An Experimental Study of Radiative Muon Decay
Brent Adam VanDevender
Poquoson, VA
B.S., University of Virginia, 1996
M.A., University of Virginia, 2002
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
January, 2006
Abstract
Experimental measurements of the Michel parameter η can be used, along with the
other Michel parameters appearing in the description of muon decays, to set limits on possible violations of the V − A form of the weak interaction. All of the
Michel parameters, save for η, can be measured by analyzing the ordinary muon
decay µ+ → e+ νe ν µ . To measure η, the radiative decay µ+ → e+ νe ν µ γ must be
observed. This work is based on more than 4 × 105 radiative muon decays observed
at the Paul Scherrer Institute meson factory using a large acceptance spectrometer.
Based on these events we measure the branching ratio for the radiative decay, with the
restrictions Eγ > 10 MeV on the photon energy and θ > 30◦ on the positron/photon
opening angle, to be B = [4.40 ± 0.02 (stat.) ± 0.09 (syst.)] × 10−3 . The best fit for the
branching ratio is found to occur for η = −0.084 ± 0.050(stat.) ± 0.034(syst.), to be
compared to the V − A Standard Model value η SM = 0. We interpret our result as an
upper limit on the allowed value: η ≤ 0.033 (68 % confidence). Combined with other
measurements of η, this reduces the known upper limit to η ≤ 0.028 (68 % confidence).
Contents
1 Introduction
1.1 The Standard Model of Particle Physics . . . .
1.2 Muons and The Weak Interaction . . . . . . . .
1.2.1 Muon Decay . . . . . . . . . . . . . . . .
1.2.2 Michel Decay: µ+ → e+ νe ν µ . . . . . . .
1.2.3 Radiative Michel Decay: µ+ → e+ νe ν µ γ
1.3 Motivation for This Work . . . . . . . . . . . .
2 The
2.1
2.2
2.3
2.4
2.5
PIBETA Apparatus
Introduction . . . . . . . . . . . . . . .
PSI Proton Cyclotron . . . . . . . . . .
PIBETA Detector . . . . . . . . . . . .
2.3.1 πE1 Beam Line . . . . . . . . .
2.3.2 Thin Tracking Detectors . . . .
2.3.3 Calorimeter . . . . . . . . . . .
Electronics . . . . . . . . . . . . . . . .
2.4.1 Triggers . . . . . . . . . . . . .
2.4.2 Front-End Computer Efficiency
Data Analysis Software . . . . . . . . .
2.5.1 Calorimeter Clumps . . . . . .
2.5.2 Track Finding Algorithm . . . .
3 Michel Decay Analysis
3.1 Introduction . . . . . . . . . . . . . . .
3.2 Event Selection . . . . . . . . . . . . .
3.2.1 Kinematic Cuts . . . . . . . . .
3.2.2 Time Structure of Muon Decays
3.3 Results . . . . . . . . . . . . . . . . . .
3.3.1 Branching Ratio . . . . . . . .
3.3.2 Michel Parameter ρ . . . . . . .
3.4 Conclusions . . . . . . . . . . . . . . .
i
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1
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55
55
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68
ii
4 Radiative Michel Decay Analysis
4.1 Introduction . . . . . . . . . . .
4.2 Strategy . . . . . . . . . . . . .
4.2.1 Branching Ratio . . . .
4.2.2 Parameter Optimization
4.3 Event Selection . . . . . . . . .
4.3.1 Time Window . . . . . .
4.3.2 Time Coincidence . . . .
4.3.3 Kinematic Cuts . . . . .
4.4 Results . . . . . . . . . . . . . .
4.4.1 The Parameters η and ρ
4.4.2 Branching Ratio . . . .
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71
71
72
72
77
82
82
83
87
90
90
97
A The Functions fi (x, y, θ)
104
B Radiative Michel Decay Event Statistics
106
List of Figures
differential decay rate for µ+ → e+ νe ν µ . . . . . .
Standard Model contribution to the µ+ → e+ νe ν µ γ
sensitivity of µ+ → e+ νe ν µ γ to the parameter η . .
sensitivity of µ+ → e+ νe ν µ γ to the parameter ρ . .
1.1
1.2
1.3
1.4
The
The
The
The
. .
ratio
. .
. .
11
15
16
17
2.1
2.2
2.3
2.4
The accelerator facilities at PSI . . . . . . . . . . . . . . . . . . . . .
The PIBETA detector in cross-section parallel to the beam direction .
The PIBETA calorimeter in relief . . . . . . . . . . . . . . . . . . . .
The PIBETA target and tracking detectors in cross-section perpendicular to the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Event signal pileup in the target . . . . . . . . . . . . . . . . . . . . .
The effect of target pileup on the measured energy spectrum . . . . .
Calibrated energy spectra for each target . . . . . . . . . . . . . . . .
Muon decay vertex distributions . . . . . . . . . . . . . . . . . . . . .
The calibrated energy deposited in the PV hodoscope for one-arm lowthreshold trigger events. . . . . . . . . . . . . . . . . . . . . . . . . .
Tracking detector efficiencies shown to be independent of particle energy
The spectrum of positrons with 40 < ECsI < 76 MeV in the one-arm
low-threshold trigger . . . . . . . . . . . . . . . . . . . . . . . . . . .
The spectrum of positrons with 0 < ECsI < 60 MeV in the one-arm
low-threshold trigger . . . . . . . . . . . . . . . . . . . . . . . . . . .
Energy deposited in the CsI veto crystals . . . . . . . . . . . . . . . .
The individual ingredients of a pion stop signal . . . . . . . . . . . .
A sketch of the beam trigger logic . . . . . . . . . . . . . . . . . . . .
The angular separation between wire-chamber tracks and calorimeter
clumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The identification of particles based on EPV vs. EPV + ECsI . . . . . .
22
23
23
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
3.1
3.2
3.3
. . . . . .
branching
. . . . . .
. . . . . .
The relative difference in the µ+ → e+ νe ν µ decay rate when ρ = ρSM ±
0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The calculated time spectrum of muon decays . . . . . . . . . . . . .
The measured time spectrum of muon decays . . . . . . . . . . . . .
iii
24
26
27
29
30
33
35
38
39
40
43
44
52
54
56
62
63
iv
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
A graphical definition of “thrown” and “detected” cuts . . . . . . . . 78
Muon gate fraction cancellation in the nine-piece target data set . . . 84
Muon gate fraction cancellation in the one-piece target data set . . . 85
Radiative muon decay event timing signal-to-background . . . . . . . 88
χ2 (η, ρ) for the nine-piece target data set . . . . . . . . . . . . . . . . 93
χ2 (η, ρ) for the one-piece target data set . . . . . . . . . . . . . . . . 94
χ2 (η, ρSM ) for the nine-piece target data set . . . . . . . . . . . . . . 95
χ2 (η, ρSM ) for the one-piece target data set . . . . . . . . . . . . . . . 96
The simulated opening angle distribution of misidentified nonradiative
decay events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.10 The radiative Michel decay kinematic spectra for the nine-piece target
data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.11 The radiative Michel decay kinematic spectra for the one-piece target
data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
List of Tables
1.1
1.2
1.3
Muon decay modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
γ
Experimental limits on the coupling constants |gαβ
| . . . . . . . . . .
The primary decay modes registered by the PIBETA experiment. . .
5
8
19
2.1
2.2
2.3
2.4
Scale factors for calorimeter energy calibration
Hardware prescaling factors . . . . . . . . . .
Members of a “clump” data structure . . . . .
Members of a “track” data structure . . . . .
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40
46
49
51
3.1
3.2
3.3
3.4
Various time scales involved in the PIBETA experiment.
Tracking efficiencies and prescale factors . . . . . . . . .
µ+ → e+ νe ν µ results for the nine-piece target data set . .
µ+ → e+ νe ν µ results for the one-piece target data set . .
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59
67
69
70
4.1
4.2
4.3
4.4
4.5
Time windows within which muon decay events are accepted .
Statistics for the normalizing, nonradiative decay µ+ → e+ νe ν µ
Optimal values of η and ρ . . . . . . . . . . . . . . . . . . . .
Gain factor correction for photons . . . . . . . . . . . . . . . .
Results for the radiative muon decay branching ratio . . . . .
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. 101
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B.1 Radiative Michel decay total event statistics . . . . . . . . . . . . . . 107
B.2 Event statistics for the nine-piece target data set in the first bin of cos θ.107
B.3 Event statistics for the nine-piece target data set in the second bin of
cos θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.4 Event statistics for the nine-piece target data set in the third bin of
cos θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.5 Event statistics for the one-piece target data set in the first bin of cos θ.109
B.6 Event statistics for the one-piece target data set in the second bin of
cos θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.7 Event statistics for the one-piece target data set in the third bin of cos θ.110
v
Chapter 1
Introduction
1.1
The Standard Model of Particle Physics
The Standard Model of Particle Physics is one of the great triumphs of modern
science. It is a powerful theory of the fundamental laws of nature supported by
extensive experimental evidence. Nearly all of its predictions have been fulfilled and
no discordant measurements have yet withstood scientific scrutiny.
Nevertheless, the particle physics community is currently at a stalemate with the
Standard Model. In spite of its many successes and the absence of any apparent
shortcomings, the Standard Model is clearly not the ultimate theory which we seek.
It is not truly fundamental as there is structure in the Standard Model which is not
understood. The situation is similar to that of the Periodic Table of the Elements in
the time of Mendeleev. The Periodic Table was (and is) a powerful organizational tool
1
Chapter 1: Introduction
2
for the known elements. It facilitated the prediction of several new elements which
were subsequently discovered and found to have the expected properties. However,
the organization into rows and columns was just a mnemonic device that arranged
elements according to their observed properties. Only after the advent of quantum
mechanics and the discovery that atoms are actually composite objects could the
structure of the Periodic Table be understood. In that case, Quantum Mechanics
provided the more basic understanding. In the case of the Standard Model, it is
unclear where the answers lie. Current experiments push technology to its limits to
search for shortcomings of the Standard Model that could indicate the direction our
inquiries should take. This work describes one of those experiments.
1.2
Muons and The Weak Interaction
According to our current understanding, there are four fundamental interactions
which occur in nature: electromagnetic (EM), weak nuclear, strong nuclear (or simply
weak and strong), and gravitational. These completely describe the behavior of the
fundamental particles: the leptons, quarks and gauge bosons. The gauge bosons mediate the interactions between the quarks and leptons and even among themselves.
The Standard Model encompasses the electromagnetic, weak and strong interactions.
It does not describe gravity, which in any case is of negligible strength compared to
the other interactions at the microscopic scale.
Chapter 1: Introduction
3
In this work, we will be concerned only with the weak interaction. All fundamentals particles interact weakly. In principle, we could therefore use any particle we
wished to do our experiment. However, any experiment involving hadrons, which are
composed of quarks, would be complicated by the presence of strong interactions, for
which effects are very difficult to account. It is most convenient to use leptons, which
are impervious to strong interactions to very good approximation. The additional
presence of electromagnetic interactions introduces no great complications, as these
are very well understood. The ideal lepton is the muon. It is heavy enough to decay
into lighter leptons (electrons and neutrinos) and photons but not heavy enough to
decay into hadrons. The lightest hadron, the pion, is heavier than the muon and such
decays are therefore prohibited by the conservation of energy.
The muon entered particle physics history in 1937 when Neddermeyer and Anderson unwittingly discovered it in cosmic rays [27]. At the time it was believed to
be the pion, which had been predicted by Yukawa in 1935 [33] as the mediator of
the strong nuclear force. A decade later however, experiments demonstrated that the
new particle did not participate in the strong interaction and therefore could not be
Yukawa’s pion [5]. The discovery of a new and unexpected particle caused Rabi famously to exclaim “Who ordered that?”. The notion that the newly discovered muon
was simply a heavy electron was also discounted by the low rate of the decay mode
µ+ → e+ γ, which was found to be B < 10−1 in 1948 [19]. The continuous energy
spectrum of the electrons from muon decay was established the same year, indicating
Chapter 1: Introduction
4
that a muon decayed into an electron and two neutral particles [31]. The discovery
of parity violation [32] prompted Feynman and Gell-Mann to suggest that the weak
interaction proceeded via the exchange of charged intermediate vector bosons [14].
This mechanism however, predicted a branching ratio for µ+ → e+ γ larger than the
known upper-limit [12]. This led several authors to hypothesize that the neutrino
which coupled to the muon was different from that coupled to the electron thus forbidding the decay µ+ → e+ γ [28, 30] (now known to be B < 1.2 × 10−11 [2]) and
furthermore to the idea that neutrinos were massless fermions with only one possible
spin state [23]. An experiment at Brookhaven National Laboratory [7] verified the
two neutrino hypothesis a few years later, implying that lepton flavors were conserved
separately.
Present experiments take advantage of the muon’s indifference to strong interactions and the relative ease with which they are produced at modern facilities to make
very clean and precise measurements. It is hoped that these measurements will eventually reveal the inevitable signal of the theory which underlies the Standard Model.
Reference [22] gives a comprehensive review of these experiments and their prospects.
5
Chapter 1: Introduction
Table 1.1: Muon decay modes.
decay mode
branching ratio
µ+ → e + ν e ν µ
100 %
µ+ → e+ νe ν µ γ (Eγ > 10 MeV)
µ+ → e + ν e ν µ e + e −
µ+ → e + γ
1.2.1
reference
(1.4 ± 0.4) %
[6]
(3.4 ± 0.4) × 10−5
[1]
< 1.2 × 10−11
[2]
Muon Decay
As noted above, muons decay into electrons and neutrinos and possibly also a photon,
which may internally convert to an electron/positron pair:
µ+ → e+ νe ν µ (γ).
(1.1)
Table 1.1 lists the decay modes and measured values of their branching ratios, or upper
limits on the branching ratio in the case of unobserved modes. We shall use positively
charged muons (µ+ ) in our notation throughout this work. The corresponding decays
of negatively charged muons are related by charge conjugation, which is known to be
a very good symmetry of nature. Reference [11] discusses current limits on possible
violations of charge symmetry.
The dominant process, µ+ → e+ νe ν µ , also referred to as the Michel decay, is the
fate of all muons. Technically, this decay also includes µ+ → e+ νe ν µ γ, as is implied
6
Chapter 1: Introduction
by the photon in parentheses in (1.1). The vast majority of the latter decays involve
a photon of very low energy emitted collinearly with the positron. These decays
are thus indistinguishable from the former decay for all practical purposes. There
is a significant probably however, that the decay is accompanied by a hard photon
emitted at a large angle with respect to the positron. We shall treat this decay
separately below (section 1.2.3). In any case, the term Michel decay refers to the
process (1.1) with photons of any energy and will be denoted by µ+ → e+ νe ν µ in
order to distinguish it from the case of µ+ → e+ νe ν µ γ, with an explicit hard photon.
