fulltext

ACTA UNIVERSITATIS UPSALIENSIS
Uppsala Dissertations from the Faculty of Science and Technology
118
Measurement of the Dalitz Plot
Distribution for η→π+π−π0 with KLOE
Li Caldeira Balkeståhl
Dissertation presented at Uppsala University to be publicly examined in Häggsalen,
Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 22 January 2016 at 09:15 for
the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty
examiner: Professor Simon Eidelman (Budker Institute of Nuclear Physics and Physics Div.,
Novosibirsk State University).
Abstract
Caldeira Balkeståhl, L. 2015. Measurement of the Dalitz Plot Distribution for η→π+π−π0
with KLOE. Uppsala Dissertations from the Faculty of Science and Technology 118. 146 pp.
Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9427-8.
The mechanism of the isospin violating η→π+π−π0 decay is studied in a high precision
experiment using a Dalitz plot analysis. The process is sensitive to the difference between up
and down quark masses. The measurement provides an important input for the determination of
the light quark masses and for the theoretical description of the low energy strong interactions.
The measurement was carried out between 2004 and 2005 using the KLOE detector at the
DAΦNE e+e− collider located in Frascati, Italy. The data was collected at a center of mass energy
corresponding to the φ-meson peak (1019.5 MeV) with an integrated luminosity of 1.6 fb−1.
The source of the η-mesons is the radiative decay of the φ-meson: e+e−→φ→ηγ, resulting in the
world’s largest data sample of about 4.7·106 η→π+π−π0 decay events.
In this thesis, the KLOE Monte Carlo simulation and reconstruction programs are used to
optimize the background rejection cuts and to evaluate the signal efficiency. The background
contamination in the final data sample is below 1%. The data sample is used to construct the
Dalitz plot distribution in the normalized dimensionless variables X and Y. The distribution
is parametrized by determining the coefficients of the third order polynomial in the X and Y
variables (so called Dalitz plot parameters). The statistical accuracy of the extracted parameters
is two times better than any of the previous measurements. In particular the contribution of
the X2Y term is found to be different from zero with a significance of approximately 3σ. The
systematic effects are studied and found to be of the same size as the statistical uncertainty. The
contribution of the terms related to charge conjugation violation (odd powers of the X variable)
and the measured charge asymmetries are consistent with zero.
The background subtracted and acceptance corrected bin contents of the Dalitz plot
distribution are provided to facilitate direct comparison with other experiments and with
theoretical calculations.
Keywords: Hadron physics, Quark masses, Hadronic decays, Light mesons, Meson-meson
interactions
Li Caldeira Balkeståhl, Department of Physics and Astronomy, Nuclear Physics, Box 516,
Uppsala University, SE-751 20 Uppsala, Sweden.
© Li Caldeira Balkeståhl 2015
ISSN 1104-2516
ISBN 978-91-554-9427-8
urn:nbn:se:uu:diva-266871 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-266871)
To my parents, for always believing in me
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Chiral Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Ingredients for the ChPT Lagrangian . . . . . . . . . . . . . . . . . . . . . .
1.1.3 ChPT Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Quark Masses from ChPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Dalitz Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Dalitz Plot Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Dalitz Plot Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 More Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Electromagnetic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Dispersive Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Previous Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
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2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 DAΦNE Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 KLOE Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Drift Chamber (DC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Electromagnetic Calorimeter (EMC) . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 DAΦNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 KLOE-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Event Reconstruction and Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 FILFO: Background Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Event Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Analysis Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Background Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Data-MC Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Dalitz Plot and Variable Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Fit Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Phase Space Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Fit Test on MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Minimum Photon Energy Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Background Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Choice of Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.4 Track-Photon Angle Cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.5 Time-of-Flight Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.6 Opening Angle Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.7 Missing Mass Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.8 Event Classification Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.9 Summary of Systematic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Final Results for Dalitz Plot Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Charge Asymmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Acceptance Corrected Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Diagonality of the Smearing Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Acceptance Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Comparison with Smearing Matrix Method . . . . . . . . . . . . . . . . . . . . . . .
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Dalitz Plot Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Acceptance Corrected Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Charge Asymmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Introduction
Physics as a science is about understanding the world around us, everything
from the big scale of the universe to very small objects like atoms, nucleons
and elementary particles, and everything in between. The goal of physics is to
describe all these things and to predict how they react, but not necessarily to
describe all things with just one equation. Among the many fields in physics
dedicated to different aspects of the world around us, this thesis fits into the
field of subatomic physics.
Subatomic physics aims to describe nuclei, the nucleons that make up the
nuclei and also other particles. The description of nuclei is a whole sub-field
in itself, but let us focus on things smaller than this, on the particles.
The nucleons, the proton and the neutron, are examples of particles called
hadrons. In contrast to the electron, hadrons are not elementary particles, that
is, hadrons are composed of other particles. These particles are called quarks.
Electrons and quarks are, as far as physics has managed to determine, elementary particles. There are two well-established kinds of hadrons: baryons and
mesons. Baryons, such as protons and neutrons, are made up of three quarks,
while mesons are composed of quarks and antiquarks1 [1]. Some of the lightest mesons are the three pions (π + , π − , π 0 ) and the eta-meson (η), all of which
feature prominently in this thesis.
The current understanding of the elementary particles and their interactions
is expressed in the so called Standard Model (of particle physics) [2, 3]. According to this model, the known particles are grouped depending on how they
interact. The interactions correspond to three of the known forces in nature:
the strong force, the weak force, the electromagnetic force; and are mediated
by particles called gauge bosons. The forth known force in nature, gravity, acts
very weakly for the elementary particles and is not included in the Standard
Model.
An illustration of the particles in the Standard Model and their interactions is shown in figure 1. The quarks are classified according to their flavor
(the name of the quantum number used, not at all related to taste) as: up (u),
down (d), charm (c), strange (s), top (t) or bottom (b). They possess a type
of charge called color charge (also nothing to do with how the quarks look),
which means they feel the strong force. The strong interaction is mediated by
particles called gluons, and the shaded area in figure 1 surrounding the quarks
1 Antiquarks are the antiparticles of the quarks.
Antiparticles are the same as particles, except for
their charge. The most well-known example is the positron, the electron’s antiparticle, having
the same mass as the electron but positive instead of negative electric charge.
9
Figure 1. The elementary particles of the Standard Model. The interactions felt by
different particles are indicated by the lightly shaded areas. Image credit: [4].
10
and gluons indicates that these particles interact via the strong force. The particles possessing electric charge: the quarks, the electron, muon, tau and the
W bosons all feel the electromagnetic force and interact with and via photons.
All quarks and leptons feel the weak force, that is, they interact with and via
the W and Z bosons. The Higgs boson is different from the other bosons. It
does not mediate any force, but it interacts with the other elementary particles
with a strenght depending on their mass. The Higgs boson is part of the mechanism that gives mass to the other elementary particles in the Standard Model
[2, 3].
The mathematical formulation of the Standard Model describes how and
with which strength all these particles interact. It is formulated as a quantum
field theory, where the particles are described as excitations of fields in spacetime. The electromagnetic and weak forces are combined in the electroweak
theory while the strong interaction is described separately in the quantum field
theory called Quantum Chromodynamics (QCD). In the mathematical formulation of the Standard Model, some free parameters appear [2, 3]. These represent constants that are not predicted by the theory, but must be measured by experiment. There are 19 such parameters in the Standard Model, like the masses
of the quarks, the charged leptons and the Higgs, and the coupling strengths
of the interactions. To test the Standard Model and to be able to build other
theories that address some of its shortcomings, the Standard Model needs to
be quantitatively understood very well. This includes knowing the parameters
of the Standard Model with high precision.
QCD has two features that make it different from the other quantum field
theories in physics: confinement and asymptotic freedom. These two properties can be seen as two related extremes of the theory. Asymptotic freedom
means that quarks and gluons at high energies, or at small distances, interact
weakly [5, 6]. In the limit of the quarks and gluons having infinite relative momenta, they would not interact at all and would behave as free particles. On the
other extreme, confinement concerns quarks and gluons at low energies or at
large distances. The strength of the strong interaction increases as the distance
gets larger, and if it gets large enough, new quarks and antiquarks are created.
This implies that the quarks can never be separated and always appear inside
hadrons [2]. If one tried to break a hadron, say a proton, by dragging it apart,
instead of seeing the separate quarks, one would get more hadrons. These
two features illustrate that the strong interaction changes strength depending
on the energy of the particles concerned, getting weaker at high energies and
stronger at lower energies. This is referred to as the running of the strong
coupling constant.
Since the quarks are always strongly bound inside hadrons, measuring their
mass is much more complicated than for example for the electron. The masses
have to be extracted from other quantities or processes which depend on the
quark masses. Especially the masses of lightest quarks, the u and d quarks,
present a challenge and are still under investigation [7, 8, 9, 10, 11].
11
Strong CP problem and the light quark masses
One of the motivations for measuring the light quark masses precisely is the so
called strong CP problem. One proposed solution to this problem is a massless
up quark [12]. Although this is not favored by experimental evidence so far, it
increases the interest in the precise measurement of the u quark mass.
The strong CP problem is related to the CP transformation, a combination of two transformations: one obtained by exchanging particles with their
antiparticles (C, charge conjugation) and the other by mirroring the physical
system (P, parity). A transformation is said to be a symmetry if the physics
description is the same for the process and the transformed process. The process is said to obey the symmetry. C and P were first thought to be symmetries
obeyed by all particles and their interactions, until it was shown in an experiment conducted in 1956 that P symmetry was violated in weak interactions
[13]. The combined CP symmetry was then proposed as a symmetry that
would be conserved also by the weak interaction, but this was also shown not
to be true when an experiment measured C and CP violation with K mesons
[1].
With CP shown not to be a strict symmetry for the weak interaction, the
question arises of why no CP violation has been seen in the strong interaction.
In the formulation of QCD, it is possible to include a parameter that, being different from zero, would imply P and CP violation [12]. Since no experimental
evidence exists for CP violation in the strong interaction, this suggest that this
parameter should be either zero or very small (so small that the experiments
so far were not precise enough to see the CP violation). To have a parameter
equal to zero or very small just by chance is not intellectually satisfying and is
known as a fine-tuning problem. Therefore, other explanations have been proposed for the non-observation of CP violation: for example, that there exists a
new type of particle, called axion, that would make CP violation unobservable
no matter the size of the parameter; or that the up quark would be massless,
which would also make CP violation unobservable [12]. So far no axion has
been found [7].
Thesis outline
This thesis concerns the experimental measurement of the decay of the etameson to three pions, η → π + π − π 0 . This is a process sensitive to the difference between the u and d quark masses, and thus this thesis contributes to
the determination of the quark masses. This contribution is not direct, but the
results presented in this thesis, the most precise measurement of the Dalitz
plot distribution of the η → π + π − π 0 decay to date, can be used together with
theoretical calculations to constrain the masses of the light quarks.
The thesis is divided into 6 chapters as follows. Chapter 1 gives a motivation for the η → π + π − π 0 measurement, including a more detailed theoretical
12
background and previous experimental results. The Dalitz plot and Dalitz plot
distribution are also explained in this chapter. Chapter 2 presents an overview
of the accelerator facility where the experiment was conducted and of the detector used. In chapter 3, the analysis is described: event selection, reconstruction and background rejection. Chapter 4 concerns the results from this
thesis: the Dalitz plot parameters, as well as the charge asymmetries, and their
systematic uncertainties. Chapter 5 gives an alternative stating of the results,
in the form of the acceptance corrected, arbitrarily normalized Dalitz plot distribution. This alternative stating does not include systematic uncertainties,
but it can directly be used for comparison with other experiments or theoretical calculations. In chapter 6, the results are discussed, with emphasis on
the comparison with previous experiments, and a conclusion about the work
performed in this thesis is presented.
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1. Motivation for Studying η → π +π −π 0
This chapter gives the motivation for the η → π + π − π 0 Dalitz plot measurement. It starts with a short introduction to Chiral Perturbation Theory (ChPT),
and how this theory can be used to relate the decay width of η → π + π − π 0
to the quark masses. The next part introduces the Dalitz plot, the kinematic
variables used and how to calculate the boundary. Then come some theory
updates for the η → π + π − π 0 that go beyond ChPT and finally a summary of
previous experimental results.
1.1 Chiral Perturbation Theory
The introduction on chiral perturbation theory presented here follows Scherer
and Schindler’s lecture notes [14], although in a simplified and condensed
way.
QCD is the quantum field theory of the strong interaction, but due to the
running of the strong coupling constant and to the confinement of quarks at
low energies, it is impractical for use at low energies. The perturbative methods of calculating QCD processes, which are successful at high energies, cannot be applied since the strong coupling constant cannot be regarded as a small
expansion parameter. Instead, one can use Effective Field Theories (EFTs),
and the low-energy EFT of QCD is chiral perturbation theory.
In general, EFTs approximate a fundamental theory at low energies, and
simplify calculations since the full theory need not be used. The fundamental
theory needs to have one (or more) energy scales (usually denoted Λ), and the
EFT works for energies that are small compared to this scale. The physics
of the fundamental theory at higher energies is included in the constants of
the EFT, which in principle can be calculated from the full theory. The EFT
uses only degrees of freedom relevant for the energy regime in question. The
correspondence of the physical observables calculated in the EFT to the ones
from the fundamental theory is guaranteed by Weinberg’s conjecture [15]. According to this, for the correspondence to be true, one needs the most general
Lagrangian consistent with all symmetries of the fundamental theory. This
could mean a Lagrangian with infinitely many terms, which would make predictions impossible. But if one is only interested in a certain accuracy of the
EFT, i.e., the results from the EFT need only be the same as the fundamental
theory up to a certain numerical digit, then not all the terms in the general Lagrangian need be taken into account (note: the energies and momenta involved
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must also be small compared to the scale Λ). Which terms are important is determined by the relevant power counting.
Chiral perturbation theory builds on the chiral symmetry of massless quark
QCD, and the ChPT Lagrangian also obeys Lorentz invariance, charge conjugation and parity invariance. There are two variants: 2- and 3-flavor ChPT.
In the first case, the u and d quarks are considered massless and the s quark
as heavy, and the relevant degrees of freedom are the pions. In the second
case, the u, d and s quarks are considered massless and the degrees of freedom are the pions, the kaons and the eta-meson (π, K, η). ChPT can thus be
used to describe interactions between π’s, K’s and η’s, and including the weak
or electromagnetic interaction appropriately (as external fields coupling to the
ChPT degrees of freedom) also processes like π 0 → γγ or π + → μ + νμ . In
both the 2- and 3-flavor case, the quark masses are actually taken into account
as a perturbation, and lead to an explicit chiral symmetry breaking of the Lagrangian. The limit of zero quark masses is referred to as the chiral limit.
The power counting in ChPT is done in powers of energy, momenta and quark
masses. The scale of chiral symmetry breaking Λχ ∼ 1 GeV determines the
region of applicability of ChPT, but also the appearance of other particles not
included as degrees of freedom signals the breakdown of ChPT. For example
the ρ-meson, with a mass of mρ = 770 MeV ∼ Λχ indicates that ChPT will
not work at these energies. 2-flavor ChPT in general converges better than 3flavor ChPT, which is expected since the s quark is significantly heavier than
the u and d quarks [7], and thus approximating its mass to zero will require
more corrections.
1.1.1 Chiral Symmetry
The quark part of the QCD Lagrangian can be written as:
LQCD, quarks =
∑
f =u,d,s,c,b,t
/ − m f )q f
q̄ f (iD
(1.1)
where q̄ f and q f are the quark fields (with implicit color and spinor indices),
/ the gauge derivative. The gauge derivative
m f the mass of quark flavor f and D
/ = γ μ Dμ = γ μ ∂μ + igγ μ Aμ , where
includes the gluon-field matrix Aμ in D
μ
μ = 0, 1, 2 or 3, γ are the gamma matrices, g is the strong coupling constant
and repeated indices are summed over.
The quark flavors can be divided into three light quarks (u, d, s) and three
heavy quarks (c, b,t), with the light quarks all having masses smaller than
ΛQCD . Concentrating on the light quarks, define a quark flavor vector q† =
(u† , d † , s† ) and consider the projection operators:
1
PR = (I + γ 5 ) = PR† ,
2
1
PL = (I − γ 5 ) = PL†
2
(1.2)
15
where I is the identity matrix and γ 5 the fifth gamma matrix. These operators
project the quark fields into right- and left-handed fields:
qR = PR q
such that
q̄R = q†R γ 0 = (PR q)† γ 0 = q† PR γ 0 = q† γ 0 PL = q̄PL
(1.3)
qL = PL q
such that
q̄L = q̄PR
(1.4)
where the anti-commutation relation for γ 5 is used ({γ 5 , γ μ } = 0).
The light-quark Lagrangian can then be written in terms of the right- and
left-handed quarks:
/ − M)q
LQCD, light quarks = q̄(iD
/ R + q̄L iDq
/ L − q̄R MqL − q̄L MqR
= q̄R iDq
(1.5)
where M is a 3 × 3 diagonal matrix with the quark masses. As can be seen,
the right- and left-handed quarks are only coupled by the mass part of the Lagrangian. Since the quark masses are light compared to ΛQCD , they can be
approximated to zero (chiral limit). In this case, the Lagrangian is invariant
under transformations of the right- and left-handed quarks separately, according to:
qR → UR qR
qL → UL qL
(1.6)
where UR ,UL are 3×3 special unitary matrices (i.e., UR ,UL ∈ SU(3)), acting in
0
/
/
/
flavor space. So the Lagrangian LQCD,
light quarks = q̄iDq = q̄R iDqR + q̄L iDqL
is invariant under transformations of the group SU(3)R × SU(3)L . This invariance is called chiral symmetry.
The 2-flavor case of ChPT corresponds to considering only the u and d
quarks as light, i.e., in equation 1.5 only the u and d quarks are included. In
this case, in the limit where both these quark masses go to zero, the Lagrangian
is invariant under SU(2)R × SU(2)L .
1.1.2 Ingredients for the ChPT Lagrangian
Before introducing the ChPT Lagrangian, the concepts of spontaneous symmetry breaking and Goldstone bosons are needed. Spontaneous symmetry
breaking is when the ground state of a theory is not symmetric under the full
symmetry group of the Lagrangian. According to the Goldstone theorem [2],
a broken continuous symmetry, i.e., a continuous symmetry of the Lagrangian
that is not a symmetry of the ground state, gives rise to massless, spin-less
bosons called Goldstone bosons. There is one Goldstone boson for each generator of the broken symmetry, and these bosons have the same quantum numbers as the generators.
16
Even in the case of a spontaneously broken approximate symmetry of the
Lagrangian, spin-less bosons appear, but in this case they are not massless
(but usually light) and are instead called pseudo-Goldstone bosons. An approximate symmetry of the Lagrangian implies a symmetry which is explicitly broken in the Lagrangian, but only by a small parameter. For example,
the Lagrangian of equation 1.5 has an approximate chiral symmetry, since it
would have full chiral symmetry if the quark masses were zero, but these are
nonzero and small, i.e., the quark masses are the small parameters explicitly
breaking the chiral symmetry.
In the case of QCD, the broken symmetry is suggested by the low lying
hadron spectrum to be SU(3)A [3]. The symmetry group SU(3)R × SU(3)L is
equivalent to SU(3)V × SU(3)A , where transformations according to SU(3)V
imply:
qR → UqR
qL → UqL
(1.7)
where U ∈ SU(3) (i.e., the left- and right-handed quarks are transformed in
the same way); and transformations according to SU(3)A imply:
qR → UqR
qL → U † qL
(1.8)
where U ∈ SU(3).
From the symmetry of the spectrum one can infer the symmetry of the
ground state. In the hadron spectrum, one can identify octets (for mesons and
baryons) and decuplets (for baryons) consistent with SU(3)V flavor symmetry
and the assumption that mesons consist of quark and anti-quark while baryons
consist of three quarks. If the full SU(3)R × SU(3)L symmetry was realized
in the spectrum, one would expect degenerate octets (or decuplets) with opposite parity. The fact that this is not realized in the spectrum, e.g. there is
no low-lying octet of negative parity 12 -spin baryons, implies a breaking of the
full symmetry, in fact, a breaking of the SU(3)A symmetry.
The broken SU(3)A symmetry implies 8 pseudo-Goldstone bosons, which
are spin-less, nearly degenerate low-mass states. These can be identified with
the octet of light pseudo-scalar mesons: the three π’s, the four K’s and the η.
The pseudo-Goldstone bosons are the degrees of freedom used in ChPT, and
they appear in the Lagrangian in the SU(3) matrix [14]:
φ (x)
i F
0
U(x) = e
√ +
√ +⎞
π 0 + √13 η
2π
2K
√ 0⎟
⎜ √2π −
1
0
−π + √3 η
2K ⎠
with φ = ⎝
√ −
√ 0
2K
2K̄
− √23 η
(1.9)
⎛
(1.10)
17
where the bosonic fields π 0 , π + , π − , K + , K − , K 0 , K̄ 0 and η all depend on the
space-time coordinate x and have dimension of energy, and F0 is the pion decay
constant in the chiral limit (which makes the exponential dimensionless).
In order to get the most general Lagrangian, the globally chiral invariant
0
Lagrangian LQCD,
light quarks is upgraded to a locally chiral invariant one by introducing external fields vμ , aμ , s and p. These fields transform under Lorentz
transformation as vector, axial-vector, scalar and pseudo-scalar respectively.
In fact, instead of these fields, the combinations rμ = vμ + aμ , lμ = vμ − aμ ,
M = s + ip and M † = s − ip are used. The extended Lagrangian:
Lext, light quarks = q̄R iγ μ Dμ qR + q̄R γ μ rμ qR
+q̄L iγ μ Dμ qL + q̄L γ μ lμ qL
(1.11)
−q̄R M qL − q̄L M qR
†
is invariant under the local SU(3)R × SU(3)L transformation:
qR (x) → UR (x)qR (x),
q̄R (x) → q̄R (x)UR (x)† ,
qL (x) → UL (x)qL (x),
q̄L (x) → q̄L (x)UL (x)† ,
rμ (x) → UR (x)rμ (x)UR (x)† +UR (x)i(∂μ UR (x)† ),
(1.12)
lμ (x) → UL (x)lμ (x)UL (x)† +UL (x)i(∂μ UL (x)† ),
M → UR (x)M UL (x)† ,
M † → UL (x)M †UR (x)† ,
where UR (x),UL (x) ∈ SU(3) and depend on the space-time coordinate x. Note
that putting vμ = aμ = p = 0 and s = diag(mu , md , ms ) = M one recovers the
Lagrangian of equation 1.5.
The Lagrangian of the effective field theory, ChPT, will use the same external fields rμ , lμ , M and M † , with the same transformation properties under
the local SU(3)R × SU(3)L transformation, as well as the Goldstone boson
field matrix U(x), which transforms as U(x) → UR (x)U(x)UL (x)† . The definition of the chiral gauge covariant derivative of an object A, which transforms
as A(x) → UR (x)A(x)UL (x)† , is:
Dμ A = ∂μ A − irμ A + iAlμ
(1.13)
and transforms as Dμ A(x) → UR (x)(Dμ A(x))UL (x)† . The field strength tensors:
fRμν = ∂μ rν − ∂ν rμ − i[rμ , rν ],
fLμν = ∂μ lν − ∂ν lμ − i[lμ , lν ]
18
(1.14)
are also needed, and they transform as:
fRμν (x) → UR (x) fRμν (x)UR (x)† ,
(1.15)
fLμν (x) → UL (x) fLμν (x)UL (x)† .
