A Study of the Structure of the Pi Meson
via the Radiative Pion Decay π + → e+ νγ
Maxim Aleksandrovich Bychkov
Protvino, Russia
B.S., Moscow State University, 1998
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
August, 2005
Abstract
Two of the most important parameters describing pi meson structure are FV and FA ,
the vector and axial vector form factors, respectively. Their values can be determined
primarily by measuring the π + → e+ νγ radiative pion decay. This study is based on
the 20,182 radiative pion decay events collected at the Paul Scherrer meson factory
using a large acceptance detector and a stopped pion beam. A detailed analysis of the
dependence of the decay rate on kinematic parameters has resulted in the following
best fit values: FV =0.0262(15), FV =0.0118(3) and a=0.241(93), where the parameter
a describes the momentum dependence of the form factors. This is the first experimental determination of a ever. The reported values of FV and FA represent nearly
six- and twofold improvements in accuracy over previously published results, respectively, and are in an excellent agreement with predictions of the Standard Model.
i
Acknowledgments
There are so many people that made this work possible who I would like to thank.
I would like to thank my supervisor Professor Dinko Počanić for his continuous support and guidance throughout my years in graduate school. His help stretched beyond
the realm of physics and his advise was always timely. I would also like to thank Dr.
Emil Frlež. His relaxed attitude goes along with enormous amount of tasks accomplished. His work and support were constant reminders that everything is possible.
Special thanks to my wife Evgenia Baryshnikova. There is not enough room here
to express how thankful I am for having her as my life companion. She makes it feel
like home anywhere we go.
Great deal of gratitude goes to my friends and colleagues Alexander Stolin, Andrea
Soddu, Brent VanDevender and Tamaz Brelidze. Thank you for discussing matters
on and off the subject with me. Life would not be the same without you guys. Finally
my endless gratitude to the staff and professors of the University of Virginia Physics
Department and members of the PIBETA collaboration. There are so many things
these people helped me with and taught me.
ii
Contents
1 Introduction
1
2 Theoretical description of π + → e+ ν and π + → e+ νγ decays
6
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
π + → e+ ν decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
π + → e+ νγ decay . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3.1
Kinematics of the decay . . . . . . . . . . . . . . . . . . . . .
10
2.3.2
The decay rate . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.3.3
The Inner Bremsstrahlung contribution . . . . . . . . . . . . .
14
2.3.4
The structure-dependent contribution . . . . . . . . . . . . . .
17
2.3.5
Inner Bremsstrahlung-structure-dependent interference contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.6
Total decay rate . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.7
Radiative corrections . . . . . . . . . . . . . . . . . . . . . . .
21
Acknowledgments
3 Experimental Method
iii
22
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.2
Alignment of the detector elements . . . . . . . . . . . . . . . . . . .
26
3.2.1
Alignment of the wire chambers with CsI crystals . . . . . . .
27
3.2.2
Relative alignment of the target and the degrader . . . . . . .
35
3.2.3
Alignment of the plastic veto and the wire chambers . . . . .
37
Components of the PIBETA trigger system . . . . . . . . . . . . . . .
40
3.3.1
Pion stop signal . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.3.2
Pion gate signal . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.3.3
Prescaled pion gate signal . . . . . . . . . . . . . . . . . . . .
45
3.3.4
The ADC gate and the CsI signal . . . . . . . . . . . . . . . .
48
Gain matching of the CsI and PV modules . . . . . . . . . . . . . . .
48
3.4.1
CsI gain match . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.4.2
PV gain match . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3
3.4
4 Data Analysis
61
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.2
Basis of the PIBETA data analysis . . . . . . . . . . . . . . . . . . .
63
4.2.1
MIDAS Data Acquisition System and analyzer . . . . . . . . .
63
4.2.2
Data “packing” and PAW . . . . . . . . . . . . . . . . . . . .
65
4.2.3
Simulation of the experiment . . . . . . . . . . . . . . . . . .
65
iv
Acknowledgments
4.3
4.2.4
Normalizing decay . . . . . . . . . . . . . . . . . . . . . . . .
67
4.2.5
Particle identification . . . . . . . . . . . . . . . . . . . . . . .
68
4.2.6
Assignment of the time variables . . . . . . . . . . . . . . . .
70
π + → e+ ν decay analysis . . . . . . . . . . . . . . . . . . . . . . . .
73
4.3.1
Time offsets and gate fraction . . . . . . . . . . . . . . . . . .
74
4.3.2
Degrader energy and prompt cuts . . . . . . . . . . . . . . . .
78
4.3.3
Energy calibration and background subtraction.
π + → e+ ν
ADC spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4.3.4
π + → e+ ν TDC spectrum . . . . . . . . . . . . . . . . . . .
87
4.3.5
ADC method vs. TDC method of counting events . . . . . . .
92
4.3.6
Absolute normalization and related quantities . . . . . . . . .
93
4.3.7
Extraction of the normalization parameters from the ODB . .
96
4.3.8
Muon contamination of the beam . . . . . . . . . . . . . . . .
99
4.3.9
SCX in the target . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3.10 π + → e+ ν acceptance . . . . . . . . . . . . . . . . . . . . . . 103
4.4
π + → e+ νγ analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.2
π + → e+ νγ DST and preparation of the sample . . . . . . . 110
4.4.3
Background subtraction . . . . . . . . . . . . . . . . . . . . . 114
4.4.4
π + → π 0 e+ ν and µ+ → e+ ννγ background subtraction . . . 116
4.4.5
CsI energy calibration . . . . . . . . . . . . . . . . . . . . . . 119
Acknowledgments
v
4.4.6
CsI TDC discriminator inefficiency . . . . . . . . . . . . . . . 127
4.4.7
Total positron energy: energy in the PV and target . . . . . . 130
4.4.8
Acceptance calculations for the radiative decays . . . . . . . . 132
4.4.9
The acceptance for π + → e+ νγ events. . . . . . . . . . . . . 133
4.4.10 Global fit to the data . . . . . . . . . . . . . . . . . . . . . . . 134
4.4.11 Radiative corrections to π + → e+ ν acceptance . . . . . . . . 139
4.4.12 Form factor dependence on the invariant mass of the lepton pair 145
4.4.13 Analysis of the synthetic data . . . . . . . . . . . . . . . . . . 152
5 Results
164
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.2
Global fit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.3
Compilation of uncertainties . . . . . . . . . . . . . . . . . . . . . . . 169
5.4
5.3.1
Statistical uncertainties . . . . . . . . . . . . . . . . . . . . . . 169
5.3.2
Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . 172
Stability of the results . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.4.2
π + → e+ ν energy cut . . . . . . . . . . . . . . . . . . . . . . 175
5.4.3
π + lifetime measurements from the π + → e+ νγ data . . . . . 176
5.4.4
Two subsets of π + → e+ νγ data . . . . . . . . . . . . . . . . 178
5.4.5
Extension of Regions B and C in π + → e+ νγ . . . . . . . . . 179
Acknowledgments
5.4.6
vi
Subdivision of the experimentally accessible phase space into
more bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.5
Comparison with the previous experimental and theoretical results . . 190
5.6
Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . 196
Appendices
197
A Alignment of wire chambers
197
B On the question of “shape” fits
199
1
Chapter 1
Introduction
Pi mesons (pions) hold a special place in subatomic physics. The history of pions
started with the theoretical work of Yukawa in 1934 who introduced it in order to
explain the strong interaction which holds the nucleus together. At present, pions are
regarded as bound states of two elementary particles called quarks. Pions come in
three types: negatively and positively charged (π − and π + ) and neutral (π 0 ). They
are made of different combinations of the lightest quarks u and d, and energetically
disallowed to decay into other mesons or barions. In fact, their low masses limit pions
to decay exclusively via weak or electromagnetic interactions. Along with the decays
of muons (another elementary particle of the so-called lepton family), charged pions
create an opportunity to study weak interactions in a very precise way.
This work is a part of a larger effort to study rare decays of pions and muons, the
PIBETA project. The PIBETA program encompasses 8 international institutions.
2
CHAPTER 1: Introduction
Table 1.1: Summary of the pion and muon decays.
Mode
π + → µ+ νµ
Fraction(Γj /Γ)%
Subsequently
(99.98770 ± 0.00004) µ+ → e+ ν e νµ (≈ 100%)
µ+ → e+ ν e νµ γ (1.4%)(kin.)
π + → µ+ νµ γ
π + → e+ νe
π + → e+ νe γ
π + → π 0 e+ ν
∼ 10−4 (kin.)
(1.230 ± 0.004) × 10−4
∼ 10−7 (kin.)
(1.036 ± 0.006) × 10−8 π 0 → 2γ (98.8%)
π 0 → e+ e− γ (1.2%)
π + → e + ν e e+ e−
(3.2 ± 0.5) × 10−9
Its experimental base located at the Paul Scherrer Institute (PSI) in Switzerland. We
study a variety of physical processes which are summarized in table 1.1. 1 .
The PIBETA experimental setup is sensitive to all of the aforementioned processes
and allows for a wide range of self-consistency cross-checks.
The topic of this dissertation is the radiative pion decay π + → e+ νγ . This
decay is a radiative counterpart of the regular leptonic decay of the pion π + → e+ ν .
The emission of a photon is governed by several distinct physical processes. The
case when a photon is emitted by a charged propagating particle (either π + or e+ )
1
The abbreviation kin. signifies that the branching ratio depends on the kinematic region being
studied; low energy photons are excluded in practical measurements
CHAPTER 1: Introduction
3
is well understood in the framework of the quantum electro dynamics (QED) and is
calculable precisely. On the other hand, direct emission of a photon from the weak
vertex is determined by the internal structure of the pion and is associated with
both vector and axial-vector currents. The structure-dependent emission rate of the
photon is described by a pair of a priori unknown form factors parameterized as the
vector form factor FV , and the axial-vector form factor FA . These two parameters
can be calculated independently using various models of the strong interaction. The
precise knowledge of the form factors helps to discriminate among the models and to
improve the accuracy of related theoretical calculations.
In addition to the above, studying π + → e+ νγ decay may shed light on processes
outside the Standard Model (SM) . The SM uses the vector and axial-vector currents
exclusively to describe the weak interaction. Firstly, the conserved-vector-current
hypothesis (CVC) put forth by Feynman and Gell-Man [25] is tested by relating
the vector form factor to the neutral pion decay rate [62]. Even though the current
experimental precision of the neutral pion lifetime is not satisfactory, a competitive
measurement of FV from an independent source is of considerable interest. Secondly,
the π + → e+ νγ decay rate should be sensitive to the presence of other types of interactions such as scalar and tensor weak currents. If detected, these interactions would
indicate the existence of previously unknown particles (see references [32, 18, 19]).
Finally, recent developments in neutrino physics raise important questions regarding
lepton universality, which in turn, is strongly constrained in a precise measurement
CHAPTER 1: Introduction
4
of the π + → e+ ν decay. This work does not provide an improved value of the
π + → e+ ν branching ratio, but confirms the feasibility of its accurate measurement
using the PIBETA experimental setup.
There have been a series of experimental studies of the π + → e+ νγ decay (see
chapter 5 for a comprehensive comparison). As can be expected, this decay mode
has proved to be a challenge due to its low branching fraction. Early experiments did
not achieve the required precision, while later experiments were mutually inconsistent.
Moreover, the two latest results, obtained by the ISTRA and PIBETA collaborations,
have indicated deviations from the Standard Model description of the process. The
two results are also somewhat incompatible with each other.
This state of affairs indicates a strong need for a more thorough experimental
study of the π + → e+ νγ decay, which is presented in this dissertation.
This thesis is organized as follows. The theoretical description of the π + → e+ νγ
decay within the framework of Standard Model is given in chapter 2. Chapter 3
briefly describes the PIBETA experimental setup and certain aspects of the data
acquisition system (DAQ) . Chapter 4 is dedicated to the data analysis of the π + →
e+ νγ decay. It comprises three sections. Section 4.2 relates to the general structure
of the analysis and touches on general issues pertinent to the analysis of the data
recorded by the PIBETA detector. Next, section 4.3 deals with the normalizing decay
mode π + → e+ ν. Finally, section 4.4 concludes the description of the data analysis
by summarizing the procedures and techniques used for the reconstruction of π + →
CHAPTER 1: Introduction
5
e+ νγ decay events. Chapter 5 combines different aspects of the analysis and presents
results, discussion and comparison with previous experimental and theoretical results.
6
Chapter 2
Theoretical description of
π + → e+ν and π + → e+νγ
decays
2.1
Introduction
The π + → e+ ν and π + → e+ νγ decays are theoretically very well described in the
Standard Model of elementary particles and interactions. We examine the theoretical
description of these decays in detail below. More information on the topic could be
found in [15] with references therein.
CHAPTER 2: Theory of π + → e+ ν (γ) decay
2.2
7
π + → e+ν decay
The differential decay rate for π + → l+ νl decay (where l = e or µ) can be written as:
1
d 3 pl d 3 pν
2 1
|M |
(2π 4 )δ 4 (q − pl − pν ) ,
dΓ =
3
3
2mπ
El Eν (2π) (2π)
(2.1)
where mπ is the mass of the pion, q, pl = (El , pl ) and pν = (Eν , pν ) are the fourmomenta of the pion, lepton (l) and neutrino, respectively, and the matrix element
M is given in the (V − A) theory [49] as:
iG
M = √ h0|{Vλ (0) − Aλ (0)}|πiul γλ (1 − γ5 )vν ,
2
(2.2)
where G = 1.16639 × 10−11 (MeV)−2 is the Fermi coupling constant. When the sum
over final spin states is performed, the differential decay rate becomes:
dΓ =
G2 fπ2 m2l pl pν 3 3 4
d pl d pν δ (q − pl − pν ) .
2mπ (2π)2 El Eν
(2.3)
After integrating equation ( 2.3) over lepton momenta, the total decay rate for the
π → lνl decay is:
Γ=
G2 fπ2 m2l
(m2π − m2l )2 ,
3
8πmπ
(2.4)
where fπ =130.7 MeV is a pion decay constant defined as follows:
h0|Aλ (0)|πi = ifπ (q 2 )qλ
(2.5)
In equation (2.4) it is seen that Γ ∝ m2l . This can be interpreted as a physical
consequence of the term (1 − γ5 ) in equation ( 2.2), which is a helicity projection
CHAPTER 2: Theory of π + → e+ ν (γ) decay
8
operator for massless leptons. It allows only left-handed massless particles and righthanded massless anti-particles. If ml = 0, angular momentum conservation would
prohibit the π → lνl decay when mν = 0. However, for the electron and muon, both
positive and negative helicity states are mixed by an amount proportional to the
mass, resulting in non-zero decay rates. Using equation (2.4), the branching ratio
R0 = Γ(π → eνrme )/Γ(π → µνµ ) can than be calculated:
2
R0 =
2
fπe m2e (m2π − m2e )2
fπe
(1.283 × 10−4 ) .
=
µ2 2 (m2 − m2 )2
µ2
fπ mµ π
fπ
µ
(2.6)
The principle of electron-muon universality in pion decay holds under the assumption
that the basic interaction current is of the V − A type if fπe = fπµ . In the framework
of the universal V − A theory it was shown by Berman [6] and Kinoshita [40] that
the branching ratio, equation (2.6), is substantially modified by radiative corrections
which depend on ml . Feynman diagrams for the radiative corrections involved are
shown in figure 2.2. These diagrams depict the loop radiative corrections which
involve emission-reabsorbtion of a virtual photon by a charged propagator. The
diagrams that involve the emission of the real photon (IB) are similar to the ones
shown in figure 2.3.
Adding the IB and virtual corrections, the radiative corrections to R0 of equation
(2.6) assuming electron-muon universality, we obtain:
R
π + →e+ ν
=
Γ(π → eνe (γ))
= R0 (1 + δ)(1 + ε) = 1.233 × 10−4 ,
Γ(π → µνµ (γ))
where the largest correction is δ = −(3α/π)ln(mµ /me ) and ε = −0.92(α/π).
(2.7)
CHAPTER 2: Theory of π + → e+ ν (γ) decay
9
e
γ
π
e
γ
π
ν
ν
γ
e
π
ν
Figure 2.1: Processes leading to corrections of the π + → e+ ν decay due to virtual emission
and reabsorbtion of photons.
Two most recent results of the Standard Model based calculations are by Marciano
and Sirlin [48] and Decker and Finkemeier [21]. They give, respectively,
R
π + →e+ ν
=




 (1.2352 ± 0.0005) × 10−4 , ref. [48]
Γ(π → eνe (γ))
=
Γ(π → µνµ (γ)) 


 (1.2356 ± 0.0001) × 10−4 , ref. [21]
(2.8)
We note the far greater precision of these results than that of the best experimental
CHAPTER 2: Theory of π + → e+ ν (γ) decay
10
value given in [31]
R
π + →e+ ν
= (1.230 ± 0.004) × 10−4 .
2.3
π + → e+νγ decay
2.3.1
Kinematics of the decay
(2.9)
We will consider the positive pion only, since the experiment was conducted on positive pions: k = (Ek , k), pe = (Ee , pe ), pν = (Eν , pν ) and pπ = (Eπ , pπ ) are the
four-momenta of the photon, positron,neutrino and pion respectively. The equations
of energy-momentum conservation read:
Eπ = E k + E e + E ν , p π = k + p e + p ν .
(2.10)
The masses of the particles involved in the decay are [10]:
me = 0.511 MeV, mπ = 139.57 MeV .
(2.11)
The neutrino will be considered massless (a small mass of the order of a few electronvolts would have a negligible influence on the kinematics). We will extensively use
the variables x and y defined in references [12, 3]:
x=
2pπ · k
2pπ · pe
, y=
.
2
mπ
m2π
(2.12)
CHAPTER 2: Theory of π + → e+ ν (γ) decay
11
The use of these two variables is convenient because, in the rest frame of the pion, x
and y are the photon and positron energies in units of mπ /2:
x=
2Ee
2Ek
, y=
.
mπ
mπ
(2.13)
These energies are two of the tree quantities directly measurable in the decay. The
third observable is the angle θeγ between the positron and photon momenta. In the
pion rest frame these three quantities are related by:
y(x − 2) + 2(1 − x + r 2 )
√
,
x y 2 − 4r2
cos θeγ =
(2.14)
where r2 = m2e /m2π = 1.34 × 10−5 .
The ranges of definition of x and y are easily established:
1
1
1 q 2
1q 2
2
1 − y − y y − 4r ≤ x ≤ 1 − y +
y − 4r2
2
2
2
2
(2.15)
2r ≤ y ≤ 1 − r 2 .
(2.16)
In the pion rest frame, it is useful to express the scalar products as follows:
1
pe · pν = m2π (1 − x − r 2 )
2
(2.17)
1
pe · k = m2π (x + y − 1 − r 2 )
2
(2.18)
1
pν · k = m2π (1 − y + r 2 ) .
2
(2.19)
The smallness of r 2 allows us to neglect it in the following formulae. We also introduce
the variable:
λ = y · sin2
θeγ
x+y−1
=
.
2
x
(2.20)
CHAPTER 2: Theory of π + → e+ ν (γ) decay
12
Occasionally we will make use of the variable z, which is the energy of the neutrino
in the pion rest frame (in units of mπ /2):
z = 2Eν /mπ ,
(2.21)
with the conservation of energy relation x + y + z = 2.
Figure 2.3.1 shows the kinematically allowed regions in a triangular plot (x, y, z).
Three plots depict kinematic Regions A, B and C described in section 4.4.2. The
kinematically allowed regions are bounded by the lines x = 1 (photon of maximum
energy), y = 1 (positron of maximum energy) and z = 1 (neutrino of maximum
energy). The line x = 1 corresponds to θeγ = 180◦ (λ = +1), photon and positron
being emitted anti-parallel and the photon being left-handed. The line y = 1 also
corresponds to θeγ = 180◦ (λ = +1), photon and positron being still emitted anti
parallel but the photon being now right-handed. The line z = 1 corresponds to
θeγ = 0◦ (λ = 0), photon and positron being emitted parallel and the photon being
left-handed.
There are two methods to observe the π + → e+ νe γ decay in a complete kinematics.
One can either measure the two energies (positron and photon) or one energy (say,
the positron energy) and the angle θeγ . A combination of both measurements is
redundant, but very efficient in reducing the background.
CHAPTER 2: Theory of π + → e+ ν (γ) decay
13
Figure 2.2: The kinematic of the π → e+ νγ decay.
2.3.2
The decay rate
There are two parts in the amplitude describing π + → e+ νγ decay:
M (π + → e+ νe γ) = MIB + MSD .
(2.22)
CHAPTER 2: Theory of π + → e+ ν (γ) decay
14
MIB is the Inner Bremsstrahlung amplitude: the pion emits the positron and the
neutrino via the axial-vector current and the photon is radiated from the external
charged particles (figure 2.3, top panels). It is the “trivial” part of the process in the
sense that the effects of strong interactions are absent. It is calculated using the usual
rules of Quantum Electrodynamics. MSD is the structure-dependent (SD) amplitude,
governed by a vector and an axial-vector from factor (figure 2.3, bottom panels) and
it is the place where the strong interactions are acting. The photon is emitted from
intermediate states generated by strong interactions: hadronic states, quarks, etc.
The description of the complex mechanisms involved at that level requires models.
The structure dependent terms are of order unity and higher with respect to the
photon energy. Gauge invariance leaves only two form factors undetermined in the
structure-dependent part of amplitude: the vector (FV ) and axial-vector (FA ) form
factors. A more complete and detailed calculation of the amplitude M (π + → e+ νe γ)
along these lines may be found in a paper of De Baenst and Pestieau [2].
2.3.3
The Inner Bremsstrahlung contribution
The Inner Bremsstrahlung contribution can be expressed by:
MIB
G
= −ie √ cos θc fπ mπ me uν
2
(1
2
6 k 6 ε ∗ + p e · ε ∗ pπ · ε ∗
−
(1 + γ5 )ve .
pe · k
pπ · k
)
(2.23)
In equation (2.23), ε is the photon polarization, θc the Cabibbo angle, G the Fermi
constant. e is the proton electric charge. The units are such that α = e2 /4π = 1/137,
CHAPTER 2: Theory of π + → e+ ν (γ) decay
15
Figure 2.3: The contributions to the radiative decay M (π + → e+ νe γ); top panels represent
the Inner Bremsstrahlung contribution; bottom panels represent the structure-dependent
contributions governed by the vector form FV and the axial-vector form factor FA respectively.
G · cos θc = 1.16639 × 10−11 (MeV)−2 and fπ =130.7 MeV is the pion decay constant
defined in equation (2.5). The pion decay constant is sometimes expressed in units of
the pion mass and in such units fπ =0.936. An alternative definition of fπ in literature
√
involves a factor of (1/ 2). Taking the square of the amplitude in equation (2.23)
and summing over polarizations (denoted by the symbol Σ), one obtains:
CHAPTER 2: Theory of π + → e+ ν (γ) decay
Ã
G
Σ|MIB | = e √ cos θc fπ mπ me
2
2
!2
16
2
{4(pν ·k)(pe ·k)−V 2 pe ·pν )−2(pν ·k)(pe ·V )
2
(pe · k)
+ 2(pe · k)(pν · V )} .
(2.24)
The four-vector Vµ has been introduced in equation 2.24 for purpose of simplification.
It is defined by:
"
Vµ = 2 peµ − pπµ
Ã
pe · k
pπ · k
!#
(2.25)
and has the property:
V ·k =0 .
(2.26)
Finally, equation 2.24 becomes:
Ã
G
Σ|MIB | = e √ cos θc fπ mπ me
2
2
!2
8(1 − y)[1 + (1 − x)2 ]
.
x2 (x + y − 1)
(2.27)
Introducing the phase space factor for a three body decay, one obtains the differential transition rate with respect to the photon and positron energies for the Inner
Bremsstrahlung contribution to the π + → e+ νγ decay:
α
d2 ΓIB
=
Γ(π → eν)IB(x, y) ,
dxdy
2π
(2.28)
(1 − y)[1 + (1 − x)2 ]
;
x2 (x + y − 1)
(2.29)
where:
IB(x, y) =
Γ(π → eν) is the electronic decay rate given by:
Γ(π → eν) =
m2e (m2π − m2e )2
Γ(π → µν) = 1.283 × 10−4 · Γ(π → µν).
2
2
2
2
mµ (mπ − mµ )
(2.30)
CHAPTER 2: Theory of π + → e+ ν (γ) decay
17
Strictly speaking, the Inner Bremsstrahlung contribution diverges for x = 0 (lowenergy photons). The divergence in the total rate is canceled by the radiative corrections which is briefly explained in section 2.2. Practically, it has no consequence
on π → eνγ experiments which are limited to high-energy photons. We note that
there is no divergence for x + y − 1 = 0. This becomes clear if one does not neglect
the electron mass (factor r 2 ). Exact formulae are given in reference [12]. Being
proportional to the non-radiative decay rate, the IB contribution to the π → eν e γ
is also helicity-forbidden. This is not the case for the π → µνµ γ which is therefore
dominated by Inner Bremsstrahlung.
2.3.4
The structure-dependent contribution
The structure-dependent amplitude is the sum of the contributions of the vector and
axial-vector currents:
V
MSD
+
A
MSD
Ã
!
G cos θc ∗
εν [uν γµ (1 − γ5 )ve ] · [FV (s)εµναβ pπα kβ +
= e√
2 mπ
iFA (s)(g µν (pπ · k) − k µ pνπ )] ,
(2.31)
where FV (s) is the vector from factor and FA (s) is the axial-vector from factor. These
are real functions.
Consequently, the structure-dependent amplitude can be expressed in the form:
G cos θc
MSD = ie √
uν γµ (1 − γ5 )ve {ε∗µ [FA (s) + λFV (s)](pπ · k) − FA (s)(ε∗ · pπ )k µ }.
2 mπ
(2.32)
CHAPTER 2: Theory of π + → e+ ν (γ) decay
18
In the pion rest frame equation (2.32) reduces to:
G
MSD = ie √ cos θc mπ × uν ε∗ (1 − γ5 )ve [FA (s) + λFV (s)] .
2 2
(2.33)
Squaring equation (2.33) and summing over polarizations, one finds:
Ã
G cos θc
Σ|MSD |2 = e √
2 mπ
+|FV + FA |
2
"
!2
(
(pπ ·k) |FV + FA |2
"
2(pπ · k)(pπ · pν )(k · pe )
− (k · pν )(k · pe )
m2π
2(pπ · k)(pπ · pν )(k · pe )
− (k · pν )(k · pe )
m2π
#)
2
8m2π
.