The most generic four-fermion point interaction Hamiltonian describing muon
decay assumes only Lorentz invariance, local (i.e., derivative free) interactions and
lepton-number conservation. The point interaction permits several equivalent Hamiltonians, related to each other via Fierz transformations, which differ in the way the
fermions are grouped together. We choose the charge-exchange order, with fields of
definite handedness, for which the matrix element is given by [13]
E
ED
GF X γ D
γ
γ
M = 4√
gαβ eα |Ô |(νe )n (ν µ )m |Ô |µβ ,
2 γ=S,V,T
(1.2)
α,β=R,L
where GF is the Fermi coupling constant. The labels α and β denote left- (L) or
right-handed (R) chirality of the positron and muon respectively. The chiralities of
the neutrinos, labeled m and n, are uniquely determined for each combination of α, β
and γ. The label γ distinguishes all of the interactions Ôγ allowed by Lorentz invariance: scalar (S), vector (V ) and tensor (T ). These names indicate the behavior of
7
Chapter 1: Introduction
heα |Ôγ |(νe )n i and h(ν µ )m |Ôγ |µβ i under Lorentz transformations and parity inversions.
The explicit forms of the operators are
ÔS = 1
ÔV
= γµ
ÔT = iσµν ≡
(1.3)
i
{γµ , γν } ,
2
where the γµ s are the usual Dirac matrices satisfying the anticommutation relations
{γµ , γν } = gµν
(1.4)
and gµν is the metric tensor.
γ
There are ten complex coupling constants gαβ
in Equation (1.2). One might
T
T
naively expect twelve, but the terms involving gLL
and gRR
vanish identically due to
the algebra of the associated currents. The constants are subject to the normalization
condition [13]
¡ S 2
¢
S 2
S 2
S 2
nS |gRR
| + |gLL
| + |gRL
| + |gLR
|
¡ V 2
¢
V 2
V 2
V 2
+ nV |gRR
| + |gLL
| + |gRL
| + |gLR
|
(1.5)
¡ T 2
¢
T 2
+ nT |gRL
| + |gLR
|
= 1,
γ 2
| represents the relative probawhere nS = 14 , nV = 1 and nT = 3. Physically, nγ |gαβ
bility that a β-handed muon will decay into an α-handed electron via the interaction
Ôγ . There is no a priori reason to expect that any of these couplings vanish. However, all experimental tests are consistent with a weak interaction which has only V
8
Chapter 1: Introduction
γ
Table 1.2: Experimental limits on the coupling constants |gαβ
|, derived
from various muon decay experiments [11]. All numbers represent a
90% confidence level and are upper limits, unless specifically noted
otherwise.
The maximum values allowed by definition are 2, 1 and
√
1/ 3 for S, V and T , respectively.
γ
|gαβ
|
S
V
T
LL
0.550
> 0.960
≡0
LR
0.125
0.060
0.036
RL
0.424
0.110
0.122
RR
0.066
0.033
≡0
coupling between left-handed muons and left-handed electrons. This fact is built into
the Standard Model by setting
V
gLL
=1
(1.6)
with all other coupling constants vanishing, as they must according to Equation (1.5).
It is important to understand that although this action is consistent with experimental
evidence, experimental uncertainties still allow the possibility of small but non-zero
values of the other constants. The current experimental limits are given in Table 1.2.
Reference [13] gives a comprehensive review of the experiments which led to those
limits.
9
Chapter 1: Introduction
1.2.2
Michel Decay: µ+ → e+ νe ν µ
Beginning from the matrix element (1.2), one can arrive at the differential decay rate
for Michel decay [13]:
mµ 4 2
d2 Γ
= 3 Weµ
GF
dx d(cos θ)
4π
q
h
i
x2 − x20 [FIS (x) + Pµ+ cos θFAS (x)] 1 + P~e+ (x, θ) · ζ̂
(1.7)
where Weµ = (m2µ + m2e )/(2mµ ), x = Ee+ /Weµ and x0 = me /Weµ . Here, Ee+ is the
energy of the positron and mµ and me are the masses of the muon and positron,
respectively. The range of allowed positron energies is me ≤ Ee+ ≤ Weµ , or equivalently, x0 ≤ x ≤ 1. The variable θ is the angle between the muon polarization P~µ
and the positron momentum and 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 the
positron along the direction of its momentum. The functions FIS and FAS are the
isotropic and anisotropic parts of the positron energy spectrum. They are given by:
2
FIS (x) = x(1 − x) + ρ(4x2 − 3x − x20 ) + ηx0 (1 − x),
9
(1.8)
and
1
FAS (x) = ξ
3
q
x2
−
x20
µ
·
µq
¶¸¶
2
2
1 − x + δ 4x − 3 +
1 − x0 − 1
.
3
(1.9)
The parameters ρ, η, ξ and δ are called the Michel parameters [25].
The expression for the differential decay rate (1.7) can be simplified for the case
where polarizations are not observed. Averaging over the possible polarizations in-
10
Chapter 1: Introduction
volves integrating cos θ over the antisymmetric interval −1 ≤ cos θ ≤ 1:
Z
1
cos θ d(cos θ) = 0.
(1.10)
−1
Thus, the term in (1.7) involving FAS vanishes. An analogous argument leads to the
vanishing of the term P~e (x, θ) · ζ̂. We shall also neglect the last term in FIS since x0 is
small (x0 = 9.67 × 10−3 ) and furthermore the parameter η is measured to be within
1 % of its Standard Model value, ηSM = 0 [22]. With these modifications we get the
final form of the differential decay rate for µ+ → e+ νe ν µ when no polarizations are
observed:
dΓ
mµ 4 2
= 3 Weµ
GF
dx
π
q
x2
−
x20
·
¸
2
2
2
x(1 − x) + ρ(4x − 3x − x0 ) .
9
(1.11)
The relative rate is plotted in Fig. 1.1.
The appropriate, simplified decay rate (1.11) is explicitly dependent on the Michel
parameter ρ. This parameter is related to the coupling constants in Equation (1.2):
ρ=
¤
3 3£ V 2
V 2
T 2
T 2
S
T∗
S
T∗
− |gLR | + |gRL
| + 2|gLR
| + 2|gRL
| + Re(gRL
gRL
+ gLR
gLR
) .
4 4
(1.12)
Recalling the Standard model prescription (1.6) and the normalization condition (1.5),
one easily obtains the Standard Model value of ρ:
3
ρSM = .
4
(1.13)
A recent experiment has resulted in a very precise determination [26]:
ρ = 0.7508 ± 0.0011,
(1.14)
Chapter 1: Introduction
11
Figure 1.1: The differential decay rate (1.11) for µ+ → e+ νe ν µ . The
variable x = 2Ee /mµ is the e+ energy in dimensionless units.
in agreement with the Standard Model prediction. A measurement in contradiction
to the theory would imply scalar, vector and/or tensor coupling between left-handed
muons and right-handed electrons or vise-versa, although one would not be able to
distinguish exactly which couplings were present on the basis of this measurement
alone. However, a corroborating measurement such as (1.14) is itself insufficient
grounds to rule out the possibility of deviation from the accepted V − A interaction,
even in the idealized case of an exact measurement with no uncertainties. Inspection
S
S
V
V
of Equation (1.12) reveals that any arbitrary values of gLL
, gRR
, gRR
and gLL
can still
result in ρ = 34 .
Chapter 1: Introduction
1.2.3
12
Radiative Michel Decay: µ+ → e+ νe ν µ γ
The measurement of any individual Michel parameter is generally insufficient to determine the complete Lorentz structure of the weak interaction [13], as discussed
above for the case of ρ. To build knowledge of the interaction then, we need to
measure additional parameters. The parameter ρ exhausts the possibilities for the
Michel decay in the case where polarizations are not observed. Fortunately, we may
still proceed so long as we can observe photons by analyzing the radiative Michel
decay µ+ → e+ νe ν µ γ, where the hard photon is explicitly observed as a particle distinct from the positron. Approximately 1.2% of all muon decays are accompanied by
a photon with energy E > 10 MeV [21, 6]. The presence of the additional photon
provides more access to the parameters of the weak interaction and, since the photon couples electromagnetically to either the muon or the positron, it introduces no
new uncertainties, as discussed above. This situation is analogous to the use of deep
inelastic scattering of electrons from nuclei to study the strong interaction. There,
the electron couples to the hadronic constituents of nuclear matter predominantly
through electromagnetic interactions and therefore allows the study of the strongly
interacting quarks and gluons without introducing extraneous uncertainties.
The spectrum of the radiative Michel decay has been treated by several authors [20, 17, 9]. The differential branching ratio for the case where no polarizations
13
Chapter 1: Introduction
are observed can be written as follows [10]:
d3B(x, y, θ)
4
= f1 (x, y, θ) + ηf2 (x, y, θ) + (1 − ρ)f3 (x, y, θ)
dx dy 2π d(cos θ)
3
(1.15)
where
x=
2Ee+
2Eγ
, y=
, cos θ = p̂e+ · p̂γ
mµ
mµ
(1.16)
and each function fi is a polynomials in x, y and ∆ = 1 − β cos θ with β = |~pe+ | /Ee+ .
Appendix A gives the explicit forms of the functions fi . Energy and momentum
conservation are enforced by the inequality
∆≥
2(x + y − 1)
.
xy
(1.17)
The parameter ρ is the same as that which occurs in the Michel decay positron
energy spectrum (1.11). The parameter η is a new Michel parameter, observable only
in the radiative decay. Like the other Michel parameters, it is related to the coupling
constants in (1.2):
¡ V 2
¢ 1¡ S
¢
¡ T 2
¢
V 2
T 2
S
T 2
T 2
η = |gRL
| + |gLR
| +
|gLR + 2gLR
| + |gRL
+ 2gRL
| + 2 |gLR
| + |gRL
| .
8
(1.18)
Recalling the normalization condition (1.5) and the Standard Model prescription (1.6),
we see that η is a positive semidefinite number with the nominal value
η SM = 0.
(1.19)
The most precise measurement of η to date agrees with the Standard Model [10]:
η = −0.035 ± 0.098.
(1.20)
14
Chapter 1: Introduction
This result can be interpreted as an upper limit on the allowed value of η:
η ≤ 0.083 (68 % confidence).
(1.21)
Section 4.2 provides the details of this interpretation. Any deviation of η from the
nominal value η SM , would imply deviation from a pure V − A weak interaction. We
note however, that as with measurements of ρ, corroborating measurements of η are
not sufficient to establish the V − A form as (1.18) and (1.19) can be satisfied for
V
V
S
S
arbitrary values of gLL
, gRR
, gLL
, and gRR
.
If η = 0 and ρ =
3
4
as dictated by the Standard Model, then only f1 contributes
to the spectrum (1.15). Figure 1.2 shows that the most probable radiative decay
has a low energy photon emitted at a small angle with respect to the positron as
noted above. The greatest sensitivity to the actual physical values of η and ρ occurs
in regions of kinematic phase space for which |f2 /f1 | and |f3 /f1 | are maximized,
respectively. Figures 1.3 and 1.4 demonstrate that these ratios are significant in
large regions of the phase space, though Figure 1.2 reminds us that these regions are
relatively sparsely populated.
1.3
Motivation for This Work
The PIBETA project is an ongoing series of experiments at the Paul Scherrer Institute
in Villigen, Switzerland. The primary goal of the experiment was to make a precise
Chapter 1: Introduction
Figure 1.2: f1 (x, y, θ) for various values of θ. In the Standard Model
with η = 0 and ρ = 43 , f1 is the sole contribution to the differential
branching ratio (1.15) of the µ+ → e+ νe ν µ γ decay.
15
Chapter 1: Introduction
Figure 1.3: f2 /f1 for various values of θ. |f2 /f1 | is a measure of the
sensitivity of Equation (1.15) to the value of η.
16
Chapter 1: Introduction
Figure 1.4: f3 /f1 for various values of θ. |f3 /f1 | is a measure of the
sensitivity of Equation (1.15) to the value of ρ.
17
18
Chapter 1: Introduction
measurement of the branching ratio of pion beta decay [29],
π + → π 0 e+ νe .
(1.22)
However, several other pion decay modes were measured [16] in parallel with the
the mode (1.22) as well as the muon decay modes discussed above. The primary
decays are summarized in Table 1.3. The reason for this methodology is twofold.
On one hand it increases the return of physics results relative to the investment in
the experiment. Most important though, is that it provides independent internal
calibrations of the detector response over the broadest possible kinematic range. One
of the great challenges of experimental science is the elimination of systematic errors.
The PIBETA methodology allows for analysis to be validated by verifying internal
results for well understood and precisely measured reactions (e.g., µ+ → e+ νe ν µ or
π + → e+ νe ) against external results. This lends confidence to results obtained for
the primary modes of interest (e.g., µ+ → e+ νe ν µ γ, π + → e+ νe γ and π + → π 0 e+ νe )
so that if any unexpected phenomena are revealed, it is unlikely that they can be
ascribed to mere systematic experimental errors.
This work presents the analysis of the muon decays listed in Table 1.3 and discussed above, based on data taken by the PIBETA experiment from May through
August of 2004. The Michel decay spectrum is well understood theoretically and has
been very precisely measured [26, 18]. Therefore, we will use it to validate our analysis tools, particularly the simulation of the detector response and the calibration of
19
Chapter 1: Introduction
Table 1.3: The principle decay modes registered by the PIBETA experiment. Note that it is not meaningful to assign exact numbers to
the radiative decay modes in the absence of kinematic constraints on
the spectrum of the photon.
decay mode
branching ratio
µ+ → e + ν e ν µ
100 %
µ+ → e + ν e ν µ γ
∼ 1%
π + → e + νe
1.23 × 10−4
π + → e + νe γ
∼ 10−7
π + → π 0 e+ νe
1.04 × 10−8
the experimental data (chapter 2). When we are satisfied that our analysis is sound
we shall then progress to the radiative Michel decay. The formal condition to be met
for satisfaction is the extraction of values for the branching ratio B µ+ →e+ νe ν µ and the
Michel parameter ρ which are consistent with the best external measurements [11, 26]
(Chapter 3). We shall then lay out our strategy for analysis of µ+ → e+ νe ν µ γ events
and present the results of this analysis (Chapter 4). Our main goal is to extract the
Michel parameters η and ρ from the analysis of µ+ → e+ νe ν µ γ events, and to measure
the branching ratio Bµ+ →e+ νe ν µ γ for the largest possible region of phase space.
Chapter 2
The PIBETA Apparatus
2.1
Introduction
This chapter describes the PIBETA detector hardware and data analysis software
alongside the simulation of the detector response. The hardware and software are
described with details sufficient to understand the analysis in subsequent chapters.
Complete details of the PIBETA detector are published in Reference [15].