Locally chiral invariant Lagrangians can be built out of flavor traces of products of the form AB† , where A and B transform as A(x) above. This is easily
seen using the cyclicity of traces:
Tr(AB† ) →Tr UR (x)A(x)UL (x)† (UR (x)B(x)UL (x)† )† =
Tr UR (x)A(x)UL (x)†UL (x)B(x)†UR (x)† =
(1.16)
Tr UR (x)A(x)B(x)†UR (x)† =
Tr UR (x)†UR (x)A(x)B(x)† = Tr A(x)B(x)† .
With the fields introduced, examples of entities transforming as A(x) are:
U(x), Dμ U(x), Dν Dμ U(x), M (x), fRμν (x)U(x) and U(x) fLμν (x). There is an
infinite amount of these entities, and thus an infinite amount of different invariant traces that one could construct. To decide which terms are needed, a power
counting is introduced. Let q be a small energy or momentum, of the order of
the masses of the pseudo-Goldstone bosons. Derivatives are of order O(q), so
to be consistent, the fields rμ and lμ are also considered O(q) and thus also
the gauge covariant derivative Dμ . The field strengths fRμν and fLμν are then
of O(q2 ). The boson field matrix is considered O(q0 ), while M is of O(q2 ),
since the quark masses can be related to the square of the pseudo-Goldstone
boson masses, see section 1.1.4.
1.1.3 ChPT Lagrangian
The lowest-order Lagrangian in ChPT is of O(q2 ). At O(q0 ) only constant
terms can contribute to the Lagrangian, e.g. Tr(UU † ) = 3, and these have no
information on the dynamics of the fields. There is no term at O(q), or in
fact at any O(qn ) where n is odd. The only building block with odd order is
Dμ , but since Lorentz invariance requires Lorentz indices to be contracted, the
derivatives will always appear in pairs and thus give terms of even order. The
lowest non-trivial Lagrangian is thus of O(q2 ).
The candidate hermitian structures of the Lagrangian are:
Tr (Dμ U)† Dμ U , Tr U † M + M †U and iTr U † M − M †U . (1.17)
The last structure is forbidden by parity conservation: under the parity transiφ
−i φ
formation,
M † and U = e F0 → e F0 = U †, so that
ip → s− ip =
† M =† s +
iTr U M − M U → iTr UM † − M U † = −iTr U † M − M †U . At the
considered order, charge conjugation invariance does not impose any more
19
constraints, and the Lagrangian is:
L2,ChPT =
F2
F02 Tr (Dμ U)† Dμ U + 0 · 2B0 Tr U † M + M †U
4
4
(1.18)
where F0 and B0 are the low-energy constants at this order, F0 is related to the
pion decay and B0 to the quark condensate. Any process in ChPT O(q2 ) is
calculated by tree level diagrams with vertices from L2,ChPT . Loop diagrams
appear first at O(q4 ).
One complication that appears with loop diagrams is the fact that these
diverge. In renormalizable theories, the infinities arising from the loops are
compensated with counter terms. ChPT is in general not renormalizable, but
it is renormalizable order by order, as the higher order Lagrangians contain
the counter terms for the loops of the lower order Lagrangians. For example,
one-loop diagrams from L2,ChPT are compensated by terms in L4,ChPT , by a
suitable redefinition of the low-energy constants of L4,ChPT .
With a suitable renormalization in place, the order at which a loop diagram
contributes can be understood using the following contributions to the power
counting:
• vertices from L2n,ChPT each contribute q2n , e.g. vertices from L2,ChPT
contribute q2 , vertices from L4,ChPT contribute q4 ;
• each pseudo-Goldstone boson propagator contributes q12 ;
• each independent loop contributes q4 (because it introduces a momentum integration in four dimensions).
This can be summarized in a formula for the “chiral dimension” D of an
arbitrary diagram which contributes at order qD :
∞
D = 4Nl − 2N p + ∑ 2nNv,2n
(1.19)
n=1
where Nl is the number of independent loops, N p the number of propagators
and Nv,2n the number of vertices from L2n,ChPT . In a connected diagram, the
number of loops, propagators and vertices are not independent of each other
but obey the relation Nl − N p + ∑∞
n=1 Nv,2n = 1 and with this, equation 1.19 can
be rewritten as [15]:
∞
D = 2Nl + 2 + ∑ 2(n − 1)Nv,2n .
(1.20)
n=1
From this equation it is easy to see that the lowest value for D is 2, when
there are no loops, no vertices with n > 1 and an arbitrary number of vertices
with n = 1 (i.e., from L2,ChPT ).
As an example of equation 1.20, consider a loop diagram of 2 → 2 pseudoGoldstone boson scattering, with two L2,ChPT vertices connected by two propagators, which has one independent loop, see figure 1.1. According to the
20
Figure 1.1. Feynman diagram of 2 → 2 pseudo-Goldstone boson scattering with one
independent loop. The vertices indicated by a dot are from L2,ChPT .
power counting above, this diagram has chiral dimension D = 4, i.e., it contributes at O(q4 ).
At next to leading order (NLO), i.e., at O(q4 ), both one-loop diagrams with
an arbitrary number of vertices from L2,ChPT and tree-level diagrams with one
vertex from L4,ChPT need to be taken into account. The NLO Lagrangian
L4,ChPT has 12 low-energy constants, and can be written as [16]:
2
L4,ChPT =L1 Tr (Dμ U)† Dμ U
+ L2 Tr (Dμ U)† Dν U Tr (Dμ U)† Dν U
+L3 Tr (Dμ U)† (Dμ U)(Dν U)† Dν U
+L4 Tr (Dμ U)† Dμ U Tr χ †U + χU †
2
+L5 Tr (Dμ U)† Dμ U(χ †U + χU † ) + L6 Tr χ †U + χU †
2
+L7 Tr χ †U − χU † + L8 Tr χ †U χ †U + χU † χU †
−iL9 Tr fRμν (Dμ U)(Dν U)† + fLμν (Dμ U)† Dν U
μν +L10 Tr U † fRμν U fL
μν
μν +H1 Tr fRμν fR + fLμν fL + H2 Tr χ † χ ,
(1.21)
where M is now encoded in χ = 2B0 M . Of the 12 low-energy constants,
10 (L1 , . . . , L10 ) have physical significance. The remaining 2 parameters (H1
and H2 ) relate to terms including only external fields, so they have no physical
significance, although they are needed for the renormalization of the one-loop
diagrams.
At next to next to leading order (NNLO), O(q6 ), the Lagrangian L6,ChPT
is needed. This Lagrangian has 94 low-energy constants, of which 4 concern
only external fields and have no physical significance [17].
The low-energy constants of ChPT, as for any EFT, contain the physics of
the original theory at energies not covered by the EFT. In principle, these could
be calculated from QCD, but our inability to solve QCD at low energies is one
21
of the things prompting the use of an EFT like ChPT in the first place. Nevertheless, lattice QCD1 [18] can be used to calculate the low-energy constants.
These constants can also be fixed from experimental data, i.e., some data is
used to calculate these constants, and once they are fixed, ChPT has predictive
power for other processes. At present, the accuracy of lattice QCD is for most
low-energy constants not competitive with determinations from experimetnal
data, but it can be used as a cross-check or to determine low-energy constants
that are not easily extracted from experiment. For a recent determination of
low-energy constants using both experimental data and lattice results see [19].
1.1.4 Quark Masses from ChPT
The quark masses m f in the QCD Lagrangian (equation 1.1) are free parameters of the theory. Since the quarks are confined in hadrons, their masses cannot be measured directly. For the light quarks, the quark mass term appearing
in the ChPT Lagrangians (equation 1.18, equation 1.21, etc.) enables the calculation of quark mass ratios from the pseudo-Goldstone bosons’ masses and
interactions. More information, for example from lattice QCD, is needed to
get the absolute value of the quark masses.
At leading order, the masses of the pseudo-Goldstone bosons can be directly related to the quark masses, by looking at the mass terms of L2,ChPT .
Expanding U and U † in powers of the field matrix φ :
1
i
φ − 2 φ2 +...,
F0
2F0
i
1
U† = I − φ − 2 φ2 + ...,
F0
2F0
U =I+
(1.22)
setting rμ = lμ = p = 0, s = M (the quark mass matrix), and keeping only
terms up to φ 2 , the Lagrangian can be written as:
L2,ChPT = L2,ChPT,kin + L2,ChPT,mass
1 Lattice
(1.23)
QCD is a numerical method based on the discretization of QCD on a space-time grid,
using Monte Carlo simulations to sample from possible configurations in QCD.
22
where L2,ChPT,kin corresponds to the kinetic terms and L2,ChPT,mass to the mass
terms. The kinetic part of the Lagrangian is:
F02
i
i
μ
I+ φ
Tr ∂μ I − φ ∂
L2,ChPT,kin =
4
F0
F0
1 = Tr ∂μ φ ∂ μ φ
4
1
= 2∂μ π 0 ∂ μ π 0 + 2∂μ η∂ μ η + 4∂μ π + ∂ μ π − +
4
4∂μ K + ∂ μ K − + 4∂μ K 0 ∂ μ K̄ 0
1
1
= ∂μ π 0 ∂ μ π 0 + ∂μ η∂ μ η + ∂μ π + ∂ μ π − + ∂μ K + ∂ μ K − +
2
2
∂μ K 0 ∂ μ K̄ 0 .
(1.24)
These are the usual kinetic terms of scalar hermitian fields (π 0 and η) and
scalar non-hermitian fields (π + , π − ; K + , K − and K 0 , K̄ 0 ). The mass terms are:
F02
iφ
φ2
iφ
φ2
· 2B0 Tr(M − M − 2 M + M + M − M 2 )
4
F0
F0
2F0
2F0
B
0
= F02 B0 Tr(M) − Tr(Mφ 2 )
2
B0
η 2
0
+ −
+ −
=C−
mu
(π + √ ) + 2π π + 2K K
2
3
η 2
+ −
0
0 0
√
) + 2K K̄ md
+ 2π π + (−π +
3
4η 2
+ 2K + K − + 2K 0 K̄ 0 +
ms .
3
(1.25)
L2,ChPT,mass =
Dropping the constant term C, of no physical importance, and collecting terms
of the same fields gives
B0
2
(π 0 )2 (mu + md ) + √ π 0 η(mu − md )
L2,ChPT,mass ≈ −
2
3
+ −
+ 2π π (mu + md ) + 2K + K − (mu + ms )
(1.26)
1
1
4
+ 2K 0 K̄ 0 (md + ms ) + η 2 ( mu + md + ms ) .
3
3
3
To get the masses of the pseudo-Goldstone bosons, the normal form of the
mass term of a scalar hermitian field (− 12 m2a a2 ) and of a scalar non-hermitian
23
fields (−m2a a† a) is used. Neglecting the π 0 − η mixing, the masses can be read
directly from equation 1.26:
m2π = m2π 0 = m2π ± = B0 (mu + md ),
m2K ± = B0 (mu + ms ),
(1.27)
m2K 0 = m2K̄ 0 = B0 (md + ms ),
1
m2η = B0 (mu + md + 4ms ).
3
These equations are called the Gell-Man, Oakes, Renner relations [20]. As
can be seen, the quark masses are of the order of the pseudo-Goldstone boson
masses squared, so assigning O(q2 ) to M is consistent.
To be able to use the physical meson masses, the electromagnetic interaction and its effect on the masses also has to be taken into account. According
to Dashen’s theorem [21], the electromagnetic contribution to the mass difference of pions and kaons is the same at leading order, i.e.
(m2K ± − m2K 0 )E.M., LO = (m2π ± − m2π 0 )E.M., LO = ΔE.M.
⇔ (m2K ± − m2K 0 )E.M. − (m2π ± − m2π 0 )E.M. = O(e2 M).
(1.28)
Using also the fact that the neutral particles do not get any electromagnetic
corrections at lowest order and including the unknown ΔE.M. , equation 1.27
gives
m2π 0 = B0 (mu + md ),
m2π ± = B0 (mu + md ) + ΔE.M.,
(1.29)
m2K ± = B0 (mu + ms ) + ΔE.M. ,
m2K 0 = m2K̄ 0 = B0 (md + ms ).
With this equation, the quark mass ratios
mu
md
and
ms
md
were calculated in [22]:
2m2π 0 − m2π ± + m2K ± − m2K 0
mu
=
= 0.56
md
m2K 0 − m2K ± + m2π ±
m2 0 + m2K ± − m2π ±
ms
= K2
= 20.2.
md
mK 0 − m2K ± + m2π ±
(1.30)
Including the next order in chiral perturbation theory is of course more complicated. Gasser and Leutwyler [16] noted that the ChPT NLO corrections are
the same for the two pseudo-Goldstone bosons’ squared mass ratios
m2 0 −m2 ±
K
K
m2K −m2π
24
m2K
m2π
and
, where m2K is the isospin averaged kaon mass, i.e., the mass of the
kaons if mu = md . The ratios are:
m2K
ms + m̂ =
1 + ΔM + O(M 2 )
2
mπ
2m̂
2
2
mK 0 − mK ±
md − mu =
1 + ΔM + O(M 2 )
2
2
ms − m̂
mK − mπ
(1.31)
where the average u, d quark mass m̂ = 12 (mu + md ) is used, and ΔM is the
same NLO correction (for the exact formula see [16]). The leading order part
of this result is easily seen with equation 1.27 and setting mu = md = m̂ for the
cases when a charge of the meson is not specified.
A new ratio, Q2 , which does not receive a correction at NLO, can be constructed out of the ratios in equation 1.31:
m2 0 −m2 ±
K
K
m2K 0 − m2K ± m2π
1
m2K −m2π
=
=
2
2
2
m2K
Q
mK − m2π mK
m2π
md −mu 2)
1
+
Δ
+
O(M
M
m −m̂
= mss +m̂
2
2m̂ (1 + ΔM + O(M ))
=
=
=
=
⇔ Q2 =
md −mu
ms −m̂
ms +m̂
2m̂
1 + O(M 2 )
(1.32)
md − mu 2m̂ 1 + O(M 2 )
ms − m̂ ms + m̂
md − mu 1
2 (mu + md ) 1 + O(M 2 )
2
2
ms − m̂ 2
m2d − m2u 2
1
+
O(M
)
m2s − m̂2
m2s − m̂2 m2K − m2π m2K
2
1
+
O(M
)
=
.
m2d − m2u
m2K 0 − m2K ± m2π
With Dashen’s theorem, assuming that the mass difference between charged
and neutral pions is exclusively due to electromagnetic effects, and inserting
m2K = 12 (m2K 0 + m2K ± ) (only the QCD contribution of the masses), one can calculate Q2 from the measured meson masses:
Q2D =
(m2K 0 + m2K ± − m2π ± − m2π 0 )(m2K 0 + m2K ± − m2π ± + m2π 0 )
4(m2K 0 − m2K ± + m2π ± − m2π 0 )m2π 0
.
(1.33)
Inserting the known values of the masses from [7] gives QD = 24.3.
25
Knowing Q provides an elliptical constraint on the quark mass ratios, as can
be seen by rewriting equation 1.32:
m2d − m2u
1
=
Q2
m2s − m̂2
⎛
⎞
m2u
1
−
2
2
m ⎜
md ⎟
= d2 ⎝
⎠
ms 1 − m̂22
m
s
m2d 1 −
m2s 1
m2
1 m2
⇔ 2 2s = 1 − 2u
Q md
md
≈
⇔
m2u
m2d
(1.34)
1 m2s m2u
+
=1
Q2 m2d m2d
where one has used the fact that the u and d quarks are much lighter than
mu
plane,
the s quark. The last line is the equation of an ellipse in the mms vs m
d
d
with semi-axes Q and 1. This ellipse is called the Leutwyler ellipse [8] and is
shown for the value Q = QD in figure 1.2, together with the quark mass ratios
from equation 1.30.
25
20
15
ms
md
10
5
QD
Weinberg 77
0
0
0.2
0.4
0.6
mu
md
0.8
1
Figure 1.2. The Leutwyler ellipse [8] for Q = 24.3 and the values of the quark mass
ratios from Weinberg [22].
Weinberg’s result [22], supplemented by Leutwyler’s ellipse [8], means that
the u quark mass is non-zero, but to what accuracy? To further test the un26
derstanding of QCD and the standard model at low energies, it is useful to
determine these quantities in alternative ways. The η → π + π − π 0 decay can
be used for an alternative determination of Q.
The η → π + π − π 0 decay
The amplitude of the η → π + π − π 0 decay can be calculated in ChPT, at leading order using equation 1.18, by expanding the matrix U up to order φ 4 . The
simplified result is [23]:
3(s − s0 )
B0 (mu − md )
√
1+ 2
(1.35)
ALO (s,t, u) =
mη − m2π
3 3F02
where s,t, u are the Mandelstam variables and 3s0 = (s + t + u) = m2η + m2π 0 +
2m2π ± . The Mandelstam variables are defined similarly as for 2-to-2 scattering:
s = (Pπ + + Pπ − )2 = (Pη − Pπ 0 )2
t = (Pπ 0 + Pπ − )2 = (Pη − Pπ + )2
(1.36)
u = (Pπ + + Pπ 0 )2 = (Pη − Pπ − )2
with PX being the four-momentum of particle X. As can be seen from the
definition of s0 above, the Mandelstam variables are not all independent. A 1to-3 decay of spin-less particles has only two independent variables, and it is
enough to use two of the Mandelstam variables. The amplitude is proportional
to the quark mass difference mu − md , so this decay would not occur if mu =
md .
Equation 1.35 can be rewritten in terms of Q. Note that, at LO ChPT, using
equations 1.27 and 1.32:
B0 (mu − md ) = −(m2K 0 − m2K ± ) = −
1 m2K 2
(m − m2π )
Q2 m2π K
so that the amplitude becomes
1 m2 m2 − m2
ALO (s,t, u) = − 2 K2 K√ 2π
Q mπ 3 3F0
3(s − s0 )
1+ 2
mη − m2π
(1.37)
.
(1.38)
Using the value of Q = 24.15 and integrating over phase space gives the LO
result for the decay width ΓLO = 66 eV [23]2 . The NLO result is again more involved, and calculations with the same value of Q give ΓNLO = (160 ± 50) eV
[23], later updated to ΓNLO = (168 ± 50) eV [8]. Both results are quite far
from the experimental value Γexp = (300 ± 11) eV [7]. A full NNLO calculation has also been performed [24], and using the same value of Q gives
ΓNNLO = 298 eV 3 . These results show at best a slow convergence of the SU(3)
Q = 24.15 is the value of QD from equation 1.33 at the time [23] was written.
value for Γ is not quoted in this reference, but using Q = 24.15 and ms /m̂ = 27.4 together
with their results gives Γ = 298 eV [25].
2
3A
27
chiral expansion and that the theoretical uncertainty estimate is not under control (cf. the error in the NLO result with the NNLO result). It turns out that
the biggest part of the corrections at NLO comes from final state interactions
between the pions [23]. But Dashen’s theorem is also known to be a leading
order result, and the corresponding corrections should be taken into account
when calculating the value of Q. Instead of relying on Dashen’s theorem and
its corrections to predict the decay width of η → π + π − π 0 , the experimental
decay width can be used to extract the value of Q. For this approach, one
needs a good theoretical description of the decay dynamics together with accurate experimental knowledge of the decay width. The theoretical description
of the decay should be checked with accurate experimental measurements of
the Dalitz plot distribution.
1.2 Dalitz Plot
The physical region in a 1-to-3 body decay is called Dalitz plot [26] and is
usually defined using two of the Mandelstam variables from equation 1.36,
but it can also be defined using variables linearly related to these. The Dalitz
plot distribution is the decay amplitude squared in the Dalitz plot, and can be
written as a function of the same variables. Since there are only two independent variables in a 1-to-3 decay of spin-less particles, this distribution contains
all the information on the dynamics of the decay.
Considering four-momentum conservation, the boundary of the Dalitz plot
can be calculated. The equation for the boundary of the Dalitz plot in the s − t
plane can be written for t in terms of s as [26]:
t ± = m2π 0 + m2π ± −
1
1 (s − m2η + m2π 0 )(s + m2π ± − m2π ± )
2s
1
∓λ 2 (s, m2η , m2π 0 )λ 2 (s, m2π ± , m2π ± ) ⇔
1
1
1 t ± = m2π 0 + m2π ± −
(s − m2η + m2π 0 )s ∓ λ 2 (s, m2η , m2π 0 )λ 2 (s, m2π ± , m2π ± )
2s
(1.39)
where the Källén function λ is given by
√ √ √ √ λ (x, y, z) = x − ( y + z)2 x − ( y − z)2
= (x − y − z)2 − 4yz
(1.40)
= x2 + y2 + z2 − 2(xy + xz + yz),
mi is the mass of particle i and the ± in the superscript of t stands for which
of the equations to use (the one with − or + before the Källén functions,
respectively). The boundary is shown in figure 1.3, with different line types
for t + and t − .
28
2
t(GeV )
0.18
0.16
0.14
0.12
0.1
0.08
0.08
0.1
0.12
0.14
0.16
0.18
s(GeV2)
Figure 1.3. The Dalitz plot boundary in the s − t plane, where the dashed line corresponds to t + and the full line to t − .
1.2.1 η → π + π − π 0 Dalitz Plot Variables
For the η → π + π − π 0 decay, historically the X and Y variables are used to
construct the Dalitz plot. These dimensionless variables are defined in the η
rest frame as:
√ Tπ + − Tπ −
3
Qη
3T 0
Y = π −1
Qη
with Qη = Tπ + + Tπ − + Tπ 0 = mη − 2mπ ± − mπ 0
X=
(1.41)
(1.42)
(1.43)
and Ti the kinetic energy of particle i (in the η rest frame).
These variables are related to the Mandelstam variables defined in the previous section by calculating the energies (Ex ) of the decay particles (x) in the
η rest frame:
s = (Pη − Pπ 0 )2
⇔ s = m2η + m2π 0 − 2Pη Pπ 0
⇔ s = m2η + m2π 0 − 2mη Eπ 0
⇔ Eπ 0 =
(1.44)
m2η + m2π 0 − s
2mη
29
and similarly
Eπ − =
m2η + m2π ± − u
2mη
(1.45)
Eπ + =
m2η + m2π ± − t
.
2mη
(1.46)
Since the kinetic energy is defined as T = E − m, this can be substituted in
equations 1.41 and 1.42 for
√
3
(u − t)
(1.47)
X=
2mη Qη
3 (mη − mπ 0 )2 − s − 1.
(1.48)
Y=
2mη Qη
Dalitz plot boundary
Equation 1.39, together with equations 1.47 and 1.48, allows to calculate the
values of the X and Y variables for all t and s (which also define u) at the
Dalitz plot border, and thus to calculate the border in the variables X and Y .
For these variables though, a more intuitive way can be used to calculate the
boundary of the Dalitz plot. In the η rest frame, the pions’ three-momenta
sum to zero (pπ 0 +pπ + +pπ − = 0), and thus for pπ 0 as a function of the other
pions’ momenta (momenta and three-momenta are used interchangeably)
|pπ 0 |2 = p2π 0 = p2π + + p2π − + 2pπ + ·pπ −
p2π 0 = p2π + + p2π − + 2pπ + pπ − cos(θπ + ,π − )
(1.49)
where θπ + ,π − is the angle between the three-momenta of the charged pions
and the simplified notation for the modulus of the momenta |pπ + | = pπ + is
used. The physical region is delimited by −1 ≤ cos(θπ + ,π − ) ≤ 1, and the
border corresponds to the extreme cases, the equalities. For any values of the
modulus of the three-momenta of the three pions, it is easy to check if this
momentum configuration is inside the Dalitz plot or not, by checking if
|p2π 0 − p2π + − p2π − | ≤ 2pπ + pπ − .
(1.50)
Being interested instead in evaluating if a certain point (X,Y ) is inside the
Dalitz plot, one can invert the relations 1.41-1.43 to get the kinetic energies of
the pions:
(1.51)
Tπ +
(1.52)
Tπ −
30
Qη
(Y + 1)
3
√
Qη
=
(2 −Y + 3X)
6
√
Qη
=
(2 −Y − 3X).