(pπ · k)
(2.34)
Going to the pion rest frame and neglecting r 2 , one obtains:
Ã
G
Σ|MSD |2 = e √ cos θc m2π
2
!2
2|FV |2 (1 − x)[(1 + γ)2 (x + y − 1)2 + (1 − γ)2 (1 − γ)2 ] ,
(2.35)
where we have introduced the definition:
γ = FA /FV .
(2.36)
Equation (2.35) and three-body decay phase space integration lead to:
mπ
d2 ΓSD
α
=
Γπ→eν
dxdy
8π
me
µ
¶2 Ã
FV
fπ
2
!
[(1 + γ)2 SD+ (x, y) + (1 − γ)2 SD− (x, y)] (2.37)
with the following definitions:
SD+ (x, y) = (1 − x)(x + y − 1)2 = (1 − x)(1 − z)2
(2.38)
SD− (x, y) = (1 − x)(1 − y)2 = (1 − x)(x + z − 1)2 .
(2.39)
Due to the presence of the factor m−2
in the decay rate, the SD contribution is
e
greatly enhanced and therefore can be observed in certain kinematic configurations.
#
CHAPTER 2: Theory of π + → e+ ν (γ) decay
19
In contrast, the SD contribution is negligible in the corresponding muon decay. The
vector (FV ) and axial-vector (FA ) form factors depend on the variable:
s = (pe + pν )2 = m2π (1 − x)
(2.40)
x and y being positive, s is always very small (< m2π ) so that it is certainly a good
approximation to evaluate FV and FA at s = 0. The vector form factor FV can be
extracted from an information external to the π + → e+ νe γ decay. The ConservedVector-Current (CVC) theory [25] enables us to relate the isovector part of the
electromagnetic current and the strangeness-conserving weak vector hadronic current.
Then, via CVC., the vector form factor FV (0) is related to the π 0 → γγ amplitude d
through the relation [62, 50, 2]:
1
FV (0) = − √ d
2
(2.41)
and consequently:
1
|FV (0)| =
α
s
2Γ(π 0 → γγ (∗) )
,
πmπ0
(2.42)
where Γ(π 0 → γγ (∗) ) is the total decay rate of the neutral pion. This relation is well
understood in theory and fixes the value of FV for the practical purposes.
CHAPTER 2: Theory of π + → e+ ν (γ) decay
2.3.5
20
Inner Bremsstrahlung-structure-dependent interference
contribution
This contribution is obtained from:
(pπ · k) ∗
G
∗
[FA (s) + γFV∗ (s)]ν e (1 + γ5 )εuν .
ΣMIB MSD
= −ie √ cos θc
mπ
2
(2.43)
Introducing the variables x and y and going to the pion rest frame, we find:
d2 Γint
α FV
=
Γπ→eν [(1 + γ)F (x, y) + (1 − γ)G(x, y)] ,
dxdy
2π fπ
(2.44)
1
F (x, y) = − (1 − y)(1 − x)
x
(2.45)
where:
x2
1
.
G(x, y) = (1 − y) 1 − x +
x
x+y−1
Ã
!
(2.46)
It is worth noting that these terms are order of magnitude smaller than the IB and
SD terms, but there is no difficulty in including them into the analysis. The smallness
of these terms is the key to the re-scaling procedure described in chapter 5.
2.3.6
Total decay rate
Considering equations (2.28), (2.37) and (2.44), the total decay rate is given by
equation (2.47). We should notice that if vector form-factor FV is fixed the total
decay rate becomes quadratically dependent on a single parameter, namely γ.
d2 Γ
d2 ΓIB d2 ΓSD d2 Γint
α
=
+
+
=
Γπ→eν IB(x, y)+
dxdy
dxdy
dxdy
dxdy
2π
CHAPTER 2: Theory of π + → e+ ν (γ) decay
α
m2
Γπ→eν π2
8π
me
Ã
FV
fπ
!2
[(1 + γ)2 SD+ (x, y) + (1 − γ)2 SD− (x, y)] +
21
(2.47)
FV
α
Γπ→eν
[(1 + γ)F (x, y) + (1 − γ)G(x, y)] .
2π
fπ
2.3.7
Radiative corrections
Theoretical description of the radiative corrections to π + → e+ νγ decay was included
in the analysis. It is best described in reference [16]. At this point, it is important to
say that radiative corrections for all three regions are negative and vary in the range
of ∼(1 %-3 %) of the total yield.
22
Chapter 3
Experimental Method
3.1
Introduction
The PIBETA experiment is being conducted at the Paul Scherrer Institute Accelerator
Facility in Switzerland. The entire accelerator facility and its secondary beam areas
are shown in figure 3.1.
The PIBETA experiment uses the πE1 experimental area. Detailed information
about the yield and operational modes of this secondary beam line can be found in
reference [63]. The magnet configuration of the πE1 area is described in reference
[29].
A secondary beam of positive pions (π + ) with 113 MeV/c momentum is focused
in horizontal and vertical dimensions to a roughly circular spot of 1 cm in diameter
on the PIBETA target. Bending magnets and degrader fields along the beam line
CHAPTER 3: Experimental Method
Figure 3.1: Schematic layout of the accelerator facility at PSI
23
CHAPTER 3: Experimental Method
24
assure that our beam has low muon (µ+ ) and positron (e+ ) contamination.
After leaving the last set of focusing quadrupole magnets, the beam enters the PIBETA detector. The cross-sectional view of the PIBETA detector is shown in figure
3.2. The pion beam arrives to the beam counter (BC) and proceeds to the active degrader (AD) through a set of active collimators (AC1 and AC2) . The beam is slowed
down in the AD before it enters the active target (AT). Upon entering the target
the beam is stopped roughly in the middle of the target where the pions decay. The
subsequent decay products are tracked by a series of co-centric detectors surrounding
the beam stopping region. Charged particles (mainly positrons) are tracked by a
set of two cylindrical multi-wire proportional chambers (MWPC1 and MWPC2) and
then deposit some energy in a cylindrical plastic veto detector (PV). Neutral particles (mainly photons) traverse the wire chambers and plastic veto undetected. Both
neutral and charged particles subsequently deposit the majority of their energy in a
granulated pure CsI calorimeter. The signals from all of the detectors are detected
by the photomultiplier tubes (PMT) and transmitted into the electronics hut via a
series of the analog delays.
Entire assembly is surrounded by the thermal shields so that the temperature of
the CsI crystals can be effectively kept at a constant 22◦ C. The outer lead housing
of the detector absorbs the possible splashes of the electromagnetic showers leaking
through the back of the CsI crystals to preserve the self-vetoing of the valid events by
the the plastic cosmic veto (CV) detectors. The CVs form the exterior surroundings
CHAPTER 3: Experimental Method
25
Figure 3.2: A schematic cross-section of the PIBETA apparatus showing the main components.
CHAPTER 3: Experimental Method
26
of the detector assembly and veto the cosmic ray signals in the detector.
3.2
Alignment of the detector elements
The topological alignment of the detector elements is crucial to any experiment.
Most importantly, our geometry routines of the GEANT simulation package [13]
implement the surveyed dimensions of the detector and rely heavily on the accuracy
of the relative alignment of the set-up components. Inherently, our detector uses two
independent coordinate systems. One based on the center and alignment of the CsI
sphere, while the other one is fixed relative to the the wire chambers. During the
assembly of the detector special effort was given to make the centers of these systems
to coincide.
One of the most important quantities measured in the PIBETA experiment along
with the energies and types of the particles are the opening angles between the particles of interest. The angles between a pair of charged-neutral particles is determined
based on their track geometry. Neutral particles directions are derived solely from the
coordinates of the CsI crystals hit by the particles. For charged particles additional
information from the wire chambers is utilized. If there is a displacement between
the centers of these systems, then the opening angle distributions of such pairs would
be systematically distorted.
It is, however, possible that during the detector maintenance some elements could
CHAPTER 3: Experimental Method
27
have been displaced. Therefore, the precise knowledge of possible displacement is
necessary in order to understand the geometry of the detector.
3.2.1
Alignment of the wire chambers with CsI crystals
First, we need to find possible displacement between the coordinate system of the CsI
calorimeter and the coordinate system of the wire chambers.
One can use raw chamber data to reconstruct a track caused by a charged particle,
thus viewing a track and its intercept with the calorimeter surface (a sphere of 26 cm
radius) as “seen by the chamber”. Additionally, finding the point where track hit the
face of the calorimeter from the calorimeter data would enable us to compare these
points and deduce possible displacement between them.
In real life it is complicated to find out the exact location of track interceptions
with the face of the calorimeter from the calorimeter data. All we can say is whether
or not one particular crystal was hit by a given track. Assuming that in such a case
most of the energy was deposited in this crystal we can run the following procedure.
We first set our cuts to cleanly separate µ+ → e+ νν (Michel) events, accept and
reconstruct only one-track events and their interceptions with the calorimeter face.
We subsequently run through all the energies deposited in the calorimeter and find
out the greatest value, fill calculated value of the intercept coordinate into the histogram corresponding to the crystal with most energy deposited. An outcome of this
procedure is a set of crystal’s maps, e.g., two dimensional plots (θ vs. ϕ) of the crys-
28
CHAPTER 3: Experimental Method
tal illumination as seen by the chambers. Knowing the geometry of the crystals we
can predict how θ and ϕ projections of the maps should appear and compare them
to the experimental data. In order to construct a trial function which fits θ and ϕ
projections we need the following considerations.
• Geometry of the crystals determines the individual trial functions. Detailed
information about the geometrical shapes of the crystals and their dimensions
is given in [1]. It is only important to know that our crystals are of different
polygonal shapes varying from regular pentagons to irregular hexagons and
trapezoids. Since the inscribed radius of the calorimeter sphere is fixed at
26 cm we could effectively reduce our problem to a two dimensional one, i.e.,
everything depends only on spherical coordinates θ and ϕ of the crystals. It is
important no note that the polygonal shapes in a regular space remain so upon
the transformation into the θ − ϕ space. This assertion is illustrated in figure
3.3.
• We know that Michel events are uniformly distributed in space. Therefore, if
in θ − ϕ space our crystals were rectangular, then the probability of hitting
one particular crystal (for, say, theta projection) would be
dA
dθ
(where dA is
infinitesimal area of the θ-ϕ projection) and therefore our trial function would
be a constant. Since we have no rectangular shapes in θ-ϕ space our singlecrystal probability density is not a constant.
29
CHAPTER 3: Experimental Method
2.15
2.1
2.05
y2
1.95
1.9
1.85
1.45
1.5
1.55
x
1.6
1.65
1.7
Figure 3.3: A regular pentagon shape will remain pentagonal in θ-ϕ space. This result
is obtained by transforming the straight lines connecting the pentagon vertices into the
spherical coordinate system using a proper Jacobian. Obviously, while the curves connecting
the vertices are not straight lines they are virtually indistinguishable from straight lines for
our purposes.
• In addition, our plots are smeared by the chambers’ directional resolution.
Mathematically it means that our
dA
dθ
function is convoluted by some other
function. An educated guess would be a Gaussian convolution and therefore
our trial function becomes
f (m) =
Z
+∞
−∞
2
dA(θ) − (θ−m)
Γe σ2 dθ ,
dθ
(3.1)
30
CHAPTER 3: Experimental Method
where Γ and σ are fit parameters of the Gaussian.
An example of a hexagon in θ-ϕ space is shown in figure 3.4. As can be seen from
this figure, unlike
dA dA
,
dθ dϕ
function remains symmetric about the center of the crystal.
Thus, the center of the crystal is related to the peak value of a symmetric bell-shape
curve. It provides an additional convenience not to calculate some complicated functions for our ϕ-projections and use, instead, one of the available symmetric functions
such as a Gaussian.
Variations
dAi
dθ
of three separate areas marked as I, II and III in figure 3.4 are
considered in detailed calculations below.
The area of the triangular area I is
AI (θ) = 0.5(θ − θ2 )(ϕu − ϕd ) .
(3.2)
From a straight line equation we find
ϕ
u,d
ϕu,d
a − ϕ2
=θ
θa − θ 2
Ã
!
− θ2
Ã
ϕu,d
a − ϕ2
θa − θ 2
!
+ ϕ2 ,
(3.3)
and, therefore,
ϕu − ϕd = 2θα − 2θ2 α = 2α (θ − θ2 ) ,
(3.4)
where
α≡
ϕua − ϕ2
ϕd − ϕ 2
=− a
= 1.6814 ,
θa − θ 2
θa − θ 2
(3.5)
is the tangent of corresponding straight line. It has been shown in figure 3.3 that in
θ-ϕ space these are indistinguishable from the straight lines.
31
Theta, degrees
CHAPTER 3: Experimental Method
Hexagon type D
(0.0000, 79.1877)
O
1
III
b
(−6.9684, 75.3784)
(6.9685, 75.3784)
O
b
II
(−7.5630, 67.1491)
(7.5630, 67.1491)
a
Oa
I
O2
(0.0000, 62.6510)
Phi da
Phi d
b
Phi 1
Phi
u
b
u
Phi a
Phi, degrees
Figure 3.4: A graphical representation for the area calculations.
Thus
AI = α (θ − θ2 )2 , and
dAI
= 2α (θ − θ2 ) .
dθ
(3.6)
By analogy
AII = 0.5(θ − θa )(ϕu − ϕd + a), ϕu − ϕd = −2βθ + 2βθa + a, β ≡
ϕdb −ϕda
θb −θa
= 0.0723 .
AIII = 0.5(θ − θb )(ϕu − ϕd + b), ϕu − ϕd = −2γθ + 2γθb + b,
ϕ2 −ϕdb
θ1 −θb
= 1.8293 .
γ≡
Finally
dAII
= −2βθ + 2βθa + a, a = 15.126 .
dθ
dAIII
= −2γθ + 2γθb + b, b = 13.937 .
dθ
(3.7)
(3.8)
CHAPTER 3: Experimental Method
32
Convolution with a Gaussian was performed in Maple [47]. The outcome of this
procedure is rather cumbersome and is not given here. It leaves us with 4 parameters
of the fit to be determined. They are Γ and σ of the Gaussian and θa , θb of the
probability density. We also note that parameters θa , θb additionally enter the fitting
function as limits of integrations over corresponding regions.
For this analysis we picked only crystals that are symmetric in ϕ. These turned
out to be the following crystals: pentagons (crystals 0 through 9), hexagons type A
(10 through 20), hexagons type B (80 through 90), hexagons type C (110 through
120) and hexagons type D (170 through 179) (see reference [1]). Unfortunately, type
C hexagons are located too close to the veto crystals and their spectra are distorted
by the absence of the vetoed showers.
We note the “statistical advantage” of type D and B hexagons (chosen as an
example for all shown calculations), since the trial function contains two physically
interesting parameters (θa and θb ) along with two parameters of the convolution
Gaussian. Pentagons and type A hexagons have only one set of such vertices.
The choice of the parameters (and therefore trial functions) is not unique. For example, for crystals D and B we could have fixed one of the parameters by introducing
the theoretical difference between two points and replacing one of the parameters by
the sum (difference) of the second parameter and the theoretical difference. Careful
analysis has shown that reducing the number of parameters in such a way does not
improve our fits significantly (and sometimes makes them worse), but we loose half
33
CHAPTER 3: Experimental Method
of the information about the position of the crystals.
Typical crystal map and fitted θ projections for various crystal types could be seen
in figure 3.5. Compared to the theoretical values, all the ϕ angles look very close
to expected values, leading us to conclude that there is no significant displacement
of two systems in x or y directions. Assuming that only z displacement took place,
simple geometrical calculations shown in the figure 3.6 yield the following result:
0
∆z > 0,
sin(θ −θ)
∆z = r sin(π−θ
0
)
(3.9)
0
+θ)
∆z < 0, ∆z = r sin(−θ
,
0
sin(θ )
(3.10)
0
where θ are angles deduced from the fits and θ are theoretical angles. It is probably
worth mentioning that since sin(180o − θ) = sin(θ) for any θ between 0 and 180o ,
formulae for positive and negative displacements look absolutely the same (up to a
minus sign).
The final calculations for ∆z were done as an error weighted average over all
available crystals. The results turned out to be
∆z = (0.063 ± 0.040) mm .
This results assures us that the chambers are lined up with the CsI sphere extremely
well. Given the close fit of the detector components in the sub-mm range, this outcome
is not surprising.
CHAPTER 3: Experimental Method
Figure 3.5: The typical crystal maps and corresponding θ projections fits.
34
35
CHAPTER 3: Experimental Method
x
r
O‘
O
Z
z
Figure 3.6: Geometric setup for calculating the displacement along z-axis.
3.2.2
Relative alignment of the target and the degrader
The plastic scintillator target and active degrader detectors used in the PIBETA
experiment have been discussed in reference [29]. The target is a crucial element
of the detector system and it is positioned in the center of the CsI crystal spherical coordinate system. Knowing the exact position of the target is essential to the
understanding of the incoming beam profile as well as the beam stopping distribution.
As it has been discussed in the previous section, the track coordinates of the
neutral and charged particles are derived from different detector elements. In order
to understand the position of the target fully we need to study a process with at least
CHAPTER 3: Experimental Method
36
two charged particles in the final state. Having these particles tracked by the wire
chambers only, enables us to trace them back to their interception in the target and
reconstruct the beam profile of the incoming pions.
Dalitz decay of the neutral pion π 0 → e+ e− γ provides this opportunity. This decay
has relatively high branching ratio (∼ 10−2 ). It can be compared to the next available
two-charged tracks decay of the positive pion π + → e+ νe+ e− with branching ratio of
∼ 10−9 or a muon decay µ → e+ νe+ e− with branching ration of the order of ∼ 10−5 .
Even though neutral pions are not present in our beam, the single charge exchange
(SCX) reaction of the positive pions (π + + n → π 0 + p) with the degrader and target
material produces π 0 s in sufficient numbers. This reaction, however, has a relatively
high threshold value of the pion kinetic energy of T =18.7 MeV (see reference [28])
which is exceeded by the positive beam pions only in the the degrader. Majority of the
pions reaching the target do not have enough energy to undergo the charge exchange.
Thus, we have an ample supply of the neutral pions decaying in the degrader.
On the other hand, the β-decay of the charged pions ( π + → π 0 e+ ν ) [36, 53]
in the target is a significant source of neutral pions. Therefore, if the Dalitz decay
events were detected in a relatively wide range of the longitudinal coordinate, we
should clearly see the reconstructed vertices distributed in the degrader according to
the Bethe-Bloch distribution of energy loss in the degrader and a completely different
distribution coming from the target and peaked around the stopping point.
This information should reveal the relative positions of the target and the degrader
CHAPTER 3: Experimental Method
37
as an independent test of the positioning accuracy. Despite the apparent dominance
of the degrader Dalitz events as compared to the converted πβ events total number
of events seems to be equally divided between the two processes. The reason for that
is quite simple. If a charged beam pion experiences a SCX reaction in the degrader,
it will only produce an accidental signal needed to create a valid trigger (see section
3.3 for details). Hence, Dalitz decay events coming from the degrader are strongly
suppressed by the prompt veto logic in the trigger.
The result of this exercise is presented in figure 3.7. Nominal positions of the
degrader and the target are shown by the vertical lines. One can clearly identify the
air gap between the two where there are no SCX reactions happening.
For completeness of the picture, the peak to background histogram for the charged
tracks is shown in figure 3.8. Figure 3.9 shows the z-distribution of the Dalitz events
coming from the degrader. In a basic simulation, the Bethe-Bloch energy losses were
folded into the cross-section of the SCX in the degrader. The qualitative agreement
with the data is satisfactory and was subsequently used for the simulation of the
prompt events in our detector.
3.2.3
Alignment of the plastic veto and the wire chambers
The plastic veto (PV) detector is described in detail in reference [29]. It consists of 20
thin plastic scintillator staves arranged so as to form a cylinder around the outer wire
chamber. The main purpose of the plastic veto detector is to differentiate between
38
CHAPTER 3: Experimental Method
Number of events
70
Degrader
Target
60
50
40
30
20
10
0
-80
-60
-40
-20
0
20
40
Longitudinal z-coordinate (mm)
Figure 3.7: The z-distribution of the Dalitz events in the degrader and target.
the charged and neutral particles coming from the target region. The charged track
is identified as a track registered in two wire chambers having a close hit (within 11 ◦
in azimuthal angle) from the center of a plastic veto stave. When matched with the
MWPC coordinate frame, the azimuthal angle of the center of each stave is given by:
φveto = (
i
· 360 + 9) ,
20
(3.11)
where i is a stave number and runs from 0 to 19.
There is a relatively simple way of checking the validity of equation (3.11) using
charged particles with momenta uniformly distributed in solid angle such as positrons
from π + → e+ ν decays. If the PV assembly is positioned properly, then the variable
φtrack − φveto is symmetrically distributed around zero, with φveto being a center of
39
CHAPTER 3: Experimental Method
70
Number of events
60
P/B=26
50
40
30
20
10
0
-10
-7.5
-5
-2.5
0
2.5
5
7.5
10
te+-te- (ns)
Figure 3.8: Time difference between the electron and the positron in coincidence with a
photon.
the plastic veto stave closest to the track and having energy deposition above certain
threshold corresponding to a minimum ionizing particle. Knowing the amount of
asymmetry enables us to determine the possible coordinate offset.
Figure 3.10 shows the φtrack − φveto before any corrections were applied. The
asymmetry becomes more apparent if one plots |φtrack − φveto |. Therefore, before
the final pass of the data analysis is performed, a trial pass of the data is needed to
determine the offset between the chambers and PV and encode it in the determination
of the φcenter . In case of the year 2004 data set the offset was found to be
δφ = 0.970 ± 0.010 .
40
CHAPTER 3: Experimental Method
50
45
Simulation
Number of events
40
•
35
Data
30
25
20
15
10
5
0
-70
-60
-50
-40
-30
-20
Longitudinal z-coordinate (mm)
Figure 3.9: The z-distribution of the Dalitz events in the degrader. Dots are data and solid
line is simulation.
This procedure implies that the pair of wire chambers is in a proper relative alignment. A technical note on the relative alignment of the wire chambers is given in
Appendix A.
3.3
Components of the PIBETA trigger system
Once signals are read from the detector components via the PMTs, the data stream
is split into two parts: the data branch and the trigger branch.
The trigger system of the PIBETA experiment imposes a series of logical conditions on the signals coming from the detector, which allows the selection of certain
41
CHAPTER 3: Experimental Method
12000
Number of events
10000
8000
6000
0
5
10
|φtrack-φpv| (degrees)
4000
2000
0
-10
0
10
20
φtrack-φpv (degrees)
Figure 3.10: Azimuthal difference between the uniformly distributed charged tracks and the
center of the plastic veto staves closest to the track and with the energy deposition above
a certain threshold.
42
CHAPTER 3: Experimental Method
types of events to be processed by the data acquisition electronics and to be recorded.
The logic of all the triggers and its implementation on the hardware level is described in references [56, 29, 46]. There are, however, a few aspects of the trigger
components which have not been discussed in these papers but could be of interest to the future discussion of the radiative pion decay ( π + → e+ νγ ) analysis. The
snap-shots of the analog/digital signals in this section were recorded with a Tektronix
digital oscilloscope [61].
3.3.1
Pion stop signal
The basic component of virtually every trigger in the PIBETA experiment is a pion
stop (π-stop) signal. It is formed as a logical ’AND’ of four components, namely, of
the beam counter, active degrader, active target and the rf accelerator signals .
π−stop = BC · AD · AT · rf
(3.12)
The oscilloscope snap shot of the π-stop components is shown in figure 3.11. It is
important to note that the signal from the target comes the latest and thus determines
the timing of the π-stop in majority of the cases. Also, one can see the earlier
(about 6-8 ns) feeble rf signal coming from the positrons in the secondary beam.
They are visible in the figure 3.11 because entire collection of the π-stop components
is triggered on the target signal. This would not be the case were the rf signals chosen
as the trigger. Even at this stage we have essentially eliminated the vast majority of
CHAPTER 3: Experimental Method
43
Figure 3.11: Four components of the π-stop signal as seen in the trigger branch before
entering the LB500 trigger forming unit are 1-beam counter, 2-degrader, 3-target and 4accelerator rf signal.
the positron contaminants by timing them out of the coincidence with the rest of the
π-stop components.
The π-stop signal time serves as a reference for all the physical signals detected
in the CsI calorimeter. In other words, the time difference between the CsI and
the π-stop signals defines the physical time distribution for the events with only one
CHAPTER 3: Experimental Method
44
detectable particle in the final state such as π + → e+ ν . It requires a good fit among
all of the π-stop components which is maintained by routine checks during the run
time of the experiment.
3.3.2
Pion gate signal
Once the π-stop signal is formed, it is used to define a much longer signal called
pion gate (π-gate) . This signal serves as a coincidence gate to help define delayed
(non-prompt) events. The width of the π-gate determines the time during which the
electronics is open for collecting the information.
After generating the π-gate signal, π-stop signal is delayed so that its leading
edge appears about 35 ns after the π-gate leading edge. This arrangement enables
sampling of so-called accidental or pile-up events. Those are the events that come
“without” a π-stop that caused them and in such an event the decayed particle was
“sitting” in the target longer than the width of the π-gate before decaying. Thus
any decay registered in the gap between the beginning of the π-gate and the π-stop
signal is an accidental decay. Obviously, accidentals also occur after the π-stop and
are superimposed on the physics signals. Having a clean sample of accidental events
before the π-stop signal allows for a precise separation of the accidental background
and the physics signals.
Two important parameters are the amount of the delay between the π-stop and
the π-gate and the length of the π-gate. The former one determines the size of the
CHAPTER 3: Experimental Method
45
accidental sample and the latter one limits the extent of the physics events or the
gate fraction of the pion decays. These two parameters combined together determine
the maximal range of the reference signal (i.e., since our time-to-digital converters
(TDC) [45] operate in the common stop mode and the CsI signal is the common
stop, then the π-gate or π-stop signals provide a reference for the time of a particle
hit in the calorimeter.)
As seen in figure 3.12 the width of the gate is 185 ns and the delay is 33 ns. The
bottom panel also displays a few secondary π-stops which, however, did not work out
a new π-gate nor did they extend the current gate.
3.3.3
Prescaled pion gate signal
Events satisfying certain trigger conditions came at much higher rate compared to the
other triggers. In order to record a reasonable proportion of these events we have to
prescale the most copious signals, i.e., record only every nth event of the given type,
where n is a prescale factor. Prescaled signals had their own prescaled π-gate. It is
important to confirm that the original and the prescaled gates had the same width
and offset, that is that the events coming from the prescaled triggers have the same
gate fraction as the unprescaled ones.