2.2
PSI Proton Cyclotron
Figure 2.1 shows the layout of the PSI accelerator facilities. The cyclotron accelerates
an approximately 1.7−1.9 mA proton beam to an energy of 590 MeV. The accelerator
operates at a frequency of 50.63 MHz producing proton pulses 1 ns wide and sepa20
Chapter 2: The PIBETA Apparatus
21
rated by 19.75 ns. The primary proton beam is transported to two target stations
which produce pions and muons. These products are transported along secondary
beam lines to the experimental areas. The PIBETA detector is operated in the πE1
experimental area which is designed for 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, a full-width-half-maximum momentum resolution of less
than 2 % and an accepted production solid angle of 32 msr.
2.3
PIBETA Detector
Figure 2.2 shows a sketch of the PIBETA detector in a cross-sectional plane through
the beam axis. Figure 2.3 is a relief of the CsI calorimeter and figure 2.4 is a crosssectional view, perpendicular to the beam, of the thin tracking detectors. This section
describes the main elements shown in those figures: beam line components, multiwire-proportional-chambers (MWPC1 and MWPC2), plastic-veto hodoscope (PV)
and the segmented, pure-CsI calorimeter (CsI).
2.3.1
πE1 Beam Line
Positively charged pions entering the πE1 experimental area are first registered in a
3 mm thick active beam counter (BC). Immediately downstream of BC is a passive
lead brick collimator (PC) with a 7 mm pinhole aperture. Pions which clear the colli-
Chapter 2: The PIBETA Apparatus
Figure 2.1: The layout of the accelerator facilities in building WEHA
at PSI. The PIBETA detector operates in the πE1 experimental area.
22
23
Chapter 2: The PIBETA Apparatus
pure
CsI
PV
π+
AC1
MWPC1
beam
BC
AD
AC2
AT
MWPC2
10 cm
Figure 2.2: A cross-sectional view of the PIBETA detector showing the
main elements described in Section 2.3.
Figure 2.3: The 240-element pure-CsI calorimeter. It covers 3π sr of
solid angle. The two openings allow for the beam to enter the detector
and for maintenance access to the interior components.
24
Chapter 2: The PIBETA Apparatus
Active TGT
MWPC-1
MWPC-2
PV
array
Figure 2.4: The stopping target and cylindrical tracking detectors
shown in a cross-sectional plane perpendicular to the beam.
mator are slowed in a 40 mm thick active degrader (AD) and ultimately come to rest
in the active target (AT) positioned in the center of the detector. The active qualifier
indicates that these elements are constructed of plastic scintillator and that their light
output is detected and recorded with the rest of the data. Discriminated signals from
the beam line elements are fundamental ingredients of the various electronic triggers
described below (Section 2.4.1).
Two different stopping targets were used in the apparatus during the course of the
Summer 2004 run. Figure 2.4 shows the first of these. Note the segmentation of that
target into nine individual pieces. The segmentation aids in the reconstruction of the
Chapter 2: The PIBETA Apparatus
25
pion stopping distribution. This reconstruction is well understood based on studies
of past runs (1999-2001) [24], so it was decided that the last weeks of the 2004 run
would benefit from the improved light collection properties of a solid one-piece target.
The one-piece target is of precisely the same dimensions as the nine-piece target, only
without the segmentation. Because the target is such a crucial part of the detector,
we choose to analyze the data from 2004 as two distinct sets, corresponding to the
nine-piece target and the one-piece target.
The target is the most complicated element of the detector in terms of extracting
data and simulating its performance. This is due to the very high signal rate there.
The target bears the total event rate of the experiment (roughly 100 kHz), whereas
individual elements of the calorimeter for instance, bear less than 0.5% due to the
calorimeter’s articulated structure. Furthermore, virtually every event in the target
consists of three distinct parts: the beam pion coming to rest, the subsequent decay
π + → µ+ νµ which results in the muon coming to rest and finally, the decay of the
muon. All of these events deposit energy in the target and their light output and
subsequent photomultiplier signals pileup on each other. The situation is illustrated
in Figure 2.5.
We want to reconstruct the energy of the muon decay products. We therefore
want the target signal most nearly in coincidence with a calorimeter shower and
corresponding wire-chamber hits. Section 2.5 gives details about the algorithm that
associates calorimeter showers with wire-chamber hits. The first step in extracting the
Chapter 2: The PIBETA Apparatus
26
Figure 2.5: A signal from the target captured by the event display program. The first peak corresponds to the stopping pion. The pion decays
via π + → µ+ νµ and the resulting muon comes to a stop producing the
small shoulder on the pion stop signal. The muon then decays and the
resulting positron causes the final peak as it exits the target. The final
positron signal rides on the combined tails of two other signals.
energy deposited in the target is to subtract the pedestal energy. The pedestal energy
is just the peak of the energy distribution recorded for random trigger events which
are not correlated with pions stopping in the target (see Section 2.4.1 below). The
peak of the energy distribution in each run is then placed at the mean peak position,
obtained by averaging over all runs, by simply subtracting the difference between the
peak and the average. The peak fluctuates slightly due to low-frequency noise on the
signal lines. After these steps, the pion- and muon-stop pileup can be subtracted.
Figure 2.6 shows how the energy in the target depends on the time between the pion
Chapter 2: The PIBETA Apparatus
27
Figure 2.6: The effect of pileup in the target is clearly visible in this
plot of the energy left there by decay positrons in one-arm low-threshold
trigger events as a function of time. The events preceding the pion stop
at t = 0 are pileup free so we subtract the time dependent energy from
events and then add back the constant mean of t < 0 events, denoted
by the dashed line.
stop and the muon decay (taken from the corresponding calorimeter shower). The
increased energy at time t > 0 is due to the tail of the pion stop at t = 0. The energy
for events at t < 0 is pileup free since these events precede the pion stop. The time
dependence is removed from the target energy by subtracting the energy versus time
distribution shown in Figure 2.6 and then adding back the constant mean value of the
energy for events at t < 0, denoted in the figure with a dotted line. The last step is
simply to multiply by a calibrating factor which converts ADC channels to MeV such
28
Chapter 2: The PIBETA Apparatus
that the peak of the experimental distribution is aligned with that of the simulated
distribution. The simulated target energy is best matched to data by smearing it
with photoelectron statistics and then adding pedestal noise:


rσ=1 
sim
sim 
,
1+ q
Etgt
→ Etgt
sim
nEtgt
(2.1)
and subsequently
sim
sim
Etgt
→ Etgt
+ r σ ru ,
(2.2)
where rσ is a Gaussian distributed random number with variance σ 2 , 0 ≤ ru ≤ 1 is
a uniformly distributed random number and n is the number of photoelectrons per
MeV of energy deposited. The uniformly distributed random number in the pedestal
noise simulates the increasing uncertainty in the energy for events closer to the pion
stop. Both targets are found to generate 80 photoelectrons/MeV and the pedestal
noise is σ = 1.55 MeV for the nine-piece target and σ = 2.00 MeV for the one-piece
target.
All of the above discussion implicitly assumes that the simulation uses the correct
distribution of decay vertices. Otherwise, the energy spectra shown in Figure 2.7
would be skewed because particles would exit the target along longer or shorter paths.
Figure 2.8, showing the decay vertex distribution inferred from wire-chamber tracks,
confirms that the stopping distribution used in the simulation is indeed correct.
Chapter 2: The PIBETA Apparatus
Figure 2.7: The calibrated energy deposited in the stopping targets for
one-arm low-threshold trigger events.
29
Chapter 2: The PIBETA Apparatus
Figure 2.8: The decay vertex distributions inferred from wire chamber
tracks for one-arm low-threshold trigger events. The vertex (x0 , y0 , z0 )
is taken to be the point on the wire-chamber track closest to the z-axis.
30
Chapter 2: The PIBETA Apparatus
2.3.2
31
Thin Tracking Detectors: MWPC1, MWPC2 and PV
Two cylindrical multi-wire proportional chambers, MWPC1 and MWPC2 precisely
track charged decay products. They are highly efficient (greater than 95%) and stable
at rates of up to 107 minimum-ionizing particles per second. Each chamber has one
anode wire plane along the z-direction and two cathode strip planes in a stereoscopic
geometry. The resolution with which the chambers can track charged particles is simulated by simultaneously matching the distributions of reconstructed decay vertices
and the angular separation of wire chamber tracks and their corresponding calorimeter clumps. Figures 2.8 and 2.16 demonstrate the agreement between the simulation
and the data. These figures will be discussed in more detail after we have elaborated
on the software reconstruction of tracks in Section 2.5. The best match between the
data and the simulation is found when both chambers have the coordinates of the
simulated track hit smeared by the amounts
∆x = ∆y = 1.6 mm, ∆z = 1.0 mm.
(2.3)
This implies an angular resolution of approximately 1◦ and is consistent with the
resolution found independently from an analysis of cosmic muon events [15].
The plastic-veto hodoscope (PV) surrounds the MWPCs. It is composed of twenty
individual staves of plastic scintillator, fitted together to form a cylinder covering the
entire 2π azimuthal angle. Its length is such that any particle emanating from the
stopping target and arriving in the calorimeter must also traverse the active volume
32
Chapter 2: The PIBETA Apparatus
of the hodoscope. The scintillator pieces are supported by a very thin, cylindrical
carbon fiber shell (∆r = 1 mm = 5.3 × 10−3 radiation lengths). The light output
of each stave is registered by photomultiplier tubes on each end (denoted ±z). The
energy deposited in an individual piece is taken to be the geometric mean of the
calibrated energy registered at each end separately:
exp
EPV
=
p
E+z E−z .
(2.4)
PV energy deposition is reproduced in simulation by smearing the raw (simulated)
energy deposition with finite photoelectron statistics and applying a gain factor:
!
Ã
rσ
sim
sim
(2.5)
EPV → EPV G + p sim ,
EPV
where G is the gain factor and rσ is a Gaussian distributed random number with
variance σ 2 . The parameters G and σ were found for all 20 PV elements individually
by minimizing the difference between the recorded and simulated spectra. Figure 2.9
shows the cumulative result (i.e., all 20 elements together) for minimum-ionizing
positrons.
These same positrons can be used to measure the efficiency with which each of the
tracking detectors registers minimum-ionizing charged particle hits. Here we describe
the computation of the inner-chamber efficiency ²MWPC1 . The computation of the
outer-chamber and hodoscope efficiencies is analogous. The efficiency is the ratio of
the number of events where the positron registered in all possible tracking detectors,
including MWPC1, and the calorimeter, to the number of events where it registered
Chapter 2: The PIBETA Apparatus
33
Figure 2.9: The calibrated energy deposited in the PV hodoscope for
one-arm low-threshold trigger events.
in the other detectors regardless of whether it also registered in MWPC1:
²MWPC1 =
N (MWPC1 ◦ MWPC2 ◦ PV ◦ CsI)
.
N (MWPC2 ◦ PV ◦ CsI)
(2.6)
The efficiencies of the other tracking detectors are computed in the same way with
trivial transpositions of the symbols in Equation (2.6):
²MWPC2 =
²P V =
N (MWPC1 ◦ MWPC2 ◦ PV ◦ CsI)
,
N (MWPC1 ◦ PV ◦ CsI)
N (MWPC1 ◦ MWPC2 ◦ PV ◦ CsI)
.
N (MWPC1 ◦ MWPC2 ◦ CsI)
(2.7)
(2.8)
These numbers are computed for each run individually where there are roughly 7×104
charged tracks registering under the one-arm low-threshold trigger. The statistical
34
Chapter 2: The PIBETA Apparatus
uncertainty in each efficiency computed for an individual run is therefore
δ²i ≈
r
2
≈ 0.5%.
7 × 104
(2.9)
The statistical uncertainty in the total tracking efficiency of all three detectors together is therefore
δ²tot ≈
√
3 × 0.5% ≈ 1%.
(2.10)
We can use the efficiencies to weight the number of observed events and arrive
at the number of events which actually occurred (i.e., the number we would have
observed in the ideal case of 100 % efficiency). If ² is any generic efficiency and we
observe N obs events, then the number of events N act which actually occurred is
N act =
N obs
.
²
(2.11)
In order to account properly for inefficiencies with such a simple formula we must
be certain that ² is truly a constant rather than a function of some other observed
variable, such as particle energy. We expect this to be the case since all positrons
used in the analysis have energies E ≥ 5 MeV and are therefore minimum ionizing
(E & 1 MeV). That is, they all deposit the same amount of energy per thickness
of material traversed in the thin tracking detectors and thus have equal chances of
detection. Figure 2.10 confirms this assertion.
The efficiencies defined by Equations (2.6–2.8) represent only the instrumental
efficiencies of the detectors and their attendant electronics. The physical source of
Chapter 2: The PIBETA Apparatus
Figure 2.10: The detection efficiencies of the thin tracking detectors
MWPC1, MWPC2 and PV as a function of the energy in the calorimeter for a typical run. The dashed line represents the efficiencies computed via (2.6), (2.7) and (2.8) for this particular run. Note that any
energy dependence is statistically insignificant.
35
Chapter 2: The PIBETA Apparatus
36
this inefficiency is ascribed primarily to discriminator units. Other sources of tracking
inefficiency, namely physical processes which involve the disappearance of the particle
before it reaches the calorimeter, and the software algorithm which reconstructs tracks
from the topology of detector hits (Section 2.5), are included in the simulation and
therefore accounted for in the detector acceptance.
2.3.3
Calorimeter
The heart of the PIBETA detector is the electromagnetic shower calorimeter. Its
active volume is composed of pure CsI crystal, segmented into 240 individual pieces
fitted tightly together to cover 0.77×4π sr of solid angle. The inner radius of the assembly is 26 cm and its active volume is 22 cm thick, corresponding to 12 radiation lengths
(X0CsI = 1.85 cm). Each individual crystal is painted with a wavelength-shifting lacquer and wrapped in aluminized mylar to improve its light collection properties and to
contain showers within individual crystals to the greatest extent possible. The main
volume is composed of 200 pentagonal and hexagonal pyramids with half hexagons
used to finish the pattern at the edges. The beam openings themselves are each
bordered by 20 trapezoidal pyramids. These latter crystals allow for the vetoing of
events where some of the shower energy is likely to have spilled out of the main body
of the calorimeter. The crystal shapes can be seen in Figure 2.3.
The simulation of calorimeter showers is handled by the standard GEANT3 [3]
software package. The only custom adaptations required are to smear the energies to
37
Chapter 2: The PIBETA Apparatus
simulate photoelectron statistics and pedestal noise associated with the photomultiplier tubes and apply an overall energy scale factor g, which simulates software gain
factors used in the data analysis software. In principle we should apply this factor to
the data, but for practical reasons we choose instead to apply it to the simulation:
sim
ECsI
sim
ECsI
.