6
Tπ 0 =
(1.53)
From the kinetic energiesof the pions one can calculate the modulus of their
three-momenta by pi = Ti (Ti + 2mi ) and use equation 1.50 to check if the
point is inside the Dalitz plot. The shape of the boundary of the Dalitz plot in
the X −Y variables can be seen in figure 4.3 on page 76.
1.2.2 Dalitz Plot Parameters
To allow for a direct comparison of the Dalitz plot distribution between theory
and experiment, the amplitude squared of the decay is usually parametrized
by a polynomial expansion around (X,Y ) = (0, 0):
|A(X,Y )|2 N(1 + aY + bY 2 + cX + dX 2 + eXY + fY 3 + gX 2Y + hXY 2 + lX 3 )
(1.54)
The experimental or theoretical distribution can then be fit to this formula
to extract the parameters a, b, . . ., called the Dalitz plot parameters. Note that
c, e, h and l must be zero assuming charge conjugation symmetry which implies that the decay probability should not change if π + and π − are interchanged. This interchange will, however, change the sign of X according to
equation 1.41 and therefore all Dalitz plot parameters in terms containing odd
powers of X’s must vanish.
1.3 More η → π + π − π 0 Theory
To better understand the η → π + π − π 0 decay, one can go beyond pure pseudoGoldstone boson ChPT. One important part is the calculation of electromagnetic contributions to the decay. Another extension is the use of dispersion
relations to calculate the pion rescattering in the final state to all orders in
ChPT.
1.3.1 Electromagnetic Corrections to η → π + π − π 0
The decay η → π + π − π 0 can also occur via the electromagnetic interaction.
In fact, this was the initial hypothesis considered for this decay, but it was
shown that this electromagnetic transition is forbidden [27, 28], which obviously contradicted the comparatively large experimental decay width.
Later on, the framework of ChPT has been used, including the photons
as additional degrees of freedom, to calculate the electromagnetic corrections
at higher order in ChPT. The photons are included as fields in the covariant
derivative, and the photon field appears multiplied by the quark electric charge
matrix:
⎛
⎞
2 0
0
e
(1.55)
Qch = ⎝0 −1 0 ⎠
3 0 0 −1
31
where e is the proton charge. To keep a consistent chiral counting scheme,
i.e., the covariant derivative as O(q), the photon fields are considered as O(1)
while e is considered O(q) [29]. The leading order Lagrangian including electromagnetic effects, in addition to the photon terms included in the covariant
derivative and in a photon field strength tensor, also gets a term with the quark
charge matrix and the pseudo-Goldstone boson fields C ·Tr(QchUQchU † ) [30].
This term, for example, is responsible for the electromagnetic part of the
pseudo-Goldstone bosons’ masses: expanding U and U † up to φ 2 and looking
only at terms quadratic in pseudo-Goldstone bosons gives the electromagnetic
2
mass terms −2Ce
(π + π − + K + K − ). As can be seen, the electromagnetic con2
F
0
tribution to the charged pions’ and charged kaons’ mass is the same at this
order, in agreement with Dashen’s theorem, and the contribution to the neutral
pseudo-Goldstone bosons’ mass is zero.
For the η → π + π − π 0 , at the leading order of the electromagnetic expansion (O(e2 q0 )), the decay is forbidden. Calculations at O(e2 q2 ) in the isospin
limit, i.e, with mu = md [29], show only small differences from the pure ChPT
O(q4 ) result, both for the decay width but also for the shape in the Dalitz
plot. Calculations at order O(e2 q2 ) including the effects of O(e2 (md − mu ))
[31] show that the O(e2 (md − mu )) effects are comparable in size to other
O(e2 (mq )) effects (where mq is a typical light quark mass), in contradiction to
the assumption in the previous O(e2 q2 ) calculation [29]. The total effect of the
electromagnetic corrections, however, remains very small, and the conclusion
is that the η → π + π − π 0 decay is very sensitive to the strong isospin breaking
due to the quark mass difference mu − md .
It is worth noting that electromagnetic corrections can also enter indirectly
in the constants used for the ChPT calculations. For example, F0 can be identified with the pion decay constant Fπ . The value of Fπ changes by ∼ 1% when
including radiative corrections [32], which also changes the η → π + π − π 0
amplitude through equation 1.38.
1.3.2 Dispersive Calculations
The NLO ChPT result showed that the biggest part of the corrections relative
to the LO result arise from the rescattering of pions in the final state, specifically the 2-to-2 pion rescattering [23]. These corrections are expected to be
considerable even at higher orders and therefore it is useful to have an exact
method to calculate them.
The dispersive calculations use the decay amplitude’s unitarity, analyticity
and crossing symmetry to calculate ππ rescattering to all orders. Assuming
that these corrections are separable from other corrections in ChPT and then
matching to ChPT yields ChPT predictions corrected for ππ scattering at all
orders.
32
Scattering (or decay) amplitudes, if extended to the complex plane, are analytic functions, except for where they have singularities and discontinuities.
Without going into details, dispersion relations build on the following. Using
Cauchy’s integral formula, the value of the amplitude is related to a closed
integral of the amplitude in the complex plane, where the integral contour
avoids discontinuities. The countour of the integral extends to infinity, and assuming the integrand vanishes quickly enough there, this contribution disappears, leaving only the integral along the discontinuities. Thus, the amplitude
is related to an integral along its discontinuities in the complex plane. Crossing symmetry takes care of the fact that, if the amplitude is decomposed in
terms of amplitudes in the s,t and u channels, where s,t and u are the Mandelstam variables, these amplitudes need to be related, and in fact, singularities
in one channel appear as discontinuities in the other channels. In general,
the discontinuities are non-linearly related to the scattering amplitudes themselves via the optical theorem [2]. Therefore, a set of integral equations is
obtained, which must be solved self-consistently. If the integrand is not vanishing quickly enough at infinity, then so called subtraction is used, giving rise
to the subtraction constants of this method. These constants are free parameters and need to be determined from elsewhere, e.g. by comparison to ChPT,
by ensuring that the final amplitude matches that of ChPT in some region of
the complex energy plane where ChPT converges well. At present, the choice
of this region differs for different research groups.
Dispersion relations were first used in 1996 for the η → π + π − π 0 decay
[33, 34]. Both calculations match the amplitude to the ChPT NLO result and
find a small enhancement of the partial decay width compared to this. Newer
dispersive calculations have appeared recently, making use of the precise values for the ππ phase shifts which became available (the ππ phase shifts enter
in the integrand). The Bern-Lund-Valencia method [9, 25] and the PragueLund-Marseille method [10] differ both in the construction of the amplitude
and the determination of the subtraction constants. Both calculations have
been matched to NLO ChPT to give predictions of the η → π + π − π 0 decay
width and Daltiz plot distribution. However, both methods can instead use as
input the experimental Dalitz plot distribution data to extract some of the subtraction constants, and calculate a value for Q. Since the quantity Q appears
in the ChPT amplitude and not naturally in the dispersive amplitude, and the
experimental Dalitz plot distributions cannot easily provide the absolute normalization, the dispersive treatments still need to match to ChPT for the rest
of the subtraction constants to determine Q. This method of matching to data
and ChPT has also been used by a third dispersive method [11].
33
1.4 Previous Experimental Results
Several experiments have measured the η → π + π − π 0 decay. Here, only the
high statistics experiments which measured the Dalitz plot distribution and
which extracted at least the b parameter (see equation 1.54) will be mentioned.
For references on earlier experiments see [35].
The experiment reported in [36] was performed at the Brookhaven National
Laboratory Alternating Gradient Synchrotron (AGS). The protons from the
AGS produce a beam of π − used in the experiment in the reaction π − p → nη.
The neutron is detected in a forward counter and its momentum is determined by time-of-flight. The π + and π − from the η decay are measured
in sonic spark chambers inside a magnetic field. The π 0 is reconstructed
through missing mass techniques. For more information on the experimental setup, see [37]. The final Dalitz plot contains 30 000 events, and the results for the Dalitz plot parameters are seen in table 1.1. This experiment
found a small charge asymmetry and a corresponding non-zero value for c,
labelled as “Gormley(70)c ” in the table. The authors also performed the fit for
the Dalitz plot parameters by folding the distribution around X = 0, labeled
“Gormley(70)” in the table.
The experiment reported in [35] used a similar setup. It was performed at
the Princeton-Pennsylvania Accelerator, with a beam of π − produced from
accelerated protons. The studied reaction was again π − p → nη, with the neutron’s time-of-flight measured in scintillation detectors and the π + and π −
detected in sonic spark chambers [38]. The Dalitz plot contains 80 884 events
and the results for the Dalitz plot parameterns are seen in table 1.1, labeled
“Layter(73)”. The charge conjugation violating parameter c was assumed to
be zero. The value for b is found consistent with zero, unlike the previous
experiment.
The Crystal Barrel collaboration has measured the η → π + π − π 0 Dalitz
plot distribution from 3230 events [39]. The experiment was carried out at the
LEAR accelerator, using the reaction p̄p → ηπ 0 π 0 . The Crystal Barrel detector consists of two multiwire proportional chambers and a jet drift chamber
in a magnetic field, to measure charged particles, surrounded by an electromagnetic calorimeter comprised of 1380 CsI(Tl) crystals, to detect photons. It
covers almost the 4π solid angle. The analysis required two tracks measured
in the jet drift chamber and six photons in the calorimeter. The η was identified from the π + π − π 0 invariant mass. This analysis only considered the Dalitz
plot distribution’s dependence on Y , assuming c = 0 and different values of d.
The values of a and b were not sensitive to the assumed values of d. One such
fit is reported in table 1.1.
The previous measurement with the highest statistics, of 1.34 · 106 events in
the Dalitz plot, is by the KLOE collaboration [40]. The detector and setup is
the same as for the present analysis (see chapter 2), but a different data set was
used. The η originates from the φ → ηγ decay, and all final state particles are
34
Table 1.1. Summary of Dalitz plot parameter results, both from experiments and theoretical calculations. Row “Gormley(70)c” includes also a result for the c parameter,
c = 0.05(2). The rows BLV correspond to the Bern-Lund-Valencia dispersive calculations (both with a value for g), in the row labeled ChPT the dispersive calculation
is matched to the ChPT NLO result, while the row labeled KLOE is instead fit to the
experimental data from [40]. The row labeled “disp WASA” correspond to the dispersive calculations in [11], where the amplitude has been fit to the WASA data [41].
Experiment
−a
Gormleyc (70)[36] 1.18(2)
Gormley(70)[36] 1.17(2)
Layter(73)[35]
1.080(14)
CBarrel(98)[39] 1.22(7)
KLOE(08)[40]
1.090(5)(+19
−8 )
WASA (14)[41] 1.144(18)
BESIII(15)[42] 1.128(15)(8)
b
d
f
0.20(3)
0.21(3)
0.03(3)
0.22(11)
0.124(6)(10)
0.219(19)(47)
0.153(17)(4)
0.04(4)
0.06(4)
0.05(3)
0.06(fixed)
0.057(6)(+7
−16 )
0.086(18)(15)
0.085(16)(9)
0.14(1)(2)
0.115(37)
0.173(28)(21)
Calculations
−a
b
d
f
ChPT LO[24]
ChPT NLO[24]
ChPT NNLO[24]
dispersive[33]
BLV ChPT[43]
1.039
1.371
1.271(75)
1.16
1.266(42)
g=−0.050(7)
1.077(25)
g=−0.037(8)
1.116(32)
g=−0.042(9)
0.27
0.452
0.394(102)
0.26
0.516(65)
0
0.053
0.055(57)
0.10
0.047(11)
0
0.027
0.025(160)
−0.052(31)
0.126(15)
0.062(8)
0.107(17)
0.188(12)
0.063(4)
0.091(3)
BLV KLOE[43]
disp WASA[11]
35
measured. A kinematic fit is used to improve the resolution, mostly affecting
the photon energies. This experiment was the first to report a value for the
f parameter, and the results for the parameters are shown in table 1.1. The
Dalitz plot distribution is shown in figure 1.4.
Figure 1.4. The Dalitz plot distribution from KLOE(08) (figure from [40]). The distribution is binned with a bin width of 0.125 in X and Y , with a total of 154 bins
used.
The two most recent measurements come from the WASA-at-COSY collaboration [41] and the BESIII collaboration [42]. The WASA experiment was
carried out at the COSY accelerator, using a proton beam on a deuterium pellet
target, with the reaction pd →3 Heη. The WASA detector consists of a forward
and a central part. The forward part is comprised of plastic scintillators and a
straw tube tracker, and provide energy, time and tracking information for the
forward going particles, in this case the 3 He. The central part contains a small
drift chamber in a magnetic field, to detect momentum of charged particles,
a plastic scintillator and an electromagnetic calorimeter with 1012 CsI(Na)
crystals, to measure photon energy. The central detector is used for the decay
particles of the mesons, in this case, for the π + , π − and the photons from the
π 0 decay. The analysis requires the detection of all final state particles and a
kinematic fit is performed with the pd →3 Heπ + π − γγ hypothesis to improve
the resolution. The Dalitz plot is constructed out of 1.74 · 105 η event candidates and is binned in 0.2 wide bins in X and Y . The shape of the Dalitz plot
is shown in figure 1.5, normalized to the bin with center at X = Y = 0. The
results for the Dalitz plot parameters are shown in table 1.1.
The BESIII experiment is situated at the BEPCII e+ e− collider in Beijing. For this analysis, the radiative decay of the J/ψ is used as the source
of the η (J/ψ → ηγ). The BESIII detector consists of a drift chamber, plastic
scintillators (for time-of-flight measurements), an electromagnetic calorimeter of CsI(Tl) crystals and a counter system, all in a magnetic field. All
final state particles are detected and a kinematic fit is performed with the
36
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
X
0.5
0
−0.5
−1
1
Y
0.5
0
−0.5
−1
Figure 1.5. The acceptance corrected Dalitz plot distribution from WASA-at-COSY,
normalized to the bin at X = Y = 0 (obtained from table IV in [41]). In total 59 bins
are used.
Figure 1.6. The Dalitz plot distribution from BESIII, figure from [42].
37
J/ψ → ηγ → (π + π − π 0 )γ hypothesis. The Dalitz plot contains ∼80 000
events, with a background contamination of 0.1 − 0.2%. The Dalitz plot distribution is shown in figure 1.6. An unbinned maximum likelihood fit is used
to extract the Dalitz plot parameters seen in table 1.1.
In addition to the Dalitz plot parameters for the mentioned experiments,
table 1.1 includes also some theoretical calculations. As can be seen there is
some disagreement between the experiments, specially for the b but also for
the a parameters. Looking at the theory, both the b and the f parameters are
hard to get in agreement with experiment.
1.4.1 Asymmetries
To test C-invariance in the η → π + π − π 0 decay one can also look at asymmetries, which might be more sensitive to C-violation than the c, e, h and l Dalitz
plot parameters. The left-right asymmetry (ALR ) tests overall C-invariance
[44, 45]. The quadrant asymmetry (AQ ) is sensitive to C-violating transitions
with isospin change ΔI = 2 and the sextant asymmetry (AS ) to transitions with
ΔI = 0 [46, 47]. The asymmetries are defined as follows:
N+ − N−
N+ + N−
NI − NII + NIII − NIV
AQ =
NI + NII + NIII + NIV
N1 − N2 + N3 − N4 + N5 − N6
AS =
N1 + N2 + N3 + N4 + N5 + N6
ALR =
(1.56)
(1.57)
(1.58)
where N is the number of acceptance corrected events in the regions defined
in figure 1.7.
Some of the experiments described above have also measured the charge
asymmetries, and one additional experiment at the Rutherford Laboratory reported only the asymmetries [48]. This experiment also used the reaction
π − p → nη to produce the η, and an axially symmetric setup. Table 1.2 summarizes the results. The values quoted for WASA-at-COSY are from a PhD
thesis [49] and have not been published. All results are consistent with zero
except for ALR from [50], which most likely was due to a systematic bias (unmeasured effects in the spark chamber due to the electric and magnetic fields
[7]).
38
1
Y
Y
1
0.8
0.8
0.6
0.6
0.4
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
Y
-1
-1
0.4
0.2
0.4
0.6
0.8
-1
-1
1
X
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X
Figure 1.7. Definition of the kinematic regions used for the asymmetries ALR , AQ and
AS .
Table 1.2. Summary of charge asymmetry results in the η → π + π − π 0 decay. Systematic errors are only explicitly quoted for the KLOE(08) results.
Experiment
ALR · 102
AQ · 102
AS · 102
Gormley(68)[50]
Layter(72)[51]
Jane(74)[48]
KLOE(08)[40]
WASA(14)[49]
1.5(5)
−0.05(22)
0.28(26)
0.09(10)(+9
−14 )
0.09(33)
−0.07(22)
−0.30(25)
−0.05(10)(+3
−5 )
−0.22(33)
0.5(5)
0.10(22)
0.20(25)
0.08(10)(+8
−13 )
−0.06(33)
39
2. Experiment
This chapter gives an overview of the DAΦNE accelerator and the KLOE
detector. In the last part, the crabbed waist upgrade of the DAΦNE accelerator
is presented, as well as the recent upgrades to the KLOE detector, now named
KLOE-2.
2.1 DAΦNE Accelerator
The DAΦNE accelerator [52], Double Annular φ -factory for Nice Experiments, is an e+ e− collider located in√Frascati, Italy. The accelerator is optimized for a center of mass energy of s ∼ Mφ = 1019.461 ± 0.019 MeV, the
φ -meson mass [7].
Figure 2.1. Schematic view of the DAΦNE accelerator [53].
Figure 2.1 shows a schematic view of the DAΦNE accelerator. The linear
accelerator (LINAC) can accelerate electrons up to 800 MeV, and positrons up
40
to 550 MeV. The positrons are created in an intermediate stage of the LINAC:
a high intensity beam of 250 MeV electrons impinges on a tungsten target,
producing photons, electrons and positrons in electromagnetic showers; the
positrons are separated from the electrons by a chicane of dipoles that brings
positrons to the beam axis and electrons to a collimator, where they are stopped
[54]. The rest of the process is the same for electrons and positrons, although
not at the same time. From the LINAC, the electrons or positrons are transferred to the accumulator ring, where the longitudinal and transverse beam
emittance is damped. From the accumulator, they are then injected into one
bunch in the storage rings (DAΦNE), at an energy of ∼ 510 MeV. Due to the
short lifetimes of the beams in the storage rings, the accelerator is “topped
up” several times per hour [55]. An example of this “topping up” is shown in
figure 2.2.
Figure 2.2. Example of the DAΦNE currents under ∼1 hour of operation. The blue
line is for electrons, the red one for positrons. The peaks correspond to the “top up”.
Image from the KLOE display software.
There are two separate storage rings, one for electrons and one for positrons,
and they intersect in two points with a crossing angle of θcross = 2 · 12.5 mrad.
DAΦNE is operated with collisions at only one interaction region at a time, in
the other interaction region the beams are kept vertically separated. The peak
current in the storage rings is 2.4 A for electrons and 1.5 A for positrons. The
number of bunches is ∼ 100 and the bunch spacing is 2.7 ns. The bunch size
at the interaction point is σx = 2 mm, σy = 0.02 mm and σz = 3.0 cm, where x
is the horizontal coordinate transverse to the beam trajectory, y is the vertical
coordinate and z is the horizontal coordinate along the beam trajectory.
During the best period of operation (2005-2007), the DAΦNE collider has
reached peak luminosities of L peak ∼ 1.6 · 1032 cm−2 s−1 , and while running
with the KLOE detector, an integrated daily luminosity of ∼ 8.5 pb−1 . The
accelerator has later been upgraded and reached even higher luminosities –
see section 2.3.
41
2.2 KLOE Detector
The KLOE detector has been operating at one of the interaction regions of the
DAΦNE accelerator, from 1999 to 2006. In fact, after commissioning of the
accelerator and detector, the KLOE data taking occurred at two periods: from
2001 to 2002, with about 450 pb−1 of integrated luminosity, and 2004-2006
with about 2000 pb−1 of integrated luminosity [55]. Most
√ of these data were
taken at a center of mass energy of the φ -meson mass, √ s ∼ 1019.5 MeV, but
integrated luminosity was taken at s = 1 GeV and there
∼ 250 pb−1 of the √
s from 1010 MeV to 1030 MeV, comprising four points
was
also
a
scan
in
√
in s and integrating ∼ 10 pb−1 of luminosity.
One of the aims of the KLOE experiment is to measure with great precision
decays of the φ -meson. The main decays and their branching ratios are shown
in table 2.1. Special attention is given to the decays to neutral kaons, and
the subsequent decays of the kaons. This task requires a high acceptance and
efficiency, as well as a good resolution and the ability to detect both neutral
and charged particles.
Table 2.1. The decays of the φ meson with a branching ratio bigger than 1% [7].
Decay channel
φ
φ
φ
φ
→ K+K−
→ KL0 KS0
→ ρπ and φ → π + π − π 0
→ ηγ
Branching ratio (%)
48.9 ± 0.5
34.2 ± 0.4
15.32 ± 0.32
1.309 ± 0.024
If the φ -meson is produced at rest, then the neutral kaons from its decay
have momentum pK 0 = 110 MeV/c. For the long lived KL0 , with lifetime
τK 0 = 5.116 ± 0.021 · 10−8 s [7], it corresponds to a mean path in the laboraL
tory frame of λLF = γτβ c = 3.4 m (here γ is the Lorentz factor and β = 0.22
is the velocity of the KL0 in the lab frame, in units of c). This implies that
the detector must have a large volume in order to measure the KL0 decays, e.g.
radius of 3.4 m to be able to detect 1 − 1/e = 63% of the decays. Due to the
crossing angle of the beams, φ -mesons produced in KLOE have a small horizontal momentum towards the center of the accelerator of pφ = 13 MeV/c.
The neutral kaons are thus not monochromatic in the lab frame, and their momentum varies from 104 MeV/c to 117 MeV/c [55]. A compromise between
the size and the complexity of the detector leads to the radius of 2 m, meaning
∼ 40% of the KL0 decay within this region and can be detected.
KLOE, depicted in figure 2.3, consists mainly of two detectors: a drift
chamber (DC) to measure the momentum of charged particles, and an electromagnetic calorimeter (EMC) to mainly measure energy, time and impact
position of photons. As can be seen, surrounding both these detectors there is
a superconducting coil and an iron yoke, giving rise to an axial magnetic field
of 0.52 T. At the interaction point, the beam pipe is an Al-Be spherical shell
42
with a radius of 10 cm and thickness 0.5 mm [55]. The low atomic number
(Z), low density material minimizes the energy loss of charged particles passing the beam pipe. The spherical shape with a radius of 10 cm ensures almost
all KS0 decay in the vacuum inside the beam pipe. This minimizes KS0 → KL0
regeneration on the beam pipe. With lifetime τK 0 = 8.954 ± 0.004 · 10−11 s [7]
S
and considering the momentum pK 0 = 110 MeV/c from the φ decay at rest,
the mean path of the KS0 in the lab frame is λKS = 5.96 mm, and the radius of
10 cm corresponds to ∼ 16λKS , thus ensuring most KS0 have already decayed
in the vacuum. There is also a tile calorimeter surrounding the beam pipe
around the interaction region quadrupoles, whose main purpose is to measure
photons from the KL0 decay which would otherwise be lost in the beam pipe.
This detector is not used in the current analysis.
Figure 2.3. Vertical cross-section of the KLOE detector, showing the DC, EMC and
superconducting coil. Figure from [56].