Figure 3.13 confirms that there are no any kinds of delay or extensions of the
prescaled gate compared to the original one.
CHAPTER 3: Experimental Method
46
Figure 3.12: The π-gate and the π-stop shown together. The readings on the bottom of
each panel show the length between the vertical bars.
CHAPTER 3: Experimental Method
47
Figure 3.13: Original and the prescaled π-gates shown together. The numbers at the bottom
of the panel show the time interval between the vertical cursor lines.
CHAPTER 3: Experimental Method
3.3.4
48
The ADC gate and the CsI signal
Some of the pion decays inside the gate produce decay products that register in the
CsI calorimeter. The analog signal from the CsI detectors are sent to the FASTBUS
LeCroy Model 1882F analog-to-digital converter (FADC) [44, 45]. The CsI signal is
integrated within the separate 100 ns long ADC gate. The width of the ADC gate
was chosen to reduce the dead time of the system while fully integrating a CsI analog
signal. The gate leading edge precedes the CsI signal by about 50 ns so that CsI signal
tail appears to be outside of the ADC gate. The reason for that was that the FADC
units have an internal 30 ns delay on the input analog signal. Figure 3.14 shows the
relative position of two signals before entering the FADC unit. One can clearly see
that the fast component of the CsI signal “forwarded” by 30 ns fits completely into
the ADC gate.
3.4
Gain matching of the CsI and PV modules
In case of multichannel detectors, the response of the individual detectors may vary
with time due to a changing environment, radiational degradation of the scintillator
light output, drift of the ADC pedestals and high voltage (HV) supply output etc.
This reduces the energy resolution of a detector. Constant monitoring of the each
detector module’s gain is performed by a dedicated routine described below.
CHAPTER 3: Experimental Method
49
Figure 3.14: The ADC gate and the CsI signal to be integrated within this gate shown
together. The polarity of the ADC gate is inverted for display purposes.
3.4.1
CsI gain match
The PIBETA calorimeter is a granulated sphere of pure CsI crystals [41, 42, 43]. The
degree of the granularity was chosen such that the typical module contains over 90%
of the electromagnetic shower induced by an on-axis 70 MeV photon. High granularity
can decrease calorimeter resolution by introducing increased amount of the electronics
noise, pedestal spread and through uncorrelated HV drift.
CHAPTER 3: Experimental Method
50
The crystal ball consists of 240 CsI crystals. Geodesic division resulted in 220
truncated hexagonal and pentagonal pyramids and truncated trapezoids [39] covering
the total solid angle of 0.77×4π sr [1]. Additional 20 crystals surround the two
detector openings and act as electromagnetic shower leakage vetoes. The inner radius
of the sphere is 26 cm and the axial length of each crystal is 22 cm which corresponds
to 12 radiation lengths. The surface of each crystal was specially treated to increase
the light collection efficiency.
The scintillation light from the crystals is collected by the photomultiplier tubes
(PMT) attached to the base of each element. Large pentagonal and hexagonal crystals
are read by EMI 9821QKB 10-stage fast PMTs [17] with 75 mm diameter end windows. Smaller half-hexagonal and trapezoidal modules are viewed by EMI 9211QKA
10-stage PMTs [17] with 50 mm size windows. Each of these PMTs has a sensitivity peak at ∼ 380 nm wave length which matches the pure CsI fast scintillation light
component peaking at ∼ 310 nm [64].
All of the PMT voltage dividers (including the ones attached to the beam detectors) are powered by two LeCroy 1440 high voltage mainframes (HVM). The HVM
are controlled through the slow control computer which allows simultaneous control
of 340 channels with an accuracy of 1 V. All of the PMTs operate in their linear range
with HV set between 1500 V and 2200 V.
The response of a PMT to a mono-energetic particle depends on the HV placed
on the PMT. For a given amount of light produced in the crystal the ratio of the
51
CHAPTER 3: Experimental Method
ADC channels recorded for different HV applied to the PMT is well described by the
following formula:
g=
ADC1
HV1
=
ADC2
HV2
µ
¶n
,
(3.13)
where n is the number of the PMT’s accelerating stages, i.e., for a 10-stage PMT n
is very close to 10. The intrinsic light output of different detectors and variations
in intrinsic PMT gains lead to differences in the ADC channel reading even for the
same HV initially applied to the PMT and the same energy of the incoming particle.
This is corrected by applying a so-called software gain (SG) factor for each channel
upon converting the ADC channels to the energy units resulting in the same overall
MeV/channel conversion factor (ECF) for all the channels. The ECF was chosen so
that the response to the ∼70 MeV positron was near the middle of the ADC range and
a cosmic ray with the energy ∼200 MeV would be at the upper edge of the individual
ADC channel range.
In order to ensure a uniform response of all the PMTs to a mono-energetic particle
one needs to illuminate all crystals with a mono-energetic source and set the HV so
that all the ADC respond with pulse-height peak in the same (or nearly the same)
channel. Then, after collecting some statistics one converts the ADC readings into
the energy units and applies the SG factors so that the energy response is the same
for all the channels. This procedure was utilized for the PV staves. The physics of
pion decays provided a more elegant solution for the CsI gain matching which allowed
simultaneous calculation of the HV and SG factors.
CHAPTER 3: Experimental Method
52
The vast majority of the pions decay through the chain-decay of π + → µ+ ν and
subsequently via so-called Michel channel µ+ → e+ νν . The resulting positron
spectrum is continuous and could be seen in figure 3.15 top panel. The next most
probable decay π + → e+ ν (πe2 ) is suppressed by a factor ∼104 compared to the
Michel chain. Its positron spectrum is, however, mono-energetic as shown in figure 3.15, middle panel. The effect of finite energy resolution, energy losses in the
target and the calorimeter detection threshold of 5 MeV are folded into the spectra
so that the end point of the Michel spectrum as well as the peak of the πe2 events
are smeared. By setting a trigger energy cut around the Michel end-point, one can
enhance the contribution of the πe2 events and significantly reduce the number of
the Michel positrons (without suppressing them entirely). The combined spectrum
of such positrons in the so-called 1-arm high trigger is shown in figure 3.15, bottom
panel. It is worth noting that the first “peak” comes from the smearing of the trigger
threshold and results from a high-threshold cut of the continuous Michel spectrum.
Increasing the HV would manifest in an overall shift of the combined positron
spectrum as shown in figure 3.16, top panel. Since the trigger threshold (energy cut)
is kept the same, it lets more Michel positrons in, thus changing the ratio between the
heights of two peaks as shown in figure 3.16, bottom panel. Therefore, by keeping
the position of the πe2 peak at a fixed value by means of the SG factors and the ratio
between the peaks’ heights constant by a variation of the HV, one ensures a uniform,
or properly gain-matched, response of the CsI detectors.
53
CHAPTER 3: Experimental Method
Arbitrary normalization
600
500
400
300
200
100
0
0
10
20
30
40
50
60
70
80
60
70
80
60
70
80
Positron energy (MeV)
Arbitrary normalization
450
400
350
300
250
200
150
100
50
0
0
10
20
30
40
50
Positron energy (MeV)
Arbitrary normalization
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
Positron energy (MeV)
Figure 3.15: Monte Carlo simulated Michel spectrum with a theoretical end-point of
52.8 MeV (top), π + → e+ ν spectrum theoretically peaks at 69.3 MeV (middle) and combined spectrum in 1-arm high trigger (bottom).
54
CHAPTER 3: Experimental Method
180
Number of events
160
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
70
80
Arbitrary ADC channel
90
50
Number of events
80
70
40
60
30
50
40
20
30
20
10
10
0
40
50
60
70
80
0
Arbitrary ADC channel
40
50
60
70
80
Arbitrary ADC channel
Figure 3.16: The shift of the overall positron energy spectrum (top panel) and the resulting
spectra for the 1-arm high trigger (bottom panel).
CHAPTER 3: Experimental Method
55
During the running time of the PIBETA experiment properly defined spectra for
the 1-arm high trigger for each of the 220 crystals included in the trigger logic were
accumulated until each had at least 5000 entries and were subsequently fitted with a
double Gaussian function. The position of the πe2 peak was fixed at 67 MeV and the
ratio of the amplitudes of two Gaussian fits was analyzed. For each ∼20% discrepancy
from 1:3 ratio HV of 1 V was added (subtracted) to the corresponding channel. New
SGs were recalculated using the 10th power law as in equation 3.13. A few selected
fits are shown in figure 3.17. The overall spectrum before and after only a couple
of iterations is shown in figure 3.18. Energy resolution improves significantly after
gain-matching over a course of only 25 runs.
3.4.2
PV gain match
As it is described in reference [29] the plastic veto detector is a collection of 20 plastic
staves put together to form a cylinder with the axis collinear with the pion incident
beam direction. Each stave has a 10 stage PMT attached to both ends. The total
energy deposited in a stave is calculated as a geometric mean (GM ) of the readings
of two PMTs. It is done so to avoid the effect of the attenuation of the scintillation
light during the transmission. Assume that a charged track traversed the stave some
distance z away from the first PMT. If the length of the stave is L then the distance
to the second PMT is (L − z). Assuming exponential law of the signal attenuation
first PMT reads A1 =A·e−z/λ and the second one reads A2 =A·e−(L−z)/λ , where A is
56
CHAPTER 3: Experimental Method
40
70
35
60
30
50
25
40
20
30
15
20
10
10
5
0
40
50
60
70
80
60
0
40
50
60
70
80
40
50
60
70
80
90
80
50
70
40
60
50
30
40
20
30
20
10
10
0
40
50
60
70
80
0
Figure 3.17: π + → e+ ν spectra in the selected individual crystals of several types. Double
Gaussian fits are shown and the parameters of the fits displayed.
57
CHAPTER 3: Experimental Method
4500
4000
3500
3000
2500
2000
1500
1000
500
0
40
45
50
55
60
65
70
75
80
3500
2250
2000
3000
1750
2500
1500
2000
1250
1500
1000
750
1000
500
500
250
0
40
50
60
70
80
0
40
50
60
70
80
Figure 3.18: Overall positron spectrum in the 1-arm high trigger after only 2 iterations of
the gain match routine.
58
CHAPTER 3: Experimental Method
10
20
2
10
3
PV channel
15
10
5
0
0
0.5
1
1.5
2
2.5
3
GM energy in the plastic veto (MeV)
Figure 3.19: The geometric mean spectra of 20 PV staves.
the original amplitude of the signal and λ is the attenuation length. The value
GM =
q
A1 · A2 = Ae−L/(2λ) ,
(3.14)
does not depend on the location of the intersection of a stave and a track while
retaining proportionality to A.
On average, however, each PMT collects the same amount of light and the spectra
of all 40 of them should look alike. In contrast to the CsI crystals, PV staves are thin
counters (3.2 mm) and all the positron energy spectra look alike regardless of their
initial energy, i.e., being minimum ionizing particles (MIP) on average they deposit
the same amount of energy in a stave.
59
CHAPTER 3: Experimental Method
4.5
Number of channels
4
3.5
3
2.5
2
1.5
1
0.5
0
60
80
100
120
140
160
180
200
220
240
Peak position before HV adjustment (ADC ch.)
Number of channels
8
7
6
5
4
3
2
1
0
60
80
100
120
140
160
180
200
220
240
Peak position after the 8.2 power HV adjustment (ADC ch.)
Figure 3.20: Peak positions of 40 raw PV spectra before (top) and after (bottom) HV
adjustments.
The gain matching of the PV signals is done in a two step procedure. All the raw
ADC spectra of the PV PMTs are fitted to find out the peak positions. Next, the
HVs are changed according to a power law, equation (3.13), so that all the peaks
land in the same ADC channel. Finally, all the relevant histograms are cleared and
new statistics are collected. Fine adjustments are done by applying the SG factors.
Since the PMTs that read the staves are different than the ones on the CsI crystals,
CHAPTER 3: Experimental Method
60
we had to determine the proper power law for the HV correction. Figure 3.20 shows
the peak positions before and after applying new HV values. The actual shifts were
then analyzed. From the fit to the data the power exponent was determined to be
8.2.
The final GM spectra for all 20 staves are depicted in the figure 3.19. Due to large
variations of properties among individual staves, the remaining GM discrepancies are
taken into account in our simulation (see section 4.4.9 for details).
61
Chapter 4
Data Analysis
4.1
Introduction
The present chapter describes the analysis of the π + → e+ νγ decay (RPD) , with an
emphasis on the steps required to extract RPD events from the PIBETA data stream.
In addition, a part of the chapter is devoted to the discussion of the PIBETA detector
simulation tools. The first few sections describe the aspects of the analysis common
to any pion and muon weak decay. The overall course of the analysis is shown on the
flow chart in figure 4.1.
62
CHAPTER 4: Data Analysis
EXPERIMENT
SIMULATION
Produce Data
Summary Files, Apply
Corrections (ADC,TDC)
GEANT Monte Carlo:
6 +
+
3X10 π →e νeγ Events
C C
C T T
(Ee, Eγ, Θeγ, Ee, Eγ, ΘeγT, w)
Apply Software Cuts,
Subtract Background
E
Fill ∆λ HISTOGRAM
Apply Software Cuts,
MC
Fill ∆λ HISTOGRAM
st
1 Pass
Compare ∆λ AND ∆λ
2
E
MC 2 2
χ (ES)=(∆λ -∆λ ) /σ
Calculate Energy Scales
E
MC
Calculate Monte Carlo
Experiment Acceptances
A=A(xi,yj,α)
Use MC Acceptances And
+
+
π → e νe Normalization
E
to Calculate ∆Γ (xij)
FIT
Calculate MC Branching
T
Ratios ∆Γ (xij,α)
For α=α(FV,FA,a)
Evaluate In MINUIT
2
T
E 2 2
χ =σ (∆Γ -∆Γ ) /σ
2
Minimum χ ?
NO, Vary α
YES
Done!
2
Results: FV, FA, a, χ
Figure 4.1: The box diagram of the π + → e+ νγ analysis. The content of the individual
blocks will be addressed in the subsequent sections.
CHAPTER 4: Data Analysis
4.2
63
Basis of the PIBETA data analysis
Any given process detected in the PIBETA experiment is analyzed using three separate sets of computer programs. The actual extraction of the candidate decay events
or creation of the data summary tapes (DST) is done using standard MIDAS [55]
analyzer with specific user routines added to it. The simulation of the experiment is
done with GEANT3 code [13]. Final analysis is performed in PAW [14] with a help
of the MINUIT [35] code.
4.2.1
MIDAS Data Acquisition System and analyzer
MIDAS is a general, highly customizable data acquisition system. Thanks to its
customization tools, MIDAS allows an experiment to be run with minimal human
involvement. The entire structure of the MIDAS DAQ could be divided into two
interrelated sections.
The hardware section of the MIDAS runs the slow control computers responsible
for the control of the environmental variables (temperature, gas pressure, high voltage
etc.) as well as the front end computers which are responsible for reading of the
CAMAC, FASTBUS and VMI based electronics [44], and for storing the data on
disk or tape.
MIDAS software section is described in references [55] and [29]. Two main components of the software part of the DAQ are an on-line data base (ODB) and analyzer
CHAPTER 4: Data Analysis
64
code. ODB contains all the information relevant to the experiment such as event definitions, slow control variables, scaler readings and so on. All the information in the
ODB is easily accessible and can be changed during the running of the experiment.
Actual data (the readings of the CAMAC-based FASTBUS ADCs, TDCs etc. [44])
are recorded into separate MIDAS data files (.mid files). In the present experiment
the size of the .mid files was limited to 500,000 events of any type. The information
from the .mid files is analyzed by an analyzer written in C language and capable of
accessing the raw data from the .mid files and storing them into user defined banks.
The ODB files are attached to the .mid files during the analysis for a cross-over access
of the information. Any number of user routines can be added to the analyzer’s basic
code.
The PIBETA analyzer has a separate routine for nearly every detector described in
the previous chapter. Each of these routines converts the raw readings from a detector
into the physical quantities such as deposited energies, measured times, coordinates
of the hits etc., and stores them in a bank normally named after the detector (for
example mwpc bank or pv bank). Higher level user routines then impose a set of
logical and numerical conditions (cuts) to select a desired subset of the events. All
the relevant variables for the events that passed the cuts are then recorded in ASCII
files called data summary tapes (DST).
CHAPTER 4: Data Analysis
4.2.2
65
Data “packing” and PAW
The information from the DST is translated (“packed”) into PAW N-tuples using
FORTRAN HBOOK routines [14] (N-tuple packing). Some of the variables are
modified during the packing process and this is the reason for not packing the Ntuples directly from the analyzer. Otherwise, a small subsequent correction to a
variable would require a complete replay of the entire data set. The N-tuples are
read by PAW and the final data analysis is performed in PAW using special macros
which contain the final definition of cuts, background subtraction procedures, fit
routines and comparison to the simulated spectra. Some of the fitting routines are
cross-checked or performed independently using stand-alone MINUIT-based codes.
4.2.3
Simulation of the experiment
The simulation of the experiment is performed with a custom-written GEANT3 detector simulation package. Our user defined simulation routines can be divided into
three main groups: geometrical description of the experimental set-up, a set of reaction kinematics generators and, finally, the analyzer-like routines for the generation
of the physical quantities.
The geometrical description of the experiment is performed maximally resembling
the actual experiment. Each detector’s description adheres to its specified dimensions
and every single active component of the detector is implemented.
CHAPTER 4: Data Analysis
66
The kinematics generators describe all the decays of the pions and muons of interest. The majority of the generators have the most recent updates of the dynamics
description of the processes (matrix elements calculations) available in the physics
literature, including radiative corrections. All of the incoming particles are placed
into the target according to the measured distribution of the particles in the beam.
This is a commonly accepted technique which allows for the conservation of the processing time of an event as opposed to the tracking of all of the incoming particles
through the beam elements. The decay of a pion (or a muon) can be of a certain type
or an admixture of the several decays with relative probability distributed according
to their measured branching ratios.
The physical quantities generated by the simulation are additionally modified to
account for the finite resolution of the detector. All the energies in active elements
are smeared with a Gaussian distribution to simulate the effect of the photo-electron
statistics in the readout PMTs. Positions of the π + → e+ ν positron energy peaks
for each CsI cluster are adjusted to match the results of the gain-matching procedure
described in section 3.4.
The tracks’ coordinates are also smeared according to the chamber resolution
functions described in [46]. The pion beam stopping x-y profiles are reproduced
according to the measured x-y distributions based on the high-statistics π + → e+ ν
events. The z-stopping distribution of the beam is, again, measured for the π + →
e+ ν events and realistically describes the incoming pion beam momentum spread as
67
CHAPTER 4: Data Analysis
well as the small percentage of the pions decaying in flight.
The output of the GEANT’s routines is grouped into a series of the data banks
which resemble (where possible) the content of the analyzer banks. Finally, all the
relevant variables are recorded to the ASCII files and packed into N-tuples in a manner
similar to the analyzer DST. The acceptance calculations are done in separate macros
and then fed back to PAW for the final calculations of the branching ratios.
4.2.4
Normalizing decay
In order to reduce the systematic effects in the final result, RPD branching ratio is
normalized to the number of π + → e+ ν (πe2 ) events. In other words, the branching
ratio for a given kinematic region of the RPD decay was calculated as follows:
ΓRPD = Γπe2 ·
NRPD · Aπe2
,
Nπe2 · ARPD
(4.1)
where Γπe2 is a well known branching ratio of the π + → e+ ν decay, Ndecay is the
number of events detected in a given decay channel and Adecay is the acceptance for
the same decay mode. This method requires simultaneous detection of two decay
modes, and the branching ratio of each one of them can be calculated as follows:
Γdecay =
Ndecay
,
Npistop · Adecay · ²
(4.2)
where Npistop is the number of stopped pions in the experiment and ² is a combined
detection efficiency which includes the efficiencies of the wire chambers, detector live
time, beam contamination fraction and so on. Comparing the two above formulae
CHAPTER 4: Data Analysis
68
we note that Npistop cancels out while normalizing to a known decay. The overall
detector efficiencies of the π + → e+ ν and π + → e+ νγ are very similar. In fact
there is no an additional factor related to the detection of the radiative photon and
therefore the factor of ² cancels out as well confirming that the two decays have similar
systematic effects. The uncertainties of Npistop and ² in our experiment are higher
then the precision with which the normalizing decay Γπe2 is known. Therefore we can
reduce the overall uncertainty of the final result by using formula (4.1) rather than
(4.2).
4.2.5
Particle identification
There are several aspects of the analysis that are common for any decay mode and
can be discussed without the specifics of any particular process. One of them is a
particle identification (particle ID) and the way it is implemented in the PIBETA
analyzer.
Particle ID is performed on the basis of the energy deposited by a particle in
the PV detector. If a particle is charged, it creates a “hit” (energy loss due to the
ionization of the chambers media above certain threshold) in the MWPC and a spatial
track of a particle can be reconstructed. This track, in turn, can be associated with
hits in the CsI calorimeter and in PV staves. The energy deposited in the CsI crystal
and in the PV are differently correlated for different types of particles. If a particle
is a positron (or an electron) then for any energy detected in the CsI calorimeter
CHAPTER 4: Data Analysis
69
(as low as 5 MeV) positron is MIP and deposits the minimum possible energy in the
PV. If a particle is a proton (from a SCX reaction), for all possible proton energies
(≤ 300 MeV) the particle is not MIP and leaves significantly more energy in the PV.
This is best seen the in figure 4.2 which displays a two-dimensional scatter plot of
the energy deposited in the PV vs. summed energy deposition collected in the CsI
calorimeter and PV hodoscope. Two distinct regions are present in the plot: the
upper one corresponds to the protons and the middle one to the positrons. We note
that the positron region does not go all the way down to 0 MeV. This allows to cut
away a lot of false positrons that can be caused by noise in the PV detector or by a
back splash of low energy positrons (or electrons) from the CsI calorimeter. The two
lines separating the regions are parametrized as follows:
p
EPV
≥ 2.30 · exp[−0.007(EPV + ECsI )]
+
p
e
> EPV
EPV
≥ 0.20 · exp[−0.007(EPV + ECsI )] ,
(4.3)
(4.4)
Neutral particles do not leave any energy in the PV nor do they cause hits in the
MWPCs. Therefore, a charged particle’s track with the PV energy that falls into the
middle region of figure 4.2 is identified as a positron. A charged particle with PV
energy in the upper part of the same plot is taken to be a proton and a particle with
a corresponding hit in a CsI, no MWPC hits and PV energy displayed in the bottom
of the figure 4.2 is labeled as a photon.
CHAPTER 4: Data Analysis
70
Figure 4.2: The particle type discrimination between protons and minimum ionizing
positrons using the calibrated ADC values of PV and CsI calorimeter.
4.2.6
Assignment of the time variables
Another important issue in the analysis of our data is the assignment of the time
variables for different detectors. Signals from each active detector in our experiment
are fed into FASTBUS LeCroy Model 1877 multi-hit TDC units [44]. All of the
TDCs are operated in a common stop mode so that the time span stored in a TDC
is counted from the beginning of the event (say zero time) until the signal from the
CsI unit stops the clock and fixes the count . Multi-hit units allow multiple hits to
be resolved within one detector during a single event. In case of multiple hits in a
detector they are identified as follows. The time value closest to the CsI hit is called
71
CHAPTER 4: Data Analysis
π-stop
30 ns
ttim
CSI hit
X
π-gate
the simpliest case
Figure 4.3: The simplest case of a single π-stop hit within the π-gate.
ttim. The closest pulse preceding the ttim (if present) is labeled tpre, and, finally,
the nearest pulse after ttim is labeled tpst. The simplest case of only one π-stop
signal within a pion gate (see section 3.3 for the description of these signals) is shown
in figure 4.3. The definition of the ttim is unambiguous in this case.
An example of two π-stops in a gate is shown in figure 4.4. We note that the
time closest to the CsI signal actually occurs after the CsI hit, and if it were labeled
ttim, it would have a non-causal relation to the CsI hit. The CsI hit shown in figure
4.4 (top panel) could not have been caused by a second π-stop (pile-up) and therefore
the tpre variable should be retained as the true time reference. This procedure is
the causal correction of the time reference.
72
CHAPTER 4: Data Analysis
π-stop
30 ns
tpre
CSI hit
X ttim
π-gate
non-causal assignment
π-stop
30 ns
ttim?
CSI hit
ttim X
π-gate
unresolved case
Figure 4.4: A more complex case of two π-stop hits within the π-gate. Top panel shows a
case that leads to a non-causal assignment of the ttim variable. Bottom panel shows an
unresolved case when CsI hit could be caused by either π-stop.
CHAPTER 4: Data Analysis
73
Finally, figure 4.4, bottom panel, shows the situation when the proper value of
the ttim is ambiguous. The smallest value was kept as the chosen value of the ttim.
Due to such ambiguities, appropriate corrections are made to the gate fraction, since
the second pion (if present) “sees” a shortened pion gate (i.e., has smaller probability
to decay within the gate) than the first one. These issues will be addressed in a
section devoted to the analysis and fits of the π + → e+ ν time spectrum.
It is also worth noting that the time of the leading edge of the pion gate signal is
free of these complications. By the nature of this signal (see section 3.3) secondary
π-stops within a single pion gate do not generate an additional pion gate signal and
therefore all of the pile-up effects (the effects of two and more π-stop signals within
a single gate) can be analytically described [52].
4.3
π + → e+ν decay analysis
As discussed in section 4.2, finding the absolute π + → e+ νγ branching ratio with
increased precision requires detection and analysis of an additional normalizing pion
decay channel, in our case π + → e+ ν decay. This section describes the analysis
of the latter decay. The initial preparation of the π + → e+ ν data sample is very
similar to the procedure used for π + → e+ νγ mode.
The statistical and systematic uncertainties will be discussed in greater detail in
chapter 5.3.
CHAPTER 4: Data Analysis
4.3.1
74
Time offsets and gate fraction
Events recorded in the πe2 DST undergo a series of broad cuts in the analyzer. It is
required that an event in a 1-arm-high trigger [29] only contain a single positron,
as defined in section 4.2. If there are multiple positron candidates in an event, the
positron with energy deposition in CsI calorimeter closest to 67 MeV (see section
3.4) is recorded to the DST. In order to study systematic effects we retain two time
references: the causally corrected ttim of π-stop signal and ttim of the leading edge
of the π-gate signal (see section 4.2). The time of an event (ET) is evaluated as the
difference between the CsI time of the positron and the ttim value of the reference
counter (π-stop or π-gate).