→
g
(2.12)
The factor g is found by minimizing the difference between the simulated and recorded
energy spectra of positron showers which initiate one-arm low-threshold triggers. The
dominant source of such events is the Michel decay of the muon, µ+ → e+ νe ν µ .
However, there is a small background of positrons from the pion decay π + → e+ νe ,
as can be seen in Figure 2.11. The high energy “edge” of the muon decay spectrum
overlaps with the low energy tail of the monoenergetic pion decay spectrum. Although
the pion decay contribution will be negligible for our later purposes, satisfactory fits
cannot be obtained in this instance if the simulated spectrum neglects the pion decay
background. The simulated positron energy spectrum is formed by generating the
muon and pion decays independently and then combining them, allowing the relative
normalization of the pion decay spectrum to be a free parameter. The overall gain
factor g is also a free parameter. The fit is performed over the range 40 < ECsI <
76 MeV as shown in Figure 2.11. The results are shown in Table 2.1. Note that the fits
were also performed over the range 10 < ECsI < 76 MeV with no significant difference
in the results for g. Neglecting the pion decay contribution results in a significant
Chapter 2: The PIBETA Apparatus
38
Figure 2.11: The overall calibration between data and simulation was
found by simultaneously matching the µ+ → e+ νe ν µ “edge” and the
π + → e+ νe peak.
inflation of the uncertainty in g. The agreement between the simulation and the
data in Figure 2.11 also implicitly confirms the simulation of photoelectron statistics
and pedestal noise discussed above. Too much noise or too few photoelectrons would
result in a more gently sloping edge and a broader peak. An error in the other
direction would sharpen the edge and constrict the peak. Figure 2.12 shows the
energy spectrum over the full range of energies of interest to our muon decay study.
The veto crystals which line the openings of the calorimeter are simulated sepa-
Chapter 2: The PIBETA Apparatus
39
Figure 2.12: The energy deposited in the CsI calorimeter for one-arm
low-threshold trigger events. Virtually all of these events are positrons
from the Michel decay µ+ → e+ νe ν µ .
rately from the other crystals, since they were operated at higher voltages. Figure 2.13
shows the spectrum of total energy deposited in these veto crystals.
2.4
Electronics
An event is recorded to the data set when it satisfies the criteria that create a highlevel trigger. The triggers vary in complexity. The most basic triggers are simply
discriminated versions of analog pulses, which indicate whether the voltage in a particular channel has exceeded a predetermined discriminator threshold. More complex
40
Chapter 2: The PIBETA Apparatus
Table 2.1: Energy scale factors g which appear in Equation (2.12) and
the relative π + → e+ νe normalization Nπ for the two data sets.
data set
g
Nπ
nine-piece target
0.9337 ± 0.0023
0.0037 ± 0.0007
one-piece target
0.9355 ± 0.0019
0.0046 ± 0.0008
Figure 2.13: The energy deposited in the CsI veto crystals for one-arm
low-threshold trigger events. The gap is due to a software threshold
which zeroes any channel that reports E < 0.8 MeV.
Chapter 2: The PIBETA Apparatus
41
triggers are constructed from logical combinations of these discriminated outputs and
indicate various temporal coincidences and topological geometries of detected hits.
The ultimate, high-level triggers are formed in turn from these lower level signals
and enable the front-end computer to record data from the various channels of the
detector. This section describes the formation of the triggers apropos to the analysis
hereafter. We shall also discuss the efficiency with which the front-end computer logs
trigger events to the data set.
2.4.1
Triggers
Random Trigger
A small (190 × 20 × 8 mm) plastic scintillator radiation counter is placed a short
distance away from the main detector, but is shielded from it by a 50 mm thick lead
brick wall as well as another 500 mm of concrete. Thus, signals in the random detector
are completely uncorrelated with events in the primary detector. The purpose of the
counter is to trigger on ionizing cosmic rays which arrive at random intervals, about
1–2 sec−1 , and subsequently to record detector signals as for any other event. This
random trigger gives a sampling of the ambient electronic noise in the detector. The
information gained in this way is used at the end of every production run (every few
hours during normal running conditions) to compute and record ADC pedestals for
every channel in the detector.
Chapter 2: The PIBETA Apparatus
42
Beam Triggers
The beam triggers are a fundamental component of all physics events. They ultimately alert the rest of the system that a pion has stopped in the target. Signals
from each of the active beam line elements (BC, AD and AT) are discriminated to
produce logic pulses, set to be 10 ns wide. These logic signals along with the accelerator rf pulse RF (provided to the detector from the cyclotron) are ingredients of the
pion stop signal PS, also adjusted to be 10 ns wide:
PS = BC ◦ AD ◦ AT ◦ RF.
(2.13)
The fourfold coincidence of these signals can only occur when a pion is produced at
the target station by a proton bunch from the cyclotron (coincidence with RF) and
subsequently traverses the secondary beam line elements (BC ◦ AD) and lands in the
target (AT). Figure 2.14 shows the coincidence as viewed on an oscilloscope.
The PS signal initiates an additional signal referred to as the pion gate PG. The
pion gate is arranged via a delay unit to be opened 50 ns before PS and to remain open
for a total of 185 ns. Only events occurring within a pion gate can be recorded to the
data set. Furthermore, events occurring within the pion stop which initiated the pion
gate are rejected in order to suppress prompt, single-charge-exchange interactions
(π + n → π 0 p) between the pions and the nuclei of the target material, which would
otherwise swamp the data acquisition system with uninteresting events. Thus, the
Chapter 2: The PIBETA Apparatus
43
Figure 2.14: The four ingredients of a pion stop trigger PS as viewed
on an oscilloscope. The signals from top to bottom are BC, AD, AT
and RF.
basic beam trigger B which is used in the higher level physics triggers is
B = PG ◦ PS0 .
(2.14)
The subscript “0” on PS reminds us that the veto applies only to the PS which
initiated PG. It is possible for multiple PS signals to pileup within a PG signal since
PG encompasses several accelerator periods (Trf = 19.75 ns). These pileup pions
result in some prompt hadronic events being recorded to data. The number of such
events is not very large since the probability of pileup is quite small. Figure 2.15
illustrates the situation.
Chapter 2: The PIBETA Apparatus
44
Figure 2.15: Only events coincident with B = PG ◦ PS0 are accepted.
Note that multiple PS signals can pileup within a single PG, but only
the one at t = 0 (PS0 ) is vetoed.
Calorimeter Triggers
The individual segments of the CsI calorimeter are grouped into 60 clusters composed
of 9 crystals each. Obviously, the clusters overlap (there are only 240 crystals altogether) such that most crystals belong to more than one cluster. The members of a
cluster are physically adjacent to each other in the calorimeter. Furthermore, groups
of 6 adjacent clusters form 10 overlapping superclusters. These superclusters are the
basis of the CsI trigger logic.
The simplest CsI trigger is the one-arm trigger. There are two versions corresponding to a low (5 MeV) and a high (53 MeV) threshold denoted CSL and CSH respectively.
A cluster fires if the total energy deposition in its constituent crystals exceeds one
Chapter 2: The PIBETA Apparatus
45
of the thresholds. Furthermore, a supercluster fires if any of its constituent clusters
does. The firing of a single supercluster constitutes a one-arm trigger. Note that C SL
and CSH are distinct triggers so that an excess over the high threshold could fire both
versions simultaneously. This would always be the case if not for prescaling of events,
discussed below.
The next simplest CsI trigger is the two-arm trigger. A two-arm trigger is caused
by the coincident firing of two non-neighboring superclusters. There are three versions corresponding to the possible combinations of high and low thresholds: both
arms above high threshold, both arms above low threshold and one arm above each
H
L
HL
threshold. These triggers are denoted CSS
, CSS
and CSS
respectively. Note again
H
for instance, could also be accompanied
that these are all distinct signals so that CSS
HL
L
and CSS
and would always be if not for prescaling of triggers.
by both CSS
In order to optimize the event statistics of all decay modes of interest to the
PIBETA collaboration, it is necessary to prescale some of the triggers. This prevents
very common decay processes such as µ+ → e+ νe ν µ from usurping the detector live
time and precluding more rare events like π + → π 0 e+ νe from being recorded to the
data set. Our experimental event weights must account for this prescaling. If we
observe N obs events in a trigger with prescaling factor p then the actual number of
events which occurred N act is
N act = pN obs .
(2.15)
Prescaling is accomplished in electronics by accepting only every pth occurrence of a
46
Chapter 2: The PIBETA Apparatus
Table 2.2: Hardware prescaling factors. The low-threshold triggers are
prescaled to suppress the copious muon decays which would otherwise
swamp the electronics and prevent satisfactory statistical samples of
the more rare pion decays.
trigger
prescale factor
CSL
27 = 512
LL
CSS
24 = 16
CSH
20 = 1
HH
CSS
20 = 1
HL
CSS
20 = 1
particular trigger. Table 2.2 lists the prescale factors used in the experiment. The
prescasle factor for each trigger is computed at the end of each run from data provided
by scaler units. It is simply the ratio of the number of occurrences which passed the
prescaling unit to the number of raw occurrences. These numbers can be different
from the nominal numbers listed in Table 2.2 depending on where in the scaler cycle
the end of the run occurs. There are typically between 108 and 109 raw triggers in a
run so the statistical uncertainty in this calculation is negligible.
47
Chapter 2: The PIBETA Apparatus
2.4.2
Front-End Computer Efficiency
We shall not go into details about the computer equipment itself but we must know
the efficiency with which it records trigger events. The efficiency depends on the event
rate and therefore fluctuates throughout the experiment. Typical values are 85–90 %.
The efficiency is calculated from the recorded scaler counts for each individual run
(just as for all other efficiencies and prescale factors). It is the ratio of the number of
events recorded to data N rec to the number of triggers generated by the electronics
N gen :
²FE =
N rec
.
N gen
(2.16)
Both of these numbers are typically about 5 × 105 so the statistical uncertainty in
this efficiency is
δ²FE ≈
2.5
r
2
= 0.2%.
5 × 105
(2.17)
Data Analysis Software
This section describes the analyzer software in enough detail to understand the analysis that follows. In general, we will take the data acquisition stage for granted and
beginning from detector hits, calibrated energy depositions and times in the various
detector channels, we will work our way toward fully reconstructed tracks.
48
Chapter 2: The PIBETA Apparatus
2.5.1
Calorimeter Clumps
The PIBETA calorimeter was designed so that the majority of the shower energy
(more than 90 %) would be deposited within a single CsI crystal, if the crystal is
centrally hit. For off-axis showers, three crystals at most share the energy. In any case,
we must account for shower energy which leaks into neighboring crystals. The software
therefore identifies clumps in the CsI topology, consisting of a central crystal and its
nearest neighbors. A crystal can have 5–7 nearest neighbors depending on its shape.
Note that these clumps are distinct from the clusters which determine the calorimeter
trigger logic: clusters are hardwired in the electronics in predetermined topologies
whereas any crystal in the calorimeter can be the center of a clump. Table 2.3 lists
the relevant members of the clump data structure.
The clump-finding algorithm is straightforward. The first step is to identify the
central crystal of the first clump. The first central crystal is the one with the maximum
energy deposition. The energy of the clump ECsI is the sum of the energy in this
segment along with the energies of any of its nearest neighbors which are hit within
|∆t| < 14 ns of the central crystal or which did not register a valid time (because the
signal was below the TDC threshold). The clump time tCsI is the time of the central
crystal. The clump is also assigned angular coordinates (θCsI , φCsI ) according to the
energy weighted centers of the crystals involved:
θCsI
Pn
wi θ i
,
= Pi=1
n
i=1 wi
(2.18)
49
Chapter 2: The PIBETA Apparatus
Table 2.3: The members of a clump data structure. All members listed
below NCsI are arrays with NCsI elements.
symbol
description
NCsI
number of clumps
ECsI
total energy in the clump
tCsI
time of the central crystal
θCsI
energy-weighted polar angle of the clump center
φCsI
energy-weighted azimuthal angle of the clump center
where the sums are over the n crystals involved in the clump, θi is the polar angle of
the center of the ith crystal and the corresponding weight is
Ei
wi = a0 + ln Pn
i=1
Ei
.
(2.19)
The definition of φCsI is obtained by the transposition θ → φ in Equation (2.18).
The optimum value of the parameter a0 was found via simulation of the calorimeter
energy resolution to be a0 = 5.5. Once a clump is identified and booked into memory
as a data structure, all crystals involved in that clump have their energies zeroed
and the algorithm begins anew to find the next clump. Two neighboring crystals
can not be the centers of distinct clumps. This means that the PIBETA detector’s
ability to resolve distinct calorimeter showers is set by the typical angle subtended
Chapter 2: The PIBETA Apparatus
50
by a crystal, which is about 13◦ . However, next-nearest neighbors can be the seeds
of separate clumps and in this case some crystals will be members of both clumps.
This special case is handled by allowing the energy of the overlapping segments to be
shared in proportion to the energies of the central crystals. The algorithm repeats
this procedure, identifying clumps in order of descending energy until there are no
more crystals with more than 4 MeV of energy deposited in them to serve as a clump
center or a maximum of five clumps have been identified.
2.5.2
Track Finding Algorithm
The highest level data structure in the analyzer is a track. A track consists of a reconstructed vertex origin (x0 , y0 , z0 ), the point of entry into the Calorimeter (x1 , y1 , z1 ),
track direction cosines (p̂x , p̂y , p̂z ), energy depositions and times for the PV, target
and CsI and a flag which identifies the particle type. Energies, times and directions
are derived from lower level data structures (e.g., clump, PV, and MWPC structures).
The track finding algorithm identifies coherent topological structure in these lower
level banks and consolidates the information in a single entity. The elements of a
track structure are listed in Table 2.4.
The track algorithm begins by assembling charged particle tracks. These tracks
will have registered hits in both wire chambers (up to the inefficiencies discussed
above). The wire chambers must register azimuthal angles within 30◦ of each other
and the line through the hits must project back into the target volume and forward
Chapter 2: The PIBETA Apparatus
51
Table 2.4: The members of a track data structure. All members listed
below Ntrk are arrays with Ntrk elements.
symbol
Ntrk
(x0 , y0 , z0 )
description
number of identified tracks
point on the track closest to the origin for charged tracks
center of the stopping distribution for neutral tracks
(x1 , y1 , z1 )
track intersection with the inner calorimeter surface
(p̂x , p̂y , p̂z ) direction cosines
ECsI , EPV
tCsI , tPV
ipart
energies in associated calorimeter clump and hodoscope element
times in associated calorimeter clump and hodoscope element
particle identification code
to within α < 13◦ of a calorimeter clump with ECsI > 5 MeV. If these conditions are
met, the algorithm then finds the PV segment whose center is closest to the track
and adds the energy deposited there and the time to the structure. Figure 2.16 shows
the distribution of angles between the intersection of the wire-chamber track with the
calorimeter face (x1 , y1 , z1 ) and the angular coordinates of the CsI clump (θCsI , φCsI )
[Equation (2.18)] for reconstructed tracks. Figure 2.8 shows the corresponding decay
vertex distribution (x0 , y0 , z0 ) inferred from wire chamber tracks. The point (x0 , y0 , z0 )
Chapter 2: The PIBETA Apparatus
52
Figure 2.16: The angular separation between the charged track computed from wire-chamber information and the coordinates of the corresponding calorimeter clump [Equation (2.18]).
is taken as the point on the track closest to the z-axis.