43
2.2.1 Drift Chamber (DC)
The tracking detector of KLOE is a gas filled drift chamber, cylindrical in
shape, with length 3.3 m, inner radius 25 cm and outer radius 2 m [57]. In
a drift chamber, wires shape the electric field in the gas filled space. When
a charged particle traverses the chamber, it ionizes the gas, creating electronion pairs along its trajectory. The created electrons drift towards the positive
voltage wires, and when close to the wires, the high electric field causes an
avalanche of electrons and ions to be created out of the gas. The drifting of
the ions away from the wire induces a signal on it that can be measured at the
wire’s end [58].
In KLOE, to minimize multiple Coulomb scattering, KL0 → KS0 regeneration
and absorption of photons before reaching the calorimeter, the materials used
for the walls of the chamber and as the filling gas have low Z and low density
[59]. Carbon fiber is used for the mechanical support, i.e., the drift chamber
walls. The gas used is a mixture of helium and isobutane (90% He - 10%
iC4 H10 ). This gas mixture has a radiation length X0 ∼ 1300 m, but this is
lowered to an effective radiation lenght of X0 ∼ 900 m if the tungsten wires
are taken into account.
There are 12 582 sense wires in the drift chamber. They are made of goldplated tungsten and are 25 μm in diameter. To shape the electric field, 39 558
silver-plated aluminium field wires, with diameter 80 μm are used, of which
168 form an inner guard and 768 an outer guard to the sensitive volume. There
are in total 52 140 wires, and the voltage difference between field and sense
wires is 1800 − 2000 V [57].
The wires are arranged in cells of almost square transverse cross-section,
consisting of one sense wire surrounded by 8 field wires. Figure 2.4 shows
one example cell. The cells are arranged in cylindrical layers around the beam
pipe. To account for the higher track density close to the interaction region,
due to the usual small momenta of the charged particles coming from the φ
decays close to rest, the first 12 layers have a cell size in the transverse plane
of ∼ 2 × 2 cm2 and the remaining 46 layers of ∼ 3 × 3 cm2 [57].
In order to reconstruct the tracks in three dimensions, some wires need to
be at an angle to the drift chamber axis. Together with the requirement of
uniform efficiency, this consideration led to an all stereo geometry (i.e., all
wires have an angle to the drift chamber axis), where consecutive radial layers
have opposite signs of the stereo angle. The definition of the stereo angle is
shown in figure 2.5.
As can also be seen in figure 2.5, the stereo angle implies that the distance of
the wire from the chamber axis is not constant, with the minimum distance R0
at the middle of the chamber (z = 0) and the maximum R p at the end plates.
To ensure that the wires fill the chamber uniformly, the stereo drop is kept
constant at R p − R0 = 1.5 cm. This implies that the stereo angle changes with
the radius, increasing in absolute value from 60 mrad to 150 mrad.
44
Figure 2.4. A cell of the drift chamber, showing the definition of the angles β and φ̃ .
The filled circles correspond to sense wires, the open circles to field wires (adapted
from [55]).
Figure 2.5. Definition of the stereo angle ε (adapted from [55]).
45
The “almost square” shape of the cells comes from the stereo geometry.
Layer k of wires is defined as all the wires sharing the radius with either the
sense wire at radius Rk or the field wire just below, at Rk− . All wires at Rk
have the same stereo angle, and it is almost the same as the stereo angle for
the wires at Rk− , i.e. εk ≈ εk− . The field wires of the upper part of the cell, on
the other hand, belong to the next layer, and thus have a stereo angle with the
opposite sign. This results in a cell shape varying periodically along the axis
of the chamber, and varying also with the radius and the azimuthal angle.
Tracking
The signals from the wires of the drift chamber are the drift times. To reconstruct a track, the drift times first have to be translated into drift distances of the
electrons from the ionization to the wire. For this, 232 space-time relations are
used, depending on cell shape and track impact parameter (see calibration on
page 47). Then the track reconstruction program works in three steps: pattern
recognition, track fit and vertex fit.
A “hit” is a measured signal from a wire. First, space-time relations averaged over cell shape and track impact parameter are used (since these two
variables depend on the track, they can only be calculated after the track fitting) to get the drift distances from the drift times, the signal. For each of
the two stereo views, the pattern recognition starts at the outermost layer and
works inward, associating hits close in space to track candidates. After the
association is done, the track candidates in each view are fit and the track parameters extracted. Then tracks from both views with the same curvature and
compatible geometry are combined into a three dimensional track. The three
dimensional track is fit again, providing also information about the z coordinate. The pattern recognition program outputs tracks together with a first
estimation of their parameters.
The next step, the track fitting, refines the track parameters through a χ 2
minimization, with:
2
track
di
− di
2
(2.1)
χ =∑
σ (di )
i
where the sum goes over the hits in the track, ditrack are the drift distances
calculated from the fitted track parameters, di the drift distances as measured
from the drift time, dependent on the space-time relations, and σ (di ) is the
drift distance resolution. This is done in an iterative procedure, where the
space-time relations are first used to calculate the drift distances di . In the first
iteration, the space-time relations are calculated from the output of the pattern
recognition. The drift distances are then fit to a track by the χ 2 minimization, and new track parameters are obtained. The procedure is repeated with
these new track parameters until a sufficiently good track is obtained. Since
the space-time relations depend on the track parameters, each time new track
parameters are found, new space-time relations are used, and thus new drift
46
distances di . After the first iteration, procedures to improve the quality of the
track fit are employed: adding hits missed by the pattern recognition, rejecting
wrong hits, identifying split tracks and joining them to one track.
The vertex fit then associates track pairs to a vertex, by extrapolating the
tracks and checking their point of closest approach. Primary vertices are found
by extrapolating the tracks to the beam crossing point, after this a search for
secondary vertices is done, ignoring tracks already associated to a vertex.
Charged tracks with polar angle larger than 45◦ are reconstructed with a
momentum resolution of σ p⊥ /p⊥ 0.4%, and the spatial resolution is σxy ∼
200 μm in the transverse plane and σz = 2 mm in the axial direction. The
resolution of the vertex position is σV ∼ 1 mm.
Calibration
The calibration of the DC is performed with cosmic ray muons, selected by
requiring two calorimeter clusters separated in time and a track in the DC.
The time signal measured in the drift chamber has contributions from the
drift time, Tdri f t , the propagation time along the wire, Twire , and a time offset,
T0 . Using the time from the calorimeter cluster measurement, the time of
flight between the calorimeter and the wire, Tto f , also needs to be taken into
account, but the calorimeter information allows the determination of Twire and
Tto f event by event. The distribution of Tdri f t + T0 for each wire can then be
fitted to extract the time offset for each wire. This procedure is performed once
per run period, but it is repeated after interventions in the front-end electronics.
To relate the drift time to drift distances, 232 space-time relations are used.
These describe the dependence of the drift distance on the drift time, the cell
shape and the track impact parameter. The variable β is used to classify the cell
shape, and φ̃ for the track impact parameter. Both these variables are shown
for the example cell in figure 2.4. To parametrize the space-time relations as a
function of only drift time, six different values of β characterize the cell shape,
and 36 evenly spaced intervals of φ̃ characterize the track impact, giving a
set of 116 space-time relations for the big cells (∼ 3 × 3 cm2 ) and the same
number for the small cells (∼ 2 × 2 cm2 ). Each of the space-time relations is
parametrized by a fifth-order polynomial of the drift time, resulting in 6 · 232
calibration coefficients (Ck ).
At the start of each run, an online filter selects 80 000 [55] cosmic ray
events, fits the hits to tracks and checks the residuals of the space-time relations. The residuals are defined as the difference between the drift distance
calculated from the track parameters and the drift distance calculated from
the drift time using the space-time relations (ditrack − di ). If these are too big,
the calibration procedure for the space-time relations is started, which collects
300 000 cosmic ray events and finds new calibration coefficients for the 232
space-time relations. The starting values for the coefficients Ck are taken from
the previous calibration run. Before the track is fit, there is no information on
the β and φ̃ parameters, so at first coefficients are averaged over all cell shapes
47
and track impact angles, as for the regular track fitting procedure. After the
track fitting iterations are done, a fit for new Ck coefficients for each of the
232 space-time relations is performed, this time by minimizing the absolute
value of the residuals (|ditrack − di |, i.e., only changing di ). With these new Ck ,
the track fitting and Ck fitting is performed in an iterative procedure, until the
residuals are small enough.
2.2.2 Electromagnetic Calorimeter (EMC)
The electromagnetic calorimeter of KLOE [60] consists of a cylindrical barrel
surrounding the drift chamber, and two end-caps perpendicular to the beam
axis, see figure 2.3. It is a sampling calorimeter composed of lead and scintillating fibers. A photon with energy larger than a few MeV interacts with
matter mainly via pair-production. The created electrons and positrons in turn
radiate photons which, if their energy is high enough, create more electronpositron pairs and so on, in what is called an electromagnetic shower. In a
material with high density and high Z, these interactions are more probable
and if the material is thick enough, the original photon will deposit all its energy in the material, in the form of electron-positron pairs and photons. Measuring the energy of these created photons and e+ e− gives the energy of the
original photon. In the KLOE calorimeter, the lead serves as a passive material that due to its high density and Z accelerates the showering process, while
the scintillating fibers are the active part and convert the deposited energy into
light, which is measured by photomultiplier tubes (PMTs).
The detector is built in layers of 0.5 mm thick lead foils, with grooves to
accommodate the fibers, and 1 mm diameter, clad scintillating fibers. The
fibers are glued to the lead foils with epoxy. The final material has a volume
of 42% lead, 48% scintillating fibers and 10% epoxy, and is shaped into 23 cm
thick modules. Its radiation length X0 is 1.5 cm, so the module thickness
corresponds to ∼ 15X0 , corresponding to ∼ 99.99997% of absorbed energy.
For the barrel, 24 modules, 4.3 m long, with trapezoidal cross-section are
used, while each endcap consists of 32 C-shaped modules, 0.7 - 3.9 m long of
rectangular cross-section with variable width. The calorimeter covers 98 % of
the full solid angle, see figure 2.3. For the read-out, each module is subdivided
into ∼ 4 × 4 cm2 cells, which are matched to the circular area of the PMTs by
light guides. Each module is read-out at both ends. Cells at the same depth
(same value of r in the barrel or z in the endcap) form a so called calorimeter
plane, for a total of 5 planes in depth over the whole calorimeter.
The energy resolution of the calorimeter, σE /E = 5.7%/ E(GeV), is determined with radiative Bhabha events, i.e., e+ e− → e+ e− γ events, where
the electron and positron are measured in the DC. The time resolution is
σt (E) = √57 ps ⊕ 140 ps [55], where the second term is added in quadraE(GeV)
ture and includes calorimeter miscalibrations and trigger jitter. The time reso48
lution is determined using cosmic rays, e+ e− → γγ, radiative φ decays (i.e.
φ → γX) and φ → π + π − π 0 events. Excluding the trigger jitter contribution to the resolution results in an intrinsic calorimeter time resolution of
σt (E) = √57 ps ⊕ 100 ps. The cluster position is reconstructed with a resE(GeV)
olution of σrφ ∼ σxz ∼ 1.3 cm transverse to the fibers, and along the fibers,
from the time measurement, with a resolution of σz = σy = √1.2 cm (for the
E(GeV)
barrel the coordinate along the fibers is z, for the endcaps y).
Reconstruction
For each cell, time (T ) and energy (S) is measured at both ends (A and B) in a
time to digital converter (TDC) and an analog to digital converter (ADC), and
expressed in counts. The energy of each side (E A and E B ), for each cell (i),
is corrected for the pedestal, calibrated relative to the response to minimum
ionizing particles (mip) and multiplied by a factor accounting for the absolute
energy scale [60]:
A,B
SiA,B − S0,i
·K
(2.2)
EiA,B =
Smip,i
A is the pedestal of the amplitude scale of side A of cell i, S
where S0,i
mip,i is the
response of cell i to mip passing through its center, both in ADC counts, and
K is the absolute energy scale calibration constant.
The time at each side of the cell is converted to nanoseconds using caliA,B
, where t is the time in nanosecons and c
bration constants: tiA,B = cA,B
i · Ti
the calibration constants. The particle arrival time at the cell (ti ) and the its
position along the fiber (si ) is determined from the measured times:
t A + tiB t0,i + t0,i Li
−
− ,
(2.3)
ti (ns) = i
2
2
2v
v
A
B
si (cm) = tiA − t0,i
− (tiB − t0,i
) ,
(2.4)
2
A is the time offset for side A of cell i, L is the lenght of the cell (in
where t0,i
i
cm) and v is the velocity of light in the fibers (in cm/ns, ∼ 17 cm/ns). The
definition of si assumes si = 0 cm in the middle of the cell.
The total energy deposited in the cell (Ei ) is taken as the mean value of the
determination at both ends, corrected by a factor AA,B
i (si ) accounting for the
attenuation of light along the fiber. Note that the attenuation factor depends on
the position of the particle along the fiber:
A
B
EiA AAi (si ) + EiB ABi (si )
.
(2.5)
2
To reconstruct energy and time of incidence of a particle in the calorimeter, the information from the different cells is joined by a clustering algorithm. First, cells adjacent to each other in r − φ (barrel) or x − z (endcaps)
Ei =
49
are grouped together if they have both energy and time information from both
sides. Then, the algorithm uses the longitudinal coordinates (si above, corresponding to z for barrel and y for endcaps) and the incidence times to further
join and/or split cells to form a cluster. At this stage, cells missing time or
amplitude information are recovered if their φ (barrel) or x (endcaps) is close
enough to the clusters corresponding variable. The energy of the whole cluster
is evaluated as the sum of the energies of the cells:
Eclu = ∑ Ei
(2.6)
i
while the time of arrival of the particle and its position are calculated as energy
weighted averages:
∑ i t i Ei
,
∑i Ei
Rclu = ∑iri Ei
∑ i Ei
tclu =
(2.7)
(2.8)
where i stands for the ith cell included in the cluster and r or Rclu for the
position vector relative to the KLOE reference frame (as (r, φ , z) or (x, y, z)).
The time of flight of the particle from the interaction point to the calorimeter, tto f , can be related to the calorimeter time tclu by tclu = tto f + δC − Nbc TRF
[55], where δC is a number accounting for the electronic offset and delay due
to cable length, Nbc is the number of bunch crossings needed to start the TDCs
and TRF = 2.715 ns is the machine radio frequency period. The last term
is needed because the particles can have a big spread of arrival times at the
calorimeter. The KLOE trigger (see section 2.2.3) cannot identify the bunch
crossing related to each event. Instead, the KLOE fast trigger is used as the
start signal for the TDCs, and this trigger is phase-locked with a replica of the
machine RF, in a 4 · TRF period clock, giving rise to the Nbc TRF term.
The correct bunch crossing, and correspondingly the quantity Nbc , are offline determined event by event. The quantities δC and TRF are determined for
each run with e+ e− → γγ events. For these events, the calorimeter is detecting
photons, and their time of flight should be tto f = |Rclu |/c. The distribution of
tclu − |Rclu |/c, seen in figure 2.6, shows well separated peaks, corresponding
to diffrent values of Nbc . The distance between peaks gives TRF , and δC is chosen as the mean value of the peak with largest statistics. Note that although the
choice of δC is arbitrary, the same definition has to later be used in determining
Nbc event by event.
Calibration
A,B
The energy calibration consists of finding the pedestals S0,i
for each side A
and B and the response to mip, Smip,i , for each cell i; as well as finding the absolute energy calibration constant K. The pedestals are determined with cosmic
50
Figure 2.6. Distribution of tclu − |Rclu |/c for e+ e− → γγ events (adapted from [60]).
ray runs without circulating beams, and cross-checked with pulser triggered
runs [60]. Smip,i , which serves as a relative energy calibration between cells, is
done with cosmic ray minimum ionizing particles which cross the center of the
cells. For each cell, the peak of the energy distribution defines Smip,i . The same
data is used for determining the light attenuation in the fibers, AA,B
i (si ). This
dedicated cosmic ray run is performed before the start of each long data taking
period [55]. The absolute energy scale factor K is determined with e+ e− → γγ
events. The monochromatic 510 MeV photons are identified and used to set
the energy scale (in MeV/Smip ). The K calibration is repeated every 400 nb−1
of collected luminosity.
A,B
The time calibration determines the time offsets t0,i
, the light velocity in
the fibers v and the time calibration constants cA,B
i . The time offsets and the
light velocity v are determined with high momentum cosmic rays, which can
be collected in parallell with data taking, every few days of data taking. The
tracks are identified in the DC, and an iterative procedure minimizes the time
residuals between the tracks and the calorimeter. The quantities t0A −t0B , t0A +t0B
and v are determined from time distributions, and from these t0A and t0B are
calculated. The values of t0A − t0B and v are also checked using the s coordinate
as determined from the extrapolation of the track to the calorimeter. Equation
2.4, if rewritten to tiA − tiB as a function of s, is a straight line with slope 2/v
51
and intercept t0A −t0B , which allows the determination of these two values from
the distribution of tiA − tiB vs s [60].
The time calibration constants cA,B
have been determined in a laboratory
i
test stand, and are good up to an overall scaling factor. To get the absolute
time calibration, the value of TRF is used. The scale factor needed is the ratio between the TRF measured with e+ e− → γγ and the value of the period
obtained from the accelerator RF signal TRF,DAΦNE (i.e. TRF /TRF,DAΦNE ).
2.2.3 Trigger
The KLOE trigger uses information from both the EMC and the DC [61]. The
high events rates at DAΦNE, mostly due to background, make it desirable for
the trigger to accept all φ decays and at the same time to reject the main backgrounds: e+ e− (Bhabha) scattering at small angles and machine related background due to lost particles from the beams. The trigger should also accept
Bhabha events and e+ e− → γγ events at large angles (for detector monitoring
and calibration) and reject cosmic rays. Both cosmic ray events and Bhabha
scattering at small angles should, however, be accepted in a downscaled sample for monitoring puposes. To allow determination of the bunch crossing
corresponding to the event, the trigger should also be fast.
These considerations have led to the choice of a two level trigger: the level
1 trigger (T1, fast trigger) has minimal delay, is synchronised to the accelerator
radio frequency and starts the data aquisition at the front-end electronics; the
level 2 trigger (T2, validation trigger) uses more information from the detector,
validates the T1 and starts the whole data aquisition.
The T1 trigger is generated by either the EMC or the DC. For this purpose,
the EMC is divided into sectors of ∼ 30 calorimeter cells (30 cells in the barrel
and 20, 25 or 30 in the endcaps). The EMC level 1 trigger requires at least two
fired trigger sectors, with energy deposit greater than 50 MeV in the barrel
and 150 MeV in the endcaps [55]. If there are only two fired sectors, and these
are in the same endcap, the event is rejected, since this topology is mainly
from machine background. The DC level 1 trigger requires at least 15 hits in
the DC within 250 ns. The Bhabha rejection is done at this level with EMC
information: as in the normal EMC trigger, two fired sectors are required, but
here with a minimum energy deposit of 350 MeV, and only for topologies
where both sectors are in the barrel or in different endcaps. If this condition
is met, the T1 is vetoed, except for a downscaled amount of the events. After
each T1, there is a fixed dead-time of 2.6 μs where no new T1 can occur.
In the case where T1 is due to the EMC, the T2 occurs automatically after
a fixed time of ∼ 1.5 μs. If the T1 is due to the DC, a validation trigger
from the DC is needed: this requires ∼ 120 hits within 1.2 μs. As with the
Bhabha trigger, the T2 signal can be vetoed by the cosmic ray trigger: two
EMC sectors with signals from the outer cells of the detector, with energy
52
deposit above ∼ 30 MeV, identify the cosmic rays. If no T2 signal arrives
within the 2.6 μs of fixed dead-time from T1, all read-out is reset.
The background events from Bhabha scattering, cosmic rays and machine
background that survive the trigger are later rejected at the beginning of the
offline reconstruction by the background filter FILFO, see section 3.1.1.
2.3 Upgrades
The upgrades presented in this section have no direct impact on the results
of this thesis, since the data used is from before the upgrades. Nonetheless,
it is interesting to know what the present and near future hold for the KLOE
detector, and the DAΦNE upgrade has a direct impact on this.
2.3.1 DAΦNE
In 2007, the DAΦNE accelerator was upgraded in order to increase its luminosity. To achieve this, the beam horizontal size σx is reduced and the crossing
angle increased (this reduces the overlap region of the beams), which allows to
decrease the vertical beam size, while the crabbed waist scheme is used to suppress resonances [62]. Figure 2.7 shows the differences in the crossing angle,
which in the upgrade is doubled to θcross = 2 · 25 mrad, and in the horizontal
beam size.
The crabbed waist uses sextupoles to rotate the minimum of the vertical
beam size of a beam, such that the position of the minimum is aligned along
the central trajectory of the other beam.
The upgrade also included other hardware improvements [63]. It has been
used for the SIDDHARTA experiment [64] and since 2010 for the KLOE-2
experiment. With the first experiment, the peak luminosity reached was of
4.53 · 1032 cm−2 s−1 , with a maximum daily integrated luminosity exceeding
15 pb−1 . The more complex setup of the KLOE-2 experiment, specially the
magnetic field, makes it hard to reach the same performance. The peak luminosity of 2.0 · 1032 cm−2 s−1 has been reached, but with higher backgrounds
than during the KLOE run.
2.3.2 KLOE-2
The higher values of luminosity possible at DAΦNE after the upgrade has
prompted a new physics program [65] and upgraded detector, called KLOE2. The KLOE-2 detector consists of the DC and EMC as in KLOE, but with
several new detectors added: a high and a low energy tagger (HET and LET)
for each lepton ring, two new calorimeters CCALT and QCALT and an new
inner tracker (IT) placed between the interaction region and the DC.
53
Figure 2.7. Representation of the DAΦNE interaction region in the old interaction
scheme (upper) and the upgrade scheme (lower).
54
The purpose of the LET and HET is to detect the electrons and positrons
from e+ e− → e+ e− γ ∗ γ ∗ → e+ e− X types of processes, where X are hadrons
[66]. The HET [67] is designed to detect electrons and positrons with small energy transfer, and thus are close to the nominal orbit in the accelerator. There
is one such detector in each of the electron and positron arms, and they are
located after the first dipole following the interaction region, 11 m from the
beam crossing. The detector consists of 28 plastic scintillators, and measures
the position of the lepton passing through from which of the scintillators was
fired. The detector is located after the bending dipole which serves as a spectrometer and the position at the detector is related to the momentum of the
measured lepton, see figure 2.8. The HET can detect leptons in the energy
range ∼ 400 − 500 MeV.
Figure 2.8. The HET detector, located after the first bending dipole. The light guides
which transfer the light from the scintillators to PMTs are shown in orange. Figure
adapted from [67].
The LET [68] is an energy sensitive detector comprised of 20 LYSO crystals
(Lutetium Yttrium Orthosilicate), read out by silicon photomultipliers (SiPM).
There are two LETs, one for electrons and one for positrons, located inside
the KLOE detector, 1 m from the interaction region. The LETs cover a limited
angular region, and they are replacing the corresponding part of the QCALT.
Figure 2.9 shows the LET casing mounted with the QCALT, and the window
in the QCALT to allow the leptons to reach the LET. The LETs detect leptons
in the energy range 160 − 230 MeV.
The crystal calorimeter with timing, CCALT [70], is a small angle calorimeter with the purpose of extending the KLOE angular acceptance for photons
from the interaction region down to θ = 10◦ (the EMC only goes down to
θ = 21◦ ). There are two CCALT detectors, one on each side of the interaction
region. They each contain 48 crystals, read out by SiPMs. Figure 2.10 shows
one of four wedges of one CCALT, with four crystals visible.
55
Figure 2.9. The LET (gray casing with black label) mounted on the QCALT (with
black casing) [69].