Each of the detectors is read by a different TDC unit. Special routines are implemented in the analyzer in order to line up the CsI times in all crystals and the result
is presented in figure 4.5. The differences between the individual TDC channels can
lead to a shift in the ET since it involves the uncorrected values of the reference time.
Finding the time offsets for each TDC channel is essential for the following reason. As
described in section 3.3, the events in the PIBETA experiment are detected within a
finite width pion gate (∼185 ns). Assume that all events are recorded in an interval of
ET∈[tb ,te ], where tb and te are the beginning and the end times of the gate. Therefore,
in order to find the total number of events one has to correct for the fraction of the
75
CHAPTER 4: Data Analysis
1
10
10
2
10
3
200
CsI channel
150
100
50
0
-20
-10
0
10
Corrected CsI TDC values
Figure 4.5: CsI times were carefully aligned and shifted to zero.
20
76
CHAPTER 4: Data Analysis
gate in which the decays were observed , i.e.,
gπ =
Z
te
tb
e
−t/τπ
dt
·Z
+∞
0
e
−t/τπ
dt
¸−1
,
(4.5)
where τπ =26.03 ns is the pion life time. The integrand in the numerator in the
expression above can be slightly modified by the pile-up effects which will be addressed
in subsequent sections. The present formalism explicitly requires the ET time to be
properly offset to zero.
The time offset for a given counter is found by analyzing the prompt events detected in the PIBETA apparatus. The prompt events are primary caused by the SCX
reaction described in section 3.2. They can be misidentified photons from π 0 → γγ
or misidentified protons directly from the SCX that come into a 1-arm-high trigger.
All of them come from either the very short-lived neutral pions (τπ0 ∼ 10−16 sec) or
strongly interacting particles and for our purposes are in time with the incoming pion
beam. The pion beam comes in bunches of particles separated by an rf frequency interval T =19.750 ns. Consequently one can plot the ET for the prompt events within
a pion gate which looks like the insert in figure 4.6 and fit the coordinates of the
peaks with a linear function to determine the offset. The result is shown in figure 4.6
as well.
Time offsets have a slight run dependence which can normally be traced back to
the variations in the TDC set-up during the run time, such as lower TDC thresholds
or new trigger timing introduced by the change of the target. Time offsets for the pion
77
CHAPTER 4: Data Analysis
200
39.35
175
etc...
59.16
19.79
Position of the prompt peak (ns)
150
125
100
75
-50
0
50
100
Time of the prompt (ns)
150
50
Slope=19.743 ± 0.005
25
Offset=-0.071 ± 0.021
0
-25
-50
-2
-1
0
1
2
3
4
5
6
7
Multiple of the rf frequency
Figure 4.6: The time of the prompt events (insert) and a linear fit to the peak positions for
a subset of 2004 runs.
78
CHAPTER 4: Data Analysis
gate and the π-stop signal for the 1-arm-high trigger for the year 2004 are summarized
in the table 4.1 below .
Table 4.1: Time offsets for different run groups.
Run number
π-gate offset
π-stop offset
[50076,50496)
(-0.7066±0.2082)·10−1
-1.400±0.2170·10−1
[50496,50600)
0.9978±0.3592·10−1
0.3939±0.3390·10−1
[50600,51015)
(0.8866±0.2130)·10−1
0.1722±0.1950·10−1
[52000,52471]
0.9559±0.1897·10−1
0.8531±0.1992·10−1
It is important to note that π-gate and π-stop signals have an advantage over
other counters such as degrader or target due to their very small offset variations.
Compared to, for example, the degrader, which had the offsets varying from
∼ −3 to ∼ +1 ns, the systematic uncertainties that come from the zero time determination are much smaller for the π-gate (less so for the π-stop) signals.
4.3.2
Degrader energy and prompt cuts
Discussion of the time offsets in section 4.3.1 implies that the DST for the π + → e+ ν
events contains relatively large portion of prompt events. Unlike the Michel positrons
that come as a natural background of the πe2 events, prompt events have a very different time signature (they are instantaneous), abundant and hard to model properly.
CHAPTER 4: Data Analysis
79
A majority of the prompt events were eliminated by introducing the prompt hardware veto: a signal of ∼10 ns duration centered around the π-stop signal that inhibits
the trigger. This, obviously, shortens the useful part of the π-gate but significantly
suppresses the copious background. There are, however, pile-up pions coming in the
gate which are not vetoed in the described manner. This is illustrated on the insert of
figure 4.6 where there is no prompt peak around zero time. By their nature, prompt
events provided the means to eliminate them.
As described in section 3.2, most of the prompt events occur in the degrader.
One can plot the ADC values of the degrader (exact energy calibration is irrelevant
for this discussion) as a function of ET (see section 4.3.1). This scatter plot is shown
in figure 4.7.
Two features are evident from the plot. Firstly, just like in case of the PV ADC
plot (figure 4.2), there are two distinct regions on the plot. One of them has lower
energy deposition and forms a “sausage”-like shape (Region 1). Another, with significantly more energy, is stripe-shaped (Region 2). Region 1 represents the π-stops
depositing their energy along the way in the degrader. Region 2 contains misidentified protons which are in time with the beam rf frequency. The reason why they are
identified as positrons lies in the fact that PV staves are very thin counters (3.2 mm)
and fluctuations of the deposited energy are high. In addition, protons coming from
the SCX span quite a wide range of energies (up to 300 MeV) and can mimic the
MIP particles. The degrader is, however, a relatively massive detector (∼4 cm di-
80
CHAPTER 4: Data Analysis
160
140
Degarader ADC channel
120
100
80
60
40
20
0
-20
-50
-25
0
25
50
75
100
125
150
Event time tCsI-tπ-gate (ns)
Figure 4.7: Degrader ADC vs. event time for the π + → e+ ν candidates clearly shows the
prompt contamination (Region 2) that can be easily eliminated.
CHAPTER 4: Data Analysis
81
ameter) and the separation between the nearly MIP pions and SCX protons is more
pronounced.
A graphical cut separating these regions can be defined and applied to the data
to eliminate the unwanted prompts. In the analysis this cut is referred to as the
“sausage” cut.
An important feature of this procedure is that, unlike CsI calorimeter ADC counts,
the raw degrader counts are not gain matched as thoroughly. There is a run dependence of the position of the ADC peak for the degrader caused by the change of HV
and diminished light output over prolonged periods of time. This run dependence is
corrected and the mean positions of the degrader ADC as a function of a run number
before and after the gain match are illustrated in figure 4.8. If proper gain-matching
has not been performed it would have resulted in the “piercing” effect, i.e., the stripes
of the prompt events with lower gain would have penetrated the useful events in the
“sausage” and could not have been cleanly separated.
4.3.3
Energy calibration and background subtraction.
π + → e+ ν ADC spectrum.
The energy spectrum of the πe2 candidate events, properly time adjusted and promptfree, looks like what is shown on the bottom panel of figure 3.15. It is assumed that
there were only 2 major contributions to this spectrum, namely, positrons coming from
the Michel decay (and its pile-up), and the πe2 positrons (and corresponding pile-up).
82
CHAPTER 4: Data Analysis
50
45
40
35
30
25
20
15
10
5
0
50100
50200
50300
50400
50500
50600
50700
50800
50900
51000
50100
50200
50300
50400
50500
50600
50700
50800
50900
51000
75
70
65
60
55
50
45
40
35
30
25
Run number
Figure 4.8: Degrader ADC vs. run number before(top panel) and after (bottom panel).
Note that ADC gain-matching is done on the peak positions while the plots show the mean
energies.
83
CHAPTER 4: Data Analysis
Arbitrary units
25
no pile-up
ƒπe2
20
Early gate
Late gate
15
ƒπ→µ→e
10
5
0
0
25
50
75
100
125
150
Time (ns)
Figure 4.9: Graphical interpretation of theoretical probability distributions of Michel and
π + → e+ ν events.
The timing signatures of these decays are very different. The Michel positrons are
products of the decay chain: π → µ → e, with the probability function given by:
fπ→µ→e =
³
´
1
e−t/τµ − e−t/τπ ,
τµ − τ π
(4.6)
while πe2 positrons come simply from the exponential decay:
fπe2 =
1 −t/τπ
e
,
τπ
(4.7)
where τµ = 2197.03 ns and τπ = 26.03 ns are the life-times of the muon and pion, respectively. These distributions are shown in figure 4.9 within the π-gate time range.
If one plots the energy spectra of the positrons for the early (10 ns to 70 ns) and
late (70 ns to 130 ns) fractions of the π-gate, the early part contains both Michel and
84
CHAPTER 4: Data Analysis
30000
Number of events
40000
Early gate
Late gate
30000
20000
20000
10000
0
10000
40
60
80
0
40
60
80
Positron Energy (MeV)
Figure 4.10: Positron spectra for the early (left) and late (right) gate portions. Software
cut of 50 MeV imposed on both spectra.
πe2 positrons. The late fraction contains almost exclusively Michels (see figure 4.10).
Thus, the late gate events give us a relatively clean sample of the Michel background
which is to be subtracted from the early events.The limits for the gates are chosen so
that the length of the early and late gates are of equal length and the end points are
far from the smeared edges of the π-gate.
The energy spectrum of πe2 positrons in the PIBETA detector is also simulated in
our GEANT code. The difference of two experimental histograms shown in figure 4.10
is fitted to the simulated shape of πe2 events by minimizing the χ2 function which
is built as error weighted difference between the content of the experimental and
simulated histograms in every bin. The subtraction factor (fsub ) of two experimental
85
CHAPTER 4: Data Analysis
Number of Events
250
•
200
Simulation
Data
150
100
50
0
40
50
60
70
80
CSI Energy (MeV)
Figure 4.11: Background subtracted π + → e+ ν positron spectrum with GEANT simulation
superimposed.
histograms is a freely varying parameter and the energy calibration factor which
matches the peak positions of the final spectra is the second parameter of the fit.
Two parameters are assumed to be uncorrelated and are treated independently, i.e.,
for each energy scale the best value of the subtraction factor is found and subsequently
the χ2 distribution is minimized with respect to the energy scale. The result is shown
in figure 4.11.
The entire data set from the year 2004 is divided into two parts. The first set
corresponds to the experimental set up with the 9-piece target [29] (run numbers
50000 to 51017). The second set corresponds to the set up with the single piece target
constructed at the University of Zürich (run numbers 52000 to 52471) . Two sub-sets
86
CHAPTER 4: Data Analysis
also average different π-stop intensities which can be used to detect systematic effects
related to changes in the beam intensity. The results for the subtraction factors and
positron energy scales can be seen in the table 4.2 below.
Table 4.2: Subtraction factors and energy scales for π + → e+ ν .
Variable name
Set 1
Set 2
Subtraction Factor 0.82905±0.00190
Energy Scale
0.9429±0.00180
0.81222±0.00186
0.9411±0.00180
The difference in the subtraction factors is related to different intensities of the
pion beam for two sets of data and will be discussed in section 4.4.11.
Finally, the total integral under the experimental curve shown in figure 4.11 is
the number of detected π + → e+ ν events. The total number of events should be
corrected by the gate fraction which is calculated as follows:
gπ =
µZ
70 ns
10 ns
e
−t/τπ
dt − fsub ·
Z
130 ns
70 ns
e
−t/τπ
dt
¶ ·Z
+∞
0
e
−t/τπ
dt
¸−1
.
(4.8)
The fsub factor in the numerator takes into account the portion of the πe2 events that
remain in the late fraction of the gate. We have labeled this method of extracting
the total number of π + → e+ ν events is called the ADC method since it is based on
the ADC spectra of the detected particles.
87
CHAPTER 4: Data Analysis
4.3.4
π + → e+ ν TDC spectrum
Applying the same set of cuts as in section 4.3.3 one can book the TDC spectrum
of the πe2 candidates. Using the same premises as before, one can assume that there
are only 2 contributions to the spectrum: (a) Michel decays including pile-up in the
target and (b) original πe2 events including pion pile-up in the target. The probability
distribution for these events could be written as follows:
i
h
h
i
pu
(t) , (4.9)
F (t) = α(1) · fπe2 (t) + α(3) · fπpue2 (t) + α(2) · fπ→µ→e (t) + α(3) · fπ→µ→e
where α(i) (i=1..3) are free parameters of the fit to be determined. Individual components of the fit correspond to the original πe2 events and their pile-up, Michel
events and their pile-up, respectively. The first and the third terms are described in
equations (4.7) and (4.6) of section 4.3.3. The pile-up terms are described by the
following formulae:
pu
fπ→µ→e
(t) =
n=+∞
X
n=−∞
fπpue2 (t) =
n=+∞
X
n=−∞
fπ→µ→e (t − nT )θ(t − nT )
fπe2 (t − nT )θ(t − nT ) ,
(4.10)
(4.11)
where T =19.75 ns is the separation between the cyclotron beam bunches and θ(t) is
the Heaviside step-function. The probability of a pile-up is embedded in the value of
parameter α(3). The values of parameters α(1) and α(2) are the contributions coming
from any kind of πe2 and Michel respectively. In the numerical implementation of F (t),
the upper limit of the sums is limited to 10 since there cannot not be more than 10
bunches of beam particles within a 185 ns long π-gate.
88
CHAPTER 4: Data Analysis
Number of Events
π → eν +
10
background
Data
Monte Carlo
4
µ
10 3
-50
π
SUM
µp
0
50
100
150
tCsI - tπ-gate (ns)
Figure 4.12: TDC spectrum of the π + → e+ ν candidates with individual components of
the fit superimposed on top of it.
The result of the fit is shown in figure 4.12. It is worth noting that the pion pileup is negligible and, therefore, is off the scale in he plot. The ratio of the resulting fit
function to the data curve is shown in figure 4.13. A linear fit with slope consistent
with zero is shown as a solid line.
By integrating the πe2 component of the fit (α(1) · fπe2 (t)) in the range between
10 ns and 130 ns, one gets the detected number of π + → e+ ν events. The gate
fraction in this case is defined by equation (4.5) with tb =10 ns and te =130 ns.
This method is not entirely applicable in the case when ET is defined by the πstop signal because of the “unresolved” ttim combination described in section 4.2.6.
This effect results in a slight overpopulation of the early bins of the TDC spectrum
89
CHAPTER 4: Data Analysis
Fit to data ratio
1.25
1
0.75
0.5
0.25
0
0
50
100
tCsI - tπ-gate (ns)
Figure 4.13: Ratio of the TDC fit to the data. Solid line represents a linear fit with a slope
consistent with zero and constant consistent with 1.
since events with smaller value of ttim are preferred in our algorithm.
A Monte Carlo simulation was written to simulate this effect for each component
of the fitting function. Just as expected, the simulated time spectra for the π-gate
and for the π-stop look slightly different. It is especially pronounced in the Michel
positrons time spectrum and its pile-up component, shown in the figure 4.14. The
simulated intensity of π-stops is 130 KHz which is slightly higher than the rates in
the experiment in order to magnify the effect.
The experimental π-stop time spectrum is fitted with a sum of two simulated
histograms with the individual multipliers as free fit parameters. The main difference
compared to the π-gate fit is that the fitting function is not an analytical expression
90
CHAPTER 4: Data Analysis
•
Arbitrary normalization
90
80
70
tCsI-tπ-stop
tCsI-tπ-gate
60
50
40
30
20
10
0
-40
-20
0
20
40
60
80
100
120
140
Time (ns)
Figure 4.14: TDC spectrum of the simulated Michel events for different reference counters.
but rather a numerical simulation. The πe2 component of the fit multiplied by a proper
weight is integrated to find the detected number of πe2 events. The gate fraction
for these events is also calculated numerically; it is equal to the integral under the
properly normalized πe2 contribution. As was mentioned above, the π-stop reference
is kept as a consistency check and the result is not used in the final calculation of the
branching ratio. The two methods provide results that are consistent on a sub-percent
level.
The advantage of the analytical representation of the terms of the time fit components is illustrated by the following example. Once the time spectrum is fitted
using equation (4.9), the positive time portion of the π-gate contains the original π e2
events as well as the pile-up events produced by the stopped pions and decayed after
91
CHAPTER 4: Data Analysis
w/ pile-up
no pile-up
Number of events
10 4
10
3
0
25
50
75
100
125
150
tCsI-tπ-gate (ns)
Figure 4.15: Clean π + → e+ ν spectrum with and without pile-up events.
the original pion opened the gate. The time spectra are shown in figure 4.15.
Integrating either spectrum gives slightly different number of events. For example,
for the set 1 of our data, integration yields 489,708 events with no pile-up pions
and 498,640 with them. The gate fractions for these events are, however, different.
Events without pile-up described by a pure exponential decay have the gate fraction
as in equation (4.5). For the limits between 10 ns and 130 ns the integration yields
gπ =0.67423. The events that include the pileup would on average experience different
fraction of the gate before they decay, since some of the decays come from the later
pions. Integration of the analytical expression gives gπ =0.68647. However, the total
number of events divided by the gate fraction comes out to be 726,314 and 726,387
92
CHAPTER 4: Data Analysis
events without and with pile-up respectively. The two numbers are in remarkable
agreement due to the high precision of the analytical gate fraction calculations.
Another cross-check of the validity of the time fit can be made by estimating the
value of the fit parameter α(3), the probability for a pile-up to occur. This parameter
can be calculated independently. If the π-stop rate is rπ and the separation between
the beam bunches is T , then the probability of a pile-up is
η = rπ · T = α(3)
(4.12)
The value of parameter α(3) from the fit gives π-stop rate of 115 kHz. The mean
nominal recorded stopping rate for the same data set was ∼123 kHz. Analogously
good agreement is observed for the lower intensity runs.
4.3.5
ADC method vs. TDC method of counting events
Sections 4.3.3 and 4.3.4 describe two independent methods of finding the total
number of π + → e+ ν events. The total yield with statistical uncertainties, split into
two regions, is given in table 4.3 below.
Table 4.3: Total number of π + → e+ ν events.
Method type
Set 1
Set 2
ADC method
720 191±1 131.65
449 152±892.87
TDC method
726 259±1 037.86
452 264±819.01
93
CHAPTER 4: Data Analysis
The TDC method is the more precise of the two since it only depends on the
understanding of the background. The ADC method also depends on the line-shape
produced by our simulation, therefore the ADC method should only be considered as
a check of our GEANT simulation. In fact, if the line-shape of the energy spectrum
were incorrect (say, if it underestimated the low energy “tail” of the distribution),
it would invariably lead to the incorrect value of the subtraction factor (it would be
overestimated in order to bring the experimental tail in agreement with the GEANT
simulation) and would provide an incorrect number of πe2 events. The agreement
between the ADC and TDC is on a sub-percent level.
4.3.6
Absolute normalization and related quantities
An independent cross-check of the πe2 results was performed because of the most
importance of the π + → e+ ν decay to our final analysis of the π + → e+ νγ decay.
Equation (4.2) describes the π + → e+ ν branching ratio. Including all known
efficiency factors explicitly into this formula, one gets the following expression:
Γπe2 =
Nπ−gate · Aπe2
Nπe2 · s
,
· fd · fπ · τlive · ²MWPC1 · ²MWPC2 · ²PV
(4.13)
where the additional factors that appear above are the following:
• s is the software prescale factor which allows to keep only every sth candidate
event in order to reduce the size of the DST files. The value of s=30 is kept
throughout the analysis.
CHAPTER 4: Data Analysis
94
• Nπ−gate is the total number of pion gates for the given data sample. Normalizing to the pion gates takes into account the proper gate fraction since the
analysis is done relative to the π-gate time. Calculated from the ODB data.
• Aπe2 is the acceptance of the πe2 events in our detector. Calculated using
GEANT simulation.
• fd is the fraction of the pions that produced the π-stop signal and did not
interact strongly in the target. Calculated with the GEANT code.
• fπ is the fraction of the actual pions that produced the π-stop signal. Some
of counts could have been caused by muons. Calculated by comparing the
z-stopping distribution in the data and in GEANT.
• τlive is the live time fraction of the detector. Not every trigger fired by the
electronics was recorded on tape. Some of them were ignored if the electronics
was busy analyzing a previous trigger. Calculated from ODB data.
• ²MWPC1 , ²MWPC2 and ²PV are the efficiencies of the wire chambers and plastic
veto respectively. Calculated by analyzing MIP tracks.
Each of these items will be addressed in more detail in the following sections. The
final results for each subset of the data with all the factors taken into account are
shown in table 4.4. An estimate of the total error, including the systematic effects are
95
CHAPTER 4: Data Analysis
given in the table 4.5. The final uncertainty is calculated by summing the individual
components in quadratures.
Table 4.4: π + → e+ ν branching ratio.
Variable
Γ
π + →e+ ν
·10−4
Set 1
Set 2
1.245 ± 0.010
1.237 ± 0.010
Table 4.5: Uncertainties for the π + → e+ ν branching ratios.
Variable
Uncertainty (%)
Set 1
Set 2
0.143
0.181
TDC method 0.198
0.250
Nπe2
τlive
0.100
0.100
²MWPC1
0.100
0.100
²MWPC2
0.100
0.100
²PV
0.100
0.100
fd
0.500
0.500
fπ
0.500
0.500
Aπe2
0.224
0.224
96
CHAPTER 4: Data Analysis
These experimental results can be compared with the best measured value of
Γ
π + →e+ ν
= (1.235 ± 0.005) · 10−4 published in [20] and
Γ
π + →e+ ν
= (1.2265 ± 0.0034(stat) ± 0.0044(syst)) · 10−4 in reference [11].
4.3.7
Extraction of the normalization parameters from the
ODB
Many parameters used in the absolute normalization of the π + → e+ ν branching
ratio are extracted from the ODB during the analysis process, following a simple
procedure. The value of a variable at the end of each run is recorded into a data
file. These data are then booked into an N-tuple in order to have a more flexible
environment for excluding runs from the analysis at the later stage. Different methods
are used to calculate different quantities stored in the ODB:
• Pion gates are summed directly for the set of runs in question. The top left
panel of figure 4.16 shows pion gates as a function of the run number for a data
sub-set.
• The detector live fraction is calculated as the ratio of the accepted triggers to
the number of total triggers recorded in the ODB. In addition, it is weighted
by the number of πe2 events collected in each run so that:
τlive
i
Naccept
0d
=
· Nπi e2
i
N
all
i=allruns
Ã
X
!Ã
X
i=allruns
Nπi e2
!−1
.
(4.14)
97
CHAPTER 4: Data Analysis
x 10 2
x 10 5
1200
9000
1000
Number of events
Number of π-gates
8000
7000
6000
5000
4000
3000
2000
800
Bin1
Bin2
600
Bin3
400
200
1000
0
Number of normalized Nπe2
x 10
50200
-5
50400
50600
50800
51000
0
0
Run number
1
2
Random units
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
50100
50200
50300
50400
50500
50600
50700
50800
50900
51000
Run number
Figure 4.16: Various parameters for the absolute normalization calculations. Number of
π-gates as a function of run (top left), MWPC1 efficiency histogram for 1-arm-high trigger
(top right) and normalized number of πe2 events (bottom).
98
CHAPTER 4: Data Analysis
The weighting by the number of πe2 events in a run is implemented to ensure
that the time when the detector operated in a different mode does not bias the
value of τlive , since there will be no πe2 events collected for these sets of runs.
• Efficiencies of the MWPC and PV detectors are calculated assuming that they
are mutually independent, so that the inefficiency of one detector does not lead
to a missed hit in another. In case of MWPC1, for example, if an MIP track
has a hit in all three detectors then a count is recorded in the second bin of a
dedicated histogram. In case when the hits are only detected in the MWPC2
and PV then the first bin of the histogram is filled. The third bin is filled
when all three detectors fire and the track is identified as a positron (see section
4.2.5). This procedure accounts for inefficiencies that arise from the analysis
cuts for MIP positrons. The ratio between the content of two bins (either 2:1
or 3:1) is recorded as efficiency of MWPC1 for a given run and a given trigger.
The same procedure is applied for all three detectors. An example of such a
histogram is shown in the upper right panel of figure 4.16. Finally, after the
efficiencies are collected for all the runs, the average efficiency is weighted with
the number of πe2 events in each run as described in equation (4.14).
Another consistency check is depicted in figure 4.16, bottom panel. The plotted
variable is the normalized number of πe2 events defined as follows:
Nπnorm
=
e2
Nπe2
Nπ−gate · τlive · ²MWPC1 · ²MWPC2 · ²PV
,
(4.15)
99
CHAPTER 4: Data Analysis
The plot displays a good consistency as a function of run number. The final accounting of the quantities is summarized in table 4.6.
Table 4.6: Efficiencies for the π + → e+ ν branching ratio.
4.3.8
Variable
Set 1
Set 2
Nπ−gate
3.7921·1011
2.3199·1011
τlive
0.84576
0.85841
²MWPC1
0.93320
0.93315
²MWPC2
0.95968
0.97049
²PV
0.97719
0.97542
rπ (kHz)
122.58
91.178
Muon contamination of the beam
Some of the particles that produce the π-stop signal could have been muons. Most
of the muon contamination in the beam comes from the pions decaying into muons
in flight in front of the target and thus producing valid π-stop signals. A smaller
fraction of the muons come as original beam particles. These muons can be easily
discarded on the basis of their shorter time of flight (TOF) of the muons between the
BC and the target.
This method relies on the fact that on average muons lose less energy in the mate-
CHAPTER 4: Data Analysis
100
rial than pions with the same momentum and stop deeper in the target. Therefore, if
one reconstructs the z-vertex distribution of the positrons in a wide range of energies
it has a small secondary peak that corresponds to the positrons coming directly from
beam muon decays. We label this histogram “main” (Fig. 4.17, top left).
One can also simulate the π-stop distribution in GEANT by propagating only π + s
through the degrader and the target. For this purpose the incoming beam momentum
profile is reconstructed in the TURTLE beam transport program for the πE1 beam
line [57] and it is shown in figure 4.18 (top panel). The resulting z-distribution is
shown in figure 4.17, top right, and labeled as a “pion contribution”. The difference
between the main histogram and the pion contribution yields the muon component of
the beam which is called “muon contamination” (figure 4.17, bottom left). The muon
contamination correction to the number of π-stops is calculated as a ratio of the area
under the “muon contamination” histogram to the area under the main histogram.
Figure 4.17, bottom right, shows the superposition of all three components on the
same plot.
Finally, as an independent check of the validity of our π-stop simulation one can
compare the z-stopping distribution of the “pion contribution” and the simulation
of the z-stops that describes the µ+ → e+ νν data (figure 4.18, bottom). Michel
decays are chosen over the π + → e+ ν decays because 100% of the GEANT pions
decay this way. In the simulation pions were placed in the target according to the
beam xy-profile [46] without tracking the primary pions through the beam elements.
