of Least Squares that is used in estimating
parameters, given a set of experimental data. Data set will consist of a group of points
xi , y(xi ) and σi , where xi is the independent variable, y(xi ) is the dependent variable,
and σi is the standard deviation of y(xi ). The subscript i denotes a particular element
of N data points. This data set is to be fit to an equation f (xi , a) which is a function
of the independent variable and the the vector a of parameters to be evaluated. The
fit is to be performed such that the dependent variable y(xi ) can be approximated by
the fitting function evaluated at the corresponding independent variable xi , and the
parameter values having the maximum likelihoof of being correct: that is
y(xi ) u f (xi , a).
103
(B.1)
Appendix B: The Least Squares Method
104
Least Squares analysis is comprised of a group of numerical procedures that can be
used to evaluate the optimal values of the parameters in vector a for the experimental
data. In general, Least Squares procedure consist of an algorithm that uses an initial
approximation vector of parameters to be used in determining which one of them
has the optimal value. It is iterative process which is continued until the vector
of parameters converges to value such that the weighted sum of the squares of the
differences between the fitted function and the experimental data weighted by the
inverse of variances is minimal.
The values of parameters are taken as those that minimize the quantity
2
χ (a) =
2
N X
y(xi ) − f (xi , a)
i=1
σi
.
(B.2)
To demonstrate that the Least Squares method is appropriate and will yield the
parameters a having the maximum likelihood of being correct, several interrelated
assumptions must be made. We must assume: (1) that all of the experimental uncertainty can be attributed to the dependent variables y(xi ), (2) that the experimental
uncertainties of the data can be described by a Gaussian distribution, (3) that no
systematic errors exist in the data, (4) that the functional form f (xi , a) is correct,
(5) that there are enough data points to provide a good sampling of the experimental
uncertainties, and (6) that the observations (data points) are independent of each
other.
Appendix B: The Least Squares Method
105
Least Squares method is also connected to another method of parameter estimation called the Maximum Likelihood. This can be shown as follows: by assuming
(1) that the experimental uncertainties of the data are all in the dependent variables
y(xi ), (2) that the experimental uncertainties of data follow a Gaussian distribution,
(3) that no systematic errors exist, and (4) that the fitting function is correct, then
the N measurements of y(xi ) can as well be taken as a single measurement of an
N -dimensional random vector, whose joint p.d.f is the product of N Gaussians,
P (a; xi ; y(xi ); σi2 )
=
N
Y
i=1
1
p
exp
2πσi2
−(y(xi ) − f (xi , a))2
2σi2
(B.3)
If we take the logarithm of the joint p.d.f we get a log-likelihood function,
N
1X
logL(a) = −
2 i
y(xi ) − f (xi , a)
σi
2
(B.4)
This is maximized by finding the values of parameters a that minimize the quantity
2
χ (a) =
2
N X
y(xi ) − f (xi , a)
i=1
σi
.
(B.5)
Bibliography
[1] D.A. Greenwood W.N. Cottingham. An Introduction to the Standard Model of
Particle Physics. Cambridge, 2003.
[2] B. Muller W. Greiner. Gauge Theory of Weak Interactions. Springer, fourth
edition, 2009.
[3] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson. Experimental Test of Parity Conservation in Beta Decay. Phys. Rev., 105:1413–1414,
1957.
[4] Michael E. Peskin and Daniel V. Schroeder. An Introduction to Quantum Field
Theory. Addison-Wesley, 2002.
[5] L. Michel. Interaction Between Four Half Spin Particles and the Decay of the
Mu Meson. Proc. Phys. Soc., A63:514–531, 1950.
[6] K. Nakamura et al.(Particle Data Group). 2011 Review of Particle Physics.
Journal of Physics, G37:1, 2011.
106
bibliography
107
[7] C. Fronsdal and H. Überall. µ-Meson Decay with Inner Bremsstrahlung. Physical
Review, 113:654–657, January 1959.
[8] S. G. Eckstein and R. H. Pratt. Radiative Muon Decay. Annals of Physics,
8:297–309, 1959.
[9] W. Eichenberger, R. Engfer, and A. Van Der Schaaf. Measurement of the Parameter η̄ in the Radiative Decay of the Muon as a Test of the V-A Structure of
the Weak Interaction. Nucl. Phys., A412:523–533, 1984.
[10] B. A. VanDevender. An Experimental Study of Radiative Muon Decay. PhD
thesis, University of Virginia, Charlottesville, Virginia, 2006.
[11] Frlež et al. Design, Commissioning and Performance of the PIBETA detector at
PSI. Nucl. Instrum. Meth., pages 300–347, 2004.
[12] R. Brun, F. Bruyant, M. Maire, A. C. McPherson, and P. Zanarini. GEANT
3.21. CERN, Geneva, 1994.
[13] J. R. Musser et al. Measurement of the Michel Parameter ρ in Muon Decay.
Phys. Rev. Lett., 94:101805, 2005.
[14] Charles A. Rey. Inner bremsstrahlung in muon decay. Phys. Rev., 135:1215–1224,
1964.
[15] Glen Cowan. Statistical Data Analysis. Oxford, 1998.