Figure 2.10. Part of the CCALT, with 4 crystals showing. Figure from [71].
56
The QCALT [72] (tile quadrupole calorimeter) is a new calorimeter surrounding the beam pipe and covering the quadrupoles inside KLOE. There are
two QCALTs, one on each side of the interaction region, see figure 2.11. These
new detectors were needed because of the changes done to the beam crossing
region for the DAΦNE upgrade. The main purpose of the QCALTs is to improve detection efficiency of photons coming from KL0 decays, which might
otherwise be lost in interactions with the beam pipe and the quadrupoles. The
QCALT is composed of alternating layers of scintillator plates and tungsten
plates, for a total depth of 4.75 cm (∼ 5.5X0 ) and 1 m in lenght. The scintillating plates are divided in tiles, 20 per plane, each with a fiber embedded to
transmit the light to SiPMs for read-out.
Figure 2.11. The detectors around the new KLOE-2 interaction region: the IT (in the
middle with the Italian flag) and the two QCALTs (to the left and right of the IT). One
can also see the space left in the QCALTs for the LET. The CCALT is inside the IT,
not visible. Figure from [56].
The IT (inner tracker) is a tracker for the inner region, and fits between the
beam pipe and the DC. It is composed of four layers of cylindrical triple gas
electron multipliers (GEM) [73] with the purpose of improving acceptance of
low transverse momentum tracks and improving the vertex resolution. One
layer of a cylindrical tripple GEM consists of concentrical electrodes: a cathode, three GEM foils (for multiplication) and an anode, which also functions
as the read-out. The read-out is done in two coordinates, but instead of the
usual XY read-out for planar GEMs, it uses a XV view, where the V strips are
at an angle of ∼ 40◦ . The IT mounted on the interaction region can be seen in
figure 2.11.
All these new detectors have been installed and the KLOE-2 detector is
operating since November 2014.
57
3. Event Reconstruction and Selection
This chapter explains how the events in the KLOE detector are reconstructed
and how the selection for the η → π + π − π 0 decay is made. First, the event
reconstruction and selection are reviewed. Second, the Monte Carlo (MC)
simulation is explained. Then, the analysis steps for background rejection are
listed and last a data-Monte Carlo simulation comparison is done.
In the analysis, the data from the whole 2004-2005 data taking period is
used, in total 1595 pb−1 .
3.1 Reconstruction
In the following description, tracks from the Drift Chamber (DC) and clusters
in the Electromagnetic Calorimeter (EMC) are assumed known, see section
2.2. In the first part of this section the common KLOE reconstruction and
background rejection tools used are reviewed, while the second part deals with
the decay specific selection.
3.1.1 FILFO: Background Filter
The FILFO routine (FILtro de FOndo) implements the rejection of machine
background and of cosmic ray background [74]. This filter implements two
routines: machine background rejection routine and the cosmic ray background rejection routine. To pass the FILFO filter an event must pass both
these routines, or be selected by the μ + μ − selection routine.
Machine background rejection
The machine background rejection filter is described in detail in the KLOE internal documentation [75]. Events with less than 200 hits in the drift chamber,
with two to six clusters in the calorimeter with total calorimeter energy, Etot ,
less than 1.7 GeV are considered as machine background candidates for further processing. Studies of data and MC simulations were used to identify the
signatures of the machine background, resulting in a series of six conditions
to reject the machine background:
• since machine background usually is forward peaked, a cut on the plane
θ1 vs θ2 is used, where the θ ’s are the angles of the two most energetic
calorimeter clusters. The cut removes the two forward-backward corners, only for events with Etot < 500 MeV to reduce its inefficiency for
φ events.
58
• a cut on the plane θav vs Etot , where θav is the energy averaged polar
Nclu
θ i Ei
angle θav = ∑
. Events with Etot < 80 MeV OR (|θav − 90◦ | > 30◦
E
tot
i
AND Etot < 250MeV) are rejected.
• if the most energetic cluster in the event has deposited less than 80 MeV
in the calorimeter, and this energy is all deposited in the first or last plane
of the calorimeter the event is rejected as machine background.
• for events with less than 50 hits in the drift chamber, an upper limit
for Etot depending on the number of calorimeter clusters (from one to
eight here, instead of two to six as before), is used to reject machine
background. The upper limit is between 250 MeV and 350 MeV.
• using the ratio R between the numbers of cells hit in the DC small cells
and the total number of cells hit, the plane R vs Etot is used with a diagonal, linear cut, but only if Etot < 300 MeV (in this case, the number of
clusters is less than or equal to five).
• a condition using the asymmetry in calorimeter energy upstream and
downstream, when more than one cluster, and Etot and number of drift
chamber hits also rejects machine background.
Cosmic ray background rejection
The cosmic ray background rejection routine finds cosmic ray events at the
FILFO level by looking at the calorimeter information. In the most recent implementation [76], several criteria are used. As shown in [77], cosmic muon
events usually have only few clusters in the calorimeter, so this filter is implemented for events with less than six clusters. Looking at energy and time
of the different planes of the EMC, this algorithm rejects cosmic events. An
important part of the rejection is the presence of deposited energy in the fifth
plane of the EMC, the outer plane, for example used in calculating the difference between the time on the fifth and first planes, which is negative for
particles coming from the outside of the detector. When this is not available,
further rejection criteria are needed, like cluster position and crossing velocity
(the velocity calculated assuming the first and last cluster to be due to the same
particle), and depth profile of the energy release in the calorimeter planes.
μ + μ − selection
This routine selects μ + μ − candidates to pass the FILFO routine even if they
are rejected by the other two routines. For events with 2 to 10 clusters in the
calorimeter, a double-loop over the clusters checks if there are at least two
clusters both having energy between 100 and 350 MeV and transverse radius
less than 65 cm, the time between the two clusters is less than 2 ns and the
cosine of the angle between them is less than -0.7 (i.e., the angle is close to
180◦ ). It also requires that the number of hits in the drift chamber is larger
than 20.
59
3.1.2 Event Classification
Following the FILFO routine, the events are reconstructed and the event classification routine classifies them according to the probable main physics channels, producing separate files for the following physics channels identified at
KLOE [74]:
• large angle electron-positron scattering and e+ e− → γγ events
• tagged K + or K − from φ -decays
• tagged KL or KS from φ -decays
• φ → π + π − π 0 decays
• fully neutral photon final states and π + π − + photon final states, from
e.g. φ → ηγ, φ → η γ, e+ e− → π + π − γ, φ → f0 (980)γ , φ → a0 (980)γ.
An event can be classified in several physics channels, and the selection is
not mutually exclusive. The algorithms for the different channels are described
briefly in [74], and in more detail in the internal documentation [78, 79].
The source of η mesons for decay studies at the KLOE experiment is the
radiative decay of the φ meson produced in the electron-positron collision,
i.e., e+ e− → φ → ηγφ . For the η decay into π + π − π 0 , e+ e− → π + π − γγγφ ,
there are thus two charged tracks of opposite curvature and three photons in
the final state. The photon from the φ decay, γφ , has an energy E ∼ 363 MeV.
In the KLOE event classification scheme, these events are found in the
π + π − + photons channel. They are identified by the PPFILT [79] algorithm,
which requires one charged vertex in the interaction region, i.e., a vertex with
2 + y2 < 8 cm and |z | < 15 cm.
two tracks connected to it with R = xvtx
vtx
vtx
Before describing the PPFILT algorithm, definitions of a few variables are
needed.
Let the two tracks connected to the vertex have three-momenta p1 and p2 at
the vertex, reconstructed by the track fitting algorithm, see section 2.2.1. The
two variables PΣ and ΔEγ are defined as:
PΣ = |p1 | + |p2 | and
2
2
2
2
ΔEγ = |p1 + p2 −pφ | − Eφ − mπ + |p1 | − mπ + |p2 | ,
with Eφ = m2φ + |pφ |2
(3.1)
(3.2)
(3.3)
where mφ is the mass of the φ -meson, mπ the mass of the charged pions and
pφ , the three momentum of the φ , is only in the x direction.
In addition to these track variables, some EMC information, see section
2.2.2, is also used. Rclu is the distance from the calorimeter cluster to the
origin of the KLOE coordinate system (which is approximately the interaction
Rclu
, with c the speed of
point) and tclu is the time of the cluster. Then β = c·t
clu
light, is the “velocity” of the cluster. A prompt neutral cluster is defined as a
cluster with 0.8 < β < 1.2 and that is not associated to a track. For each event,
the number of prompt neutral clusters nγ , the total calorimeter energy Etot , the
60
total prompt energy Eγ (sum of energy from the nγ clusters) and the maximum
prompt neutral energy Emax are defined.
There are three different algorithms included in PPFILT, but only algorithm
2 is relevant for the η → π + π − π 0 events.
Algorithm 1
This algorithm looks for ππγ-like events in which the photon does not reach
the EMC but instead goes into the quadrupole calorimeter, i.e., this algorithm
requires that no photons were detected in the EMC. Since for the η → π + π − π 0
one needs three (detected) photons in the final state, this algorithm is not applicable for this set of events.
Algorithm 2
This algorithm identifies mainly φ → Y γ with Y = η, a0 or η and requires a
logical AND between the following conditions:
• 0.15 < PΣ < 0.55 GeV/c,
• −0.7 < ΔEγ < −0.05 ,
• 1.73(PΣ − 0.2) − 0.95 < ΔEγ < 1.73(PΣ − 0.2) − 0.45,
• 350 < Etot < 800 MeV and
• nγ > 3 OR 1 < nγ < 4 AND Emax > 250 MeV.
Algorithm 3
For φ → η γ with η → ργ a third algorithm is needed. This algorithm requires, among other things, that there are exactly two detected photons in the
final state. It is therefore not applicable for the η → π + π − π 0 events.
3.1.3 Analysis Selection
For the signal selection, a few more requirements are made at this stage:
• at least three prompt neutral clusters, i.e., clusters in the electromagnetic
calorimeter
– with polar angle 23◦ < θ < 157◦ , where θ is calculated using the
cluster position, relative to the interaction point (rcluster ). The same
angle requirement is applied relative to the KLOE reference frame
(using Rclu defined above).
– prompt: within the time window
particles,
for massless
2
|
√ 57 ps
| < 5σt = 5 ·
+ (140 ps)2
|tclu − |rcluster
c
Ecluster (GeV)
– neutral: not associated to a track in the Drift Chamber (a cluster gets
associated to a track if it is the closest cluster to the extrapolation
of that track to the EMC, and within a defined maximum distance)
– the cluster energy is at least 10 MeV
61
• at least one of the clusters has energy greater than 250 MeV1 , figure 3.1
illustrates why this value was chosen.
1800
×103
1600
1400
1200
1000
800
600
400
200
0
0
50
100
150
200
250
300
350
400
450
500
energy of the 3 selected photons (MeV)
Figure 3.1. Monte Carlo signal distribution of the calorimeter energies for the three
selected clusters. The peak to the right corresponds to the photon from the φ decay,
with energy 363 MeV. The choice of the selection cut requiring a photon with at least
250 MeV (vertical line) selects this peak, while avoiding the left peak (photons coming
from the π 0 decay).
For the charged tracks reconstructed in the drift chamber, the momentum
at the point of closest approach is used. The two tracks with opposite curvature closest to the interaction point are chosen. In their four-momenta (Pπ +
and Pπ − ), the energy is calculated assuming the mass of the charged pion
mπ + = 139.57018 MeV [7], and the measured momenta. No requirement of
a measured vertex is made here. A check is made which rejects tracks reconstructed as two separate tracks but actually belonging to the same physical
track.
The four-momentum of the η particle is calculated from the φ -meson decay
kinematics: Pη = Pφ − Pγφ . The variable Pφ is determined from the total beam
energy and momentum, which is measured by Bhabha scattering for each run.
For the photon, the position of the cluster with the highest energy is used.
In order to improve the energy resolution, see figure 3.1 and figure 3.2, the
photon energy is calculated by imposing the η meson mass constraint on the
two-body decay:
Eγ =
1 There
m2φ − m2η
2 · Eφ − |pφ | cos θφ ,γ
is a correction factor of 1.014 needed for the EMC measured energy in the MC, this is
implemented at a later stage, so the energy selected here is removing some events in the MC
which should not be removed
62
where m are masses, E energy and p momenta of the particles, and θφ ,γ is the
angle between the φ and the γ momenta. The four-momentum of the photon,
Pγφ , is calculated from the position information and the energy value Eγ .
The π 0 four-momentum is determined from the η and the charged pions
Pπ 0 = Pη − Pπ + − Pπ − .
×103
5000
4000
3000
2000
1000
0
300
310
320
330
340
350
360
370
380
390
400
energy of γ φ (MeV)
Figure 3.2. The calculated energy of the photon from the φ decay, for signal Monte
Carlo.
3.2 Simulations
The official KLOE Monte Carlo production with GEANFI (GEANT 3 based)
is used for the simulation of signal and background events [74]. This simulation includes machine parameters and background conditions, which are
measured on-line and stored for each run to be used in the simulation. The
simulation of production and decay of the φ -meson includes initial state radiation, and final state radiation is included for all simulated channels. The simulation of e+ e− → ωπ 0 , an important background in this analysis, assumes a
cross section of σ = 0.008 μb and takes into account the ω width.
The signal MC is generated with a luminosity scaling factor of 10 (corresponding to 10 times the integrated luminosity of data), to ensure that the
errors from the simulation are small compared to the errors in the data. The
signal events are saved even if they do not pass the event classification or background rejection criteria, so that the evaluation of the efficiency can be made
directly from the simulation. The background simulations, including other
φ decays and e+ e− → hadrons processes, have instead a luminosity scaling
factor of close to one: 1588.5
1595 .
Although not used as a background at the final stage of the analysis, since
it has a negligeble contribution, a Bhabha scattering MC production is used to
check the background from e+ e− elastic scattering. This simulation includes
63
initial and final state radiation and is based on the BABAYAGA generator [80].
The luminosity scaling factor of this production is 0.5, but the sample used
corresponds to only 565 pb−1 of integrated luminosity for data, so compared
to the whole data used the effective luminosity scaling factor is ∼ 0.2. This
background is correspondingly scaled when compared to the data in the next
section.
3.3 Background Rejection
To increase the signal to background ratio, several cuts are made in sequence:
• a cut on the angle between the γ’s and charged π’s momenta,
• a cut on the time-of-flight of the charged π’s,
• a cut on the opening angle between the photons from the π 0 in the π 0
rest frame and
• a cut on the missing mass |Pπ 0 |.
A summary of the efficiency of each cut and the signal to background ratio
after each cut, is shown in table 3.1. The total signal efficiency after the above
cuts is 37.6% and the signal to background ratio is 133.
Table 3.1. Summary of the effect of the cuts on the signal, on the background and on
the signal to background ratio. The background events here include the Bhabha simulation. All simulated events are scaled to correspond to the same integrated luminosity
as data.
Cut
Signal events
(S)
Background
events (B)
Signal
efficiency
S to B ratio
Before
the
cuts
Track-photon
angle
TOF
π 0 γ’s angle
Missing mass
5.895(1) · 106
4.164(3) · 106
44%
1.42
5.862(1) · 106
3.763(2) · 106
99%
1.56
5.757(1) · 106
5.252(1) · 106
4.992(1) · 106
2.902(2) · 106
5.139(7) · 105
3.74(2) · 104
98%
91%
95%
1.98
10
133
Track-photon angle cut
This is a graphical cut on the two dimensional plot: the minimum angle between the momentum of one of the tracks and the π 0 decay photons vs the
minimum angle between the momentum of the other track and the π 0 decay
photons.
The cut is shown in figure 3.3. As can be seen, it does not have a large effect
on signal (preserves 99% of these events), and mostly reduces the Bhabha
background (it rejects 50% of these events).
64
103
π
2.5
2
102
1.5
2
102
1
10
0.5
0.5
1
1.5
2
2.5
3
0
0
1
∠min(pπ-, pγ ) (rad)
0.5
1
1.5
2
2.5
3
1
∠min(pπ-, pγ ) (rad)
Data
Bhabha MC
∠min(p +, pγ ) (rad)
103
2.5
10
0.5
3
103
π
2.5
2
102
1.5
104
3
2.5
103
2
102
1.5
1
10
0.5
0
0
3
1.5
1
0
0
∠min(pπ+, pγ ) (rad)
background MC
3
∠min(pπ+, pγ ) (rad)
∠min(p +, pγ ) (rad)
Signal MC
1
10
0.5
0.5
1
1.5
2
2.5
3
∠min(pπ-, pγ ) (rad)
1
0
0
0.5
1
1.5
2
2.5
3
1
∠min(pπ-, pγ ) (rad)
Figure 3.3. The track-photon angle cut for signal MC, background MC, Bhabha MC
and data. The MC data is scaled to correspond to the same luminosity as the experimental data. The rejected regions mostly affect the Bhabha events.
Time-of-flight cuts
For tracks that have an associated cluster, i.e., for tracks in the DC that have
a cluster in the EMC within a maximum distance of the track position extrapolated to the EMC, a time difference Δt can be calculated. This is the
time difference between the track time, ttrack , and the time of the EMC cluster, tcluster , i.e. Δt = ttrack − tcluster . The track time is calculated in the assumption of a certain mass: it is the time it would take for a particle of that
mass, with the measured momentum, to travel the distance to the calorimeter
√ pc . Two hypothesis for the particle
ttrack = Lv , with v = β · c = c pc
E =c
2
2
p +m
masses are used, giving Δte = ttracke − tcluster assuming an electron or positron
and Δtπ = ttrackπ − tcluster assuming a pion.
Figure 3.4 shows a correlation plot for the two time-of-flight hypotheses,
Δte vs Δtπ , for MC signal, MC background and the data. Positrons and electrons should give a horizontal band around Δte = 0, as is the case for the
Bhabha simulation. The charged pions should give a vertical band around
Δtπ = 0 and this can be seen to be, more or less, reproduced in the signal simulation. This cut mostly rejects Bhabha scattering (rejects more than 99%) and
the signal efficiency is 98%.
65
background MC
Δ t_e (ns)
5
10
104
103
102
10
−2
0
2
4
6
8
10
1
Δ t_π (ns)
Bhabha background
105
104
103
102
10
−2
0
2
4
6
8
10
1
Δ t_π (ns)
Data
Δ t_e (ns)
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
104
103
102
10
−2
0
2
4
6
8
10
Δ t_π (ns)
1
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
Δ t_e (ns)
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
Δ t_e (ns)
Signal
105
104
103
102
10
−2
0
2
4
6
8
10
1
Δ t_π (ns)
Figure 3.4. The time-of-flight cuts, shown in Δte vs Δtπ plots, after the track-photon
cut. Events above the dotted line or above the full line are rejected. The MC data is
scaled to correspond to the same luminosity as the experimental data.
Opening angle cut
The photons from the π 0 decay should be back to back, i.e., 180◦ between
them, in the π 0 rest frame. These photons are selected as the pair of prompt
neutral clusters with the largest opening angle in the π 0 rest frame, but skipping the cluster identified as the radiative photon (γφ ). The calculated angle
is the azimuthal angle, i.e. the angle in the xy plane in the KLOE reference
frame.
Figure 3.5 shows the two photon opening angle. As can be seen, the signal
is peaked at 180◦ and the value 165◦ is chosen as cutoff, corresponding to an
efficiency for signal of 91%. The rejection of background, from φ decays or
e+ e− → hadrons, is at this stage ∼ 80% of the events surviving the previous
cuts (∼ 85% of these background events rejected with the cuts up to this point).
Missing mass cut
The missing mass is calculated as MM = |Pπ 0 | = Pπ20 , where Pπ 0 = Pφ −
Pγφ − Pπ + − Pπ − . For the signal events, this missing mass should correspond
to the mass of the π 0 . Figure 3.6 shows the missing mass squared distribution,
and the applied cut ||Pπ 0 | − mπ 0 | < 15 MeV can also be seen in the figure.
The signal efficiency of this cut is 95% and it rejects ∼ 90% of the hadronic
background surviving the previous cuts.
66
105
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
104
103
102
0
20
40
60
80
100 120 140∠(γ160
180
,γ ) (°)
π
π
0
1
0
2
Figure 3.5. Opening angle between the π 0 decay photons in the π 0 rest frame, for
data, signal MC and background MC. The MC data is scaled to correspond to the
same luminosity as the experimental data. Events to the left of the line at 165◦ are
rejected.
106
105
104
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
103
102
10
1
−250 −200 −150 −100
−50
0
50
P2π0 (MeV2)
×103
Figure 3.6. Pπ20 , for data, signal MC and background MC. The MC data is scaled to
correspond to the same luminosity as the experimental data. Only events between the
two vertical lines are kept.
67
3.4 Data-MC Comparison
As shown in figure 3.5, the MC simulation does not describe exactly the height
of the data distribution. To get a better agreement between data and Monte
Carlo, needed for the background subtraction, we introduce a global scaling
factor for the MC which is fit to the data.
Two linear fits are performed: on the distribution of the angle between the
π 0 decay photons, figure 3.5; and on the missing mass squared Pπ20 , figure 3.6.
Both distributions after the Bhabha rejection cuts are used, the track-photon
angle and time-of-flight cuts described in section 3.3. Three simulated components are fit to the data distribution: signal, ωπ 0 background and the rest
of the background (from the simulation of hadronic background). The result
for the two fits is shown in table 3.2. As can be seen, the χ 2 is quite bad for
both, but much better for the opening angle fit. The resulting scaling factors
are significantly different for each of the fits, and the difference is much bigger
than the errors from the fit. This indicates that the errors are underestimated,
or that the used model is not correct (e.g using four separate background contributions instead of two lowers the χ 2 ). For these reasons, the values of the
scaling factors are taken from the opening angle fit (better χ 2 ) and the errors
in the scaling factors are estimated as the difference between the results of the
two fits.
Table 3.2. Summary of the results for the fits for the scaling factors.
Scaling
factors for
Opening
angle
Pπ20
Signal
ωπ 0
background
rest
background
χ 2 /dof
0.1109(1)
1.530(6)
1.222(3)
7.2 · 103 /497
0.1131(1)
1.839(5)
0.973(3)
7.8 · 104 /497
Figures 3.7 and 3.8 show the resulting distributions where the MC components have been scaled by the factors resulting from the opening angle fit. To
facilitate comparison, figure 3.9 shows the same plots in linear scale and with
a narrower range in the x-axis. As can be seen, there is a better data-MC agreement after applying the scaling factors. An explanation for this could be that
some of the cross-sections and decay probabilities used in the MC simulation
are not accurate enough, but this is corrected for, in the selected region, with
the scaling factors.
The resulting good data MC agreement after all the analysis cuts can be
illustrated with some distributions. Figure 3.10 shows the opening angle and
missing mass squared of figure 3.9, with the corrected scaling of the MC, but
now after all the cuts, including the cuts on the opening angle and missing
mass. Distributions of some other kinematic variables can be seen in figures
3.11, 3.12 and 3.13 and show a good data-MC agreement.
68
106
105
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
104
103
102
0
20
40
60
80 100 120 140
160 180
∠(γ ,γ ) (°)
π
π
0
0
1
2
Figure 3.7. Opening angle between the π 0 decay photons in the π 0 rest frame. Events
to the left of the line at 165◦ are rejected.
106
105
104
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
103
102
10
1
−250 −200 −150 −100
−50
0
50
P2π0 (MeV2)
×103
Figure 3.8. Missing mass squared Pπ20 , where the MC contributions are scaled with the
values from the fit to the π 0 photons’ opening angle histogram. Only events between
the two vertical lines are kept.