After all possible charged particle tracks are constructed, the algorithm associates
any remaining calorimeter clumps with neutral tracks. Neutral particles will not have
left any hits in the MWPCs or PV so it must be assumed that the particle came from
the center of the stopping distribution. This point is assigned as (x0 , y0 , z0 ). The
direction cosines and the point (x1 , y1 , z1 ) are based on the angles assigned by the
clump-finding algorithm. These are both with respect to the center of coordinates
and not the center of the stopping distribution. The energy and time recorded in the
Chapter 2: The PIBETA Apparatus
53
nearest hodoscope segment is recorded to the structure. Neutral particles are very
unlikely to deposit a significant amount of energy in the thin hodoscope.
Particle identification is inferred from hits in the calorimeter and thin tracking
detectors. Neutral particles such as photons will leave very little, if any, energy in the
PV and will not register at all in the MWPCs and are therefore easily identified. The
only charged particles which can be produced in decays or scatters in the PIBETA
detector are electrons, positrons and protons. Electrons are quite rare relative to
positrons. This is fortunate as there is no way to distinguish between the two without
magnetic spectroscopy. Any electrons in the PIBETA detector will be identified as
positrons. The minimum ionizing positrons will leave hits in both chambers and a
small fraction of their total energy in the PV. Non-relativistic charged particles, such
as protons, will track in both chambers and leave a relatively large amount of their
kinetic energy in the hodoscope. Figure 2.17 shows the clean separation between
positrons and protons on a scatter-plot of the PV energy EPV versus the sum of the
PV and CsI energies EPV + ECsI . The functions which determine the boundaries
between the particle types are
fγ = 0.2e−0.007(EPV +ECsI ) and fe = 2.3e−0.007(EPV +ECsI )
such that
(2.20)
Chapter 2: The PIBETA Apparatus
54
Figure 2.17: Positrons and protons are clearly separated by the PV.
The plotted functions are the boundaries of the particle identification
regions parameterized in Equation (2.20).
EPV < fγ ⇒ photon (γ),
fγ ≤ EPV < fe ⇒ positron (e+ ),
EPV ≥ fe ⇒ proton (p).
(2.21)
Chapter 3
Michel Decay Analysis
3.1
Introduction
The PIBETA experiment recorded large samples of well understood pion and muon
decay modes in order to calibrate the apparatus and the analysis methods, as well
as to set the normalization for the more interesting decay modes. The mode of
interest to this work is the radiative decay µ+ → e+ νe ν µ γ. The appropriate calibrating/normalizing mode is the related nonradiative decay µ+ → e+ νe ν µ . This chapter
presents results for the total branching ratio and the Michel parameter ρ based on
PIBETA’s sample of µ+ → e+ νe ν µ decays. Our results are not as precise as those of
dedicated µ+ → e+ νe ν µ experiments [8, 26]. However, our results are consistent with
those experiments and this fact lends credence to our methodology.
Before we undertake the extraction of ρ, it is instructive first to see how the rate
55
Chapter 3: Michel Decay Analysis
56
Figure 3.1: The relative difference in the differential decay rate (3.1)
of µ+ → e+ νe ν µ for two values of ρ different from the Standard Model
value ρSM = 0.75. The variation of δρ = ±0.01 is roughly the limit
which can be resolved with the PIBETA detector using the current
calibration method.
depends on this parameter. Figure 3.1 shows the relative difference in rate
R(x; ρ) = 1 − dΓ(x; ρ0 )/dΓ(x; ρ)
(3.1)
for two values of ρ 6= ρSM : ρ = ρSM ± 0.01. Figure 1.1 shows the rate itself, with the
Standard Model value ρSM = 0.75. The PIBETA detector can measure the spectrum
with a statistical uncertainty of about 1 % in energy bins which are 1–2 MeV wide,
so Figure 3.1 portends a result for ρ with a precision of about δρ = ±0.01.
Chapter 3: Michel Decay Analysis
3.2
3.2.1
57
Event Selection
Kinematic Cuts
Trigger discriminator thresholds in the PIBETA experiment were chosen so that the
vast majority of positrons from muon decays are detected below the calorimeter high
threshold: (EeCsI
+ ≤ HT ≈ 53 MeV). Muon decays are so prevalent in the low threshold triggers that any background from pion decays is negligible compared to other
uncertainties. The most abundant pion decay involving a final state e+ , π + → e+ νe ,
has a branching ratio four orders of magnitude less than that of µ+ → e+ νe ν µ and
a monoenergetic energy spectrum that puts the e+ above the endpoint of the michel
decay. There are some π + → e+ νe events detected far below the peak because of
the calorimeter’s energy response function. However, simulation of this background
indicates that the relative number of these background events in the range in which
muon decays are to be analyzed, 10 < EeCsI
+ < 56 MeV, is
Nπ+ →e+ νe
. 10−4 .
Nµ+ →e+ νe ν µ
(3.2)
Thus, culling µ+ → e+ νe ν µ events from the total data sample is a straightforward exercise. Any event containing a track identified as a positron with 10 < EeCsI
+ < 56 MeV
that fires the one-arm, low-threshold calorimeter trigger is a candidate. The lower
energy cut is well above the CsI low-threshold, eliminating uncertainties associated
with the simulation of hardware thresholds. The upper energy cut is between the
Chapter 3: Michel Decay Analysis
58
Michel “edge” and the π + → e+ νe peak and thus allows virtually all Michel decays
and only a negligible background from π + → e+ νe . The only additional kinematic cut
rejects events with more than 5 MeV of energy deposited in the calorimeter veto crystals, which line the perimeters of the beam openings. Such events are likely to have
had some of their energy lost by leaking through the boundaries of the calorimeter.
Their inclusion could therefore skew the positron energy spectrum.
3.2.2
Time Structure of Muon Decays
The PIBETA experiment was conceived primarily as a study of rare pion decays. The
ideal methodology for such an experiment is to stop charged pions in a target at a
high rate in order to maximize the occurence of rare events. Fortuitously, a stopped
pion beam is for all practical purposes equivalent to a stopped muon beam, since
more that 99.9 % of pions decay into muons via π + → µ+ νµ . The small phase space
available to this reaction, along with a pion-stop distribution which is well centered
in the target and very small with respect to the target dimensions, ensures that the
muon can not escape the stopping target and will itself come to rest within a short
distance of the pion decay vertex. A typical muon resulting from π + → µ+ νµ will
travel a few mm at most before coming to rest. The radius of the active target is
much larger, at 2 cm.
Studying muon decays recorded during the course of a pion decay experiment
requires some attention to details which arise because of the disparate lifetimes of
59
Chapter 3: Michel Decay Analysis
Table 3.1: Various time scales involved in the PIBETA experiment.
Note that τµ ∼ 102 τπ . The pion stop period is based on 105 π + /sec
which is typical for the 2004 run.
time scale
symbol
value (ns)
pion lifetime
τπ
26.02
cyclotron period
Trf
19.75
pion gate width
TπG
180
τµ
2197.03
1/rπ
∼ 10000
muon lifetime
Pion Stop Period
the two particles: τµ ∼ 100 × τπ . The time scales in the PIBETA experiment were
optimized based on the the pion lifetime as can be seen in Table 3.1. For instance,
the pion gate is open for approximately five pion lifetimes after the pion stop so
that virtually all pions stopped in the target decay within the gate. Muons, on the
other hand, pileup in the target and their subsequent decays can occur within a later
pion gate with which they are not causally connected. For these reasons, the time
spectrum of decaying muons is more complicated than that of pions. This section will
elucidate the muon decay time structure and explain how the seemingly paradoxical
situation where muon decays are their own background can be exploited to maximize
event statistics without introducing additional systematic uncertainties.
As mentioned previously, the fate of virtually all pions stopped in the target is
60
Chapter 3: Michel Decay Analysis
the decay chain
π + → µ+ νµ , µ+ → e+ νe ν µ (γ).
(3.3)
The probabilities per unit time for these decays individually are
fπ (t) =
1 −t/τπ
1
e
and fµ (t) = e−t/τµ .
τπ
τµ
(3.4)
These are the usual exponential decay spectra. The joint probability for the sequential
decay is therefore
fs (t) =
Z
t
0
fµ (t − t0 )fπ (t0 )dt0 =
¡ −t/τµ
¢
1
e
− e−t/τπ .
τµ − τ π
(3.5)
The function fs (t) is the probability per unit time that the decay chain will complete
(i.e., the muon will decay) at time t after the pion stop with which it is causally
connected.
The analysis must also account for events resulting from old pileup muons which
decay inside the current pion gate. The probablitity that a pion is delivered in a
particular beam pulse is
p = rπ Trf .
(3.6)
The rate of pileup muon decays is the sum over all past beam pulses of the causal
decay rate (3.5), weighted with the probability (3.6) that a beam pulse results in a
pion in the target:
fpu (t) =
∞
X
n=1
pfs (t + nTrf ).
(3.7)
61
Chapter 3: Michel Decay Analysis
The series (3.7) can be summed in a straightforward manner with the result
p
fpu (t) =
τµ − τ π
µ
e−t/τµ
e−t/τπ
−
eTrf /τµ − 1 eTrf /τπ − 1
¶
.
(3.8)
It is informative to see how this function behaves when the pion lifetime is neglected
compared to that of the muon. The first term in (3.8) dominates the second so
we neglect the latter. At pion stop rates employed in the PIBETA experiment, the
remaining term is very well approximated by its constant value at t = 0. With these
approximations we have that
fpu (0) ≈ rπ .
(3.9)
The rate of pileup events is as large as the total event rate!
The full probability distribution, valid for all times, is given by the piecewise
function
f (t) =



 fpu (t)
t<0
(3.10)


 fpu (t) + fs (t) t ≥ 0
where fs (t) and fpu (t) are given by Equations (3.5) and (3.8) respectively. Equation (3.10) is plotted in Figure 3.2 for a typical pion stop rate of 100 kHz. The actual
time structure of the µ+ → e+ νe ν µ decays used in the analysis is shown in Figure 3.3.
The gap arount t = 0 is caused by the beam veto. The peak there represents a small
proportion of prompt events which are allowed by discriminator inefficiency.
Some pileup muons decay when the data-acquisition system is not in a state which
can receive events, i.e., whenever the pion gate is not open. In order to extract a
62
Chapter 3: Michel Decay Analysis
Figure 3.2: The time spectrum of muon decays computed from Equation (3.10) at a typical pion stop rate of 100 kHz. Time t = 0 corresponds to the pion stop. The “porch” (t < 0) results from muon pileup
in the target. The rate for t ≥ 0 is due to both causal and pileup muon
decays.
branching ratio for muon decay, we need to know what fraction of all events actually
occur within a pion gate. The answer, to first approximation, is the integral of (3.10)
over times of interest:
g1 (t1 , t2 ) =
Z
t2
f (t)dt.
(3.11)
t1
The fraction given in (3.11) actually results in a slight overcounting of events by about
1 %. Once an event is registered within the pion gate, any subsequent events in the
same pion gate will be lost. The exact gate fraction must have the proportion of such
events subtracted. The expression for this second order correction is
1
g2 (t1 , t2 ) =
t2 − t 1
Z
t2
t1
·Z
t
00
f (t )dt
t1
00
Z
t2
0
0
¸
f (t )dt dt.
t
(3.12)
Chapter 3: Michel Decay Analysis
Figure 3.3: The time distribution of µ+ → e+ νe ν µ decays used in the
analysis. The time is the difference between that of the e+ shower in the
calorimeter (tCsI ) and the pion stop (tPG ). Only events in the shaded
region (−40 < t < −10) ∪ (10 < t < 140) are used in the analysis to
avoid uncertainties in the timing near the gate edges. Note also that
some prompt events “leak” through the veto at t ≈ 0. The average
rate rπ is found by fitting the data with the function (3.10) in the same
region, allowing the rate to be a free parameter. The quality of the fits
is reflected in the value of the reduced χ2 : χ2ν ≈ 1.
63
Chapter 3: Michel Decay Analysis
64
In words, Equation (3.12) represents the average joint probability to register an event
at time t > t1 followed by another event at time t0 , such that t < t0 < t2 . The total
muon gate fraction then, correct to second order in f (t), is
g(t1 , t2 ) = g1 (t1 , t2 ) − g2 (t1 , t2 ).
(3.13)
To avoid uncertainties in the timing that occur near the edges of the beam veto, we
choose to accept events with t = tCsI − tPG such that
t ∈ (−40, −10) ∪ (10, 140) ns.
(3.14)
The quantity tPG is the “zero time” of the pion gate PG corresponding to the time of
the pion stop PS. There is no complication associated with using a disjointed time interval since the gate fraction (3.13) is a linear function. One simply adds the fractions
obtained for each interval separately. Using the average rates and intervals indicated
on the plots of Figure 3.3 we arrive at the muon gate fraction, listed separately for
each data set:
g9pc. = 0.06967 ± 0.00062, g1pc. = 0.06447 ± 0.00066.
(3.15)
The gate fraction uncertainties in (3.15) are proportional to the uncertainties in the
pion stop rates extracted from the fits in Figure 3.3.
65
Chapter 3: Michel Decay Analysis
3.3
3.3.1
Results
Branching Ratio
We are now ready to compute our experimentally measured branching ratio:
B exp =
N sc
,
gANµ+
(3.16)
where N sc is the number of scaled experimental µ+ → e+ νe ν µ events, Nµ+ is the
number of decaying muons, which is identical to the number of stopped pions Nπ+ to
very good approximation, A is the detector acceptance and g is the muon gate fraction (3.15). The number of scaled events is the number of observed events weighted
with the hardware and software prescale factors and efficiencies that were effective
when each event was recorded:
N
sc
=
obs
N
X
i=1
obs
N
X
p1L psoft
≡
wi .
1L
1L
1L
²WC1 ²WC2 ²PV flive
i=1
(3.17)
This method mitigates spurious uncertainties induced by the slight variations of the
detection efficiencies and prescale factors over time. Otherwise, the final uncertainty
is inflated by several percent. See Table 3.2 for the mean values and their root-meansquare uncertainties. The uncertainty in the number of scaled events as computed
with (3.17) has three independent components. The first is just the uncertainty due
to Poisson counting statistics:
δN obs
1
δNPoisson
.