101
x 10
x 10
10000
10000
MAIN historam
Number of events
Number of events
CHAPTER 4: Data Analysis
8000
6000
4000
Secondary muon peak
2000
0
-2
-1
0
1
2
3
4
5
6
7
PION CONTRIBUTION histo
8000
6000
4000
2000
8
0
-2
-1
Z-stopping coord. of the pion/muon beam (cm)
0
1
2
5
6
7
8
x 10
Stopped beam
split into µ and π components
10000
2500
Number of events
4
Z-stopping coord. of the clean pion beam (cm)
3000
MUON CONTAMINATION histo
8000
2000
κ=Nµ/Ntot=0.968
6000
1500
1000
4000
500
2000
0
3
-2
-1
0
1
2
3
4
5
6
Z-stopping coord. of the muon component (cm)
7
8
0
-2
-1
0
1
2
3
4
5
6
7
8
Z-stopping coord. of the pion/muon beam
Figure 4.17: Reconstructed z-vertex of the positrons exhibits two peaks: pion and muon
(top left). GEANT simulation of z-stopping distribution for pion decays only (top right).
Secondary muon peak is amplified after the pion peak is subtracted (bottom left). Combined
z-stopping spectrum with double Gaussian fits superimposed (bottom right).
102
CHAPTER 4: Data Analysis
Arbitrary normalization
700
pπ+ = 113.4 ± 0.5 MeV
600
500
400
300
200
100
0
112
112.5
113
+
113.5
114
114.5
115
π momentum (MeV/c)
–
Arbitrary normalization
0.08
µ→eνν simulation
0.07
• π-stop simulation
0.06
0.05
0.04
0.03
0.02
0.01
0
-2
-1
0
1
2
3
4
5
6
7
8
Z-stopping coord. of the clean pion beam (cm)
Figure 4.18: Incident momentum of the simulated pion beam (top) and comparison between
two GEANT simulations of the z-stopping distribution (bottom). Note that the simulation
on the bottom plot represented by the solid histogram reproduces the data very well.
103
CHAPTER 4: Data Analysis
For the 2004 data set the final result is:
fπ = 0.968 ± 0.005 .
4.3.9
SCX in the target
As described in section 4.3.8, the π + beam has some momentum spread allowing
some pions to have enough energy to undergo SCX reaction even in the target. This
effect changes the number of π-stops that have been assumed to decay rather than
interact strongly. The effect is simulated in GEANT with the same beam momentum
profile as given in section 4.3.8. If a pion reaches the target and deposits more than
a certain amount of energy in it, it is counted as a valid π-stop. After the pion has
disappeared, the mechanism through which it happened is analyzed. If the process is
a decay or a particle goes below the GEANT energy detection threshold, the status
of the π-stop remains unchanged. If the mechanism is a strong interaction, the π-stop
is discarded. The ratio of the discarded events to the total number of π-stops is
fd = 0.9747 ± 0.0050 .
4.3.10
π + → e+ ν acceptance
Acceptance of a detector for a given decay measures the fraction of events that miss
the detector due to detector design (beam openings, limited detector size etc.), and
the fraction of events that do not appear in the data sample because of the analysis
104
CHAPTER 4: Data Analysis
cuts (requirements of certain energy thresholds, limits on the opening angles etc.)
imposed on the sample.
If one simulates the experimental setup and describes the physical properties of
the detector elements adequately (correctly defined geometry and material properties,
proper description of energy losses in the materials, smearing of the signal by the
electronics etc.), one should achieve a realistic description of the detector with one
great advantage: the total number of events of any kind occurring in the detector is
controllable and easily accounted for.
Every event generated in the simulation has a certain weight associated with it.
The weight may come from either the relative probability of the event according to
the phase space distribution and/or from the value of the reaction matrix element
for the given set of kinematic parameters. We can separately sum the weights of all
the generated events (total sum), as well as the weights of all the events that pass
the series of cuts (detected sum). Then, the acceptance of the detector to a decay is
given by the ratio of the detected sum to the total sum. We note that the detected
sum should be calculated over the events that pass the cuts imposed on the simulated
values of the parameters and not the theoretical or “thrown” values.
Table 4.7: π + → e+ ν acceptance.
Variable
Set 1
Set 2
Aπe2
0.6953±0.0008
0.6958±0.0008
CHAPTER 4: Data Analysis
105
One way of checking the validity of the simulation is to compare the distributions
of the kinematic variables calculated from the data with similar quantities generated
by the simulation. Figures 4.19 through 4.22 provide the comparison of the distributions for the selected variables. The agreement is very satisfactory on all accounts.
The acceptances for two data sub-sets with statistical uncertainties is given in table
4.7. We note that the two values are in very close agreement with each other and
using either value in the final calculations is acceptable. Further corrections to the
π + → e+ ν acceptances due to the emission of a real photon will be discussed in
section 4.4.11.
106
CHAPTER 4: Data Analysis
Number of Events
200
Simulation
• Data
150
100
50
0
0
0.5
1
1.5
2
Number of Events
PV Energy (MeV)
Simulation
• Data
300
200
100
0
-2
-1
0
1
2
3
4
Z-stopping dst. (cm)
Figure 4.19: Comparison of the plastic veto energy (top) and pion’s z-stopping (bottom)
distributions. Dots represent data points, solid lines are simulation.
107
CHAPTER 4: Data Analysis
Number of Events
600
Simulation
• Data
500
400
300
200
100
0
0
0.5
1
1.5
2
2.5
3
Minimal track-Z distance (cm)
Number of Events
200
Simulation
• Data
150
100
50
0
0
5
10
15
20
Track-CSI angle (dgr)
Figure 4.20: Comparison of the minimal distance between a back-projected track and the
beam direction (top) and angle between a track and a CsI clump (bottom) distributions.
Dots represent data points, solid lines are simulation.
108
CHAPTER 4: Data Analysis
Simulation
• Data
Number of Events
150
125
100
75
50
25
0
0
2.5
5
7.5
10
12.5
15
Number of Events
Track-PV angle (dgr)
Simulation
• Data
200
150
100
50
0
0
100
200
300
φ of a clump (dgr)
Figure 4.21: Comparison of the angle between a track direction and the center of the PV
stave (top) and the azimuthal angle of the tracks (bottom) distributions. Dots represent
data points, solid lines are simulation.
109
CHAPTER 4: Data Analysis
Number of Events
200
Simulation
• Data
150
100
50
0
40
45
50
55
60
Number of events
CSI Energy (left) (MeV)
Simulation
• Data
30
20
10
0
0
50
100
150
200
CSI crystal number
Figure 4.22: Comparison of the left tale of the CSI energy deposition (top) and the map of
the CSI crystals illumination (bottom). Dots represent data points, solid lines are simulation.
110
CHAPTER 4: Data Analysis
4.4
π + → e+νγ analysis
4.4.1
Introduction
We begin by describing the selection of the candidates for π + → e+ νγ (RPD)
decay, additional modifications performed on the DST sample, separation of the signal
from the background, energy calibration and adjustments to the GEANT simulation.
Having defined the necessary terminology we present the analysis of the synthetic
Monte Carlo data in order to verify the overall validity of the method. Finally, we
present results of the RPD analysis along with a discussion of the statistical and
systematic effects.
4.4.2
π + → e+ νγ DST and preparation of the sample
Being a three-body decay, π + → e+ νγ offers a wide range of kinematic parameters
to study experimentally. Since all three particles in the final state have continuous
energy spectra (as opposed to the mono-energetic π + → e+ ν
decay), the final
analysis is performed for three overlapping kinematic regions shown in table 4.8,
defined by the theoretical values of the kinematic variables.
The choice of kinematic regions is dictated the experimental constrains. One would
like to have both positron and photon detectable energies to be as low as possible.
In doing so, the experimental sample would be completely dominated by the lowenergetic muon decays. Therefore one should compromise by requiring that at least
111
CHAPTER 4: Data Analysis
Table 4.8: Definition of theoretical kinematic regions.
Region
Theoretical kinematic range
A
Eγ and Ee+ > 50 MeV
B
Eγ > 50 MeV, Ee+ > 10 MeV, θeγ > 40◦
C
Eγ > 10 MeV, Ee+ > 50 MeV, θeγ > 40◦
one particle in the detected decay should have energy greater than Michel endpoint.
On the other hand, kinematic regions are selected based on the physics requirements
of the final analysis. Top of the figure (4.23) shows the contour plots of the structure
dependent terms SD + and SD− (see section 2 for details) as a function of the scaled
photon and positrons energies (x and y). The inner bremsstrahlung contribution IB
(shown on the bottom of figure (4.23)) is nearly four orders of magnitude larger than
any of the structure dependent terms but it peaks strongly along the line x+y = 1, i.e.,
when the neutrino energy is maximal. It is imperative for the study of the structure
dependent terms to have a region of phase space relatively free of the “uninteresting”
bremsstrahlung photons. In our case it is Region A. We note, however, that for any
experimentally accessible region the SD − term will be masked by the radiation of the
bremsstrahlung photons since they peak in the same region of the kinematic space.
The decay rate in Region A is proportional to the (1 + γ)2 term which leads to the
dual solution for γ. Regions B and C are complementary and are required if one is
to solve this ambiguity.
112
CHAPTER 4: Data Analysis
1
1
18
15
0.1
0.9
12
0.3
9
0.8
0.6
6
y=2Ee/mπ
y=2Ee/mπ
0.8
0.7
3
0.6
0.5
SD−×104
0.2
0.4
0.3
0.9
1.2
1.5
1.8
0.4
1
SD+×104
0.6
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x=2Eγ /mπ
0.6
0.8
x=2Eγ /mπ
1
y=2Ee/mπ
0.8
0.6
0.4
IB≈1.5 at (x,y)=(0.2,0.8)
0.2
0
0
0.2
0.4
0.6
0.8
1
x=2Eγ /mπ
Figure 4.23:
Contributions of the structure dependent (SD + ,SD− ) and inner
bremsstrahlung (IB) matrix elements to the total π + → e+ νγ decay rate. Dashed box
comprises theoretical Region A. Note different y scales in the three panels.
1
CHAPTER 4: Data Analysis
113
Events recorded to the DST undergo a series of relatively broad cuts. Only events
in the 1-arm-high trigger are analyzed (that is, at least one particle in the event
had energy above Michel’s endpoint of 52.8 MeV). A candidate event has to have a
positron-gamma pair (as defined in section 4.2.5) with the time difference between
the TDC readings of the corresponding CsI clumps not greater than 15 ns. In case
when two or more candidates are present, the pair with the smallest time difference is
recorded. Additional cuts on the lowest allowed summed energy of the two particles
are implemented to reduce the size of the sample. Region A is contained in either B
or C and does not require any additional modifications.
Since the π + → e+ νγ events are detected with the same trigger as the normalizing
π + → e+ ν decay, all the preliminary procedures such time offsets and degrader ADC
linearization described in sections 4.3.1 through 4.3.2 can be directly applied to the
RPD data sample. π-gate time is chosen as the time reference as in the πe2 analysis
so that the same prompt “sausage” cuts could be used as well.
A series of cuts are applied to the sample to split the available data into three
regions defined above. The trigger logic conditions for these regions are defined and
shown in table 4.9.
The energy of a positron is calculated as the sum of the energies deposited in the
CsI, PV and the target detectors:
Eetotal = EeCsI + Eetarget + EePV .
(4.16)
114
CHAPTER 4: Data Analysis
Table 4.9: Definition of experimental kinematic regions.
Region
Experimental kinematic range
A
Eγ and Ee+ > 51.7 MeV, θeγ > 40◦
B
Eγ > 55.6 MeV, Ee+ > 20 MeV, θeγ > 40◦
C
Eγ > 20 MeV, Ee+ > 55.6 MeV, θeγ > 40◦
Photons include only the CsI calorimeter energy. An additional 40◦ restriction on
the opening angle is implemented to suppress so-called split clumps with low energy.
The clumping routine in the PIBETA analyzer [46] sums only the energies of the
nearest neighbors around the crystal with the largest energy deposition. If the energy
leaks further to the second nearest neighbor it is considered as an independent neutral
particle. The angular cut eliminates this possibility since the separation between the
centers of the second nearest neighbors does not exceed 40◦ .
For each series of cuts as defined in table 4.9 with additional prompt cuts and the
size of the pion gate limited between 10 ns and 130 ns (same as for the πe2 events) the
time difference between the positron and gamma is plotted. These so-called signal
histograms are shown in figure 4.24.
4.4.3
Background subtraction
The signal histograms provide the information about the background as well. The
signal region is chosen to be ∆tsig ≡ (te − tγ ) ∈ (−5.0, 5.0) ns. The background
115
CHAPTER 4: Data Analysis
200
Region A
P/B = 144
100
Number of events
0
1000
-10
-5
0
Region B
5
10
P/B = 9
500
0
-10
-5
0
Region C
5
10
P/B = 28
1000
500
0
-10
-5
0
5
10
t(e)-t(γ) (ns)
Figure 4.24: Difference between the positron and photon times for three kinematic regions.
116
CHAPTER 4: Data Analysis
region is defined as ∆tbgr ∈ (−10.0, −5.0) ∪ (5.0,10.0) ns. The width of the signal
and background regions |∆tsig |= |∆tbgr | are equal, therefore, the distributions of the
physical quantities of the RPD decay are obtained by subtracting two histogram: one
with ∆t=∆tsig and another with ∆t=∆tbgr . The detected number of events in each
region is a total integral under any of these distributions. The total number of events
is calculated by dividing the detected number of events by the gate fraction, equation
(4.5), between 10 ns and 130 ns. The total yield in all three regions with statistical
uncertainties is shown in table 4.10.
Table 4.10: Total number of π + → e+ νγ events.
4.4.4
Region
Number of events
A
4 404.8 ± 85.45
B
10 314.1 ± 157.8
C
13 365.5 ± 155.0
π + → π 0 e+ ν and µ+ → e+ ννγ background subtraction
Several substantial sources of background to π + → e+ νγ events that cannot be easily
eliminated by the above method. One reliable test of a well performing background
subtraction would be a measurement of the life time of the final events in every
CHAPTER 4: Data Analysis
117
kinematic region (see chapter 5). Contamination attributed to the more dominant
radiative decay of a muon µ+ → e+ ννγ should have much longer life time and can be
detected on this basis. In addition, decay products of µ+ → e+ ννγ have a completely
different kinematics than that of π + → e+ νγ . Using the GEANT simulation package
we have simulated one million µ+ → e+ ννγ events and reconstructed them as
π + → e+ νγ decays. Since we measure our π + → e+ νγ decay in a redundant
configuration (both energies and opening angles between the final state particles),
it strongly suppresses events with different kinematical parameters. For example, in
Region A not a single µ+ → e+ ννγ event has passed the imposed cuts, thus assuring
us that our sample is free of the radiative Michel pollutants.
However, the π + → π 0 e+ ν decay with one photon converting into an electronpositron pair has the time structure of a pion decay with a valid time coincidence
between a real photon and a real positron.
Again, GEANT simulation helps to resolve this ambiguity. One million π + →
π 0 e+ ν events for each type of the target are generated in GEANT and processed as
π + → e+ νγ events to see how many events pass the cuts. The results are shown in
table 4.11.
An independent analysis of the π + → π 0 e+ ν events is performed and the sample
distribution of the photon energy could be seen in figure 4.26. Total number of
π + → π 0 e+ ν events in each kinematic region of the π + → e+ νγ events is calculated
118
CHAPTER 4: Data Analysis
Table 4.11: πβ contamination percentiles.
Region
Misidentified
events (%)(set1/set2)
A
3.36±0.02/3.18±0.02
B
3.57±0.02/3.40±0.02
C
3.29±0.02/3.11±0.02
as follows:
³
´
Ncont = Nπβ /Aπβ · ξ ,
(4.17)
where Nπβ is the gate-corrected number of detected π + → π 0 e+ ν events, Aπβ is the
corresponding acceptance and ξ are the percentile contaminations indicated in table
4.11. The ratio Nπβ /Aπβ gives total number of π + → π 0 e+ ν events. Table 4.12
summarizes these findings. These events are subtracted from the total yield of the
π + → e+ νγ events. The uncertainty of each number in the table is 5%.
Table 4.12: Number of background πβ events.
Data set
Total πβ # Region A Region B
Region C
Set 1
3 020.89
101.43
107.80
99.36
Set 2
2 065.75
70.16
64.24
65.79
119
CHAPTER 4: Data Analysis
4.4.5
CsI energy calibration
Energy deposition reported by the GEANT simulation is taken as an absolute energy
scale. Energy calibration involves finding the overall coefficients that match the data
and simulated energy distributions. In general the energy calibration constants for
the positrons and photons differ slightly. Positrons interact instantaneously upon
entering the detector and typically produce a hard bremsstrahlung photon. Photons,
on the other hand, travel on average one radiation length (1.8cm in CsI) before
interacting. The showers produced by photons typically distributed deeper into the
volume of a crystal, while the showers produced by positrons are distributed closer to
the front edge of the crystal. This subtle difference in the shower formation between
the positrons and photons leads to a slightly higher leakage probability of the photoninitiated showers to leak through the back of the crystal. Hence, on average, a positron
with the same physical energy as the photon will deposit slightly more energy in the
crystal.
In case of the π + → e+ ν
reaction the energy spectra of the simulated and
measured positrons are compared directly. The energy scale is varied to minimize the
difference between those two histograms (see section 4.3.3). In case of the π + →
e+ νγ , the values of the matrix element vary with the final state kinematics. Moreover,
the matrix element depends on the value of the unknown parameters of the vector
and axial vector form factors (see section 2) FA and FV . Therefore, the shape of
the energy distribution (for either e+ or γ) depends on the value of the unknown
120
CHAPTER 4: Data Analysis
Number of events
800
γ=0.5
γ=0.6
600
400
200
0
0
20
40
60
Ee Region B (MeV)
Figure 4.25: Simulated positron spectrum in region B for two different values of γ=F A /FV .
parameters that are to be determined in this analysis. The illustration of this idea
is shown in figure 4.25. Thus, comparing the energy spectra to get energy scales is
unreliable and we used other methods.
We have implemented two independent methods of determining the energy scales.
The first one obtains the scales from the reactions other then π + → e+ νγ where a
positron or a photon are detected and the matrix element does not depend on any
unknown factors (except for an overall multiplier). In this case the direct comparison
of the GEANT simulation and the data can be performed. Section 4.3.3 describes
the positron energy spectrum of the π + → e+ ν reaction and the energy scales for two
subsets of the data. An analogous procedure can be followed with the π + → π 0 e+ ν
reaction for the spectrum of the γ energy since the value of the matrix element only
121
CHAPTER 4: Data Analysis
depends on the value of Vud of CKM matrix [46] which only multiplies the rate. The
Number of events
results for two subsets of the data is shown in table 4.13.
Simulation
300
•
Data
200
100
0
40
50
60
70
80
90
γ energy (MeV)
Figure 4.26: Energy spectrum of the high energy photon in the π + → π 0 e+ ν decay followed
by π 0 → γγ decay.
Table 4.13: Photon energy scales from πβ analysis.
Variable
Set 1
γ Energy Scale 0.9506±0.0095
Set 2
0.9580±0.0096
The resulting spectrum of the high energy photon is shown in figure 4.26. This
method is, however, not very precise for the case of the γ energy scale. First, the
overall agreement between the GEANT simulation and the data is not excellent in
122
CHAPTER 4: Data Analysis
detail, which can be attributed to the limited number of detected π + → π 0 e+ ν
events (under 3,500 gate-corrected events for the full year 2004 data set). Second,
the selection criteria of the π + → π 0 e+ ν
events in principle introduces a bias
related to the fact that two photons detected in π 0 decay do not have the same mean
values of energy due to the selection algorithm [53, 46]. This limits the usefulness of
information in table 4.13.
The second method is based on the background-subtracted distributions of the
kinematic variable λ which is defined in two different ways:
λ1 ≡ y · sin2 (θeγ /2) ,
(4.18)
λ2 ≡ (x + y − 1)/x ,
(4.19)
where
x≡
2Eγ
2Ee
and y ≡
.
mπ
mπ
In the pion’s rest frame two definitions are equivalent:
λ1 = λ 2 .
(4.20)
As can be seen from the definition of λ, it involves two sets of measurable quantities,
namely energies and opening angles. These measurements are made independently in
our experiment. The definition of λ in equation (4.19) involves positron and photon
energies on the equal footing. The alternative definition in equation (4.18) involves
only the positron energy. If one constructs a variable ∆λ ≡ λ1 − λ2 , its distribution
is sensitive to any asymmetries in the energy calibration of either particle.
123
CHAPTER 4: Data Analysis
It is important to note that the experimental positron energy included the energy
in the target and the PV which will be discussed in section 4.4.7. It may seem that
including the target energy might introduce some unknown bias. That is, however,
easily eliminated if one observes how symmetric the ∆λ distribution is around the
origin. The absence of (or a mis-calibrated) target energy will shift the distribution
to a non-zero mean.
In order to determine correct energy calibration coefficients, the energy scales for
the positron and photon are varied, and the values of λs are recalculated for every
pair of these parameters. The background-subtracted experimental distribution of
∆λ is compared with the corresponding GEANT simulated distribution and the χ 2
difference [7] is constructed as follows:
χ2 =
X
i=all bins
³
´
xiGEANT − xiDATA )2 /σ 2 ,
(4.21)
where xiGEANT and xiDATA are contents of the ith bin of the properly normalized simulated and experimental histograms respectively, and σ 2 is the total statistical error of
the simulation and the data, combined in quadratures. The summation is performed
over bins with combined relative statistical uncertainty under 12%. It is important to
note that the simulated histogram is weighted with the value of the matrix element in
each bin. However, ∆λ is not a theoretical observable, i.e., theoretically it is supposed
to be zero, and therefore it depends very weakly on the value of the matrix element.
Thus, in this procedure the values of the parameters FA and FV do not have to be
124
CHAPTER 4: Data Analysis
1200
Number of events
1000
•
γ=0.5
γ=0.6
800
600
400
200
0
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
∆λ Region B
Figure 4.27: Simulated ∆λ spectrum in region B for two different values of γ=F A /FV .
known precisely without causing any significant effect in the final values of the energy
scale parameters within the accuracy of the method which will be discussed in section
4.4.13. Figure 4.27 illustrates that the ∆λ distribution is virtually independent of
the matrix element.
The χ2 values generated for all three kinematic regions are summed in order to
minimize the χ2 simultaneously without a kinematic bias. Figure 4.28 shows the
contour plots of constant χ2 for two subsets of the data. These plots are fitted with
a two-dimensional paraboloid with the minimum left as a free parameters of the fit.
The final results are summarized in table 4.14.
We note the good agreement between the positron energy scales when compared
to the values in table 4.2. These results additionally confirm that the target energy
125
CHAPTER 4: Data Analysis
Photon energy scale
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.93
0.94
0.95
0.96
0.97
Positron energy scale
Photon energy scale
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.93
0.94
0.95
0.96
0.97
Positron energy scale
Figure 4.28: Contour plot of χ2 for the ∆λ difference as a function of the positron and
photon energy scales. Set 1 is shown on top panel and set 2 is on bottom. Dashed lines are
parabolic fits to the data shown with solid lines.
126
CHAPTER 4: Data Analysis
Table 4.14: Energy scales from π + → e+ νγ analysis.
Variable
Set 1
Set 2
e Energy Scale
0.9478±0.7318·10−4
0.9412±0.1645·10−3
γ Energy Scale
0.9576±0.8884·10−4
0.9652±0.2112·10−3
(which is added only to the positrons) is calculated correctly, since the π + → e+ ν
data did not have target energy added to the positron energy. Also, photon energy
scales agree well with the values in table 4.13. Our suggestion about the positron
and photon energy scale difference came true: γ energy scale is somewhat higher,
which suggests a systematic reduction of the ADC readings related to the photons.
To summarize, the positron energy scales obtained from π + → e+ ν procedure
are used for the final calculations. The photon energy scales are taken from the table
above. The energy scale factors are summarized in table 4.15.
Table 4.15: Final energy scales.
Variable
Set 1
Set 2
e Energy Scale
0.9429
0.9411
γ Energy Scale 0.9576
0.9652
CHAPTER 4: Data Analysis
4.4.6
127
CsI TDC discriminator inefficiency
Section 4.4.2 describes selection criteria for the π + → e+ νγ decay candidates. It
involves finding the minimum time e+ − γ difference registered by the CsI’s TDC. If
a TDC unit fails to register a valid signal, there will not be a valid hit recorded for
one (or both) particles and an event will fall out of the consideration. The analog
signals from the CsI crystals pass through voltage discriminators that do not allow
signals with amplitudes below a certain threshold. Because of variations in pulse
shapes among CsI detector modules and due to noise on the pulse baselines, the
discriminator thresholds will appear different from one counter to another, and the the
efficiency is not a Heaviside step function at threshold. The discriminator efficiency
dependence on the signal amplitude is more pronounced in the 1-arm-low trigger
where the minimal detected energy of a particle is ∼5 MeV.
The positron energy spectrum in the 1-arm-low trigger is dominated by the Michel
positrons. After eliminating the prompt contamination, one can compare the experimental line shape of the positron spectrum to the simulated one. The matrix element
for the Michel events is known precisely and is reliably simulated in our experiment.
As can be seen in figure 4.29 (top panel), the two spectra coincide very well in the
high energy region. At the lower energies (below ∼30 MeV) the experimental curve
falls short of the simulation. Since the main requirement for the Michel candidates is
to have a valid time recorded in the CsI calorimeter, this effect can be attributed to
the inefficiency in the discriminator threshold at low energy.
CHAPTER 4: Data Analysis
128
In order to correct for these inefficiencies the following procedure is used. The
experimental and MC energy spectra are booked with a bin size of 1 MeV as shown
in figure 4.29 (top panel). The ratio of these two histograms in every bin is applied
to correct the experimental yield of the events in the corresponding bin. The same
correction is utilized for all reactions simultaneously. The corrected Michel spectrum
is shown in figure 4.29 (middle panel). It is important to note that the positron
energy scale used in this exercise was the same as for π + → e+ ν positrons which
additionally confirms the accuracy of our energy calibration.