69
600 ×10
×103
3
500
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
500
400
400
300
300
200
200
100
100
0
150
155
160
165
170
175
180
∠(γ ,γ ) (°)
π
π
0
1
×103
1600
1400
1200
800
600
400
200
10000
20000
160
165
170
175
180
∠(γ ,γ ) (°)
π
π
0
1
×103
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
0
155
0
2
1000
0
0
150
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
30000
40000
P2π0 (MeV2)
1600
1400
1200
1000
800
600
400
200
0
0
2
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
0
10000
20000
30000
40000
P2π0 (MeV2)
Figure 3.9. Comparing without (left) and with (right) the scaling factors: the opening
angle between the π 0 decay photons in the π 0 rest frame (upper) and missing mass
squared, Pπ20 (lower). The vertical lines show where the cuts are applied.
×103
300
250
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
200
150
100
50
0
160 162 164 166 168 170 172 174 176
178 180
∠(γ ,γ ) (°)
π01 π02
×103
450
400
350
300
250
200
150
100
50
0
10000
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
15000
20000
25000
P2π0 (MeV2)
Figure 3.10. The distributions of the opening angle between the π 0 decay photons in
the π 0 rest frame (left) and the missing mass squared, Pπ20 (right), after all the analysis
cuts. The vertical lines show where the cuts are applied.
70
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
50000
40000
40000
30000
30000
20000
20000
10000
10000
0
0
50
100
150
200+
250
p π (MeV/c)
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
50000
0
0
50
100
150
T
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
50000
30000
30000
20000
20000
10000
10000
50 100 150+ 200 250
pz π (MeV/c)
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
50000
40000
0
250
p π (MeV/c)
T
40000
0
-250-200-150-100 -50
200 -
0
-250-200-150-100 -50
0
50 100 150 - 200 250
pz π (MeV/c)
Figure 3.11. Charged pion momenta, showing data and MC simulation of signal and
background. In the top panels the transverse momentum, in the bottom longitudinal
momentum. On the left the distributions for the π + and on the right for the π − .
×103
180
160
140
120
100
80
60
40
20
0
−1 −0.8 −0.6 −0.4 −0.2 0
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
0.2 0.4 0.6 0.8
1
cos(∠(pπ+, pπ-))
Figure 3.12. Cosine of the angle between the charged pions.
71
35000
×103
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
30000
25000
300
250
200
20000
150
15000
10000
100
5000
50
0
0
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
50
100
150
200
250
300
350
|p 0| (MeV/c)
π
0
300 310 320 330 340 350 360 370 380 390 400
|pη| (MeV/c)
Figure 3.13. Modulus of momenta of the neutral particles: on the left the π 0 , on the
right the η.
72
4. Results
The main purpose of this η → π + π − π 0 study is to present the Dalitz plot distribution of this decay, and to perform a fit to extract the Dalitz plot parameters.
In this chapter, the experimental Dalitz plot and the resolution in the X and Y
variables are presented, then the fit procedure is explained. Results of a test
fit using simulated data are described and shown to produce results consistent
with the input. Next the results from the fit to data are shown, the systematic
effects are studied and the final result is presented. This data can also be used
to extract the charge asymmetries, discussed at the end of this chapter.
4.1 Dalitz Plot and Variable Resolution
The bin width of the Dalitz plot is selected considering both the resolution of
the variables (X and Y ) and the bin content. The width should be significantly
bigger than the resolution to minimize migration between the bins. The number of events in each bin should also be large enough that a χ 2 fit is justified.
For a valid background subtraction with the employed method, the number of
simulated background events in each bin should also be large enough.
The resolution of the X and Y variables is evaluated with the MC simulation. For the signal, the distribution of the difference between the true and
reconstructed values, Xrec − Xtrue shown in figure 4.1(a) and Yrec −Ytrue shown
in figure 4.1(b), is fit with a double Gaussian. The standard deviation of the
inner Gaussian is taken as the resolution of the variable, resulting in:
δ X = 0.021
δY = 0.032
(4.1)
The bin width is chosen to be ∼ 3δX in X and ∼ 3δY in Y , which gives 31
bins in X and 20 bins in Y . With both variables between -1 and 1 this results in
2
2
= 3.07 · δ X and ΔY = 20
= 3.125 · δY . With the selected bin width,
ΔX = 31
the minimum bin content in the data is (3.31 ± 0.06) · 103 events and also the
requirement of a valid background subtraction is ensured.
Figure 4.2 shows the background subtracted experimental Dalitz plot distribution, including only bins fully inside the kinematic boundaries. There are
(4.699±0.007)·106 events in the final background subtracted data Dalitz plot.
73
3
3
×10
×10
χ2 / ndf
p2
7000
3.991e+05 / 94
0.02117 ± 0.00001
6000
χ2 / ndf
p2
4500
3.839e+05 / 94
0.03171 ± 0.00001
4000
3500
5000
3000
4000
2500
3000
2000
1500
2000
1000
1000
0
−0.4
500
−0.3
−0.2
− 0.1
0
0.1
0.2
0.3
0.4
0
−0.4
(a) Resolution of X
−0.3
−0.2
− 0.1
0
0.1
0.2
0.3
0.4
(b) Resolution of Y
Figure 4.1. Resolution for the Dalitz plot variables. The full line is the fit double
Gaussian function whose two components are: the dotted line (inner Gaussian) and
the dashed line (outer Gaussian). The inserts show the relevant results from the double
Gaussian fit, the χν2 and the standard deviation of the inner Gaussian, p2. p2 is used
as the resolution.
4.2 Fit Description
The experimental Dalitz plot distribution is fit to a polynomial expansion in X
and Y of the amplitude squared:
|A(X,Y )|2 N 1 + aY + bY 2 + cX + dX 2 + eXY
+ fY 3 + gX 2Y + hXY 2 + lX 3
(4.2)
to obtain the coefficients a, b, . . ., the Dalitz plot parameters, see section 1.2.2.
Note that c, e, h and l must be zero assuming charge conjugation invariance.
The fit is performed by minimizing the χ 2 like function
χ2 =
Nbins
∑
i=1
Ni − ∑Nbins
j=1 ε̃ j Si j Ntheory, j
σi
2
(4.3)
where:
• Ntheory, j = |A(X,Y )|2 dPh(X,Y ) j , with |A(X,Y )|2 given above. The integral is over X and Y in the allowed phase space for bin j, see section
4.2.1.
• Ni = Ndata,i − s1 Bi1 − s2 Bi2 is the background subtracted data content in
Dalitz plot bin i. The scaling factors s are calculated as explained in
section 3.4. Bi1 is the ωπ 0 background in the bin i and Bi2 is the same
for the rest of the background.
• ε̃ j is the acceptance of Dalitz plot bin j and Si j the smearing matrix
from bin j to bin i in the Dalitz plot. In reality ŝi j = Si j · ε̃ j is used,
i.e., a matrix which includes the acceptance and smearing, calculated
N
j
from signal MC by ŝi j = rec,i;gen,
Ngen, j , where Nrec,i;gen, j denotes the number
74
25000
20000
15000
10000
5000
0
1 0.8
0.6 0.4
0.2 0
−0.2−0.4
X
−0.2−0.4
0.2 0
0.6 0.4
−0.6−0.8
0.8
−1 1
−0.6−0.8
−1
Y
Figure 4.2. The experimental background subtracted Dalitz plot distribution.
of events reconstructed in bin i which were generated in bin j. This
N
∑ N
j
and Si j = ∑ Nrec,i;gen, j .
corresponds to ε̃ j = k Nrec,k;gen,
gen, j
k
rec,k;gen, j
• the error in bin i, σi2 = σN2i + σŝ2i j , assumes independent variables Ni and
ŝi j
– σN2i = Ndata,i + s21 · Bi1 + σs21 · B2i1 + s22 · Bi2 + σs22 · B2i2 by error propagation assuming independent variables, where the errors in number
of events are given by Poisson statistics.
ŝi j ·(1−ŝi j )
2
– σŝ2i j = ∑Nbins
j=1 Ntheory, j · Ngen, j : the error in the acceptance and
smearing matrix is calculated in the same way as an efficiency error
(using binomial distribution), assuming negligible error in Ntheory, j .
In the fit, for i only the bins completely inside the physical border are used,
the bins shown in figure 4.3. For Ntheory , i.e., for the bins j, all the bins at least
partly inside the physical border are taken into account, and MC integration is
used to account for the Dalitz plot boundary.
4.2.1 Phase Space Integrals
For each bin j, the integral needed for equation 4.3 is:
Ntheory, j =
N 1 + aY + bY 2 + cX + dX 2 + eXY
+ fY 3 + gX 2Y + hXY 2 + lX 3 dXdY
(4.4)
where the integration is only inside the physical boundary. This means that
for the bins crossing the border, the integral is not bounded by the bin but
by the kinematic border. In order to use the same method for calculating the
75
Y
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0
25000
20000
15000
10000
5000
0.2 0.4 0.6 0.8
1
0
X
Figure 4.3. The experimental Dalitz plot distribution (color) and the line of the physical border in red.
integral for all bins, and to ensure that the correct kinematic border is taken
into account, Monte Carlo integration is used.
The fit parameters (N, a, b, . . .) are constants with respect to the integration,
they can therefore be taken outside the integral and equation 4.4 becomes:
Ntheory, j = N
+ Nd
+ Nh
dXdY + Na
X 2 dXdY + Ne
XY dXdY + Nl
Y dXdY + Nb
XY dXdY + N f
2
Y dXdY + Nc
2
Y 3 dXdY + Ng
XdXdY
X 2Y dXdY
X 2 dXdY
= N(O j + aA j + bB j + cC j + dD j + eE j + f Fj + gG j + hH j + lL j )
(4.5)
with the short-hand notation for the integrals
Oj =
dXdY j ,
Aj =
Y dXdY j ,
Bj =
Y 2 dXdY j ,
. . . (4.6)
The subscript j indicates that the integrals are dependent on the bin j.
This way, the integrals O j , A j , B j , . . . need only be evaluated once, for each
bin j, and not in each iteration of the fit.
The Monte Carlo integration is done using the program ROOT [81]. For
each bin j at least partially inside the Dalitz plot boundary, random (X,Y ) pairs
are generated until 1 · 108 pairs are accepted as inside the physical border. The
generated number of pairs is saved in a histogram. Then if the pair is inside the
physical boundary (checked as described in section 1.2.1), other histograms
are filled: one with weight one, one with weight Y , one with weight Y 2 and
so on. These histograms are then divided by the histogram of generated pairs,
76
and later multiplied by the bin size, so that each bin in the new histograms
contains the corresponding integral over that bin.
Integration errors
As mentioned on page 75, the error in Ntheory, j is neglected in equation 4.3.
Since the parameters a, b, . . ., as well as their errors, are calculated in the fit,
the only contribution to the error in the fit, from Ntheory, j , would come from
the integrals. If an analytical integration was used, the error arising from the
integrals would be non-existent. The non-trivial Dalitz plot boundary makes
an analytical integration unpractical, therefore a numerical integration is used
instead, and the errors due to the integration procedure should be evaluated
and shown to be small enough to neglect.
The integrals and the smearing matrix have uncorrelated errors, so for each
pair of generated and reconstructed bins (i, j), the error becomes:
2
2 2
2 2
σteor,i
j = σNt ŝi j + σŝ Ntheory, j
(4.7)
ŝi j · (1 − ŝi j )
Ngen, j
and σN2t = N 2 σO2 j + a2 σA2 j + b2 σB2 j + . . .
with σŝ2 =
(4.8)
(4.9)
(cf. σŝ2i j above), assuming also that the errors in the different integrals (σO2 j ,
2
σA2 j , . . .) are uncorrelated. A sum over j then gives the error σteor,i
for each
experimental bin i. Obviously, σN2t will depend on how many Dalitz plot parameters are fit and on the fit result.
For each bin j, the error on the integral is saved as the error in the corresponding histogram bin. The error is calculated as a binomial distribution
error with ROOT. The integration errors can then safely be neglected, if for
2
8
each bin pair (i, j) it is true that σN2 ŝ2i j << σs2 Ntheory,
j . For example, for 1 · 10
accepted events per bin in the integral calculation, and using the results of the
a, b, d, f , g parameter fit, the maximum value of
σN2 ŝ2i j
2
σs2 Ntheory,
j
is 0.000905. Includ-
ing all the parameters (a, b, c, d, e, f , g, h, l) the maximum value of
σN2 ŝ2i j
2
σs2 Ntheory,
j
is
2
0.000905013. So the σN2 ŝ2i j << σs2 Ntheory,
j condition is fulfilled for both the
tested cases and the errors arising from the numerical integration can be safely
neglected.
4.3 Fit Test on MC
The fit procedure was tested with a MC simulation, using as data the signal
included in the MC background production, i.e., the simulation of φ decays
and e+ e− → hadrons, with luminosity scaling factor close to one – see section
77
3.2. The signal in this production is omitted in the standard analysis. The
background subtraction is ignored by setting the background scaling factors
to zero. The same MC production is used as signal as in the final fit, with
luminosity scaling factor 10.
Table 4.1 shows the result for five sets of parameters tested, as well as the
generated values in the simulation. As can be seen, sets one to four have a
good χ 2 and the fit parameters are well reproduced. This indicates that the
fitting procedure is working as desired. As an extra check, a fit with fewer
parameters is done, set five. This results in a bad fit, which is expected since
the parameter d is set to zero while its true value in the MC is not zero, which
further shows the reliability of the fit procedure.
Table 4.1. Test of the Dalitz plot parameter fit with simulated events instead of data.
The table is presented in two parts, and the rows are numbered. The rows in red
indicate the true values used in the simulation. The number of degrees of freedom
is 371 − n param , where n param is the number of parameters of the fit, including the
normalization parameter N. A “-” indicates that the parameter was fixed at 0.
1
2
3
4
5
1
2
3
4
5
a
b · 101
c · 103
d · 102
e · 103
−1.040 ± 0.002
−1.039 ± 0.003
−1.038 ± 0.003
−1.038 ± 0.003
−1.022 ± 0.001
−1.040
1.417 ± 0.028
1.420 ± 0.028
1.423 ± 0.030
1.423 ± 0.029
1.196 ± 0.026
1.400
1.89 ± 3.32
0
6.02 ± 0.26
6.00 ± 0.26
6.08 ± 0.31
6.08 ± 0.31
6.00
−2.78 ± 3.55
0
f · 101
g · 102
h · 102
l · 103
χ2
Prob
−0.03 ± 0.06
−0.04 ± 0.07
−0.04 ± 0.07
0
−0.40 ± 0.86
−0.40 ± 0.87
0
0.20 ± 0.87
0
−6.89 ± 6.36
0
333
333
332
329
878
0.90
0.89
0.89
0.88
10−43
As the parameters are correlated, as an example the covariance matrix for
set four (all parameters included) is presented in table 4.2.
4.4 Fit Results
The fit, see section 4.2, has been performed for different sets of Dalitz plot
parameters for data. The normalization parameter N is always included, as
are the a, b and d parameters, but which of the other parameters are included
varies. Specifically, the charge conjugation violating parameters c, e, h and l
78
Table 4.2. The covariance matrix for the MC test fit set four from table 4.1.
a
b
c
d
e
f
g
h
b
c
d
e
f
g
h
l
-0.075
0.009
0.005
0.033
0.403
0.004
-0.006
-0.001
0.141
-0.004
-0.854
-0.251
-0.001
-0.157
-0.002
-0.538
-0.238
-0.005
-0.534
0.006
0.405
-0.006
-0.005
-0.649
-0.002
-0.502
0.008
0.002
-0.010
-0.007
-0.886
-0.005
-0.182
0.002
0.006
0.447
can in this way be tested as for their consistency with zero. The results can be
seen in table 4.6.
The first block of table 4.6 serves to compare the result for all the parameters
up to order three in the polynomial with the results when one parameter is
fixed at zero, also referred to as excluded from the fit. The parameters a, b
and d are well established, from previous experiments, to deviate from zero
and are therefore always free in the fit. As can be seen from the χ 2 , fixing
the parameters c, e, h and l to zero has almost no impact on the goodness of
the fit. When not fixed to zero, these parameters are consistent with zero
within 2σ (except for parameter c when l is excluded and vice versa, but these
are consistent with zero within 2.3σ ). The full covariance matrix when all
parameters are left free is given in table 4.3.
Table 4.3. The covariance matrix when all Dalitz plot parameters up to order three
are left free in the fit.
a
b
c
d
e
f
g
h
b
c
d
e
f
g
h
l
-0.113
0.002
-0.014
0.018
0.383
0.015
0.000
-0.009
0.161
0.017
-0.849
-0.215
0.016
-0.126
0.025
-0.524
-0.213
-0.015
-0.540
-0.026
0.373
-0.024
0.031
-0.651
-0.017
-0.505
-0.012
0.042
0.008
0.009
-0.882
-0.021
-0.214
-0.026
0.011
0.443
As can be seen, the parameters c and l are strongly correlated, and excluding one of them will have a big impact on the other, which may explain the
observation above.
It can also be noted that the f parameter is needed in the fit, since fixing
it at zero drastically worsens the χ 2 and significantly shifts the values of the
parameters which are not consistent with zero. The inclusion of the g param79
eter also improves the fit result, and this parameter differs from zero at a 4.9σ
level (4σ if h is excluded).
The second block in table 4.6 can be seen as the opposite of block one,
as here the parameters are added to the fit one by one. The first reconfirms
that the f parameter really is needed as it drastically changes the result of
the fit and its χ 2 . The other rows then include a, b, d and f and one of the
remaining parameters. Even here it is seen that c, e, h and l are consistent with
zero within 2σ and that including them neither improves the fit nor changes
the other parameters. Parameter g, on the other hand, improves the probability
of the fit from 0.24 to 0.56 and is consistent with zero only at the 4.9σ level.
The third block is a last check on the charge conjugation violating parameters. Here, the first row is just the best fit repeated to ease comparison, and
then the parameters c, e, h and l are added one at a time. As can be seen from
these rows, these parameters do not change the goodness of the fit much, they
are all consistent with zero within 2σ , and including them does not change the
a, b, d, f and g parameters in any significant way.
Since c, e, h and l are, for all our tests, consistent with zero and do not
alter the result of the other parameters, our final results are reported as the fits
without these parameters. As for the g parameter, both the fits with a, b, d and
f and with a, b, d, f and g have good χ 2 and while g is deviates from zero at
the 4.9σ level, this is not conclusive evidence for the need of this parameter.
So for completeness, both results and also the covariance matrices for both fits
are presented in tables 4.4 and 4.5.
Table 4.4. The covariance matrix for the fit with parameters a, b, d and f .
a
b
d
b
d
f
-0.269
-0.365
0.333
-0.832
-0.139
0.089
Table 4.5. The covariance matrix for the fit with parameters a, b, d, f and g.
a
b
d
f
b
d
f
g
-0.120
0.044
0.389
-0.859
-0.201
-0.160
-0.534
-0.225
-0.557
0.408
Figures 4.4-4.7 show a comparison of the data and the fit. In figure 4.4,
slices in the X variable are shown, hence one sees the Dalitz plot as a function
of Y for each bin in X. The figure shows the background subtracted data and
80
the smeared theoretical function calculated using the parameters reported in
table 4.6 and the integrals from section 4.2.1, for both parameter sets a, b, d, f
and a, b, d, f , g. Figure 4.5 shows the same for slices in the Y variable. From
both these figures one sees that the agreement between the data and the two
fits is good and the two fits are practically not distinguishable.
To better see the differences between the data and the fits, one can look at
the difference between the data and the smeared fit instead, so called residuals.
This is done in figures 4.6 and 4.7, where the first corresponds to the difference
between data and the fit functions in slices in X and the second the same but
in slices of Y . The errors are calculated as the sum in quadrature of the data
error and the error coming from the smearing matrix, but ignoring the error
in the parameters. These figures show that the data and fit are in agreement
within 3.3σ for every bin, for both sets of parameters (although the agreement
is better than so for most bins).
81
82
1.533 ± 0.028
1.420 ± 0.029
1.420 ± 0.029 −1.66 ± 1.08
1.420 ± 0.029
1.454 ± 0.030
1.420 ± 0.028
1.420 ± 0.029
-
1.454 ± 0.030
1.454 ± 0.030 −1.66 ± 1.09
1.454 ± 0.030
1.454 ± 0.032
1.454 ± 0.030
-
1.104 ± 0.002
1.104 ± 0.003
1.104 ± 0.003
1.104 ± 0.003
1.095 ± 0.003
1.104 ± 0.003
1.104 ± 0.003
1.095 ± 0.003
1.095 ± 0.003
1.095 ± 0.003
1.095 ± 0.003
1.095 ± 0.003
−4.34 ± 3.39
−4.68 ± 3.44
−4.29 ± 3.45
−4.33 ± 3.39
−1.66 ± 2.54
−3.84 ± 1.66
1.454 ± 0.030
1.454 ± 0.031
1.454 ± 0.031
1.598 ± 0.029
1.419 ± 0.031
1.454 ± 0.030
1.454 ± 0.030
1.095 ± 0.003
1.095 ± 0.003
1.095 ± 0.003
1.035 ± 0.002
1.104 ± 0.003
1.095 ± 0.004
1.095 ± 0.003
c · 103
b · 101
−a
2.52 ± 3.20
3.20 ± 3.71
2.45 ± 3.62
2.46 ± 3.67
4.69 ± 3.25
2.64 ± 3.55
e · 103
8.11 ± 0.33
8.11 ± 0.32
8.11 ± 0.32 1.53 ± 2.77
8.11 ± 0.36
8.11 ± 0.32
-
6.75 ± 0.27
7.26 ± 0.27
7.26 ± 0.27
7.26 ± 0.27 1.49 ± 2.70
8.11 ± 0.33
7.26 ± 0.27
7.26 ± 0.27
-
8.11 ± 0.32
8.12 ± 0.33
8.11 ± 0.33
9.14 ± 0.33
7.26 ± 0.28
8.11 ± 0.34
8.11 ± 0.32
d · 102
−4.37 ± 0.89
−4.37 ± 0.89
−4.37 ± 0.89
−11.66 ± 0.84
−4.37 ± 1.10
−4.37 ± 0.90
g · 102
1.07 ± 0.90
0.33 ± 0.68
1.37 ± 0.84
1.06 ± 0.90
1.07 ± 0.89
1.00 ± 0.82
h · 102
1.08 ± 6.54
−6.22 ± 3.03
1.96 ± 6.61
1.03 ± 6.72
1.09 ± 6.46
−2.43 ± 5.72
-
l · 103
1.41 ± 0.07
1.41 ± 0.07
1.41 ± 0.07
1.41 ± 0.07
1.41 ± 0.07
−4.37 ± 0.89
−4.37 ± 0.88
−4.37 ± 0.88
−4.37 ± 0.88 0.07 ± 0.49
−4.37 ± 0.88
−4.02 ± 2.57
1.54 ± 0.06
1.54 ± 0.06
1.54 ± 0.06
1.41 ± 0.07 −4.37 ± 0.89
1.54 ± 0.06
0.07 ± 0.48
1.54 ± 0.06
−4.00 ± 2.59
1.41 ± 0.07
1.41 ± 0.07
1.41 ± 0.07
1.54 ± 0.06
1.41 ± 0.08
1.41 ± 0.07
f · 101
Prob
360
358
360
360
358
1007
385
383
385
360
385
383
0.56
0.58
0.55
0.55
0.58
10−60
0.24
0.25
0.28
0.56
0.23
0.25
354 0.60
356 0.58
354 0.60
792 10−34
379 0.26
355 0.59
354 0.61
χ2
Table 4.6. Result of the fit for the Dalitz plot parameters. In red the best fit, chosen as standard result. In blue the fit for a, b, d and f , which
also has a good χ 2 and can be directly compared to previous KLOE and WASA results. There are for all rows 371 bins used, so the degrees of
freedom are 371 − n param , where n param is the number of parameters of the fit, including the normalization parameter N.