=
=√
sc
obs
N
N
N obs
(3.18)
66
Chapter 3: Michel Decay Analysis
There is also the root mean square (r.m.s.) fluctuation in the event weight w i under
the sum in Equation 3.17. The uncertainty in a single term is the r.m.s. fluctuation
1
δNrms
=
q
hw2 i − hwi2 .
(3.19)
Since we add N obs terms, all with the same uncertainty, the total r.m.s. uncertainty
is
δNrms =
√
N obs
q
hw2 i − hwi2 .
(3.20)
The last piece is the statistical uncertainty associated with the computation of the
efficiencies [Equations (2.10) and (2.17)]. This uncertainty is approximately 0.9 % of
the total event weight:
δN² = 0.009 × N sc .
(3.21)
These components are assumed to be independent and added in quadrature:
(δN sc )2 = (δNPoisson )2 + (δNrms )2 + (δN² )2 .
(3.22)
The results of the Michel event analysis, performed independently for each data
set, are summed up in Tables 3.3 and 3.4. Combining the two results for the branching
ratio, we obtain
Bµexp
= 0.996 ± 0.010,
+ →e+ ν ν
e µ
consistent with our expectation, Bµ+ →e+ νe ν µ = 100 %.
(3.23)
67
Chapter 3: Michel Decay Analysis
Table 3.2: Average efficiencies and prescale factors and their r.m.s.
fluctuations, listed separately for each data set.
quantity
nine-piece target
one-piece target
description
²1L
WC1
0.9509 ± 0.0041
0.9443 ± 0.0046
MWPC1 efficiency
²1L
WC2
0.9644 ± 0.0123
0.9731 ± 0.0017
MWPC2 efficiency
²1L
PV
0.9893 ± 0.0113
0.9843 ± 0.0006
PV efficiency
flive
0.845 ± 0.049
0.857 ± 0.041
front-end efficiency
p1L
512.3 ± 19.4
509.2 ± 14.0
hardware prescale factor
psoft
50
50
software prescale factor
3.3.2
Michel Parameter ρ
The experimental value of the Michel parameter ρ is found by minimizing the χ2
difference between the observed and simulated positron energy spectra in the range
26 < Ee+ < 56 MeV. The simulated spectrum depends implicitly on the parameter ρ
via the weight (1.11) assigned to each simulated event. The energy spectra are taken
from the calorimeter only. The plastic veto hodoscope and target energies are not included in the spectra. The simulated spectrum is normalized to the observed number
of events in the energy range. The uncertainty in this normalization is proportional
to the uncertainty in the branching ratio calculated above. As before, we compute
Chapter 3: Michel Decay Analysis
68
the result for each data set separately. These results are listed in Tables 3.3 and 3.4.
The combined result is
ρ = 0.758 ± 0.005.
3.4
(3.24)
Conclusions
Our results (3.23) and (3.24) from Michel decay analysis are encouraging. They
confirm that we are proceeding along the right path. In particular, our understanding
and simulation of the PIBETA detector are correct. We should have some reservations
however. We estimated uncertainties in the quantites that went into these results,
but we can not independently confirm these numbers. In particular, we said nothing
at all of the uncertainty in the number of decaying muons, implicitly neglecting it. In
fact, there is no way to get a reliable estimate of this uncertainty. This means that
the quoted uncertainties in (3.23) and (3.24) are underestimates. We are therefore
not alarmed in the least that our value of ρ is more than one standard deviation larger
that we expected. We are also not concerned about our inability to account rigorously
for these uncertainties. When we do our final analysis of the radiative Michel decay
branching ratio, we will normalize to the nonradiative decay branching ratio. All of
these quantities will cancel out along with their uncertainties, whatever they may be.
69
Chapter 3: Michel Decay Analysis
Table 3.3: Results for analysis of the nine-piece target data sample
symbol
value
(%)
remark
N µ+
2.73677 × 1011
¿ 0.01
rπ +
(132.05 ± 1.17) × 103
0.89
pion stop rate (Hz)
N obs
406426 ± 638
0.16
number of observed µ+ → e+ νe ν µ
number of decaying muons
decays w/ 10 < Ee+ < 56 MeV
N sc
(1.3527 ± 0.0122) × 1010
0.90
number of scaled µ+ → e+ νe ν µ
decays
gµ
0.06967 ± 0.00062
0.89
muon gate fraction
t ∈ (−35, −15) ∪ (20, 140) ns
A
0.7131 ± 0.0006
0.08
acceptance for Michel events
B
0.995 ± 0.013
1.27
Michel BR (Nπ+ normalization)
ρ
0.754 ± 0.007 (χ2ν = 1.7)
0.93
Michel parameter
(fit range 26 < Ee+ < 56 MeV)
70
Chapter 3: Michel Decay Analysis
Table 3.4: Results for analysis of the one-piece target data sample
symbol
value
(%)
remark
N µ+
2.35574 × 1011
¿ 0.01
rπ +
(97.97 ± 1.00) × 103
1.02
pion stop rate (Hz)
N obs
327184 ± 572
0.17
number of observed µ+ → e+ νe ν µ
number of decaying muons
decays w/ 10 < Ee+ < 56 MeV
N sc
(1.0776 ± 0.0098) × 1010
0.91
number of scaled µ+ → e+ νe ν µ
decays
gµ
0.06447 ± 0.00066
1.02
muon gate fraction
t ∈ (−35, −15) ∪ (20, 140) ns
A
0.7106 ± 0.0006
0.10
acceptance for Michel events
B
0.998 ± 0.014
1.37
Michel BR (Nπ+ normalization)
ρ
0.761 ± 0.007 (χ2ν = 1.0)
0.92
Michel parameter
(fit range 26 < Ee+ < 56 MeV)
Chapter 4
Radiative Michel Decay Analysis
4.1
Introduction
Encouraged by the results of chapter 3 that our understanding of the PIBETA detector response described in chapter 2 is correct, we now proceed to analyze radiative
Michel decay events. Our goal is to extract the Michel parameters η and ρ and to
measure the branching ratio for a large region of phase space. The experimentally
measured values of η and ρ are to be taken as those values which minimize the difference between the branching ratio measured by the experiment and that calculated
from Equation (1.15). The two parameter values affect both the measured and the
calculated branching ratios. The effect on the calculated value is explicit as can be
seen by inspection of Equation (1.15). The parameters influence the measured value
in a subtle and indirect way via the simulation of the detector acceptance. The ac71
72
Chapter 4: Radiative Michel Decay Analysis
ceptance is computed via Monte Carlo simulation of the apparatus, with each event
assigned a weight according to Equation (1.15). Accounting for this dual dependence
is straightforward in principle, but in practice it is computationally intensive. Past
µ+ → e+ νe ν µ γ measurements did not account for it. With modern computers however, the problem is tractable and we account for parameter dependence wherever it
occurs.
4.2
Strategy
4.2.1
Branching Ratio
Calculation of the µ+ → e+ νe ν µγ Branching Ratio
The theoretical branching ratio of the µ+ → e+ νe ν µ γ decay for a particular part of
the kinematic phase space is simply the differential branching ratio (1.15) integrated
over that region of the space:
B
theo
(η, ρ) = 2π
Z
x2
x1
Z
y2
y1
Z
cos θ2
cos θ1
·
¸
4
dx dy d(cos θ) f1 + ηf2 + (1 − ρ)f3 .
3
(4.1)
The integration limits and the kinematic constraint (1.17) define the phase space.
It is not meaningful to talk about the total branching ratio for a radiative decay
because such decays are infrared-divergent. Besides, photons with very low energies
are invisible to even the most sensitive detectors and decays in the corresponding
region of phase space are indistinguishable from the nonradiative decay. One must
73
Chapter 4: Radiative Michel Decay Analysis
therefore qualify any radiative decay branching ratio by stating the limits of the phase
space as above.
The integral (4.1) must be computed numerically. Because the infrastructure for
Monte Carlo computations must be developed in any case to compute the detector
acceptance, it is convenient also to compute the theoretical branching ratio by Monte
Carlo methods. The phase space is sampled at N uniformly distributed points within
the chosen bounds x1 ≤ x ≤ x2 , y1 ≤ y ≤ y2 , cos θ1 ≤ cos θ ≤ cos θ2 and consistent
with (1.17). The branching ratio is thus given by
B
theo
V X
(η, ρ) =
F ±V
N
r
¢
1 ¡ 2
hF i − hF i2
N
(4.2)
where V is the volume of the sampled phase space region and
4
F ≡ F (x, y, cos θ; η, ρ) = f1 + ηf2 + (1 − ρ)f3
3
(4.3)
is the integrand. The uncertainty can be made negligible by sampling a sufficiently
large number of points.
Measurement of the µ+ → e+ νe ν µγ Branching Ratio
The experimental branching ratio for a generic muon decay µ+ → X is given by
B exp =
Nµ+ →X
p
Nµ+ ²Aµ+ →X g(t1 , t2 )
(4.4)
where Nµ+ is the number of decaying muons, Nµ+ →X is the number of observed
µ+ → X decays, Aµ+ →X is the detector acceptance for µ+ → X decays, p is the
74
Chapter 4: Radiative Michel Decay Analysis
hardware prescaling factor, ² is the total detection efficiency which includes both the
tracking detectors and the front-end computer live time, while g(t1 , t2 ) is the gate
fraction accounting for muon decays occurring outside of the pion gate.
Because of systematic uncertainties in the counting of initial muons Nµ+ , the gate
fraction g(t1 , t2 ), and efficiency ², it is advantageous to evaluate the experimental
branching ratio of µ+ → e+ νe ν µ γ relative to that for the nonradiative decay µ+ →
e+ νe ν µ . The exact cancellation of Nµ+ and ² follows immediately. The cancellation
of the gate fraction g(t1 , t2 ) will also follow so long as care is taken to determine
consistently the time offsets for each event type. The two decays are recorded under
two different hardware triggers which do not necessarily have the same zero time. A
LL
radiative decay is a two-arm low threshold (CsS
) event while a nonradiative decay
is a one-arm low threshold (CSL ) event. Table 2.2 shows that prescaling factors for
these triggers are very different (16 and 512, respectively) so they do not cancel. The
experimental branching ratio is thus
³ P obs
N
1
A
i=1
B exp = ³ P obs
N
1
A
i=1
pi
´
pi
µ+ →e+ νe ν µ γ
´
.
(4.5)
µ+ →e+ νe ν µ
Detector Acceptance
The detector acceptance is computed via Monte Carlo simulation. Radiative Michel
decay events are thrown with randomly assigned kinematics and then assigned a
weight according to Equation (1.15), based on those kinematics. Here we define
75
Chapter 4: Radiative Michel Decay Analysis
the acceptance. The details of the detector simulation are described alongside the
description of the detector itself in Chapter 2.
The appropriate region of phase space is uniformly sampled as for the theoretical
branching ratio calculation. The coordinates of each sampled point are denoted with
the superscript “th” (e.g., xth ) to indicate that they are the thrown values. These
quantities represent actual physical values of the variables in the simulation. Thrown
values are to be distinguished from detected values of the variables. Detected variables
are denoted with a “det” superscript (e.g., xdet ). They represent the values which the
simulated detector apparatus registers. The simulation essentially takes points in the
phase space of thrown values and maps them to detected values by smearing them
with the response function of the detector.
If the simulation is a faithful representation of the real apparatus then we can
apply the same event selection criteria to the simulated events as we do to the actual
observed events in the data, and compute the acceptance as the ratio of events which
pass the cuts to the total of all thrown events. The events must be weighted according
to Equation (1.15) based on thrown values. The acceptance is given by
A(η, ρ) =
X
th·det
F
,
X
F .
(4.6)
th
The parameter dependence enters via the form of F given in Equation (1.15). The sum
in the numerator includes all terms which pass both the thrown and detected cuts,
while the sum in the denominator includes all events passing thrown cuts, regardless
Chapter 4: Radiative Michel Decay Analysis
76
of whether or not they also pass detected cuts. There is a subtle but crucial point to
be made here. The cuts must be chosen such that no event can pass the detected cuts
while simultaneously failing the thrown cuts. Figure 4.1 sketches the situation. This
possibility exists because of the resolution smearing of the detector apparatus. Some
events will inevitably be registered with values far from their actual physical values,
corresponding to the extreme tails of the resolution functions which are typically very
difficult to model properly. This problem is self-correcting for the simulation where
the thrown values of an event are known but is problematic for the actual apparatus
where the true thrown values are obscured by resolution smearing. Such events will
artificially increase the experimentally measured branching ratio because their true
values are outside of the phase space region for which the branching ratio is desired
and are thus not included in the acceptance or the theoretical branching ratio. This
caveat must be kept in mind in the final analysis but it does not impose any serious
difficulties or restrictions. Appropriate cuts can be chosen easily and it is simple to
test whether this condition is met.
The uncertainty in the detector acceptance is computed by standard statistical
methods for weighted sums. Since this uncertainty also depends on the parameters
via the weighting function (1.15), we derive them explicitly here. The uncertainty in
a single sampling of F is just the variance of the whole sample:
q
δF = hF 2 i − hF i2 .
(4.7)
Chapter 4: Radiative Michel Decay Analysis
77
The total uncertainty in the sum of several terms is just the individual uncertainties (4.7) (which are all the same, of course) added in quadrature. Thus, the total
variance in the sum is
³X ´ √ q
δ
F = N hF 2 i − hF i2 ,
(4.8)
where N is the number of points sampled. Equation (4.8) applies to both the numerator and denominator of (4.6) with the same conditions on the terms included in the
sums (and averages in the case of (4.8)). The overall uncertainty in the acceptance
is then obtained by the usual formula for the uncertainty of a ratio.
4.2.2
Parameter Optimization
The experimental values of η and ρ are those values which minimize the disparity
between the experimental and theoretical branching ratios. This difference is formally quantified by the standard statistical χ2 value. Since there are two parameters
and the normalization to the nonradiative decay gives one additional constraint, at
least four independent data points (i.e., four independently determined experimental
branching ratios) are required for a meaningful computation of χ2 . This is accomplished by dividing the phase space into bins along the three independent variables of
the reaction: x, y and cos θ. The binning must be chosen such that there are at least
four total bins. Care should also be taken to ensure that all bins have a statistically
significant number of observed events. This can be achieved, if necessary, by using
Chapter 4: Radiative Michel Decay Analysis
Figure 4.1: The cuts must be chosen in such a way that the region
encompassing thrown events (yellow) completely contains all detected
events (green). Furthermore, no event with thrown values outside of
the light shaded region can be mapped into the dark region by the
simulation. For example, events A and B are properly accounted for
by the definition of acceptance but event C is not. The grid implies the
possibility that the phase space can be divided into several bins. Only
x and y are shown here but the concept is easily extended to any other
variable (e.g., cos θ).