An additional check of the efficiencies can be performed solely using experimental
data. The positrons at given energies are MIP and their energy spectra are very
similar in all PV staves independent of the positron incident energy. The timing of
these positrons is also registered by the PV TDCs modules, but the PV discriminator
efficiencies have a negligible dependence on the positrons energy. By requiring a
valid TDC hit in either the CsI or in PV associated with the positron track, one can
reconstruct the Michel positrons energy spectra separately with valid TDC records
in PV and CsI. The resulting positron spectrum with valid PV hits is shown in the
bottom panel of figure 4.29. The efficiencies deduced from this method are consistent
with the results obtained by comparing with the MC spectrum.
129
Number of events
CHAPTER 4: Data Analysis
Simulation
• Data
300
200
100
0
0
10
20
30
40
50
60
50
60
50
60
Number of events
Positron energy (MeV)
Simulation
• Data
300
200
100
0
0
10
20
30
40
Number of events
Positron energy (MeV)
Simulation
• Data
300
200
100
0
0
10
20
30
40
Positron energy (MeV)
Figure 4.29: Michel spectrum shown against the GEANT simulation before (top panel) and
after (middle panel) discriminator efficiency corrections were applied. Bottom panel shows
the agreement between the simulation and the experimental spectrum cut on the PV time.
CHAPTER 4: Data Analysis
4.4.7
130
Total positron energy: energy in the PV and target
Section 4.4.5 describes the energy of the positron deposited in the CsI calorimeter.
The total positron energy is calculated as described in equation (4.16) The assignment
of the PV energy (EePV ) is done on the tracking stage of the analysis and is described
in the particle identification section 4.2.5. The entire energy in the PV stave that
corresponds to a track is assigned to the track. In cases when more than one track is
associated with a given veto detector, the energy is divided equally among the tracks.
Most of the energy deposited in the target (Eetarget ) comes from stopping pions and
only a small part from the decay positrons traversing the target. Both signals are
integrated within the same ADC gate and appear inseparable in the ADC readings.
The exponential nature of the decay provides some useful information about the time
structure of the signal in the target.
The event time dependence of the total energy deposited in the target is shown
in figure 4.30. The early time slope corresponds to the sum of the π-stop and
the positron energy, while at later times energy levels off and corresponds almost
exclusively to the positron energy deposited in the target.
In order to eliminate the pion energy in the target, an exponential function is
fitted to the scatter plot in figure 4.30 and its contribution subtracted out. The
result is shown on the bottom panel of figure 4.30. The overall energy conversion
factor is subsequently applied to match the peaks of the experimental and simulated
spectra. This procedure of linearization of the target energy broadened the target
131
CHAPTER 4: Data Analysis
160
140
ADC channel
120
100
80
60
40
20
0
-20
0
20
40
60
80
100
120
140
160
100
120
140
160
Time (ns)
100
ADC channel
80
60
40
20
0
-20
0
20
40
60
80
Time (ns)
Figure 4.30: Target energy (in ADC channels) vs. event time before (top panel) and after
(bottom panel) subtraction of the π-stop energy, a.k.a. linearization procedure.
CHAPTER 4: Data Analysis
132
energy distribution at earlier times. This effect can be simulated by smearing the
Monte Carlo spectrum of the total target energy by a Gaussian without shifting the
mean of the distribution. The comparison between the GEANT simulation of the
target and the data is discussed in section 4.4.9.
4.4.8
Acceptance calculations for the radiative decays
As it is discussed in section 4.4.5, the calculation of the acceptance for the radiative
pion decay is more complicated due to the dependence of the matrix elements (and
therefore weights in each bin) on the value of the initially unknown weak form factors.
A second obstacle arises from the fact that radiative decays are dominated by the
emission of low energy bremsstrahlung photons as described in chapter 2. Therefore,
all calculated partial branching ratios become divergent below a certain photon energy
threshold. In case of numerical calculations the problem manifests itself by producing
arbitrarily large values of the matrix element for photons with very low energy (below
or close to the theoretical threshold). The upper limit on the maximum weight, even
though finite, will depend on the size of the simulated sample (for the larger sample
probability of a larger maximum weight is increased). Therefore, one can not calculate
the acceptance in a way similar to that for π + → e+ ν discussed in section 4.3.10. If
one sums the weights of all the events that pass the experimental cuts and normalizes
it to the total sum of all the weights in the sample, it will erratically depend on the
size of the sample, i.e., the maximum weight of the sample. The acceptance for a
133
CHAPTER 4: Data Analysis
region R should, therefore, be calculated as follows [27]:
AR
RPD =
X
i∈(Rtheor ∩ Rexp )

W Ti 
X
i∈Rtheor
−1
W Ti 
,
(4.22)
where W Ti is a weight if the ith event, Rtheor were defined in table 4.8 and Rexp in
table 4.9 are the theoretical and experimental definitions of the regions A, B and C
respectively. In other words, the sum in the denominator contains only the events
that pass the theoretical cuts for the given region and does not extend over all thrown
events.
4.4.9
The acceptance for π + → e+ νγ events
RPD acceptances for each kinematic region are calculated using equation (4.22)
which implicitly assumes that the agreement between the data and the simulation is
satisfactory. In order to provide reasonable agreement, special care is given to the
simulation of the energy deposition in the PV hodoscope and the target.
As discussed in section 3.4, individual PV staves exhibit a non-uniform response
to the MIP positrons and this effect is accounted for in our simulation. Initially,
in our GEANT code, all of the PV staves had equivalent responses. In order to
simulate the differences between the staves, the simulated energy deposition in each
PV stave is smeared with a Gaussian distribution (2 independent parameters for each
of the 20 staves) to match the experimental Geometric Mean energy distribution
for a corresponding stave. In addition, the entire year 2004 data set is divided into
134
CHAPTER 4: Data Analysis
4 intervals. The means of GM spectra stay fairly stable within each one of these
intervals, and vary slightly from one interval to another. This adds up to a total of
160 parameters to be determined and is not reproduced here.
Figures
4.31 through
4.34 depict the comparison of simulated spectra and
background-subtracted experimental distributions for Region B, the hardest one to
reproduce faithfully. For all distributions the agreement is very good.
4.4.10
Global fit to the data
As specified in section 2, the value of the parameter FV is known theoretically, and
is related to the neutral pion lifetime. Therefore, for practical purposes, the value of
the parameter FV was initially fixed at its historical value of
FV = 0.0259 ,
and the matrix element was parametrized by only one unknown variable, namely
γ=FA /FV .
One can vary this parameter and calculate the theoretical branching ratio (BR)
for each kinematic region for a given value of the parameter γ. At the same time,
one can calculate the detector acceptance at each step and in every kinematic region
and calculate the experimental branching ratio as indicated by equation (4.1). The
result is a quadratic numerical equation in variable γ. The solution of this equation
135
Number of events
CHAPTER 4: Data Analysis
Simulation
• Data
600
400
200
0
0
0.2
0.4
0.6
0.8
1
1.2
λ=(2Ee/mπ)sin (Θeγ/2)
2
Number of events
1000
Simulation
• Data
800
600
400
200
0
50
55
60
65
70
75
80
85
Eγ Region B (MeV)
Figure 4.31: Comparison of the variable λ1 (top) and photon energy (bottom) distributions.
Dots represent data points, solid lines simulation.
136
CHAPTER 4: Data Analysis
Number of events
800
Simulation
• Data
600
400
200
0
20
40
60
80
Ee Region B (MeV)
Number of events
1200
1000
Simulation
• Data
800
600
400
200
0
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
δλ Region B
Figure 4.32: Comparison of the positron energy (top) and ∆λ (bottom) distributions. Dots
represent data points, solid lines simulation.
137
Number of events
CHAPTER 4: Data Analysis
Simulation
• Data
600
400
200
0
0
2.5
5
7.5
10
12.5
Target Energy Region B (MeV)
Number of events
800
Simulation
• Data
600
400
200
0
0
0.5
1
1.5
2
PV Energy Region B (MeV)
Figure 4.33: Comparison of the target energy (top) and PV energy (bottom) distributions.
Dots represent data points, solid lines simulation.
138
Number of events
CHAPTER 4: Data Analysis
Simulation
• Data
600
400
200
0
80
100
120
140
160
180
Opening angle Θ Region B (dgr)
Figure 4.34: Comparison of distributions of θeγ , the positron-photon opening angle. Dots
represent data points, solid lines simulation.
is found by minimizing the χ2 function [7] defined as follows:
χ2 (γ) =
X
i=RA ,RB ,RC
³
BRitheor (γ) − BRiexp (γ)
σi2
´2
,
(4.23)
where BRitheor (γ) and BRiexp (γ) are the ith theoretical and experimental branching
ratios for the given value of γ, σi2 is the total uncertainty of the branching ratios.
The sum is minimized with respect to the parameter γ. Once again, it is important
to note that experimental branching ratios change with different values of γ via the
acceptance dependence on γ. Numerical solutions of equation (4.23) are presented
in chapter 5.
139
CHAPTER 4: Data Analysis
4.4.11
Radiative corrections to π + → e+ ν acceptance
Before final results and conclusions on the π + → e+ νγ findings are presented, there
is another issue that needs to be addressed. It concerns the radiative corrections
to the π + → e+ ν acceptance. As discussed in section 4.3.10, the acceptance is
calculated under the assumption that the πe2 process has two particles in the final
state. In reality, associated photon is always present. Since πe2 events are selected
regardless of the presence of a photon, some of these events are actually π + → e+ νγ
decays with part of the positron energy carried away by a photon. This results in a
change of the line-shape of the non-radiative πe2 energy spectrum. The low energy
fraction of the total number of events increases due to the radiative photon presence,
thus reducing the acceptance of high energy positrons. Simply put, there are not as
many positrons in the energy range between 50 MeV and 80 MeV in case of RPD as in
case of an ideal πe2 decay. Figure 4.35 demonstrates the effect and sets the premises
for the calculations below. If the numbers of events in each region were as shown in
the figure 4.35, then corrected acceptance to π + → e+ ν events can be calculated as
follows:
N2 − x
Acor
,
πe2 = P
(4.24)
total
where
P
total
is the total number of all thrown events and
N2
(p − g) ,
1+p
(4.25)
N1r
N1
and g =
.
r
N2
N2
(4.26)
x=
p=
140
Number of events
CHAPTER 4: Data Analysis
10 4
10 4
10 3
10 3
10 2
10 2
10
10
25
50
75
25
50
75
Figure 4.35: Comparison of the ideal (left) π + → e+ ν simulation and the effect of the
radiative photon (right) on the line-shape of the positron energy spectrum.
An indication that the radiative effect like this takes place is the fact that the value
of the subtraction factor in the ADC method of extracting the number of πe2 events
(see section 4.3.3) is higher compared with the theoretical non-radiative value. The
theoretical value can be extracted from the Michel fit component of the π + → e+ ν
time curve (see section 4.3.4). It is simply the ratio of the areas under the Michel
component of the fit in the early and late gate portions as illustrated in figure 4.36.
For example, for set 2 of the year 2004 data the theoretical value is 0.7955±0.0022
while the experimental subtraction factor is 0.81222±0.00186. This leads to a slightly
smaller number of events determined by the ADC method compared to the TDC
method (see section 4.3.5).
141
CHAPTER 4: Data Analysis
6000
Number of Events
5000
g=A1/A2=0.7955±0.0022
4000
3000
A1
2000
A2
1000
0
0
50
100
Michel time (ns)
Figure 4.36: Michel component of the π + → e+ ν fit divided into early and late gate
parts. The ratio of the integrated areas of these parts should give a theoretical value of the
subtraction factor.
One way of estimating the size of the low-energy tail of the positron energy distribution is to use a GEANT simulation. One can generate actual π + → e+ νγ events
and process them as π + → e+ ν decay. Then, all the cuts used in πe2 analysis can
be applied, and the energy distribution of the simulated positrons can be used to
calculate the new acceptance using equation (4.24).
There are two principal methods that can be used to generate radiative events
with variable weights. The RPD sample can be weighted (where all the bins are
weighted with the value of a matrix element for the given set of kinematic parameters)
or unweighted (when all events in the sample have the same weight). Since the
entire positron energy tail is to be evaluated, the photon energy should go to the
CHAPTER 4: Data Analysis
142
theoretical infrared cut-off (the minimal photon energy used in the calculations). In
a weighted sample it leads to large fluctuations in the value of the matrix element and
consequently causes the calculations to depend on the size of the sample leading to
very large statistical errors. In case of an unweighted sample, the maximum weight
value should be known a priori. For large Monte Carlo samples this value is orders
of magnitude higher than the median value of the matrix element. This ultimately
leads to excessively long processing times to generate a reasonable size sample. In
either case one has to extrapolate the result to the value of the infrared cut-off.
This problem is best illustrated in the following plots. Figure 4.37 shows a sample
of 6×106 simulated events weighted by the values of the matrix element. The infrared
cutoff is λcut =100 keV and energy line-shape exhibits very large fluctuations. These
fluctuations obviously increase for smaller values of λcut .
On the other hand, figure 4.38 shows a sample of merely 1×105 simulated unweighted events with roughly the same value of λcut . The processing time required
to generate these two samples was, however, comparable. The line-shape of the unweighted sample is smooth and remains so for the smaller values of λcut .
Analysis of the energy tail corrections is performed independently for the weighted
and unweighted samples. For the former, six values of λcut are implemented and the
percentage of the events in the tail (parameter p in equation (4.26)) is calculated
for six different samples. The average value of p as a function of λcut is fitted with a
parabola and the value of p extrapolated to λcut =0. The fit is shown in figure 4.39.
143
CHAPTER 4: Data Analysis
Arbitrary normalization
8
7
s
on
Weighted sample
ti
tua
luc
f
rge
6
La
5
4
3
2
1
0
0
10
20
30
40
50
60
70
80
Positron energy (MeV)
Figure 4.37:
π + → e+ ν positron line shape with weighted sample and infrared cutoff
Eγ =100 keV.
This extrapolated value of p leads to new value of acceptance of π + → e+ ν events:
Acor
πe2 = 0.6662 ± 0.004855 ,
(4.27)
where the large uncertainty is attributed to the large statistical fluctuations in determining the variable p in each subset.
For the unweighted sample a similar procedure takes place. Six different samples of
radiative pion events are generated with different values of maximum weight (W max ).
Parameter p is plotted against the natural logarithm of the maximum weight (due to
orders of magnitude variances in the latter) and fitted with a parabola. This procedure
is illustrated in figure 4.40. The value of p is extrapolated to the minimum of the
144
CHAPTER 4: Data Analysis
Arbitrary normalization
30000
Unweighted sample
25000
20000
15000
10000
5000
0
0
10
20
30
40
50
60
70
80
Positron energy (MeV)
Figure 4.38: π + → e+ ν positron line shape with unweighted sample and infrared cutoff
Eγ ∼ 100 keV.
parabola and the value of the corrected acceptance converges to the following value:
Acorr
πe2 = 0.6606 ± 0.001481 .
(4.28)
The two values agree well within the quoted standard deviations. The unweighted
sample result has a significantly smaller uncertainty and is used in all of the subsequent calculations. The branching ratios for the π + → e+ ν quoted in table 4.4 are
calculated using the corrected values of acceptances. Since non-radiative acceptances
for πe2 events are nearly indistinguishable for the two data subsets, the correction
obtained above is used uniformly through out the data analysis.
145
CHAPTER 4: Data Analysis
0.13
0.12
p = N1r/N2r
0.11
0.1
0.1
0.09
0.08
0.07
0.06
0.05
0.04
-0.5
0.05
0
0.5
1
1.5
2
2.5
λcut (MeV)
Figure 4.39: Percentage of the events in the π + → e+ ν tail as a function of the minimally
accepted photon energy. The solid curve represents a parabolic fit to the data points while
the dashed line is an extrapolation to λcut =0.
4.4.12
Form factor dependence on the invariant mass of the
lepton pair
The final goal of our radiative pion decay ( π + → e+ νγ ) analysis is to extract the
experimental branching ratio of this decay (for a given set of kinematic constrains on
the final state energies and angles) along with the values of the weak form factors
of the structure-dependent terms of the differential branching ratio. We define the
kinematic variable
s = (pe + pν )2 = m2π (1 − 2Eγ /mπ ) ≡ m2π (1 − x) ,
(4.29)
146
CHAPTER 4: Data Analysis
0.13
0.125
0.12
p = N1r/N2r
0.11
0.1
0.1
0.09
0.08
0.075
0.07
0.06
-6
-5
-4
-3
-2
-1
0
1
ln(Wmax)
Figure 4.40: Percentage of events in the π + → e+ ν tail as a function of the logarithm of
the maximum event weight. The Solid curve represents a parabolic fit to the data points
while the dashed curve is an extrapolation to the minimum.
which is an invariant mass of the lepton pair in question (LPIM) . The vector (F V )
and axial (FA ) form factors should, in principle, depend on this variable. We also
define a variable q which is the same as s but is expressed in units of the pion mass
squared:
q = (1 − x) .
(4.30)
Traditionally, radiative pion decays were studied in kinematic regions limited to the
high energies of the final state lepton and photon. This was a good approach to evaluate (FV ) and (FA ) at q=0. The assumption of the small lepton pair invariant mass
remains valid for Regions A and B but does not hold true for Region C. Theoretical
147
CHAPTER 4: Data Analysis
Region A
50
Number of Events
0
-0.05
200
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1
0.15
0.2
0.25
0.3
Region B
100
0
-0.05
0
0.05
Region C
200
0
0
0.2
0.4
0.6
0.8
q=(1-2Eγ/mπ)
Figure 4.41: Theoretical lepton pair invariant mass for three kinematic regions.
values of q are shown in figure 4.41.
The above-mentioned dependence can be parametrized using a variety of theoretical approaches. All of them have in common an increase of the value of the form
factor as variable q rises. Our data in Region C provide unique opportunity to ob-
148
CHAPTER 4: Data Analysis
serve and quantify this effect. We can split Region C into three subranges according
to the value of q and calculate the value of γ=(FA )/(FV ) for each range. Following
tradition, in this analysis we keep the value of (FV ) fixed at the CVC prediction.
Hence if there is a dependence of the form factors on the LPIM, it will manifest itself
in an increase of γ for larger values of LPIM.
The sub-ranges of q are chosen as follows:
Table 4.16: Definition of Region C subranges.
Range 1
0 < q < 0.3
Range 2 0.3 < q < 0.55
Range 3 0.55 < q < 1.0
For each range the number of events and corresponding acceptances are calculated
in the regular manner. Some sample distributions are shown in figure 4.42. For
each sub-range, solutions of the BRtheory (γ)=BRexper (γ) are found using a global
fit procedure. The uncertainties are estimated based on the statistical uncertainties
of π + → e+ νγ
decay. The systematic uncertainties for the normalizing decay
π + → e+ ν are fully implemented.
The corresponding graphical solutions are illustrated in figure 4.43 and are summarized in table 4.4.12.
The effect of the growth of the value of γ with q is statistically significant among
all three ranges. The smaller change between Range 1 and Range 2 (low and medium
149
CHAPTER 4: Data Analysis
1000
Range 1
500
Number of Events
0
500
-0.2
-0.1
0
0.1
0.2
-0.1
0
0.1
0.2
-0.1
0
0.1
0.2
Range 2
250
0
-0.2
Range 3
200
0
-0.2
δλ Region C
Figure 4.42: ∆λ distributions for three sub-ranges of Region C. Solid lines represent simulation, dots are data.
150
CHAPTER 4: Data Analysis
60
Range 1
γ=0.47
40
20
0
0.2
0.4
BRx108
60
0.6
γ=0.53
0.8
1
Range 2
40
20
0
0.2
0.4
0.6
0.8
1
50
Range 3
40
γ=0.58
30
0
0.2
0.4
0.6
0.8
1
γ=FA/FV
Figure 4.43: Graphical solutions for the parameter γ for three sub-ranges of Region C.
Rising curves are theoretical BRs and descending are experimental BRs.
151
CHAPTER 4: Data Analysis
Table 4.17: Values of the parameter γ for different subranges of Region C.
Range 1
γ=0.475±0.016
Range 2
γ=0.529±0.018
Range 3
γ=0.577±0.024
values of q) may be understood in terms of the so-called dip in the q dependence at
small values of LPIM [59].
The results shown above qualitatively agree with the results we obtained for the
three kinematic regions: if no LPIM dependence is implemented (assuming that the
form factors were F (q) = F (0)), we consistently have a surplus of experimental events
in Region C, while Region A and Region B are in much closer agreement with the
theory. It is also worth noting that the value of γ for Range 1 is in very close agreement
with the overall result reported in chapter 5, just as expected for the small values of
LPIM.
It is therefore imperative to incorporate the branching ratio dependence on LPIM
in our minimization routines. In this analysis it is done by the following parametrization:
F (q) = F (0)(1 + a · q 2 ) ,
(4.31)
where a is a free fit parameter and the dependence is the same for the vector and
axial-vector form factors. More complicated schemes could be entertained as well.
152
CHAPTER 4: Data Analysis
Number of Events
140
120
πβ background
100
80
60
40
The end of Range 1
20
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
q=(1-2Eγ/mπ)
Figure 4.44: GEANT reconstructed lepton pair invariant mass for the π β background.
For example, chiral perturbation theory (χPT) calculations [8, 30] suggest a different
dependence of FV and FA on LPIM.
Additional remarks can be made about the πβ background subtraction. As shown
in figure 4.44, the πβ events that appear as radiative pion decays in our detector
have very small values of q. In fact, almost 98% of these background events fall
into Range 1. Therefore, the entire πβ background is subtracted from Range 1 alone
(which constitutes about 3% of the total yield in this range).
4.4.13
Analysis of the synthetic data
Sections 4.4.2 through 4.4.10 describe the basis of the π + → e+ νγ analysis starting
from the selection of the events, to the global fit procedure which provides the values
CHAPTER 4: Data Analysis
153
of the sought parameter γ. One has to analyze actual data with the most precise
knowledge of many factors such as time offsets for the various counters, energy calibration constants, dependence of the energies and times on the run number as well as
a thorough understanding of the prompt contamination in the signal. On the other
hand, we rely heavily on the GEANT simulation package in understanding of our
experimental acceptances and details of the various backgrounds.
The two tasks are tangled together, since precise values of the experimental parameters (such as energy calibration constants, for instance) are derived from the
simulation, which in turn should be carefully tuned to the observed parameters (PMT
photoelectrons statistics, actual values of the trigger threshold energies, etc.) It is
therefore a task of a great importance to be able to cross-calibrate the two branches
of analysis with a well understood data sample.
One way of creating such a sample is to generate it using the Monte Carlo (MC)
technique. We can create MC data with all the required variables and with the known
values of the physical parameters, and analyze these data following default routines
and check if we reconstruct the initial values of the parameters.
In order to reproduce the physics of the required decay properly, one should create
an unweighted sample of events. In this sample each event has the same weight as
opposed to the matrix element varying with the values of the kinematic parameters.
This obviously creates a realistic distribution of events, but significantly reduces the
size of the analyzable sample since now our events are as rare as they are in real life.
154
CHAPTER 4: Data Analysis
Since the PIBETA GEANT simulation mirrors the PIBETA analyzer, the higherlevel analyzer banks are easily overwritten by the Monte Carlo generated values of
physical observables. While generating the unweighted sample of MC events, the
matrix element for the synthetic π + → e+ νγ decay has the following constants
embedded in it:
FV = 0.0259,
γ=
FA
= 0.0500,
FV
a = 0.0 .
(4.32)
All of the energies recorded into the synthetic version of the analyzer banks are
unmodified, i.e., the overall energy scale factor for every counter is 1. The numbers
of synthetic events generated in each of the three analyzed regions reflect theoretical
branching ratios and are shown in the table 4.18 .
Table 4.18: Total number of generated events in synthetic data set.
Region
Generated events
A
10 788
B
57 552
C
148 646
The synthetic data are analyzed in exactly the same manner as the experimental
data with the exception of prompt events. Properly smeared CsI calorimeter times
and degrader TDC hits are also simulated, which allows realistic gate fraction cuts in
the synthetic data set. Signal-to-background distributions (virtually background-free)
155
CHAPTER 4: Data Analysis
are shown in figure 4.45. The Gaussian fits to these spectra yield widths comparable
with the actual data (see figure 4.24).
Figures 4.46 through 4.49 demonstrate the agreement between the synthetic data
and the default GEANT PIBETA detector simulation. The agreement is satisfactory
and can be compared with similar displays for the real data shown in figures 4.31
through 4.34. Throughout the analysis, the value of FV remains fixed at 0.0259 and
only statistical errors are taken into account.
The global fit procedure described in section 4.4.10 is performed and the resulting
parabolic fit to the χ2 (γ) for positive values of γ is shown in figure 4.50. From this
minimization procedure parameters γ and a are determined to be:
γ = 0.498 ± 0.0143
(4.33)
a = 0.133 ± 0.075 .
For these best values of γ and a, all the acceptances are recalculated and total number
of events (branching ratios) in each of the three regions is summarized in table 4.19.
The last column of the table specifies the fractional difference from the expected
(“theoretical”) number of events in each region.
The value of γ as well as the values of the experimental branching ratios are within
one standard deviation which only includes statistical errors. There is a small concern
about the value of the parameter a which is 1.8 standard deviations away from the
enclosed value of zero. In the analysis of the real data, however, the uncertainty
of the parameter a is higher due to inclusion of the systematic uncertainties. This
156
CHAPTER 4: Data Analysis
200
P/B > 500
Region A
100
Number of events
0
500
0
-10
-5
0
10
P/B > 500
Region B
-10
5
-5
0
5
10
1000
500
0
P/B > 500
Region C
-10
-5
0
5
10
t(e)-t(γ) (ns)
Figure 4.45: Difference between the positron and photon times for three kinematic regions
displayed for the synthetic data.
157
CHAPTER 4: Data Analysis
Number of events
800
Simulation
• Data
600
400
200
0
-0.2
0
0.2
0.4
0.6
2
0.8
1
1.2
λ=(2Ee/mπ)sin (Θeγ/2)
Number of events
1000
Simulation
• Data
800
600
400
200
0
50
60
70
80
90
Eγ Region B (MeV)
Figure 4.46: Comparison of the variable λ1 (top) and photon energy (bottom) distributions.
Dots represent synthetic data points, solid histograms simulation.
158
Number of events
CHAPTER 4: Data Analysis
Simulation
• Data
600
400
200
0
0
20
40
60
80
Ee Region B (MeV)
Number of events
1000
800
Simulation
• Data
600
400
200
0
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
δλ Region B
Figure 4.47: Comparison the positron energy (top) and ∆λ (bottom) distributions. Dots
represent synthetic data points, solid histograms simulation.