83
0
0.5
Y
1
0.5
0
0.5
Y
1
0
0.5
−0.5
0
0.5
0.55 < X < 0.61
−0.5
Y
1
−1
5000
10000
15000
−1
5000
10000
0
0.5
0
0.5
0
0.5
−0.5
0
0.5
0.61 < X < 0.68
−0.5
0.16 < X < 0.23
−0.5
-0.29 < X < -0.23
−0.5
-0.74 < X < -0.68
Y
Y
Y
Y
1
1
1
1
0
0.5
0
0.5
0
0.5
0.5
Y
1
1
20000
−1
5000
10000
15000
20000
25000
−1
5000
−1
10000
15000
−1
5000
−0.5
0
0.5
0.68 < X < 0.74
Y
1
−1
8000
10000
12000
14000
16000
−1
−0.5
0
0.5
0.74 < X < 0.81
Y
1
8000
−1
10000
12000
−1
15000
5000
0
Y
1
10000
−0.5
0.29 < X < 0.35
−0.5
Y
5000
1
0.5
10000
Y
0
-0.16 < X < -0.10
−0.5
10000
−1
5000
10000
15000
20000
25000
−1
5000
15000
0.5
1
1
10000
15000
20000
15000
0
Y
Y
-0.61 < X < -0.55
20000
−0.5
0.23 < X < 0.29
−0.5
-0.23 < X < -0.16
−0.5
10000
15000
20000
20000
−1
5000
10000
15000
20000
25000
−1
5000
10000
15000
-0.68 < X < -0.61
0
0.5
0
0.5
0
0.5
−0.5
0
0.5
0.81 < X < 0.87
−0.5
0.35 < X < 0.42
−0.5
-0.10 < X < -0.03
−0.5
-0.55 < X < -0.48
Y
Y
Y
Y
1
1
1
1
−1
5000
10000
15000
20000
−1
5000
10000
15000
20000
25000
−1
5000
10000
15000
20000
0
0.5
0
0.5
0
0.5
abdf
abdfg
Data
−0.5
0.42 < X < 0.48
−0.5
-0.03 < X < 0.03
−0.5
-0.48 < X < -0.42
Figure 4.4. Dalitz plot distribution in slices in X, with experimental data and two fit results.
−1
5000
−0.5
5000
−1
10000
15000
20000
−1
10000
15000
20000
0.5
0.48 < X < 0.55
0
5000
5000
−0.5
10000
10000
−1
15000
15000
20000
−1
15000
1
1
5000
10000
15000
20000
−1
20000
Y
Y
1
20000
1
0
0.10 < X < 0.16
−0.5
Y
25000
Y
0.5
10000
15000
25000
0.03 < X < 0.10
−1
0
-0.35 < X < -0.29
−0.5
-0.81 < X < -0.74
25000
−1
5000
5000
20000
−1
10000
0.5
1
10000
0
Y
15000
−0.5
-0.42 < X < -0.35
−0.5
8000
10000
12000
14000
16000
15000
20000
8000
−1
10000
12000
-0.87 < X < -0.81
Y
Y
Y
1
1
1
84
−1
7000
7500
8000
8500
−1
16000
16500
−1
24000
24500
25000
25500
0
0.5
0
0.5
0
0.5
1
X
1
X
1
X
−1
6000
6500
7000
−1
14000
14500
15000
−1
23500
24000
24500
25000
0
0.5
0
0.5
−0.5
0
0.5
0.40 < Y < 0.50
−0.5
-0.20 < Y < -0.10
−0.5
-0.80 < Y < -0.70
1
X
1
X
1
X
−1
5000
5500
6000
−1
12500
13000
13500
−1
21000
22000
23000
0
0.5
0
0.5
−0.5
0
0.5
0.50 < Y < 0.60
−0.5
-0.10 < Y < 0.00
−0.5
-0.70 < Y < -0.60
1
X
1
X
1
X
−1
4000
4500
−1
11000
11500
12000
−1
20500
21000
21500
0
0.5
0
0.5
−0.5
0
0.5
0.60 < Y < 0.70
−0.5
0.00 < Y < 0.10
−0.5
-0.60 < Y < -0.50
1
X
1
X
1
X
−1
3200
3400
3600
−1
9500
10000
10500
11000
−1
19000
19500
20000
0
0.5
0
0.5
−0.5
0
0.5
0.70 < Y < 0.80
−0.5
0.10 < Y < 0.20
−0.5
-0.50 < Y < -0.40
1
X
1
X
1
X
−1
8000
8500
9000
9500
−1
17500
18000
0
0.5
0
abdf
abdfg
Data
−0.5
0.5
0.20 < Y < 0.30
−0.5
-0.40 < Y < -0.30
Figure 4.5. Dalitz plot distribution in slices in Y , with experimental data and two fit results.
−0.5
0.30 < Y < 0.40
−0.5
-0.30 < Y < -0.20
−0.5
-0.90 < Y < -0.80
1
X
1
X
85
-0.87 < X < -0.81
−100
0
100
0.55 < X < 0.61
−1 −0.8 −0.6 −0.4 −0.2
−400
−200
0
200
400
0.10 < X < 0.16
−1 −0.8 −0.6 −0.4 −0.2
0
0
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
Y
1
Y
1
Y
0
100
200
300
0.61 < X < 0.68
−400
−1 −0.8 −0.6 −0.4 −0.2
−300
−200
−100
0
100
200
300
0.16 < X < 0.23
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
300
-0.29 < X < -0.23
−1 −0.8 −0.6 −0.4 −0.2
−200
0
0
0
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
Y
1
Y
1
Y
1
Y
50
100
150
200
250
0.68 < X < 0.74
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
300
400
0.23 < X < 0.29
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
300
-0.23 < X < -0.16
−1 −0.8 −0.6 −0.4 −0.2
−300
−200
−400
−1 −0.8 −0.6 −0.4 −0.2
−300
−200
−1 −0.8 −0.6 −0.4 −0.2
−300
−200
−100
−1 −0.8 −0.6 −0.4 −0.2
−200
−150
−100
0
0
0
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
Y
1
Y
1
Y
1
Y
-0.61 < X < -0.55
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
0.74 < X < 0.81
−1 −0.8 −0.6 −0.4 −0.2
−400
−300
−200
−100
0
100
200
300
400
0.29 < X < 0.35
−1 −0.8 −0.6 −0.4 −0.2
−400
−200
0
200
400
-0.16 < X < -0.10
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
0
0
0
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
Y
1
Y
1
Y
1
Y
-0.55 < X < -0.48
−300
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
300
0.81 < X < 0.87
−1 −0.8 −0.6 −0.4 −0.2
−500
−400
−300
−200
−100
0
100
200
300
0.35 < X < 0.42
−1 −0.8 −0.6 −0.4 −0.2
−400
−300
−200
−100
0
100
200
300
-0.10 < X < -0.03
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
0
0
0
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
Y
1
Y
1
Y
1
Y
-0.48 < X < -0.42
0
0
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
abdf
abdfg
−1 −0.8 −0.6 −0.4 −0.2
−200
−100
0
100
200
300
0.42 < X < 0.48
−400
−1 −0.8 −0.6 −0.4 −0.2
−200
0
200
400
600
-0.03 < X < 0.03
−1 −0.8 −0.6 −0.4 −0.2
−300
−200
−100
0
100
200
300
1
Y
1
Y
1
Y
Figure 4.6. Difference between the experimental Dalitz plot distribution and each of the two fits, slices in X.
−500
−1 −0.8 −0.6 −0.4 −0.2
−400
−300
−50
1
Y
1
Y
1
Y
1
Y
−100
0
100
200
300
−200
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
−100
0
100
-0.68 < X < -0.61
400
500
0
0
0
0
−400
−200
0
200
400
0
-0.74 < X < -0.68
200
300
−100
0
100
200
300
0.48 < X < 0.55
−1 −0.8 −0.6 −0.4 −0.2
−400
−300
−200
−100
0
100
200
300
400
0.03 < X < 0.10
−1 −0.8 −0.6 −0.4 −0.2
−300
−200
−100
0
100
200
300
-0.35 < X < -0.29
−300
−1 −0.8 −0.6 −0.4 −0.2
-0.42 < X < -0.35
−200
1
Y
−300
−1 −0.8 −0.6 −0.4 −0.2
0.2 0.4 0.6 0.8
−200
0
100
200
−100
0
-0.81 < X < -0.74
300
−100
0
100
200
300
86
-0.90 < Y < -0.80
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
X
1
X
1
X
-0.80 < Y < -0.70
−1 −0.8 −0.6 −0.4 −0.2 0
−300
−200
−100
0
100
200
300
0.40 < Y < 0.50
−400
−1 −0.8 −0.6 −0.4 −0.2 0
−300
−200
−100
0
100
200
300
-0.20 < Y < -0.10
−1 −0.8 −0.6 −0.4 −0.2 0
−400
−200
0
200
400
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
X
1
X
1
X
-0.70 < Y < -0.60
−300
−1 −0.8 −0.6 −0.4 −0.2 0
−200
−100
0
100
200
0.50 < Y < 0.60
−300
−1 −0.8 −0.6 −0.4 −0.2 0
−200
−100
0
100
200
300
-0.10 < Y < 0.00
−1 −0.8 −0.6 −0.4 −0.2 0
−400
−200
0
200
400
600
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
X
1
X
1
X
-0.60 < Y < -0.50
−250
−1 −0.8 −0.6 −0.4 −0.2 0
−200
−150
−100
−50
0
50
100
150
200
0.60 < Y < 0.70
−1 −0.8 −0.6 −0.4 −0.2 0
−300
−200
−100
0
100
200
300
0.00 < Y < 0.10
−500
−1 −0.8 −0.6 −0.4 −0.2 0
−400
−300
−200
−100
0
100
200
300
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
X
1
X
1
X
−1 −0.8 −0.6 −0.4 −0.2 0
−100
−50
0
50
100
0.70 < Y < 0.80
−300
−1 −0.8 −0.6 −0.4 −0.2 0
−200
−100
0
100
200
300
0.10 < Y < 0.20
−1 −0.8 −0.6 −0.4 −0.2 0
−400
−200
0
200
400
-0.50 < Y < -0.40
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
1
X
1
X
1
X
-0.40 < Y < -0.30
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
abdf
abdfg
−1 −0.8 −0.6 −0.4 −0.2 0
−300
−200
−100
0
100
200
300
0.20 < Y < 0.30
−1 −0.8 −0.6 −0.4 −0.2 0
−400
−300
−200
−100
0
100
200
300
1
X
1
X
Figure 4.7. Difference between the experimental Dalitz plot distribution and each of the two fits, slices in Y .
−1 −0.8 −0.6 −0.4 −0.2 0
−200
−100
0
100
200
300
0.30 < Y < 0.40
−1 −0.8 −0.6 −0.4 −0.2 0
−400
−300
−200
−100
0
100
200
300
400
500
-0.30 < Y < -0.20
−1 −0.8 −0.6 −0.4 −0.2 0
−500
−400
−300
−200
−100
0
100
200
300
4.5 Systematic Uncertainties
To quantify and account for systematic effects, several checks have been made.
For the most part, these checks consist of changing one cut in the analysis and
running the analysis once more, arriving at a new result for the Dalitz plot
parameters (a, b, d, f , g or a, b, d, f ).
4.5.1 Minimum Photon Energy Cut
The effect of the cut on the minimum energy of the photons is studied by
making this cut more stringent. In the standard analysis, the minimum energy
of photons is 10 MeV, and this is varied to 15 and 20 MeV. The results are
shown in figures 4.8 and 4.9.
Figure 4.8. Variation for the Dalitz plot parameters a, b, d and f with varying the
minimum photon energy. The point at 10 MeV is the standard analysis.
The systematic error from the minimum photon energy cut is calculated as
a symmetric error of ± half the difference between the standard result and the
20 MeV result. For the numerical values see section 4.5.9.
87
Figure 4.9. Variation for the Dalitz plot parameters a, b, d, f and g with varying the
minimum photon energy. The point at 10 MeV is the standard analysis.
88
4.5.2 Background Subtraction
The background is calculated in each bin of the Dalitz plot from MC simulation and scaled by a factor obtained from a fit to data as described in section
3.4. The fit uses the distributions of the opening angle between the π 0 photons
and the squared missing mass, Pπ20 , for all events, without any constraints on
the X and Y variables. The standard background subtraction is done by simply
subtracting the background content in each bin from the data.
To check the effect of the chosen background subtraction method, another
method has been used. It consists of essentially the same steps, except that the
fit to the two distributions is made separately for each bin included in the Dalitz
plot. The data signal content in bin i is then calculated as Ni = Ndata,i −si1 Bi1 −
si2 Bi2 , where now the scaling factors si1 and si2 are calculated for each bin.
Figure 4.10 shows the two distributions after the fit, for two test bins. The bins
chosen as examples are extremes in the number of events, bin NX , NY = 16, 2
being the bin with largest number of events, and bin NX , NY = 15, 18 the one
with smallest number of events, at the final Dalitz plot. As it can be seen, the
agreement is quite good, but the statistics used are, by necessity, much worse
than in the global fit.
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
104
3
10
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
3
10
102
2
10
10
10
0
20
40
60
80 100 120 140
160 180
∠(γ ,γ ) (°)
0
π01 π02
104
103
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
40
60
80 100 120 140
160 180
∠(γ ,γ ) (°)
π01 π02
103
102
20
DATA
MC SUM
Signal
ω π0 bkg
sum other bkg
102
10
1
10
−1
10
−60000−40000−20000
0
20000 40000
P2π0 (MeV2)
−60000−40000−20000
0
20000 40000
P2π0 (MeV2)
Figure 4.10. Top: azimuthal opening angle between the π 0 decay photons in the π 0
rest frame. Bottom: missing mass squared, Pπ20 . Left: bin NX , NY = 16, 2, the bin with
the largest data content. Right: bin NX , NY = 15, 18, the bin with the smallest data
content. The MC is scaled with the scaling factors from the fit to the opening angle
distribution, for the corresponding bin.
89
Figure 4.11 shows the results from the scaling factor fit for each bin, and
also for the global fit. The scaling factor is taken from the π 0 photons’ opening
angle fit, and the difference to the fit result from the squared missing mass
distribution, Pπ20 , is taken as the error, as in the global fit. Looking at the upper
plot, one sees that the signal scaling factor changes quite a lot, but this could be
because the Dalitz plot parameters used in generating the signal do not agree
with the experimental results from this thesis. The scaling factors of the two
backgrounds used also vary a bit, but are closer to the value from the global
fit.
As systematic error coming from the background subtraction we take half
the difference between the Dalitz plot parameters in the standard analysis and
the new bin-by-bin background subtraction analysis, and use this as a symmetric error, see section 4.5.9 for numerical results.
90
Figure 4.11. Scaling factors plotted against bin number. The green line shows the
result from the global fit, the blue line shows the weighted mean of all the points. Top
shows signal, middle the ωπ 0 and bottom the rest of the background.
91
4.5.3 Choice of Binning
To test the influence of our binning choice on the results, the number of bins
has been varied. The chosen bin width for the standard analysis corresponds
to ∼ 3δ in X and Y , and for this systematic test the bin width is allowed to vary
from ∼ 2δ to ∼ 5δ in both variables, for a total of 10 configurations, see table
4.7. The configurations were chosen such that they all have cX ≈ cY where c
is the number multiplying the resolution of each variable (cX δX and cY δY ).
Table 4.7. Number of bins used for the systematic check on binning and the corresponding resolution factor.
X
Y
Number of bins
Bin width
Number of bins
Bin width
47
42
38
35
31
29
26
23
21
19
2.03δX
2.27δX
2.51δX
2.72δX
3.07δX
3.28δX
3.66δX
4.14δX
4.54δX
5.01δX
31
28
25
23
20
19
17
15
14
12
2.02δY
2.23δY
2.50δY
2.72δY
3.12δY
3.29δY
3.68δY
4.17δY
4.46δY
5.21δY
For each bin configuration the analysis is repeated. The extracted Dalitz
plot parameters as a function of (cX + cY )/2 (labeled as delta) is shown in
figures 4.12 and 4.13. As can be seen the parameters are consistent over a big
range of bins.
The systematic error from the binning is calculated as the weighted standard
deviation, σa , using the weighted mean, ā, of these configurations. Using the
squared error in each parameter obtained from the fit as the inverse of the
weight, gives the formulas:
a
i
∑10
i=1 σa2
∑10
i=1 ai wi
= 10 1 i
ā = 10
w
∑i=1 i
∑i=1 σ 2
(4.10)
ai
(a −ā)2
i
∑10
2
i=1 σa2
∑10
i=1 (ai − ā) wi
i
=
σa =
10 1
w
∑10
∑
i=1 i
i=1 σ 2
(4.11)
ai
The formulas for the other parameters are simply obtained by substituting a
by the relevant parameter and the numerical results are found in section 4.5.9.
It is worth noting that, due to the long computing time, two bin settings
used a different calculation of the integrals. This concerns the bin settings
with 42 bins in X and 28 in Y and 21 bins in X and 14 in Y . For these, the
92
Figure 4.12. Variation for the Dalitz plot parameters a, b, d and f with the bin width,
in units of the resolution. The green lines show the results of the standard analysis and
the blue line the weighted mean value.
integration required 1 · 108 generated events in each bin, instead of this number
of accepted events within the physical border in each bin. This only makes a
difference for the border bins. For these bin settings, the integral errors were
checked as in section 4.2.1. For the other bin settings the errors are assumed
small enough. For the (42,28) binning the integral errors were bigger and
σN2 ŝ2i j
2
2
σs Ntheory,
j
Actually,
< 0.02, which should still be enough to safely neglect these errors.
σN2 ŝ2i j
2
2
σs Ntheory,
j
(21,14) binning
< 0.01, for all but one theoretical to data bin. For the
σN2 ŝ2i j
2
σs2 Ntheory,
j
< 0.002, which is small enough to be neglected.
93
Figure 4.13. Variation for the Dalitz plot parameters a, b, d, f and g with the bin width,
in units of the resolution. The green lines show the results of the standard analysis and
the blue line the weighted mean value.
94
4.5.4 Track-Photon Angle Cut
The track photon cut described in page 64 is a graphical cut. To study the
systematic effect, the area of each of the three graphical regions was varied
with ±10%. The cuts with the area variation are shown in figure 4.14, together
with the standard cut, for both data and signal and background simulation.
103
π
2.5
2
102
1.5
1
∠min(p +, pγ ) (rad)
103
2
102
1
10
10
0.5
0.5
1
1.5
2
2.5
3
0
0
1
∠min(pπ-, pγ ) (rad)
0.5
1
1.5
2
2.5
3
1
∠min(pπ-, pγ ) (rad)
Data
Bhabha MC
3
3
10
π
2.5
2
2
10
1.5
104
3
2.5
103
2
102
1.5
1
10
0.5
0
0
3
2.5
1.5
0.5
0
0
∠min(pπ+, pγ ) (rad)
Background MC
3
∠min(pπ+, pγ ) (rad)
∠min(p +, pγ ) (rad)
Signal
1
10
0.5
0.5
1
1.5
2
2.5
3
∠min(pπ-, pγ ) (rad)
1
0
0
0.5
1
1.5
2
2.5
3
1
∠min(pπ-, pγ ) (rad)
Figure 4.14. The track-photon angle cut varied: for signal MC, background MC,
Bhabha MC and data. The simulations are scaled to the corresponding data luminosity.
The results of the fit for the Dalitz plot parameters are shown in figures 4.15
and 4.16, the first for the parameter set a, b, d, f and the second for a, b, d, f , g.
As can be seen, this cut has no significant effect on the parameters.
The systematic error is taken as the difference between the ±10% area cut
and the standard analysis. If the differences to both tests have the same sign,
only the biggest difference is taken. For the numerical results see section 4.5.9.
95
Figure 4.15. Variation for the Dalitz plot parameters a, b, d and f with the variation of
the track-photon cut. Point zero is the standard analysis.
96
Figure 4.16. Variation for the Dalitz plot parameters a, b, d, f and g with the variation
of the track-photon cut. Point zero is the standard analysis.
97
4.5.5 Time-of-Flight Cuts
The time of flight related cuts described in page 65 have been varied to check
for systematic effects. Figure 4.17 shows how the cuts have been varied and as
can be seen both line cuts have had the intersect changed. The horizontal cut
was varied from Δte = −0.7 to Δte = −0.59 and Δte = −0.81. The diagonal
cut is of the form Δte = −Δtπ + b, where in the standard analysis b = 0 and in
these systematics checks b = 0.22 and b = −0.22.
Background MC
Δ t_e (ns)
5
10
104
103
102
10
−2
0
2
4
6
8
10
1
Δ t_π (ns)
Bhabha background
105
104
103
102
10
−2
0
2
4
6
8
10
1
Δ t_π (ns)
Data
Δ t_e (ns)
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
104
103
102
10
−2
0
2
4
6
8
10
Δ t_π (ns)
1
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
Δ t_e (ns)
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4
Δ t_e (ns)
Signal
105
104
103
102
10
−2
0
2
4
6
8
10
1
Δ t_π (ns)
Figure 4.17. The time of flight cuts varied: for signal MC, background MC, Bhabha
MC and data. Simulation is scaled to the corresponding data luminosity.
Figure 4.18 shows the projection onto the vertical axis, Δte , with the variation of the horizontal cut in figure 4.17. The choice of the variation of the cut
was made to reject as much as possible of the Bhabha scattering background,
since it is not very well described by the simulation. Note that this figure is
done before the background rejection cuts, and the agreement between data
and MC is therefore not good.
The results for the Dalitz plot parameters with these variations are shown
in figures 4.19-4.22, where the two first deal with the diagonal cut and the last
two with the horizontal cut. As can be seen, the diagonal cut does not influence
the parameters, but there is a dependence introduced by the horizontal cut.
The systematic error is taken for each of these cuts separately, as the difference between the altered cuts and the standard analysis. If the differences
to both tests have the same sign, only the biggest difference is taken. For the
numerical results see section 4.5.9.
98
3
×10
350
300
250
Data
MC Total no eeg
Signal
ω π0
sum other bkg
eeg bkg
200
150
100
50
0
−4
−3
−2
−1
0
1
2
Δ t_e (ns)
Figure 4.18. Δte for data and simulation, where the simulation for Bhabha background
is scaled to the same luminosity as data but the other background is scaled according
to the calculated scaling factors. The histograms are done before the cuts, but after
selection. The three Δte cuts are shown as black lines, the middle one is the standard
analysis.
Figure 4.19. Variation for the Dalitz plot parameters a, b, d and f with the variation of
intersect of the diagonal line. Point zero is the standard analysis.
99
Figure 4.20. Variation for the Dalitz plot parameters a, b, d, f and g with the variation
of intersect of the diagonal line. Point zero is the standard analysis.
100
Figure 4.21. Variation for the Dalitz plot parameters a, b, d and f with the variation of
the horizontal line. Point -0.7 is the standard analysis.
101
Figure 4.22. Variation for the Dalitz plot parameters a, b, d, f and g with the variation
of the horizontal line. Point -0.7 is the standard analysis.
102
4.5.6 Opening Angle Cut
The effect of the cut on the opening angle between the two photons from
the π 0 decay in its rest frame is tested by varying this cut. The resolution
of this variable is used to select an appropriate step with which to vary the
cut. Figure 4.23 is used to estimate the resolution. It shows the difference
between true and reconstructed values for this angle, after the track-photon
angle cut and the TOF cuts, evaluated from MC signal simulation. The figure
has been extended to −180◦ by mirroring along the 0◦ line. A fit with a sum of
two Gaussians is performed, in the interval −130◦ to 130◦ . The result is also
shown in the figure, and the resulting parameters in overlay. Even though the
fit is not perfect, the width of the inner Gaussian (parameter p2) can be used
as an estimate of the resolution, giving σ = 2.95◦ .