78
79
Chapter 4: Radiative Michel Decay Analysis
nonequidistant bins.
There is still the constraint that the experimental cuts must be completely enclosed
by the thrown cuts, but the binning described above introduces some subtlety to the
definition of the acceptance and therefore also to the branching ratio. The binning
is made within the detected region (dark shading in Figure 4.1) only. The thrown
region is left intact. The acceptance for a bin is computed as the ratio of events
detected in that bin to events thrown anywhere in the thrown region. A modification
of Equation (4.6) makes this explicit:
Aijk =
X
th·det(i,j,k)
F
,
X
F ,
(4.9)
th
where (i, j, k) labels the bin along the x, y and cos θ directions respectively and
det(i,j,k) reminds us that the sum in the numerator includes only events detected in
this particular bin. Note that this restriction also applies to the sum in the numerator
of (4.5). With this definition of acceptance, we get independent measurements of the
branching ratio for the entire thrown subspace from events detected in each individual
bin. A slight modification of our definition (4.5) makes this explicit:
µ
¶
obs
PNijk
1
i=1 pi
Aijk
µ+ →e+ νe ν µ γ
exp
Bijk = ³ P obs ´
.
N
1
p
i
i=1
A
(4.10)
µ+ →e+ νe ν µ
If the theoretical description is correct and the experiment has been done properly,
this definition leads to the same numerical result, up to experimental uncertainties,
in every bin.
Chapter 4: Radiative Michel Decay Analysis
80
Thus, the analytical task at hand consists of finding the parameter values which
result in a homogeneous distribution of experimental branching ratios, statistically
scattered about the single theoretical value (4.2). The optimal values are those which
minimize χ2 :
2
χ =
ny ncos θ
nx X
X
X
i
j
k
¢2
exp
Bijk
− B theo
,
¡ exp ¢2
δBijk + (δB theo )2
¡
(4.11)
where nx is the number of bins in the x-direction (with analogous definitions for ny
and ncos θ ) and δ denotes the total uncertainty in the quantity which follows. The
expectation value is equal to the number of degrees of freedom ν:
­
®
χ2 = ν = nx ny ncos θ − 2 − 1.
(4.12)
The product nx ny ncos θ is the number of independent measurements. The number
of degrees of freedom is three less than the number of independent measurements
because there are two free parameters and one normalization constraint.
The distribution of χ2 in the region of the minimum value χ20 is described by a
two-dimensional parabola:
χ2 (η, ρ) = χ20 + Cη η 02 + Cρ ρ02
(4.13)
where
η 0 = (η − η 0 ) cos φ + (ρ − ρ0 ) sin φ
(4.14)
ρ0 = −(η − η 0 ) sin φ + (ρ − ρ0 ) cos φ.
(4.15)
81
Chapter 4: Radiative Michel Decay Analysis
The parameters Cη and Cρ quantify the “sharpness” of the parabola along the two
perpendicular directions η 0 and ρ0 , φ is the angle by which the η 0 - ρ0 axes are rotated
with respect to the η - ρ axes and η 0 and ρ0 are the optimal values of the parameters
which minimize χ2 :
χ2 (η 0 , ρ0 ) = χ20 .
(4.16)
The value of φ represents the correlation between η and ρ. A quick inspection of the
expression for the differential branching ratio (1.15) reveals that η and ρ are indeed
correlated since f1 and f2 are not independent functions. The one standard-deviation
error in the parameters is determined by an ellipse about the optimal value which
defines the contour
χ2 (η 0 ± δη, ρ0 ± δρ) = χ20 + 1.
(4.17)
Alternatively, we may choose to fix one of the parameters at its optimum value
or even at its Standard Model value. In particular, a recent measurement of ρ [26]
indicates that it is very reasonable to use ρ = ρSM =
3
4
and subsequently to find
η as the lone free parameter. If the parameters are correlated, fixing one of them
reduces the overall experimental uncertainty in the other. In this case, the minimum
is simply a one-dimensional parabola about the central value η 0 with the one-standard
deviation uncertainty defined by
χ2 (η 0 ± δη, ρSM ) = χ20 + 1.
(4.18)
The parameter η is by definition a positive, semidefinite quantity with allowable
82
Chapter 4: Radiative Michel Decay Analysis
values between zero and one. It is therefore sensible to take a measurement of η ± δη
which is consistent with η = 0 and recast it as an upper limit η max on a possible
non-zero value:
η ≤ η max .
(4.19)
The upper limit η max is defined implicitly as the value which contains one standard
deviation (68.3%) of the probability via the relationship
Z
η max
0
"
1
exp −
2
µ
η − η0
δη
¶2 #
dη
,Z
1
0
"
1
exp −
2
µ
η − η0
δη
¶2 #
dη = 0.683,
(4.20)
where η 0 is the most probable value and δη is its one standard deviation error as
determined from the variational method described above.
4.3
4.3.1
Event Selection
Time Window
The Michel decay time structure described in section 3.2.2 actually applies to all
muon decays, which technically are included as Michel decays in our event sample.
Therefore, there is nothing new in that regard to discuss here. In fact, we have
designed our experiment in such a way that the time structure is irrelevant so long
as we take certain precautions. The gate fraction (3.13) is precisely the same for
radiative decay events and it therefore cancels when we normalize to the nonradiative
mode [Equation (4.5)]. The danger lies near the edges of the pion gate and the prompt
83
Chapter 4: Radiative Michel Decay Analysis
veto, and possibly with disparate time offsets associated with the different triggers.
There is a straightforward method which determines the allowable time window.
We simply take the ratio of the radiative decay time spectrum to that of the nonradiative decay as shown in Figure 3.3. If we can fit the ratio with a straight line,
y(t) = at + b,
(4.21)
such that a ± δa is consistent with zero slope, then we are assured that the gate
fraction will cancel in our final result. Figures 4.2 and 4.3 illustrate this method
for the nine- and one-piece targets respectively. Table 4.1 gives the allowable time
window for each data set. Note that the time window applies to both the radiative
decay events and the normalizing, nonradiative decay events. These windows are
different than those used when we analyzed nonradiative decay events, so we must
recompute the relevant statistics for our normalizing decay. These values are given
in Table 4.2.
4.3.2
Time Coincidence
A basic criterion for our event selection is the temporal coincidence of the photon and
positron showers. At typical experimental rates of approximately 100 π + /sec, there
will inevitably be random coincidences between uncorrelated events. The situation
is shown in Figure 4.4. There, the peak centered about 0 ns is due to the causally
connected showers from radiative Michel decay products. The width of the peak is
Chapter 4: Radiative Michel Decay Analysis
Figure 4.2: The ratio of the radiative decay time spectrum to the nonradiative decay time spectrum for the nine-piece target data set. The
plots are identical except that the lower one has excluded bins where
the cancellation is poor. The fit on the lower plot is of the form (4.21)
with a = (0.5 ± 2.9) × 10−5 sec−1 .
84
Chapter 4: Radiative Michel Decay Analysis
Figure 4.3: The ratio of the radiative decay time spectrum to the nonradiative decay time spectrum for the one-piece target data set. The
plots are identical except that the lower one has excluded bins where
the cancellation is poor. The fit on the lower plot is of the form (4.21)
with a = (0.4 ± 0.4) × 10−4 sec−1 .
85
86
Chapter 4: Radiative Michel Decay Analysis
Table 4.1: The time windows for which muon decay events are accepted.
The window is determined by the ratio of the radiative decay time
spectrum to the nonradiative decay spectrum (see Figures 4.2 and 4.3).
The time t = tCsI − tPG is the relative time between the calorimeter
+
showers tCsI = (teCsI + tγCsI )/2 and the “zero” time of the pion gate.
Data Set
Time Window
nine-piece target
(−41 < t < −6) ∪ (5 < t < 150) ns
one-piece target
(−40 < t < −7) ∪ (20 < t < 150) ns
Table 4.2: Statistics for the normalizing, nonradiative decay µ+ →
e+ νe ν µ . The quantities are the number of observed events N obs , the
detector acceptance A and the hardware and software prescaling factors
phard and psoft .
quantity nine-piece target
one-piece target
N obs
437831 ± 662
278133 ± 527
A
0.7126 ± 0.0006
0.6994 ± 0.0006
phard
≈ 512
≈ 512
psoft
50
50
Chapter 4: Radiative Michel Decay Analysis
87
indicative of the calorimeter timing resolution. The flat background is due to random
coincidences. The peak-to-background ratio (P/B) is slightly higher for the one-piece
target data set due to the lower event rate and consequent reduction of the probability
for random coincidences. We must subtract the background to arrive at the correct
number of signal events. This is accomplished by counting events that are out-of-time
and subtracting the result from the number of in-time events:
∆t ≤ 5 ns ⇒ in time
(4.22)
5 < |∆t| ≤ 10 ns ⇒ out of time
where
∆t ≡ te+ − tγ .
(4.23)
This technique also generalizes to functions of any kinematic variable. We simply
project a histogram of that variable for in-time events and subtract from it the corresponding histogram of out-of-time events. All results which follow have had the
random coincidence background subtracted.
4.3.3
Kinematic Cuts
We choose to extract the parameters η and ρ based on events in regions of the phase
space most sensitive to them. Dedicated Michel decay experiments are superior in
extracting ρ [26, 8] so we shall give priority to η, which can only be measured in
the radiative decay. Figure 1.3 indicates that the greatest sensitivity to η occurs
at large opening angles, and large values of x, and is only weakly dependent on
Chapter 4: Radiative Michel Decay Analysis
Figure 4.4: The positron/photon time coincidence spectrum for radiative Michel decay events. The flat background is due to random
coincidences between uncorrelated events.
88
89
Chapter 4: Radiative Michel Decay Analysis
y [10]. In chapter 3 we validated the response of our calorimeter for energies 26 <
ECsI < 56 MeV by extracting a consistent value of ρ from the Michel decay e+ energy
spectrum (see Tables 3.3 and 3.4). We therefore safely choose to analyze events where
both particles have total detected energies
det
27.5 < Etot
< 55 MeV
(4.24)
θdet > 105◦ ,
(4.25)
and measured opening angles
where Etot = ECsI + EPV + Etgt . We define the maximum detected energy to be
det
= 55 MeV so that our energy range corresponds to 0.5 < xdet , y det < 1.
Etot,max
These cuts also result in the complete suppression of the “splash-back” background
discussed below. Recalling the definition of detector acceptance in Section 4.2.1, we
must also choose cuts on the thrown values of the energies and opening angles. We
choose to accept events where both particles have
th
20 MeV < Etot
<
mµ
2
(4.26)
and
θth > 90◦ .
(4.27)
Since the maximum energy allowed by kinematics is mµ /2, this corresponds to 0.38 <
xth , y th < 1. The kinematic cuts on the normalizing, nonradiative decay are exactly
in Chapter 3.
the same as those used to compute Bµexp
+ →e+ ν ν
e µ
Chapter 4: Radiative Michel Decay Analysis
90
We measure the branching ratio in twelve bins – 2×2×3 divisions for x×y ×cos θ.
The divisions of x and y are nonequidistant. Nonequidistant bins mitigate purely
statistical effects by ensuring that all bins accept a similar proportion of the total
sample. The three bins of cos θ are equal in width. Details of the binning and the
appropriate event statistics for each bin are given in Appendix B.
4.4
4.4.1
Results
The Parameters η and ρ
The shape of the parameter space is shown in Figures 4.5–4.8, where for now we
still separate the data sets by target type. Note that the uncertainties denoted in
those figures are entirely statistical. We shall quantify systematic uncertainties below.
Figures 4.5 and 4.6 show χ2 (η, ρ) with both parameters varying freely. We see for both
sets that the Standard Model parameter values (η SM , ρSM ) = (0, 34 ) are consistent with
our data at the level of one to two standard deviations. Table 4.3 gives the optimal
parameters and their one standard deviation statistical uncertainties.
As all experimental evidence, including our own, is consistent with ρ = ρSM =
3
4
[26, 8], we are justified in fixing ρ at its nominal value and taking our final value of
η by allowing it to be the sole free parameter. Figures 4.7 and 4.8 show χ2 (η, ρ = 43 )
with the most probable values and one-standard deviation statistical uncertainties
clearly marked. These values are also tabulated in Table 4.3.
91
Chapter 4: Radiative Michel Decay Analysis
Table 4.3: The optimal values of η and ρ with both parameters free
and also the optimal value of η with ρ = ρSM = 34 fixed. The quoted
uncertainties are purely statistical.
data set
nine-piece target
one-piece target
η
ρ
−0.066 ± 0.070
0.750 ± 0.010
−0.065 ± 0.065
0.75 (fixed)
−0.115 ± 0.085
0.751 ± 0.011
−0.111 ± 0.077
0.75 (fixed)
Throughout our analysis so far, we have implicitly assumed that the difference in
the calorimeter response to photons is completely accounted for by our simulation,
which was calibrated to the positron response. We should allow for the possibility that
the energy calibration for photons is slightly different. A small energy gain difference
between photon and positron induced electromagnetic showers in the calorimeter
arises from small differences between the parts of the calorimeter volume sampled
by the two processes. This phenomenon is well established by prior analyses of pion
decays in the PIBETA detector [4, 15, 29, 16]. The difference is accounted for by
introducing one additional free parameter to our analysis – a small correction to
the gain factor g which appears in Equation (2.12) and is applied to photon shower
92
Chapter 4: Radiative Michel Decay Analysis
Table 4.4: Gain factor correction for photons and the corresponding
systematic uncertainty in η
data set
² (%)
δη (syst.)
nine-piece target 3.6 ± 0.4
0.046
one-piece target
0.051
1.7 ± 0.5
energies:
g → g(1 + ²).
(4.28)
Table 4.4 gives the optimal value of ² and its one standard deviation statistical uncertainty for each data set. Varying ² throughout its one standard deviation range
(with ρ fixed as before) induces a variation in the optimal value of η. This variation is
quantified as an additional systematic uncertainty in η and is also given in Table 4.4.
Figures 4.5–4.8 and the corresponding results in Table 4.3 were made with ² at its
optimal value so they are still valid. The systematic uncertainty is simply added to
the results already given.
We combine the two individual results for η with ρ =
3
4
to arrive at our final
result:
η = −0.084 ± 0.050(stat.) ± 0.034(syst.).
(4.29)
Chapter 4: Radiative Michel Decay Analysis
Figure 4.5: The statistical χ2 as a function of both η and ρ for the ninepiece target data set. The curves denote contours of constant ∆χ2 =
χ20 +n where n = 1, 2, 3 . . . and therefore represent nσ confidence levels.