159
CHAPTER 4: Data Analysis
Number of events
600
Simulation
• Data
500
400
300
200
100
0
0
2.5
5
7.5
10
12.5
Number of events
Target Energy Region B (MeV)
Simulation
• Data
600
400
200
0
0
0.5
1
1.5
2
PV Energy Region B (MeV)
Figure 4.48: Comparison of the target energy (top) and PV energy (bottom) distributions.
Dots represent synthetic data points, solid histograms simulation.
160
CHAPTER 4: Data Analysis
Number of events
600
Simulation
• Data
500
400
300
200
100
0
50
75
100
125
150
175
Opening angle Θ Region B (dgr)
Figure 4.49: Comparison of the positron-photon opening angle θeγ distributions. Dots
represent synthetic data points, solid histograms simulation.
Table 4.19: Number of reconstructed events in synthetic data analysis.
Region
Detected events Reconstructed events Difference (%)
A
4 080.2±77.792
10 989.3±211.489
+1.8
B
9 447.7±118.37
10 979.9±287.53
−0.94
C
11 350 ±129.78
148 971±1 727.42
+0.22
test proves that the overall method is working within the required precision and no
systematic biases are seen in this analysis.
Another aspect of the systematic effects can be controlled via this simulation procedure. We should be able to confirm that the energy scale factors for the positron
161
CHAPTER 4: Data Analysis
1200
Γπ→eνγ MINIMIZATION: REGIONS A+B+C
χ2(A+B+C)
1000
SYNTHETIC DATA
800
600
γMIN=0.50
400
200
0
0
0.2
0.4
0.6
0.8
1
γ=FA/FV
Figure 4.50: Combined χ2 over all three regions as a function of γ. The value of FV remained
fixed at 0.0259.
and the photon are precisely reconstructed. We can also quantitatively estimate the
effect of the change in the energy scales on the number of the detected events. A
two-dimensional parameter scan based on the discrepancy between the simulated and
“experimental” distributions in ∆λ is performed in the manner similar to the one
described in section 4.4.5. This procedure is performed in all three kinematic regions
and the results are summed, i.e, the χ2 is simultaneously minimized for all three studied regions. The two-dimensional contour plot with the parabolic fit superimposed
is shown in figure 4.51 and it returns the values of the energy scales shown in table
4.20.
The embedded value of unit gain is well within the error bars of the fit. In other
162
CHAPTER 4: Data Analysis
1.03
Photon energy scale
1.02
1.01
1
0.99
0.98
0.97
0.97
0.98
0.99
1
1.01
1.02
1.03
Positron energy scale
Figure 4.51: Contour plot of χ2 for the ∆λ difference as a function of the positron and
photon energy scale factors. Dashed lines are parabolic fit to synthetic data shown in solid
line.
Table 4.20: Energy scales for synthetic data.
Variable
Value
e Energy Scale
0.9996±0.4188·10−4
γ Energy Scale
1.000±0.4693·10−4
words, the synthetic data analysis proves that the results in all kinematic regions are
consistent with the underlying theoretical expectations.
Yet another cross-check of the validity of our method with synthetic data is possible. In previous discussions, the size of the synthetic data is comparable with the
experimental statistics. We can investigate the behavior of the final result for a
163
CHAPTER 4: Data Analysis
larger size of the synthetic sample. To this end a total of 8×106 radiative events
were generated. Table 4.21 summarizes the number of the generated, detected and
reconstructed events for each region. The statistics is quadrupled in this exercise and
one expects the factor of two reduction of the final error since only the statistical
error is reported in this analysis.
Table 4.21: Number of events in the enlarged synthetic data analysis.
Region
Generated events Detected events Reconstructed events
A
72 815±269.8
16 307.3±155.5
72 505.2±691.5
B
229 615±479.2
38 189.8±238.0
229 548±1430.6
C
596 433±772.3
44 846.2±257.9
594 474±3418.8
The global fit minimization routine with the value of FV =0.0259 fixed provides
the following values of parameters:
γ = 0.501 ± 0.0070
a = 0.278 × 10
−6
(4.34)
± 0.198 .
The result in equation (4.34) is an excellent agreement with the theoretical values of
parameters described in equation (4.32).
164
Chapter 5
Results
5.1
Introduction
The results of the π + → e+ νγ analysis are discussed in this chapter. The outcome
of the two-parameter global fit to the year 2004 data set are demonstrated. Statistical and systematical parameter uncertainties extracted from the analysis are further
summarized.
5.2
Global fit results
The global fit procedure is described in section 4.4.10. It simultaneously minimizes
the error-weighted difference between the experimental and theoretical branching ratios, allowing the parameter γ=FA /FV to vary freely. Section 4.4.12 describes the
165
CHAPTER 5: Results
need for one extra parameter in the fit, namely a–lepton pair momentum transfer
dependence of the form factors FA and FV ,equation (4.31)). The MINUIT minimization routine that varies those two parameters and recalculates both the theoretical
branching ratios and the acceptances for each kinematic region yields the following
values:
γ = 0.463 ± 0.0208 ,
(5.1)
a = 0.269 ± 0.185 .
(5.2)
In all of the subsequent formulae, unless specified otherwise, the total error represents
the systematical and statistical errors combined in quadrature. Along with this traditional analysis of three kinematic regions, two additional studies have been made.
They are described in section 5.4. For all three versions of the analysis the minimization is performed with freely varying parameters a and γ. As shown in the synthetic
data analysis, section 4.4.13, the value of the parameter a is poorly determined from
our data. Therefore, all three analyzes are utilized to determine the value of parameter a independently and then an error weighted average of these three values is taken
and fixed in all the subsequent minimization routines. The three values of parameter
a are as follows:
a = 0.269 ± 0.185 ,
a = 0.186 ± 0.221 ,
a = 0.253 ± 0.116 ,
(5.3)
166
CHAPTER 5: Results
which correspond to the 3-bin-analysis, the extended regions analysis and the 8-binanalysis respectively. The weighted average is:
a = 0.241 ± 0.093
(5.4)
From this point on, the value of a remains fixed as in equation (5.4) unless specified
otherwise.
Minimizing the χ2 for 3 bins with respect to the parameter γ alone and keeping
fixed a=0.241 and FV =0.0259 gives the following result:
γ = 0.466 ± 0.0160
(5.5)
The results for the theoretical and experimental branching ratios for the best parameters values can be seen in table 5.1.
Table 5.1: Theoretical and experimental BR for the best value of γ.
Region
Theoretical BR(×108 ) Experimental BR(×108 )
A
2.635 ± 0.000522
2.657 ± 0.0576
B
14.48 ± 0.00538
14.29 ± 0.237
C
37.88 ± 0.0283
37.98 ± 0.909
The uncertainties of the theoretical BR come from the Monte Carlo integration
of 30 million events.
Figure 5.1 depicts the χ2 distribution as a function of parameter γ. As discussed
in chapter 2, equation BRtheory (γ)=BRexp (γ), solution of which is the goal of this
167
CHAPTER 5: Results
Γπ→eνγ MINIMIZATION: REGIONS A+B+C
χ2(A+B+C)
10 4
γMIN= 0.47
10 3
10 2
10
1
10
-1
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
γ=FA/FV
Figure 5.1: Total χ2 over three π + → e+ νγ regions as a function of γ. There are two
distinct minima which correspond to two solutions to the problem.
analysis, is quadratic in γ and leads to a dual solution. Figure 5.1 clearly shows that
positive solution is preferred over the negative one with the probability of ∼ 1:600.
Figure 5.2 shows the graphical solution of the BRtheory (γ)=BRexper (γ) equation
with two solutions as intersects of the theoretical and experimental curves. It is
shown for Region A only to demonstrate that individual region’s solutions lie within
the uncertainties of the combined solution.
Since we analyze only three kinematic regions, our χ2 sum has three terms. Therefore, the maximum number of free parameters of the fit is two. In the beginning of
this section, the result is quoted for the case when parameters γ and a are freely
varied. In fact, the dependence on a is weak since only one term (Region C) has any
168
CHAPTER 5: Results
8
FV = 0.0259 (CVC)
REGION A ONLY
7
Γπ→eνγ x108
6
γMIN= 0.47
5
4
THE
3
EXP
2
1
0
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
γ=FA/FV
Figure 5.2: Graphical solution to BRthe (γ)=BRexp (γ).
significant correlation with the LPIM. We evaluated the MINUIT program fit with γ
and FV as free parameters. The result was:
γ = 0.711 ± 0.489 ,
(5.6)
FV = 0.0223 ± 0.00625 ,
(5.7)
which is in excellent agreement with the CVC predictions for the value of F V = 0.0259.
The result for the parameter γ in this case remains consistent with the previous value,
equation (5.5).
169
CHAPTER 5: Results
5.3
Compilation of uncertainties
This section summarizes the sources of uncertainties in our analysis. There are two
separate components, namely, statistical and systematic [7, 45] uncertainties. All of
these individual uncertainties are supplied to the MINUIT routine for an adequate
estimate of the precision of the final result. For the sake of convenience, the main
formulae used in this analysis are presented below. Experimental branching ratio for
each region is calculated as follows:
BRexp ≡ ΓRPD = Γπe2 ·
NRPD · Aπe2
,
Nπe2 · ARPD
(5.8)
and minimized χ2 function is constructed as follows:
χ2 (γ) =
X
i=RA ,RB ,RC
³
BRitheor (γ) − BRiexp (γ)
σi2
´2
,
(5.9)
where σi is a combined error for each region.
5.3.1
Statistical uncertainties
Statistical uncertainties are summarized in table 5.2.
In order to describe the meaning of the quoted errors the following comments
should be made:
• NRPD uncertainty is not simply the square root of the number of detected
events. As described in section 4.4.3, the detected number of events is obtained
by a subtraction of the out-of-time background from the in-time signal. Hence,
170
CHAPTER 5: Results
Table 5.2: Statistical uncertainties.
Name
NRPD
Nπe2
ARPD
Aπe2
BRtheor
Region
Value (%)
A
1.94
B
1.53
C
1.16
all
0.112
A
2.20·10−4
B
3.27·10−4
C
3.18·10−4
all
0.156
A
0.0198
B
0.0371
C
0.0747
the uncertainties are propagated to the difference of the in-time and out-of-time
number of events and, therefore, they are slightly larger than the simple square
root.
• ARPD and Aπe2 are described in sections 4.4.9 and 4.3.10 respectively. For the
latter one, the statistical uncertainty is simply the propagated error of the ratio
of two numbers: number of events that passed the cuts and the overall number
of generated events. The ARPD uncertainty is complicated by the fact that the
acceptance is calculated as the ratio of sums of weights. The uncertainties of the
171
CHAPTER 5: Results
numerator and the denominator are calculated separately using the following
formula [4]:
q
σfi = f i ·
hfi2 i − hfi i2 ,
(5.10)
where fi is the sum of all the corresponding weights and the operator h i gives
the average value of the argument. The error is then propagated to the value
of the ratio of two sums according to the following formula:
σARPD
f1
=
·
f2
v
!
uÃ
u σf 2
1
t
f1
Ã
σf2
+
f2
!2
(5.11)
• BRtheor are calculated by the Monte Carlo integration method [34, 54, 4]. Part
of the BR calculated in Born’s approximation is cross-checked by analytical
integration. The entire integrand contains radiative corrections as well, and
is performed using the MC method alone. The convergence of the method is
proportional to
√
N , where N is the number of the generated points in each
region. The theoretical BR are calculated with N =30 million thrown events
and the uncertainty is estimated using the formula:
σI = V ·
s
1
(hI 2 i − hIi2 ) ,
N
(5.12)
where I is the total sum of the weights for a given kinematic region, V is
the volume of the integrated phase space region and N is the number of the
generated events.
CHAPTER 5: Results
5.3.2
172
Systematic uncertainties
Systematic errors are summarized in table 5.3. The description of the last column
is given for each parameter individually. In most cases it represents the percentage
change in the total yield of the events of the given type, e.g., the change of the
energy scale by one standard deviation results in the reported change of the yield in
Region B.
The following abbreviations are used:
• T RRPD is the value of the simulated trigger energy threshold. In the experiment it was set by a common voltage on the CsI cluster summing unit. The
value of the trigger energy threshold in the simulation is adjusted using the
shape of high-energy particle spectra in all kinematic regions. The value of the
energy threshold is varied and the effect on the final value of the acceptance for
a given kinematic region is recorded in the last column of the table.
CsI
• ESRPD
stands for the CsI energy scale of the positron and photon. As de-
scribed in section 4.4.5, different methods of determining the absolute energy
calibration provide slightly different results. Overall accuracy of 0.2 % for the
positron scale and 0.1 % for the photon scale were deduced. Both energy scales
are varied by this amount and the resulting change in the total yield is given in
the last column of the table.
trgt
PV
• ESRPD
and ESRPD
are the energy scales of the PV staves and the target
173
CHAPTER 5: Results
Table 5.3: Systematic uncertainties.
Name
Region
Value (%)
A
0.612
B
0.240
C
1.86
A
0.626
B
0.444
C
0.719
A
0.132
B
0.120
C
0.123
A
0.184
B
0.180
C
0.538
A
0.0341
B
0.0533
C
0.0748
A
0.191
B
0.0614
C
0.0863
τπe2
all
0.0760
SFπe2
all
0.113
Acor
πe2
all
0.224
T RRPD
CsI
ESRPD
PV
ESRPD
trgt
ESRPD
τRPD
πβ
CHAPTER 5: Results
174
respectively. They are adjusted in the GEANT simulation. The coefficients are
varied by one standard deviation, i.e., 2 % and their effects on the acceptances
are recorded.
• τRPD and τπe2 are the corrections to the RPD and πe2 yields due to the limited
accuracy in the zero time offset determination as described in section 4.3.1.
The overall accuracy is ∼20 ps so the time offsets are varied by this amount and
the resulting total number of events recalculated. The boundaries for the gate
fraction are, obviously, kept fixed.
• πβ accounts for the effect of the π + → π 0 e+ ν background subtraction described in section 4.4.4. The overall accuracy for the π + → π 0 e+ ν events
reconstruction is found to be ∼5 %. The effect on the number of events that
can imitate the signal is summarized in the table.
• SFπe2 correction arises from the fact that the subtraction factor described in
section 4.3.3 is determined with a limited accuracy. The change in the πe2
yield due to a one standard deviation change in the SF is indicated in the last
column. This correction is only relevant if the ADC method of extracting the
number of events is used in the calculations.
• Acor
πe2 is the accuracy of the radiative corrections to the π e2 decay described in
section 4.4.11.
CHAPTER 5: Results
5.4
5.4.1
175
Stability of the results
Introduction
One of the most important questions one could ask is how stable the final result
is against the variations of the cuts imposed on the data sample. If all the errors
including the systematic ones were properly accounted for, the final result should
remain within the quoted error bars, if the cuts are varied.
5.4.2
π + → e+ ν energy cut
In the analysis of π + → e+ ν events the most important cut is applied to the minimum
energy of a positron. It is chosen to be at 50 MeV. Table 5.4 shows the absolute BR
for π + → e+ ν reaction as defined in equation (4.2) for different values of the energy
cuts. The results are shown for the largest subset 1 of the year 2004 data. Naturally,
the acceptances and total number of events are recalculated for every new value of the
cut. The rest of the parameters do not depend on the energy threshold and remain
unchanged. We note that despite a reduction of the detected number of events for
the higher energy cuts, the overall uncertainty becomes slightly smaller due to the
better quality of the time spectra fits and weaker dependence of the acceptance on
the radiative corrections.
The maximum discrepancy between the results (∼0.3 %) is easily explained by
the accuracy of the radiative corrections to the πe2 acceptance and the statistical
176
CHAPTER 5: Results
Table 5.4: π + → e+ ν BR for different energy cuts.
Energy cut (MeV)
Total yield
Acceptance
Γπe2 ·10−4
50
726 259 ± 1 037.86
0.6606 ± 0.001481
1.245 ± 0.010
51
722 652 ± 1 035.28
0.6584 ± 0.001331
1.243 ± 0.0099
52
717 425 ± 1 031.53
0.6548 ± 0.001347
1.241 ± 0.0099
55
690 157 ± 1 011.74
0.6292 ± 0.001260
1.243 ± 0.0099
uncertainty of the detected number of events. This means that the ratio of Nπe2 /Aπe2
used in the final calculations of this work remains within the projected error margins.
5.4.3
π + lifetime measurements from the π + → e+ νγ data
A good measure of the presence of contaminants (and thus the validity of the cuts) in
our sample is the extracted pion lifetime value in each kinematic region. The presence
of the copious Michel background causing random coincidences results in a longer
apparent pion lifetime. Figure 5.3 shows the background-subtracted distribution of
the times of the events with exponential fits shown as solid curves. The values of the
lifetime are to be compared with τπ =(26.033 ± 0.005) ns [31].
177
CHAPTER 5: Results
Number of events
250
τπ=26.29 ± 0.75 ns
200
150
100
50
0
0
25
50
75
100
125
150
tCsI - tπgate (ns)
600
τπ=25.91 ± 0.67 ns
Number of events
500
400
300
200
100
0
0
25
50
75
100
125
150
tCsI - tπgate (ns)
τπ=26.4 ± 0.47 ns
Number of events
600
400
200
0
0
25
50
75
100
125
150
tCsI - tπgate (ns)
Figure 5.3: π + lifetime from the π + → e+ νγ events in all three kinematic regions. Solid
curves represent exponential fits.
CHAPTER 5: Results
5.4.4
178
Two subsets of π + → e+ νγ data
An obvious check of the consistency of the results is a comparison of the two subsets
of the year 2004 data (see section 4.3.3). In the final analysis, the two sets are
merged, but individual results should, however, be consistent.
The default analysis, with the values of parameters a=0.241 and FV =0.0259 fixed,
produces the following values of parameter γ:
γset1 = 0.459 ± 0.0121 and γset2 = 0.461 ± 0.0146 .
(5.13)
In order to produce these results, only statistical uncertainties are taken into account,
assuming that systematic effects for two subsets are equivalent. The inclusion of
the systematic errors shifts the values of the parameter γ by as much as 2%, but
subsequently increases the uncertainty of the final results as follows:
γset1 = 0.450 ± 0.0158 and γset2 = 0.471 ± 0.0184 .
(5.14)
The results agree with each other and with the value found in the combined analysis.
The larger error of the second result comes from the larger statistical uncertainty in
the second subset. Figure 5.4 shows experimental and theoretical branching ratios
as functions of γ for two subsets for Region A, which is the most sensitive of the three
regions to the value of γ. The two solutions are nearly indistinguishable from one
another.
This assures that individual samples are unbiased and that the parameters calculated for the individual sets (energy scales, prompt cuts, TDC inefficiency etc.) are
179
CHAPTER 5: Results
Branching Ratio x108
5
Region A
4
γMIN=0.47
3
EXP
2
THE
1
0
0.2
0.4
0.6
0.8
1
γ=FA/FV
Branching Ratio x108
5
Region A
4
γMIN=0.48
3
EXP
2
THE
1
0
0.2
0.4
0.6
0.8
1
γ=FA/FV
Figure 5.4: Graphical solution for γ in Region A for two subsets of the data: subset 1 (top)
and subset 2 (bottom).
calculated correctly.
5.4.5
Extension of Regions B and C in π + → e+ νγ
The experimental definition of the kinematic regions described in section 4.4.2 can
be varied as well. The following extended versions of the Regions B and C are
180
CHAPTER 5: Results
implemented (table 5.5) with Region A unchanged.
Table 5.5: Definition of the extended experimental kinematic regions.
Region
Experimental kinematic range
A
Eγ and Ee+ > 51.7 MeV, θeγ > 40◦
Bext
Eγ > 55.6 MeV, Ee+ > 15 MeV, θeγ > 50◦
Cext
Eγ > 15 MeV, Ee+ > 55.6 MeV, θeγ > 40◦
A few select experimental distributions with superimposed simulation are shown
in figure 5.5 while total yields are summarized in table 5.6.
Table 5.6: Number of π + → e+ νγ events in extended kinematic regions.
Region
Number of events
A
4 404.8
Bext
13 193.1
Cext
16 836.3
With the values of parameters a=0.241 and FV =0.0259 fixed, the global fit procedure yields the result below (equation (5.15)). Agreement with the previous result
(equation (5.1)) is very satisfactory.
γ = 0.470 ± 0.0150 .
(5.15)
To compare these results with the values of the BR in table 5.1 , BR for all three
181
CHAPTER 5: Results
Number of events
400
•
300
Simulation
Data
200
100
0
120
140
160
180
Opening angle Θ Region A (dgr)
Number of events
1200
Simulation
Data
•
1000
800
600
400
200
0
0
0.2
0.4
0.6
0.8
1
1.2
λ=(2Ee/mπ)sin2(Θeγ/2)
Number of events
1200
Simulation
Data
•
1000
800
600
400
200
0
20
40
60
80
Eγ Region C (MeV)
Figure 5.5: Positron-photon opening angle (top), photon energy (middle) and λ1 (bottom)
distributions in Regions A, B and C respectively for extended versions of the analysis.
182
CHAPTER 5: Results
Table 5.7: Theoretical and experimental BR for the best value of γ for extended kinematic
regions.
Region
Theoretical BR(×108 ) Experimental BR(×108 )
A
2.647 ± 0.000525
2.657 ± 0.0568
Bext
14.50 ± 0.00538
14.47 ± 0.235
Cext
37.92 ± 0.0283
37.69 ± 0.979
extended kinematic regions for the value of parameter γ as in equation (5.15) are
presented in table 5.7.
For the case when both parameters FV and γ are varied, minimization yields:
γ = 0.492 ± 0.0911 ,
(5.16)
FV = 0.0255 ± 0.00166 .
(5.17)
Once again, the change in the parameter γ is well within one standard deviation
of the result in equation (5.15) and the value of parameter FV is in an excellent
agreement with the CVC model prediction.
5.4.6
Subdivision of the experimentally accessible phase space
into more bins
Yet another cross-check of the results can be obtained by dividing the available phase
space of the final state positron and photon into the larger number of bins. One
183
CHAPTER 5: Results
obvious advantage of such an approach is the fact that the χ2 functions constructed
as described in equation 4.23 will have more terms in it (depending on the binning).
The minimization of such a function is no longer limited to two free parameters.
The downside of this method is that each region contains fewer events and thus the
statistical error in each region becomes higher. Having these two orthogonal trends in
mind, the entire phase space was divided into 8 (eight) independent regions as shown
in figure 5.6. Exact definition of the regions is described in table 5.8.
Table 5.8: Definition of the multiple kinematic regions.
Region
Experimental kinematic range
1
0.2149< x <0.4, y >0.7967, θeγ > 40◦
2
0.4< x <0.6, y >0.7967, θeγ > 40◦
3
0.6< x <0.7967, y >0.7967, θeγ > 40◦
4
0.7967< x, y >0.7967, θeγ > 40◦
5
0.7967< y <0.6, x >0.7967, θeγ > 40◦
6
0.6< y <0.4, x >0.7967, θeγ > 40◦
7
0.4< y <0.2149, x >0.7967, θeγ > 40◦
8
0.7408< x, y >0.7408, θeγ > 40◦
Regions 1 through 4 are smaller subdivisions of Region C, 5 through 7 correspond
to Region B and region 8 is an exact replica of Region A as defined earlier. The
lower bound of 0.2149 on x(y) corresponds to particle’s energy of 15 MeV. The peak-
184
CHAPTER 5: Results
1
y=2(Ee+EPV+Etgt)/mπ
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
x=2(Eγ/mπ)
Figure 5.6: Definition of the 8 subregions on the y vs. x scatter plot.
to-background ratios vary from 117 in Region 4, to 6 in Region 6, and are shown
in figure 5.7. Gate fraction corrected yields in each region are given in table 5.9.
Several examples of the agreement of the MC simulation and data are shown in figure
5.8.
The default analysis with parameters a=0.241 and FV =0.0259 fixed is repeated
similarly to the previous cases. Systematic errors in each region and the πβ events
185
CHAPTER 5: Results
1
Number of events
-10
P/B=20
0
4
-10
10
2
-10
P/B=117
0
10
P/B=20
0
5
-10
10
3
-10
P/B=9
0
P/B=43
0
6
10
P/B=6
-10
0
-10
P/B=11
0
10
8
-10
P/B=49
0
t(e)-t(γ) (ns)
10
2
10
3
8
y=2(Ee)/mπ
1
7
10
4
5
6
7
0.5
1
x=2(Eγ/mπ)
Figure 5.7: Signal histograms for 8 kinematic regions. Last panel reproduces the definitions
of the regions in x − y variables.
186
CHAPTER 5: Results
3
2
1
400
300
200
400
400
200
200
100
Number of events
0
0
0.5
4
300
0
0
1
300
0.5
1
0.5
1
400
5
6
300
200
200
200
100
100
0
100
0
0.5
0.75
1
0.5
1
600
7
0
0
0.5
8
400
400
200
200
0
0
0
0.25
0.5
0.5
1
λ=(2Ee/mπ)sin (Θeγ/2)
2
Figure 5.8: Variable λ for 8 kinematic regions with simulation shown in a solid line and
dots are data.
187
CHAPTER 5: Results
Table 5.9: Total yield in the multiple kinematic regions.
Region
Number of events
1
6 103.33 ± 108.19
2
4 156.21 ± 90.116
3
4 146.99 ± 84.440
4
2 428.24 ± 58.966
5
2 315.53 ± 77.058
6
2 847.96 ± 93.122
7
5 961.76 ± 116.01
8
4 401.79 ± 85.369
subtraction for each bin are recalculated. The result is the following value of γ:
γ = 0.468 ± 0.0123 .
(5.18)
Just as in the case of three regions, the experimental branching ratios are calculated
for the best value of the parameters. Obviously in this analysis 4 values of branching
ratios for the theoretical Region C are obtained. Analogously 3 for Region B and
1 for Region A. In case of multiple definitions, the weighted average is taken with
weight provided by the total number of detected events in a region.
One can clearly see that the overall accuracy of this method is better than that
of the 3-bin analysis. The uncertainties of all the sought parameters are smaller or
188
CHAPTER 5: Results
Table 5.10: Theoretical and experimental BR for the best value of γ for multiple kinematic
regions.
Region
Theoretical BR(×108 ) Experimental BR(×108 )
A
2.641 ± 0.000523
2.655 ± 0.0582
B
14.49 ± 0.00538
14.59 ± 0.260
C
37.90 ± 0.0283
37.95 ± 0.600
comparable with the previous results and the agreement with the theory is more
satisfactory.
In case when both parameters FV and γ are varied, minimization yields:
γ = 0.401 ± 0.0946 ,
(5.19)
FV = 0.0271 ± 0.00177 .
(5.20)
Unlike the previous analysis, in this particular case the global fit can be performed
with all three parameters (FV , γ and a) varied and MINUIT minimization routine
returns the following values:
γ = 0.399 ± 0.0952 ,
FV = 0.0271 ± 0.00178 ,
(5.21)
a = 0.253 ± 0.115 .
Allowing more parameters to vary freely reduces the number of degrees of freedom of
189
CHAPTER 5: Results
the fit and leads to increased uncertainties as seen by comparing results in equation
(5.18) and equation (5.21). The results are, however, match very well within provided
uncertainties.
Combining the best available values for each parameter in question we conclude
the following results:
γ = 0.468 ± 0.0118 with FV = 0.0259 (fixed) ,
FV = 0.0262 ± 0.00154 and γ = 0.450 ± 0.0815 (both varied) ,
a = 0.241 ± 0.093 .
(5.22)
(5.23)
(5.24)
where the result in equations (5.22) is the average of the extended regions and
multiple bins results (equations (5.15 and (5.18)). Results in equation (5.23) are
the averaged results of the extended regions and multiple bins data presented in
equations (5.16), (5.17) and (5.19), (5.20). Equation (5.24) represents the error
weighted average of all three analyses.
Several physical parameters can be computed from the given value of the form factors.
One of them is pion electric polarizability αE which is defined as follows:
→
−
→
−
p = αE · E ,
(5.25)
→
−
−
where →
p is the pion electric dipole moment and E is an external electric filed.
Parameter αE is related to the form factors FA and FV by the following expression:
αE =
2α
8π 2 m
2
π fπ
·
FA
= (2.81 ± 0.07) × 10−4 fm3 .
FV
(5.26)
190
CHAPTER 5: Results
Two basic parameters l9 and l10 of the first order χPT are fixed by the following
relation:
(l9 + l10 ) =
5.5
1 FA
= (1.42 ± 0.04) × 10−3
32π 2 FV
(5.27)
Comparison with the previous experimental
and theoretical results.
To the author’s best knowledge there have been 8 experimental attempts to measure
the value of γ = FA /FV [22, 60, 5, 51, 24, 23, 9, 26] prior to this work. In all of these
published results, the value of the parameter FV was fixed by the CVC hypothesis
and the best value of the π 0 lifetime available at the time was used. This leads to a
slight incompatibility of the results, since only two papers reported an independently
measured value of FV [24, 9]. Figure 5.9 shows the re-scaled results (where possible)
with the value of FV =0.0259. Table 5.11 contains additional information about the
originally published values as well as the re-scaled results. The method used for
re-scaling the value of γ is described elsewhere in this section.
The two results preceding this work (references [9, 26]) draw a particular attention
due to the comparable uncertainties of the final results. Both of these papers reported
a deficit of events in certain kinematic regions which subsequently led to the lower
values of γ. Thus, reference [26] states that without the anomalous region, the value
of γ = 0.480 ± 0.016, which is in excellent agreement with the current result.
191
CHAPTER 5: Results
THIS WORK (2005)
FV=0.0259
FV=0.0253
①
FV=0.0255
POCANIC (2004)
②
BOLOTOV (1990)
DOMINGUEZ ② (1988)
EGLI ① (1986)
PIILONEN (1986)
BAY (1986)
STETZ (1978)
DEPOMMIER (1963)
-0.5 -0.25
0
0.25
0.5
0.75
γ=FA/FV
1
1.25
Figure 5.9: Comparison of the value of γ for different experiments. The central values
are re-scaled with FV =0.0259 except the marked entries which were not π + → e+ νγ
experiments.
192
CHAPTER 5: Results
The experimental data on FV summarized in the table 5.12 are sparse. This work
presents a nearly sixfold improvement in the determination of FV .
Theoretical predictions for γ are far more numerous. Since a LAMPF experiment
[51] ruled out negative values of γ, it significantly reduced the number of possible
theoretical interpretations. Some results are summarized in table 5.13. The majority of models predict FA and FV independently so the ratio was found by keeping
FV =0.0259.
The theoretical determination of the vector form factor FV is dominated by the
Table 5.11: Comparison of the values of γ for different experiments.
Reference Original FV
Original γ
Re-scaled result
Comments
[22]
0.0245
0.4 ± 0.2
0.36 ± 0.21
stopped π + → e+ νγ
[60]
0.0261
0.44 ± 0.12
0.45 ± 0.12
stopped π + → e+ νγ
[5]
0.0255
0.52 ± 0.06
0.50 ± 0.06
stopped π + → e+ νγ
[51]
0.0259
0.25 ± 0.12
0.25 ± 0.12
stopped π + → e+ νγ
[24]
0.0255
0.7 ± 0.5
insufficient inf.
π + → e+ νe+ e−
[23]
0.0253
0.52 ± 0.06
insufficient inf.
τ → ντ + nπ
[9]
0.0259
0.41 ± 0.23
0.41 ± 0.23
in flight π − → e− νγ
[26]
0.0259
0.443 ± 0.015
0.443 ± 0.015
stopped π + → e+ νγ
this work
0.0259
0.468 ± 0.0118
0.468 ± 0.0118
stopped π + → e+ νγ
193
CHAPTER 5: Results
Table 5.12: Comparison of the value of FV for different experiments.
Reference
[24]
FV
0.023
+ 0.015
− 0.013
[9]
0.014 ± 0.009
this work
0.0262 ± 0.00154
Table 5.13: Comparison of the value of γ for various theoretical models.
Reference
γ
Comments
[58]
0.97
Soft pion w/ updated mρ ,fπ ,hr2 iπ
[33]
0.46
χPT
[8]
0.45
χPT
[30]
0.39 – 0.43
χPT
this work
0.468 ± 0.0118
experiment
CVC theory that relates the vector part of π + → e+ νγ to the π 0 → γγ process
by making a rotation in isospin space [62]. The value of FV is then fixed by the
π 0 → γγ decay rate. For the value of τπ0 = (8.4 ± 0.6) × 10−17 s (reference [31]),
equation (2.42) gives the result quoted in table 5.14
We keep the value of FV =0.0259 throughout this analysis for the direct compatibility with the previously published experimental measurements. An accurate
re-scaling of the results can be done using the following technique. If one neglects
194
CHAPTER 5: Results
Table 5.14: Comparison of the value of FV for various theoretical models.
Reference
FV
Comments
[62]
0.0263 ± 0.0009
CVC
[30]
0.0270-0.273
χPT
this work
0.0262 ± 0.00154
experiment
the interference term between the IB and SD contributions to the π + → e+ νγ rate
(which constitutes less than 1% of the total rate), the integrated BR is written as:
Γ(γ) = a + b+ (1 + γ)2 + b− (1 − γ)2 ,
(5.28)
where
a=
Z
d2 ΓIB
dxdy ,
dxdy
(5.29)
b± =
Z
d2 ΓSD±
dxdy .
dxdy
(5.30)
and
The IB, SD ± terms are defined in equations (2.29), (2.38) and (2.38) respectively.
The integration should be performed over the observed experimental regions with the
resolution of the counters folded in it. Then the value of the sum in equation (5.28) is
nothing but the total number of observed events Ntotal in the experiment normalized
to the number of incoming pions. The entire analysis is merely the solution of a
quadratic equation in γ:
c + b+ (1 + γ)2 + b− (1 − γ)2 = 0 ,
195
CHAPTER 5: Results
c ≡ (a − Ntotal ) ,
(5.31)
which in general yields 2 independent roots γ1 and γ2 . Re-scaling to the new value of
0
FV would require the solution of another quadratic equation:
c + α · b+ (1 + γ)2 + α · b− (1 − γ)2 = 0 ,
α=
0
Ã
0
FV
FV
!2
,
(5.32)
0
with respective roots γ1 and γ2 .
Using a well known theorem of the roots of the quadratic equation one can easily
relate two sets of solutions as follows:


0
0


 γ1 + γ 2 = γ 1 + γ 2



 γ0 · γ0 = 1 +
1
2
1
α
.
(5.33)
(γ1 · γ2 − 1)
The solution of equation (5.33) is readily obtained providing us with the following
relation:
0

γ1,2 = 0.5 · −(γ1 + γ2 ) ±
s
(γ1 + γ2 )2 −

4
· (γ1 · γ2 − 1 + α)  ,
α
(5.34)
which relates new and old solutions without knowing any details of the analysis as
long as two original solutions and the value of FV are available.
196
CHAPTER 5: Results
5.6
Summary of main results.
From the combined analyses of this work we conclude the following values of the
physical parameters:
FA = 0.0118 ± 0.0003
FV = 0.0262 ± 0.0015
a = 0.241 ± 0.093
(5.35)
197
Appendix A
Alignment of wire chambers
Decoding of the coordinates of the hits from the raw wire chamber data stream is a
separate topic in itself and is discussed elsewhere [37, 38]. There are, however, two
parameters in our analyzer that must be determined before analyzing the entire data
set. They are analogous to the azimuthal offset for the plastic veto system discussed
in section 3.2.3. and must be fixed in the analyzer. The parameters compensate
for a possible rotation of the cylinder containing strips with respect to the cylinder
containing wires within a single wire chamber.
In the analyzer routine mwpc.c histograms id=55010 and 55011 are filled with
the values of azimuthal offset (δφ) between the strips and the wires as a function
of the azimuthal angle of the cluster. Two parameters called phi corr in 1 and
phi corr in 2 must be set to the value of the mean of the y projections of these two
histograms respectively.
198
APPENDIX: A
9000
12000
8000
10000
7000
6000
8000
5000
6000
4000
3000
4000
2000
2000
1000
0
-2
-1
0
1
2
δφ between the strips and the wires
δφ for chamber 1 (degrees)
0
-2
-1
0
1
2
δφ for chamber 2 (degrees)
4
3
2
1
0
-1
-2
-3
-4
0
50
100
150
200
250
300
350
φ of the cluster (degrees)
Figure A.1: Two dimensional histogram of the δφ difference between the strips and the
wires as a function of the cluster azimuthal angle (bottom) and corresponding projections
with a Gaussian fit (top).
In this work, the data set of 2004 required the values of phi corr in 1=0.03115 ◦
and phi corr in 2=0.8464◦ . As shown in Figure A.1, top panel represents the fits
to the corrected y-projections of the histograms. They are nicely centered around the
origin.
199
Appendix B
On the question of “shape” fits
Following publications by the ISTRA and PIBETA collaborations (references [9,
26]) several authors (e.g., reference [18]) pointed out that shapes of distributions
of kinematic variables observed in RPD also depend on the value of the weak form
factors. In particular, the variable λ given in equations (B.1), (B.2) drew attention.
λ1 ≡ y · sin2 (θeγ /2) ,
(B.1)
λ2 ≡ (x + y − 1)/x ,
(B.2)
where
x≡
2Eγ
2Ee
and y ≡
.
mπ
mπ
The two definitions are equivalent in the pion’s rest frame:
λ1 = λ 2 ,
(B.3)
200
APPENDIX: B
therefore, in the PIBETA experiment it is possible to measure the value of the pion
form factors solely based on the distribution of, say, λ1 , without having additional
uncertainties introduced by either absolute normalization to the number of stopped
pions or relative normalization to the number of π + → e+ ν decays. The results of
such an analysis are summarized in this appendix.
Background-subtracted (including π + → π 0 e+ ν events) distributions of the variable λ1 were fit with our GEANT simulated distributions. Both experimental and
simulated spectra were normalized to the same number of events independent of the
value of the fit parameters. In the simplest scheme only one parameter γ is allowed to
vary and the rest of the parameters are fixed such that FV =0.0259 and a=0.241. Only
statistical uncertainties of the input distribution taken into account in the analysis.
Each kinematic region is considered independently and the fits yield the following
results:
γ = 0.438 ± 0.173 ,
γ = 0.494 ± 0.0266 ,
(B.4)
γ = 0.480 ± 0.0348 ,
for Region A, B and C, respectively. The error weighted average of all three results
gives us the following value of parameter γ:
γ = 0.488 ± 0.0210 ,
(B.5)
which is in good agreement with the main result of this work. Figure B.1 shows
APPENDIX: B
201
all three distributions for the best value of the fit parameter with the fit curves
superimposed.
Several conclusions can be drawn from this exercise. Qualitatively, this method
works extremely well. It is easier to implement compared to the global fit procedure.
Quantitatively, however, this method lacks the precision of the global fit. Even using
only the statistical uncertainties, shape fits provide at least a factor of 2 larger errors
than the absolute normalization results. Secondly, this method is extremely sensitive
to the πβ background subtraction. In the case of Region A , for example, the total
πβ background yield is on the order of 4% and it is a good estimate of the largest
deviation in the value of γ were πβ events not subtracted. In case of the shape fits
the extracted value of γ = 0.957 ± 1.55 without πβ subtraction. This puts an undue
emphasis on the quality of the π + → π 0 e+ ν simulation in the region most sensitive
to the pion structure.
202
APPENDIX: B
Arbitrary normalization
0.1
•
0.08
Simulation
Data
0.06
0.04
0.02
0
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
λ=(2Ee/mπ)sin2(Θeγ/2)
Arbitrary normalization
0.07
0.06
•
Simulation
Data
0.05
0.04
0.03
0.02
0.01
0
0
0.2
0.4
0.6
0.8
1
1.2
λ=(2Ee/mπ)sin2(Θeγ/2)
Arbitrary normalization
0.1
0.08
•
Simulation
Data
0.06
0.04
0.02
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
λ=(2Ee/mπ)sin2(Θeγ/2)
Figure B.1: λ1 distribution for the shape fit shown in solid for Region A (top), B (middle)
and C (bottom).
203
List of Figures
2.1
π + → e+ ν radiative corrections diagrams . . . . . . . . . . . . . . .
9
2.2
The kinematic of the π → e+ νγ decay . . . . . . . . . . . . . . . . . .
13
2.3
Feynman diagrams of π + → e+ νγ decay . . . . . . . . . . . . . . . .
15
3.1
Schematic layout of the accelerator facility at PSI . . . . . . . . . . .
23
3.2
A schematic cross-section of the PIBETA apparatus . . . . . . . . . .
25
3.3
A regular pentagon shape in θ-ϕ space . . . . . . . . . . . . . . . . .
29
3.4
A graphical representation for the area calculations . . . . . . . . . .
31
3.5
The crystal maps and θ projections fits . . . . . . . . . . . . . . . . .
34
3.6
Geometric setup for calculating the displacement along z-axis . . . .
35
3.7
The z-distribution of the Dalitz events in the degrader and target . .
38
3.8
Time difference between the electron and the positron . . . . . . . . .
39
3.9
The z-distribution of the Dalitz events in the degrader . . . . . . . .
40
3.10 Azimuthal difference between the charged tracks and the PV center .
41
3.11 Definition of the π-stop signal . . . . . . . . . . . . . . . . . . . . . .
43
LIST OF FIGURES
204
3.12 The π-gate and the π-stop shown together . . . . . . . . . . . . . . .
46
3.13 Original and the prescaled π-gates shown together . . . . . . . . . . .
47
3.14 The ADC gate and the CsI analog signal within the gate . . . . . . .
49
3.15 Theoretical Michel spectrum . . . . . . . . . . . . . . . . . . . . . . .
53
3.16 Overall positron spectrum in 1-arm high trigger . . . . . . . . . . . .
54
3.17 π + → e+ ν spectra in the individual crystals of several types . . . . .
56
3.18 Overall positron spectrum after 2 iterations of the gain match routine
57
3.19 The geometric mean energy spectra of 20 PV staves . . . . . . . . . .
58
3.20 Peak positions of 40 raw PV spectra before and after HV adjustments
59
4.1
The box diagram of the π + → e+ νγ analysis . . . . . . . . . . . . .
62
4.2
The particle type discrimination between protons and minimum ionizing positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.3
The simplest case of a single π-stop hit within the π-gate . . . . . . .
71
4.4
A more complex case of two π-stop hits within the π-gate . . . . . . .
72
4.5
Alignment of the CsI times . . . . . . . . . . . . . . . . . . . . . . . .
75
4.6
The time of the prompt events and a linear fit to the peak positions .
77
4.7
Degrader ADC vs. event time for the π + → e+ ν candidates . . . . .
80
4.8
Average degrader ADC vs. run number before and after the gain match 82
4.9
Graphical interpretation of theoretical probability distributions of Michel
and π + → e+ ν events . . . . . . . . . . . . . . . . . . . . . . . . . .
83
LIST OF FIGURES
205
4.10 Positron spectra for the early and late gate portions . . . . . . . . . .
84
4.11 Background subtracted π + → e+ ν positron spectrum . . . . . . . .
85
4.12 TDC spectrum of the π + → e+ ν candidates . . . . . . . . . . . . . .
88
4.13 Ratio of the TDC fit to the data . . . . . . . . . . . . . . . . . . . . .
89
4.14 TDC spectrum of the simulated Michel events for different reference
counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.15 Clean π + → e+ ν spectrum with and without pile-up events . . . . .
91
4.16 Various parameters for the absolute normalization calculations . . . .
97
4.17 Reconstructed z-vertex of the positrons exhibits two peaks: pion and
muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.18 Incident momentum of the simulated pion beam and comparison of
two GEANT simulations of the z-stopping distribution . . . . . . . . 102
4.19 Comparison of the plastic veto energy and pion’s z-stopping distributions106
4.20 Comparison of the distance between a track and the beam direction
and angle between a track and a CsI clump distributions . . . . . . . 107
4.21 Comparison of the angle between a track and the center of the PV
stave and the azimuthal angle of the tracks distributions . . . . . . . 108
4.22 Comparison of the left tale of the CSI energy deposition and the map
of the CSI crystals illumination . . . . . . . . . . . . . . . . . . . . . 109
4.23 Contributions to the total π + → e+ νγ decay rate . . . . . . . . . . . 112
206
LIST OF FIGURES
4.24 Difference between the positron and photon times for three kinematic
regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.25 Simulated positron spectrum in region B for two different values of
γ=FA /FV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.26 Energy spectrum of the high energy photon in the π + → π 0 e+ ν decay 121
4.27 Simulated ∆λ spectrum in region B for two different values of γ=FA /FV 124
4.28 Contour plot of χ2 for the ∆λ difference as a function of the positron
and photon energy scales . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.29 Michel spectrum shown against the GEANT simulation before and
after discriminator efficiency corrections . . . . . . . . . . . . . . . . 129
4.30 Uncalibrated target energy vs. event time before and after subtraction
of the π-stop energy
. . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.31 Comparison of the variable λ1 and photon energy distributions . . . . 135
4.32 Comparison of the positron energy and ∆λ distributions . . . . . . . 136
4.33 Comparison of the target energy and PV energy distributions . . . . 137
4.34 Comparison of the distributions of the opening angle θeγ . . . . . . . 138
4.35 Comparison of the ideal π + → e+ ν simulation and the effect of the
radiative photon on the line-shape of the positron energy spectrum . 140
4.36 Michel component of the π + → e+ ν fit divided into early and late
gate parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
207
LIST OF FIGURES
4.37 π + → e+ ν positron line shape with weighted sample and infrared
cutoff Eγ =100 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.38 π + → e+ ν positron line shape with unweighted sample and infrared
cutoff Eγ ∼ 100 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.39 Percentage of events in the π + → e+ ν
tail as a function of the
minimally accepted photon energy . . . . . . . . . . . . . . . . . . . . 145
4.40 Percentage of events in the π + → e+ ν
tail as a function of the
logarithm of the maximum event weight . . . . . . . . . . . . . . . . 146
4.41 Theoretical lepton pair invariant mass for three kinematic regions . . 147
4.42 ∆λ distributions for three sub-ranges of Region C . . . . . . . . . . . 149
4.43 Graphical solutions for the parameter γ for three sub-ranges of Region C150
4.44 GEANT-reconstructed lepton pair invariant mass for the πβ background152
4.45 Difference between the positron and photon times for three kinematic
regions displayed for the synthetic data . . . . . . . . . . . . . . . . . 156
4.46 Comparison of the variable λ1 and photon energy distributions . . . . 157
4.47 Comparison the positron energy and ∆λ distributions . . . . . . . . . 158
4.48 Comparison of the target energy and PV energy distributions . . . . 159
4.49 Comparison of the opening angle θeγ distributions . . . . . . . . . . . 160
4.50 Combined χ2 over all three regions as a function of γ for fixed FV . . 161
4.51 Contour plot of χ2 for the ∆λ difference as a function of the positron
and photon energy scale factors . . . . . . . . . . . . . . . . . . . . . 162
LIST OF FIGURES
208
5.1
Total χ2 over three π + → e+ νγ regions as a function of γ . . . . . . 167
5.2
Graphical solution to BRthe (γ)=BRexp (γ) . . . . . . . . . . . . . . . . 168
5.3
π + lifetime from the π + → e+ νγ events . . . . . . . . . . . . . . . . 177
5.4
Graphical solution for γ in Region A for two subsets of the data . . . 179
5.5
Positron-photon opening angle, photon energy and λ1 (bottom) distributions in Regions A, B and C respectively . . . . . . . . . . . . . . . 181
5.6
Definition of the 8 subregions on the y vs. x scatter plot . . . . . . . 184
5.7
Signal histograms for 8 kinematic regions . . . . . . . . . . . . . . . . 185
5.8
Variable λ for 8 kinematic regions . . . . . . . . . . . . . . . . . . . . 186
5.9
Comparison of the value of γ for different experiments . . . . . . . . . 191
A.1 Chamber alignment histograms . . . . . . . . . . . . . . . . . . . . . 198
B.1 λ1 distributions for the shape fit . . . . . . . . . . . . . . . . . . . . . 202
209
List of Tables
1.1
Summary of the pion and muon decays . . . . . . . . . . . . . . . . .
2
4.1
Time offsets for different run groups . . . . . . . . . . . . . . . . . . .
78
4.2
Subtraction factors and energy scales for π + → e+ ν
. . . . . . . . .
86
4.3
Total number of π + → e+ ν events . . . . . . . . . . . . . . . . . . .
92
4.4
π + → e+ ν branching ratio . . . . . . . . . . . . . . . . . . . . . . .
95
4.5
Uncertainties for the π + → e+ ν branching ratios . . . . . . . . . . .
95
4.6
Efficiencies for the π + → e+ ν branching ratio . . . . . . . . . . . . .
99
4.7
π + → e+ ν acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.8
Definition of theoretical kinematic regions . . . . . . . . . . . . . . . 111
4.9
Definition of experimental kinematic regions . . . . . . . . . . . . . . 114
4.10 Total number of π + → e+ νγ events . . . . . . . . . . . . . . . . . . 116
4.11 πβ contamination percentiles . . . . . . . . . . . . . . . . . . . . . . . 118
4.12 Number of background πβ events . . . . . . . . . . . . . . . . . . . . 118
4.13 Photon energy scales from πβ analysis . . . . . . . . . . . . . . . . . . 121
210
LIST OF TABLES
4.14 Energy scales from π + → e+ νγ analysis . . . . . . . . . . . . . . . . 126
4.15 Final energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.16 Definition of Region C subranges . . . . . . . . . . . . . . . . . . . . 148
4.17 Values of the parameter γ for different subranges of Region C . . . . 151
4.18 Total number of generated events in synthetic data set . . . . . . . . 154
4.19 Number of reconstructed events in synthetic data analysis . . . . . . 160
4.20 Energy scales for synthetic data . . . . . . . . . . . . . . . . . . . . . 162
4.21 Number of events in the enlarged synthetic data analysis . . . . . . . 163
5.1
Theoretical and experimental BR for the best value of γ . . . . . . . 166
5.2
Statistical uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.3
Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.4
π + → e+ ν BR for different energy cuts . . . . . . . . . . . . . . . . 176
5.5
Definition of the extended experimental kinematic regions
. . . . . . 180
5.6
Number of π + → e+ νγ events in extended kinematic regions . . . . 180
5.7
Theoretical and experimental BR for the best value of γ for extended
kinematic regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.8
Definition of the multiple kinematic regions
. . . . . . . . . . . . . . 183
5.9
Total yield in the multiple kinematic regions . . . . . . . . . . . . . . 187
5.10 Theoretical and experimental BR for the best value of γ for multiple
kinematic regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
LIST OF TABLES
211
5.11 Comparison of the values of γ for different experiments . . . . . . . . 192
5.12 Comparison of the value of FV for different experiments . . . . . . . . 193
5.13 Comparison of the value of γ for various theoretical models . . . . . . 193
5.14 Comparison of the value of FV for various theoretical models . . . . . 194
212
List of Notations
PSI
Paul Scherrer Institute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
QED
Quantum Electro Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
FV
weak vector form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
FA
weak axial-vector form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
CVC
conserved-vector-current hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 3
SM
Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
DAQ
Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
IB
Inner Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
SD
Structure Dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
BC
beam counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
AD
active degrader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
AC
active collimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
AT
active target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
MWPC
multi-wire proportional chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
PV
plastic veto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
LIST OF NOTATIONS
213
PMT
photomultiplier tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
CV
cosmic veto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
SCX
single charge exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
πβ
pion beta decay π + → π 0 e+ ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
rf
radio frequency of the accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . 42
π-stop
pion stop signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
π-gate
pion gate signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
TDC
time-to-digital converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
ADC
analog-to-digital converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
HV
high voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
SG
software gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
ECF
energy conversion factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Michel
µ+ → e+ νν decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
πe2
π + → e+ ν decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
GM
geometric mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
MIP
minimum ionizing particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
RPD
radiative pion decay π + → e+ νγ . . . . . . . . . . . . . . . . . . . . . . . . . . 61
DST
data summary tapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
ODB
on-line data base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
particle ID
particle identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
ET
time of an event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
LIST OF NOTATIONS
214
f sub
subtraction factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
TOF
time of flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
BR
branching ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
LPIM
lepton pair invariant mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
a
parameter of the weak form factors dependence on LPIM . 151
χPT
chiral perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
MC
Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
215
LIST OF CONSTANTS
List of Constants
Quantity
Symbol
Value
Reference
charged pion mass
mπ +
139.57018 MeV
[31]
charged pion life time
τπ +
26.033 ns
[31]
neutral pion mass
mπ 0
134.9766 MeV
[31]
neutral pion life time
τπ 0
8.4×10−17 s
[31]
positron mass
m e+
0.5109989 MeV
[31]
pion decay constant
fπ
130.7 MeV
[31]
1.2352×10−4
[48]
1.16639×10−11 (MeV)−2
[31]
π + → e+ ν BR
Fermi coupling constant
Γ
π + →e+ ν
G
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