93
Chapter 4: Radiative Michel Decay Analysis
Figure 4.6: The statistical χ2 as a function of both η and ρ for the onepiece target data set. The curves denote contours of constant ∆χ2 =
χ20 +n where n = 1, 2, 3 . . . and therefore represent nσ confidence levels.
94
95
Chapter 4: Radiative Michel Decay Analysis
Figure 4.7: The statistical χ2 as a function of η only with ρ =
for the nine-piece target data set.
3
4
fixed,
96
Chapter 4: Radiative Michel Decay Analysis
Figure 4.8: The statistical χ2 as a function of η only with ρ =
for the one-piece target data set.
3
4
fixed,
Chapter 4: Radiative Michel Decay Analysis
97
Applying Equation (4.20), this is interpreted as an upper limit of
η ≤ 0.033 (68 % confidence).
(4.30)
Our results for η are similar to those found in Reference [10], which are given
by Equations (1.20) and (1.21). We can further reduce the known upper limit by
combining our central value (4.29) with the one found there to arrive at the combined
result
η = −0.071 ± 0.051.
(4.31)
Applying Equation (4.20), this is interpreted as a combined upper limit of
η ≤ 0.028 (68 % confidence).
4.4.2
(4.32)
Branching Ratio
It is also of interest to measure the branching ratio for radiative Michel decay over
the broadest possible kinematic range and to examine the energy and opening angle
spectra. We recall that it is not possible to measure the total branching ratio for
a radiative decay nor is it even strictly meaningful theoretically, due to the infrared
divergences. We must therefore set a lower cutoff in both photon energy Eγ and
photon/positron opening angle θ. The positron energy need not be restricted, so we
allow all possible energies Eeth+ > me (x > x0 ). We choose to allow Eγth > 10 MeV
(y th > 0.189).
Chapter 4: Radiative Michel Decay Analysis
98
The PIBETA detector itself imposes a natural opening angle cutoff. A small
fraction of nonradiative Michel decays will masquerade as radiative decays due to
shower development in the calorimeter. The situation arises when a component of
the positron shower, probably a photon, travels laterally through the calorimeter a
significant distance before beginning its subsequent shower. This subsequent shower
is far enough away that the clumping algorithm identifies it as a distinct clump.
Since such a shower will not have corresponding wire chamber hits regardless of what
kind of particle actually initiated it, the tracking algorithm identifies it as a photon
emanating from the center of the stopping distribution. These events therefore pass
the kinematic cuts that identify radiative Michel decays. Since the showers will be in
close temporal coincidence, these events are not removed by the simple subtraction
algorithm described above. These “split-clump” events constitute the high peak at
large values of cos θ in Figure 4.9. Events of this type occur only at small apparent
opening angles: θ det < 45◦ (cos θ > 0.7071). So long as we limit ourselves to events
with detected angles larger than this we will avoid this rather large background. We
safely choose to limit the thrown values of the opening angle to θ th > 30◦ .
There is still a small but manageable background from misidentified Michel decays at larger apparent opening angles. Such events make up the tail which extends
all the way to cos θ = −1 in Figure 4.9. These “splash-back” events are analogous
to those described above but several orders of magnitude less probable. They occur
when a shower component is emitted at a very large angle with respect to the direc-
Chapter 4: Radiative Michel Decay Analysis
99
Figure 4.9: The simulated opening angle distribution of misidentified
nonradiative decay events.
tion of the initiating particle. In these cases, the secondary particle actually exits the
active volume of the calorimeter, traverses the space inside and reenters the calorimeter somewhere far from the original shower. Although the secondary particle may
register hits in the wire chambers, these are very unlikely to result in a track which
passes close enough to both the target and the calorimeter shower to be identified as a
charged track. The secondary shower is therefore identified as a photon by the tracking algorithm and the event passes the requirements for a radiative decay as above.
The “branching ratio” for these events can be estimated by simulation. It is the ratio
of the number of nonradiative decay events which pass the radiative decay cuts, to the
Chapter 4: Radiative Michel Decay Analysis
100
number of nonradiative events which are properly identified. The physical cause of
these events is bremsstrahlung interactions which result in either the emitted photon
or the original charged particle to scatter at a large angle with respect to the incoming particle. The simulation software uses approximations which are known to be
inaccurate in this extreme situation [3]. With this in mind, we conservatively correct
our measured branching ratio by subtracting the simulated background contribution,
while also adding this contribution (in quadrature) to our overall uncertainty.
Figures 4.10 and 4.11 show the cosine of the opening angle cos θ and the energy
det
The simulated spectra inspectrum of the two detected particles Eedet
+ and Eγ .
clude the contribution from misidentified background events. The branching ratio is
measured for the region of phase space bounded by the thrown cuts
θth > 30◦
Eeth+ > me
(4.33)
Eγth > 10 MeV,
based on events detected in the region defined by
θdet > 45◦
Eedet
+ > 12 MeV
(4.34)
Eγdet > 20 MeV.
The theoretical branching ratio computed by the methods of Section 4.2 with Stan-
101
Chapter 4: Radiative Michel Decay Analysis
Table 4.5: Results for the radiative muon decay branching ratio in the
region defined by the limits (4.33) and (4.34).
Quantity
nine-piece target
one-piece target
240144 ± 530
184018 ± 463
0.0790 ± 0.0002
0.0824 ± 0.0002
(4.53 ± 0.03) × 10−3
(4.45 ± 0.03) × 10−3
B bkg
0.09 × 10−3
0.09 × 10−3
B exp
(4.44 ± 0.09) × 10−3
(4.36 ± 0.09) × 10−3
N
0
obs
A
B
0
exp
Comment
number of observed events
detector acceptance
uncorrected branching ratio
background “branching ratio”
corrected branching ratio
dard Model values of η and ρ is
B theo = 4.30 × 10−3 .
(4.35)
Table 4.5 sums the results for the measured branching ratio in each data set. The
final uncertainties are obtained by adding the effect of the background to the original
uncertainty in quadrature. We combine the individual results to arrive at the final
value:
B exp = [4.40 ± 0.02 (stat.) ± 0.09 (syst.)] × 10−3 .
(4.36)
The statistical uncertainty (stat.) is the result of combining the two values of B 0exp in
Table 4.5 while the systematic uncertainy (syst.) is the magnitude of the splash-back
background “branching ratio” discussed above.
Chapter 4: Radiative Michel Decay Analysis
Figure 4.10: The radiative Michel decay kinematic spectra for the ninepiece target data set.
102
Chapter 4: Radiative Michel Decay Analysis
Figure 4.11: The radiative Michel decay kinematic spectra for the onepiece target data set.
103
Appendix A
The Functions fi(x, y, θ)
α
(1 − RDalitz ) n0V
16π 2 y
¡
¢
α
f2 (x, y, θ) =
(1 − RDalitz ) 2n0S − 2n0V + n0T
2
16π y
¢
¡
α
(1 − RDalitz ) 2n0S + n0V − n0T
f3 (x, y, θ) =
2
16π y
f1 (x, y, θ) =
RDalitz
¶
· µ
¸
2α
19
ymµ
=
−
ln
3π
2me
12
(A.1)
(A.2)
(A.3)
(A.4)
is the probability that the photon will internally convert to an e+ /e− pair resulting
in the decay µ+ → e+ νe ν µ e+ e− [9].
n0V
£
= 4(1 − β 2 ) 2∆−2 x(x + y)(2(x + y) − 3)
+∆−1 x2 y(3 − 4(x + y)) + x3 y 2
104
¤
Appendix A: The Functions fi (x, y, θ)
105
+∆−1 GV−1 + GV0 + ∆GV1 + ∆2 GV2
(A.5)
£
¤
2n0S − 2n0V + n0T = −8xy 2 ∆−1 x(1 − β 2 ) + 2(1 − y − 2x) + ∆x(1 + y)
(A.6)
©
£
¤
2n0S + n0V − 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)
£
+∆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
¤ª
£
¤
GV−1 = 8 y 2 (3 − 2y) + 6xy(1 − y) + 2x2 (3 − 4y) − 4x3
GV0
GV1
GV2
£
¤
= 8 −xy(3 − y − y 2 ) − x2 (3 − y − 4y 2 ) + 2x3 (1 + 2y)
£
¤
= 2 x2 y(6 − 5y − 2y 2 ) − 2x3 y(4 + 3y)
= 2x3 y 2 (2 + y)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
Appendix B
Radiative Michel Decay Event
Statistics
The tables in this Appendix summarize the measured numbers of the radiative Michel
decay events which were analyzed in the extraction of the Michel parameters η and
ρ. In particular, the number of observed events N obs , the acceptance A for Standard
Model values of η and ρ and the resulting branching ratio B exp are given for every
bin. The results are given separately for each data set and are furtheremore divided
into a total of three tables per data set, corresponding to the three bins of cos θ. Each
individual table gives the data for the corresponding 2 × 2 bins of x and y with those
bin limits denoted at the bottom (x) and left (y) edges of the table.
106
107
Appendix B: Radiative Michel Decay Event Statistics
Table B.1: Radiative Michel decay total event statistics. The Standard
Model predicts the branching ratio for this region of phase space to be
B theo = 2.035 × 10−4 .
quantity
nine-piece target one-piece target
total
N obs
19728 ± 172
13191 ± 137
32919 ± 220
A (10−2 )
9.852 ± 0.008
10.175 ± 0.008
10.014 ± 0.006
2.032 ± 0.021
2.030 ± 0.014
B exp (10−4 ) 2.028 ± 0.018
Table B.2: Event statistics for the nine-piece target data set in the first
bin of cos θ.
0.5 < y det ≤ 0.625
0.625 < y det ≤ 1
−1 ≤ cos θ < −0.7529 (138.8◦ < θdet ≤ 180◦ )
N obs = 2195 ± 57
N obs = 1891 ± 65
A = (1.118 ± 0.002) × 10−2
A = (0.916 ± 0.002) × 10−2
B exp = (2.00 ± 0.05) × 10−4
B exp = (2.10 ± 0.07) × 10−4
N obs = 1552 ± 46
N obs = 2125 ± 63
A = (0.755 ± 0.002) × 10−2
A = (1.013 ± 0.003) × 10−2
B exp = (2.09 ± 0.06) × 10−4
B exp = (2.14 ± 0.06) × 10−4
0.5 < xdet ≤ 0.625
0.625 < xdet ≤ 1
Appendix B: Radiative Michel Decay Event Statistics
Table B.3: Event statistics for the nine-piece target data set in the
second bin of cos θ.
0.5 < y det ≤ 0.625
0.625 < y det ≤ 1
−0.7529 ≤ cos θ < −0.5059 (120.4◦ < θdet ≤ 138.8◦ )
N obs = 1797 ± 50
N obs = 658 ± 33
A = (0.909 ± 0.002) × 10−2
A = (0.297 ± 0.001) × 10−2
B exp = (2.01 ± 0.06) × 10−4
B exp = (2.27 ± 0.11) × 10−4
N obs = 2203 ± 52
N obs = 2044 ± 56
A = (1.123 ± 0.003) × 10−2
A = (1.024 ± 0.003) × 10−2
B exp = (2.00 ± 0.05) × 10−4
B exp = (2.03 ± 0.06) × 10−4
0.5 < xdet ≤ 0.625
0.625 < xdet ≤ 1
Table B.4: Event statistics for the nine-piece target data set in the
third bin of cos θ.
0.5 < y det ≤ 0.625
0.625 < y det ≤ 1
−0.5059 ≤ cos θ < −0.2588 (105.0◦ < θdet ≤ 120.4◦ )
N obs = 1028 ± 37
N obs = 68 ± 10
A = (0.550 ± 0.002) × 10−2
A = (0.0310 ± 0.0004) × 10−2
B exp = (1.90 ± 0.07) × 10−4
B exp = (2.24 ± 0.34) × 10−4
N obs = 2806 ± 58
N obs = 1361 ± 42
A = (1.461 ± 0.004) × 10−2
A = (0.656 ± 0.003) × 10−2
B exp = (1.96 ± 0.04) × 10−4
B exp = (2.11 ± 0.07) × 10−4
0.5 < xdet ≤ 0.625
0.625 < xdet ≤ 1
108
Appendix B: Radiative Michel Decay Event Statistics
Table B.5: Event statistics for the one-piece target data set in the first
bin of cos θ.
0.5 < y det ≤ 0.625
0.625 < y det ≤ 1
−1 ≤ cos θ < −0.7529 (138.8◦ < θdet ≤ 180◦ )
N obs = 1487 ± 45
N obs = 1272 ± 51
A = (1.159 ± 0.002) × 10−2
A = (0.957 ± 0.003) × 10−2
B exp = (2.02 ± 0.06) × 10−4
B exp = (2.09 ± 0.08) × 10−4
N obs = 992 ± 36
N obs = 1287 ± 49
A = (0.755 ± 0.002) × 10−2
A = (0.988 ± 0.003) × 10−2
B exp = (2.07 ± 0.08) × 10−4
B exp = (2.05 ± 0.08) × 10−4
0.5 < xdet ≤ 0.625
0.625 < xdet ≤ 1
Table B.6: Event statistics for the one-piece target data set in the
second bin of cos θ.
0.5 < y det ≤ 0.625
0.625 < y det ≤ 1
−0.7529 ≤ cos θ < −0.5059 (120.4◦ < θdet ≤ 138.8◦ )
N obs = 1307 ± 41
N obs = 453 ± 27
A = (0.944 ± 0.002) × 10−2
A = (0.326 ± 0.001) × 10−2
B exp = (2.18 ± 0.07) × 10−4
B exp = (2.19 ± 0.13) × 10−4
N obs = 1533 ± 43
N obs = 1361 ± 44
A = (1.115 ± 0.003) × 10−2
A = (1.085 ± 0.003) × 10−2
B exp = (2.16 ± 0.06) × 10−4
B exp = (1.97 ± 0.06) × 10−4
0.5 < xdet ≤ 0.625
0.625 < xdet ≤ 1
109
Appendix B: Radiative Michel Decay Event Statistics
Table B.7: Event statistics for the one-piece target data set in the third
bin of cos θ.
0.5 < y det ≤ 0.625
0.625 < y det ≤ 1
−0.5059 ≤ cos θ < −0.2588 (105.0◦ < θdet ≤ 120.4◦ )
N obs = 755 ± 31
N obs = 49 ± 8
A = (0.598 ± 0.002) × 10−2
A = (0.0361 ± 0.0005) × 10−2
B exp = (1.98 ± 0.06) × 10−4
B exp = (2.13 ± 0.36) × 10−4
N obs = 1874 ± 46
N obs = 821 ± 33
A = (1.493 ± 0.004) × 10−2
A = (0.720 ± 0.003) × 10−2
B exp = (1.97 ± 0.05) × 10−4
B exp = (1.79 ± 0.07) × 10−4
0.5 < xdet ≤ 0.625
0.625 < xdet ≤ 1
110
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