3
×10
5000
χ2 / ndf
p0
p1
4000
p2
p3
p4
4.508e+06 ± 6.556e+02
−7.008e−10 ± 4.760e−05
2.947 ± 0.000
1.236e+05 ± 8.928e+01
7.597e−10 ± 6.145e−05
p5
21.49 ± 0.01
3.521e+06 / 716
3000
2000
1000
0
−40
−20
0
20
40
Figure 4.23. Difference between true and reconstructed value of opening angle between the π 0 decay photons in the π 0 rest frame, for MC signal simulation, and two
Gaussian fit (full line), with parameters in overlay. The two component Gaussians of
the fit are shown by the dotted and dashed line.
The opening angle variable is varied in steps of 3◦ , from 159◦ to 171◦ , and
the results are shown in figures 4.24 and 4.25. As can be seen, most parameters
are stable with the variation of this cut.
The systematic error from the cut on the opening angle between the π 0
photons is calculated by taking the difference between the standard result and
the two results ∼ 1σ away (162◦ and 168◦ ). If both differences have the same
sign, the largest deviation is taken as the systematic error. For the numerical
values see section 4.5.9.
103
Figure 4.24. Variation for the Dalitz plot parameters a, b, d and f with varying the π 0
photon opening angle cut. The point at 165◦ is the standard analysis.
104
Figure 4.25. Variation for the Dalitz plot parameters a, b, d, f and g with varying the
π 0 photon opening angle cut. The point at 165◦ is the standard analysis.
105
4.5.7 Missing Mass Cut
The effect of the cut on the missing mass |Pπ 0 | is checked by varying this cut.
×103
χ2 / ndf
p3
3000
5.571e+05 / 133
4.817 ± 0.001
2500
2000
1500
1000
500
0
-40
-30
-20
-10
0
10
20
30
40
Figure 4.26. Difference between true and reconstructed missing mass |Pπ 0 |, evaluated
with signal MC, and fit with a Lorentzian function plus a straight line. The overlay
shows the relevant fit information, the χ 2 and the Γ of the Lorentzian function.
Figure 4.26 is used to estimate the resolution of the missing mass. It shows
the difference between true and reconstructed values of |Pπ 0 | for signal, after
the track-photon angle cut and the TOF cuts, as well as a fit with a Lorentzian
peak function plus a straight line background function. The FWHM (Γ =p3)
is 4.8 MeV, and using the relation for a Gaussian, the resolution is estimated
to σ = 2.04 MeV.
Figures 4.27 and 4.28 show how the Dalitz plot parameters vary when varying the missing mass cut. As it can be seen, for most parameters there is a
quite strong dependence on this cut. To further investigate this effect and to
determine if this change with the cut is intrinsic to the fit or related to the interdependence of the parameters, more tests have been performed. Keeping
all parameters except one (plus the normalization parameter) fixed at the value
for the standard result, the fit is performed with only one parameter (plus the
normalization parameter) left free, for the same range in the cut. The results
are shown in figures 4.29 and 4.30. As can be seen, the dependence is reduced
when fitting just one parameter, and thus comes mostly from the covariance of
the parameters.
The systematic error is calculated from the difference between the standard
analysis and the two with the cut moved by ∼ 1σ (13 MeV and 17 MeV),
where all parameters are left free in the fit. For the numerical results see
section 4.5.9.
106
Figure 4.27. Variation for the Dalitz plot parameters a, b, d and f when varying the
missing mass cut. The point at 15 MeV is the standard analysis.
107
Figure 4.28. Variation for the Dalitz plot parameters a, b, d, f and g when varying the
missing mass cut. The point at 15 MeV is the standard analysis.
108
Figure 4.29. Variation for the Dalitz plot parameters a, b, d and f when varying the
missing mass cut, in green the result with all parameters free, in red when only the
current parameter (and the normalization) is allowed free. The point at 15 MeV is the
standard analysis.
109
Figure 4.30. Variation for the Dalitz plot parameters a, b, d, f and g when varying the
missing mass cut, in green the result with all parameters free, in red when only the
current parameter (and the normalization) is allowed free. The point at 15 MeV is the
standard analysis.
110
4.5.8 Event Classification Procedure
To evaluate the effect of the event classification procedure, described in section 3.1.2, a prescaled data sample without event classification is used. This
data is reconstructed without the event classification requirements (although
these requirements can be imposed later in the analysis) and scaled by 1/20
with respect to the standard data. This data is used to compare results with
and without the event classification requirements, but of course the statistical
errors are bigger.
An important thing to note is that when removing the requirement of the
event classification, more background contributions enter compared to the
standard analysis. Here the contributions from φ → KS KL and φ → ρπ decays
have been considered. To keep the analysis as close as possible to the standard
analysis, the background from ωπ 0 is treated separately as before. The other
background contribution, called “rest of background” previously, is now a sum
of the φ → KS KL and φ → ρπ background as well as the other background not
coming from any of these three considered background contributions.
The available MC and prescaled data files for this study correspond to a
slightly different integrated luminosity of data. The prescaled data corresponds to a data integrated luminosity of 1703 pb−1 , the same for the signal
MC (but with luminosity scaling factor 10). The MC for the ωπ 0 background
corresponds to a data integrated luminosity of 1696.5 pb−1 , for the φ → KS KL
background 1694.3 pb−1 and for the φ → ρπ background 1707.1 pb−1 . The
background not coming from these three processes is usually taken from the
same MC files as the ωπ 0 background, and thus has the same corresponding
integrated luminosity. All background contributions are first scaled to correspond to the data, or prescaled data, integrated luminosity.
Test with different background subtraction
As a first check on the effect of the event classification, the same analysis was
performed on the prescaled data as in the standard data, that is, including also
the event classification. Then the analysis excluding the event classification
requirement was performed on the prescaled data. This is shown in figure
4.31. The first three points for each parameter correspond to these: the full
data with the event classification requirement, the prescaled data with the same
requirement and the prescaled data without this requirement. As can be seen,
the results agree within errors of each other. Because of the reduced statistics
of the prescaled data sample, this is not a convenient measure of the systematic
effect of the event classification, since statistical fluctuations could give a big
effect.
It was also noted when redoing the analysis without the event classification
requirement that the ωπ 0 background no longer was the dominant one, but it
was background coming from φ → KS KL . This raises the question if it is relevant to treat the ωπ 0 background separately, in the background subtraction
111
#
"
"
!
!
#
"
!
#
#
"
"
!
"
#
Figure 4.31. Dalitz plot parameters a, b, d, f and g for the standard analysis and for
analysis on the prescaled data with several different background subtractions.
and in the fit for the scaling factors. Figure 4.31 investigates this by trying
other combinations of background contributions, namely using the φ → KS KL
background as the one separate background, using both ωπ 0 and φ → KS KL
separately and using ωπ 0 , φ → KS KL and φ → ρπ separately. There are always some more background reactions contributing, but these are so small that
they are always treated together in what is called “rest”. The different background contributions, that are not ωπ 0 , φ → KS KL or φ → ρπ, are all taken
together, but depending on which background reactions are contributing most,
the MC simulation file to use can change. In this test, this background has been
taken either from the ωπ 0 MC file or the φ → KS KL file. The legend “combined fit” in this figure corresponds to a fit for the scaling factors where both
histograms (the opening angle between the π 0 photons in the π 0 rest frame and
the missing mass squared Pπ20 ) are used together, combined into one histogram
and this histogram is fit. As the figure shows, the different background subtractions can make a big difference in the Dalitz plot parameters. This makes
it hard to disentangle the event classification effect from the effect related to
the knowledge of the background in the absence of event classification and the
background subtraction.
These observations have led to a different evaluation of the systematic errors
due to event classification, which is described below.
112
Data - MC agreement on the event classification procedure
If the event classification is well described in the simulation and there is good
agreement between the MC and the data, then the MC can be used to evaluate the systematic effect of the event classification. This is preferred since the
MC signal sample corresponds to more events than the prescaled data sample, reducing the statistical fluctuations which could influence the systematic
errors.
Following the selection and analysis procedure as described earlier, and
also without the requirement on the event classification, the effect of the event
classification procedure on the Dalitz plot can be studied. This is done for
three sets of events: prescaled data as they are, prescaled data with the standard background subtraction (with ωπ 0 as a separate background) and signal
MC simulation. Figure 4.32 shows the ratio between the Dalitz plot distribution with the requirement of event classification to the Dalitz plot distribution
without this requirement, for the three sets. The MC distribution is similar to
the experimental one, especially when considering the background subtracted
data. The ratio for the whole Dalitz plot is for the prescaled data 90.51 ±
0.06%, for the prescaled data with background subtraction 91.45 ± 0.05% and
for the MC signal 91.490±0.004%. The signal simulation and the background
subtracted data are in agreement.
(a) prescaled data
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
Y
Y
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
0.94
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.78
1
X
0.76
0.94
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.78
1
0.76
X
(b) background subtracted prescaled data
Y
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
0.94
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.78
1
0.76
X
(c) MC signal
Figure 4.32. Ratio between number of events in the Dalitz plot when requiring event
classification and when not requiring it.
113
As can be seen from figure 4.32, the ratio has a linear dependence on Y . To
further compare the simulation and the data, a fit to a straight line in Y , for
each bin in X was performed, for the three sets of events. Figure 4.33 shows
an example of the fit, for bin 20 in X (0.22 < X < 0.29) for each of the three
sets.
χ2 / ndf
p0
p1
1
26 / 15
0.9035 ± 0.0032
-0.04455 ± 0.00579
χ2 / ndf
p0
p1
1
0.8
5.785 / 15
0.9125 ± 0.0098
-0.03464 ± 0.01497
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0
1
Y
-1
(a) prescaled data
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Y
(b) background subtracted prescaled data
χ2 / ndf
p0
p1
418.4 / 15
0.9139 ± 0.0002
-0.02444 ± 0.00041
0.8
0.6
0.4
0.2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Y
(c) MC signal
Figure 4.33. The ratio of events with event classification required to events without,
in the Dalitz plot, and its dependence on Y , for bin 20 in X.
Figure 4.34 shows the resulting slope and intersect of the linear fit for the
three event sets. As can be seen, both the slope and the intersect from the
simulation and background subtracted data are in agreement for all bins in X.
This shows that the simulation is correctly describing the effects of the event
classification procedure and can therefore be used to calculate the systematic
errors arising from the event classification.
Figures 4.35-4.38 all show that the fit reproduces the input Dalitz plot parameters used in the simulation, for the case with parameters a, b, d; a, b, c, d, e;
a, b, d, f and a, b, d, f , g, both with and without the requirement on the event
classification. The results are also in agreement with each other.
To quantify the systematic error from the event classification, the difference
for the fit results for each parameter is used, using the fit with the approapriate
number of parameters, as shown in figure 4.37 for a, b, d, f and in figure 4.38
for a, b, d, f , g. The error is calculated as a symmetric error: the difference
between the analysis requiring the event classification and not requiring it,
divided by two. The numerical results are shown in section 4.5.9.
114
0
prescaled data
MC signal
bkg sub presc. data
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
0
5
10
15
20
25
20
25
30
(a) slope
0.94
prescaled data
MC signal
bkg sub presc. data
0.93
0.92
0.91
0.9
0.89
0.88
0.87
0.86
0
5
10
15
30
(b) intersect
Figure 4.34. The results of the linear fits to the ratio of events with event classification
required to events without. The slope (upper) and intersect (lower) from the fit, for
prescaled data (black), background subtracted prescaled data (blue) and simulated
signal (red).
115
Figure 4.35. Results for the Dalitz plot parameters a, b, d, for the MC signal simulation. The green line shows the simulation input.
!
Figure 4.36. Results for the Dalitz plot parameters a, b, c, d, e, for the MC signal simulation. The green line shows the simulation input.
116
Figure 4.37. Results for the Dalitz plot parameters a, b, d, f , for the MC signal simulation. The green line shows the simulation input. The systematic error is calculated
from the difference between the points.
!
Figure 4.38. Results for the Dalitz plot parameters a, b, d, f , g, for the MC signal simulation. The green line shows the simulation input. The systematic error is calculated
from the difference between the points.
117
4.5.9 Summary of Systematic Effects
This section collects the numerical values of all the systematic errors calculated as explained in the previous sections, as well as the total systematic error
from a sum in quadrature of each error. Table 4.8 shows the summary of all
systematic errors for the a, b, d, f parameter case. Table 4.9 shows the same
thing for the case a, b, d, f , g.
Table 4.8. Summary of systematic errors for a, b, d, f parameters.
syst err
Δa
Δb
Δd
Δf
Eγ min
±0.0009
±0.0010
±0.0006
±0
bkg sub
±0.0001
±0.0005
±0.0006
±0.0008
binning
±0.0009
±0.0013
±0.0010
±0.0025
+0
−0.0001
+0
−0.0006
+0.0001
−0.0001
+0.0007
−0
+0.0003
−0
+0.0014
−0.0008
+0.0031
−0.0026
+0.0004
−0
+0.0019
−0.0015
TOF diag
±0
π 0 γ’s angle
miss mass
+0.0006
−0
+0.0010
−0.0010
+0
−0.0002
+0.0014
−0.0006
+0
−0.0001
+0.0001
−0.0001
+0.0039
−0.0036
event class
±0.0002
±0.0009
±0.0009
±0.0013
sum
+0.0018
−0.0018
+0.0046
−0.0041
+0.0038
−0.0031
+0.0045
−0.0050
track-photon
TOF hor
±0
+0
−0.0013
+0.0028
−0.0035
4.6 Final Results for Dalitz Plot Parameters
The final results, with the total systematic error as the sum in quadrature of the
separate effects, including the g parameter, are:
a = −1.095(3)(2)
b = 0.145(3)(5)
d = 0.081(3)(+6
−5 )
(4.12)
f = 0.141(7)(+7
−8 )
g = −0.044(9)(+12
−13 )
Without the g parameter the resuls are:
a = −1.104(3)(2)
b = 0.142(3)(+5
−4 )
d = 0.073(3)(+4
−3 )
f = 0.154(6)(+4
−5 )
118
(4.13)
Table 4.9. Summary of systematic errors for a, b, d, f , g parameters.
syst err
Δa
Δb
Δd
Δf
Δg
Eγ min
±0.0006
±0.0012
±0.0010
±0.0005
±0.0016
bkg sub
±0.0008
±0.0007
±0.0011
±0.0006
±0.0038
binning
±0.0016
±0.0012
±0.0009
±0.0034
±0.0040
+0
−0.0001
+0.0006
−0.0011
+0.0002
−0.0002
+0.0018
−0.0001
+0.0003
−0.0001
+0.0021
−0.0012
+0.0049
−0.0045
+0.0003
−0
+0.0003
−0.0008
±0
+0.0005
−0.0025
+0.0057
−0.0062
+0.0003
−0.0002
+0.0026
−0.0054
+0.0002
−0.0001
+0.0026
−0.0038
+0.0100
−0.0092
TOF diag
±0
π0
miss mass
+0.0014
−0.0005
+0.0008
−0.0010
+0
−0.0002
+0.0012
−0.0001
+0
−0.0001
+0.0002
−0.0001
+0.0046
−0.0043
event class
±0
±0.0008
±0.0006
±0.0009
±0.0012
sum
+0.0025
−0.0024
+0.0052
−0.0048
+0.0059
−0.0050
+0.0068
−0.0076
+0.0122
−0.0128
track-photon
TOF hor
γ’s angle
Note that the magnitude of the systematic and the statistical errors is similar.
4.7 Charge Asymmetries
The charge asymmetries ALR , AQ and AS , defined in section 1.4.1, have been
also determined. The numbers of events used in equations 1.56-1.58 refer to
the background subtracted and acceptance corrected events, e.g. for equation
1.56:
Ndata,+ − s1 B1,+ − s2 B2,+
ε+
Ndata,− − s1 B1,− − s2 B2,−
N− =
ε−
N+ =
(4.14)
where the subscript + denotes that the variables are calculated for X > 0,
the subscript − denotes that the variables are calculated for X < 0, Ndata is the
number of events in the final data sample, the scaling factors s are calculated as
explained in section 3.4, B1 is the contribution fo the ωπ 0 background, while
B2 is the contribution of the remaining background and ε is the efficiency. The
efficiency is calculated from the signal MC simulation, e.g.
ε+ =
Nrec,+
,
Ngen,+
(4.15)
i.e, the ratio between reconstructed (Nrec ) and generated (Ngen ) number of
events, in this case for X > 0. Events are included in these equations without checking if they are inside the Dalitz plot boundary.
119
The systematic checks are analogous to the ones performed for the studies
of the Dalitz plot parameters, and the uncertainties are calculated in the same
way as in section 4.5, with the following exceptions:
Background subtraction is tested similarly, but instead of calculating scaling
factors for each bin in the Dalitz plot, they are calculated for each “bin”
in the asymmetry, i.e, for “bin” + and − for ALR , and similarly for the
other asymmetries;
Choice of binning: the binning of the Dalitz plot does not affect the asymmetries, so this test is not performed;
Missing mass cut: if both tests (with the cut at 13 MeV and 17 MeV) give the
same sign for the errors, only the largest value is taken;
Event classification procedure: the good description of the data by the MC
simulation is assumed to be valid also for the asymmetries, and the systematic error is calculated in the same way as for the Dalitz plot parameters.
Table 4.10 summarizes the different contributions to the systematic errors. The
total systematic error is the sum in quadrature of all the errors, separately for
errors contributing positively and negatively.
Table 4.10. Summary of systematic error for the charge asymmetries.
syst err
ΔALR · 104
ΔAQ · 104
ΔAS · 104
Eγ min
±0.09
±0.02
±0.40
bkg sub
±0.48
±0.28
±1.62
+0.17
−0
+4.85
−9.16
+0
−0.18
+0.14
−5.71
+0
−0.35
+0
−0.21
+4.82
−2.15
+0.25
−0
+0.26
−0.41
+0
−0.12
+0.23
−0
+0.73
−1.48
+0.02
−0.12
+0
−0.77
+0.09
−0.23
event class
±0.86
±0.24
±2.54
sum
+4.95
−10.84
+4.84
−2.23
+3.13
−3.48
track-photon
TOF hor
TOF diag
π 0 γ’s angle
miss mass
The final result for the charge asymmetries is
−4
ALR = −5.0(4.5)(+5.0
−10.8 ) · 10 ,
−4
AQ = 1.79(4.5)(+4.8
−2.2 ) · 10 ,
−4
AS = −0.44(4.5)(+3.3
−3.5 ) · 10 .
As can be seen, all charge asymmetries are consistent with zero.
120
(4.16)
5. Acceptance Corrected Data
Another way of presenting the Dalitz plot distribution is to extract the acceptance corrected bin contents. The advantage of such a presentation is that the
data can be directly compared to theoretical calculations or other experimental
results, and that it can be directly fit with other theoretical functions.
Acceptance corrected data can replace the smearing matrix method, if the
smearing matrix is close to diagonal, and the leaking of events to and from
nearby bins is approximately the same. This corresponds to the approximation
of a completely diagonal smearing matrix. It is also important that the detector
resolution is smaller than the width of the structures expected to be seen in the
data.
This chapter shows that the smearing matrix is close to diagonal and that
approximating it to a diagonal smearing matrix yields results consistent with
the full calculations. A table of the content in each bin of the Dalitz plot is
also included.
5.1 Diagonality of the Smearing Matrix
N
j
Figure 5.1 shows the matrix ŝi j = Si j · ε̃ j = rec,i;gen,
Ngen, j , that is, the smearing matrix also including the efficiency. While most of the true-reconstructed bin
combinations have very small values of ŝi j , there is a lighter band in the diagonal, corresponding to the part of the smearing matrix with higher percentage.
Note that there are more than 600 bins in each axis, so to clearly see the diagonal one might need to zoom in this figure. This is already a promising sign
of the smearing matrix being close to diagonal.
To further check the diagonality of the smearing matrix, the amount of reconstructed events on the diagonal or close to it are quantified. With the chosen
binning of 31 bins in X and 20 bins in Y , 51.3% of the reconstructed events lie
on the exact diagonal of the smearing matrix, that is, they are reconstructed in
the same bin as generated. Since events with true values close to a bin border
easily can get reconstructed in a neighboring bin, the reconstructed percentage
within a one bin ring and two bin ring of the generated bin has also been calculated. The one bin ring percentage is the quotient of the number of events
generated in (bin X, bin Y ) = (i, j) and reconstructed in the nine closest bins
(the bins from i − 1 to i + 1 and j − 1 to j + 1) to the number of generated
events in (bin X, bin Y ) = (i, j) that get reconstructed in any bin. The average
121
0.24
600
0.22
0.2
500
0.18
0.16
400
0.14
0.12
300
0.1
0.08
200
0.06
0.04
100
0.02
100
200
300
400
500
600
0
Figure 5.1. The smearing matrix used in the analysis, which includes the efficiency
and the smearing. The bin number used is calculated as binX + (binY − 1) · 31.
one bin ring percentage is 96.5% and varies with bin from 79.8% to 99.9%.
The lowest values are for bins with generated Y close to -1. These can get reconstructed with Y < −1, see figure 5.2(b), which puts them outside the Dalitz
plot bins used in the smearing matrix. The bins with lowest Y are not included
in the data since they cross the Dalitz plot boundary, and removing them gives
the one bin ring percentage average 96.9% and lowest value 94.6%.
The two bin ring percentage is calculated in the same way as the one bin
ring but considering the 25 closest bins (from i − 2 to i + 2 and j − 2 to j + 2).
The average result is 98.6% and it varies with bin from 81.6% to 100%. As
in the one bin ring case, the bins with lowest value of Y are the ones with the
lowest percentage and excluding these gives the average 99% and lowest value
96.9%.
Once can also look at the X and Y variables separately. Figure 5.2 shows
how most events lie in the diagonal of each of these variables. These histograms display the generated vs reconstructed X and Y variables, the upper
plots have finer binning than the one used in the Dalitz plot, the lower plots
the same. It is clear that most events lie in the diagonals of these two plots,
and considering the same binning as in the final Dalitz plot, for the X variable,
68.9% of the events are in the exact diagonal and 97.8% within plus-minus
one bin, while the corresponding numbers for Y are 68.9% and 97.1%.
From this it can be concluded that the smearing is not substancial and that
an approximation to a diagonal smearing matrix should yield good results.
Using acceptance corrected data should thus be a good approximation to the
full smearing matrix method.
122
450
0.6
400
0.4
350
0.2
300
0
250
-0.2
200
-0.4
150
-0.6
100
-0.8
50
-1
-1
-0.5
0
Y true
X true
×103
0.8
350
0.6
300
0.4
0.2
250
0
200
-0.2
150
-0.4
100
-0.6
-0.8
50
-1
0
0.5
×103
0.8
1
X recontructed
-1
-0.5
0.5
0
1
Y reconstructed
(b) Y
×103
1600
1
0.8
1400
0.6
Y true
X true
(a) X
0
×103
1
2500
0.8
0.6
1200
0.4
1000
0.2
0
800
-0.2
600
-0.4
2000
0.4
0.2
1500
0
-0.2
1000
-0.4
400
-0.6
-0.6
200
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0
0.8
1
X reconstructed
(c) X
500
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Y reconstructed
0
(d) Y
Figure 5.2. Number of generated Dalitz plot events vs reconstructed, for the Dalitz
plot variables X and Y . The bottom figures have the binning used in the Dalitz plot.
5.2 Acceptance Correction
The acceptance corrected Dalitz plot distribution is obtained from the content
in each bin, Ni , calculated as:
Nsub,i = Ndata,i − s1 Bi1 − s2 Bi2
Nsub,i Ndata,i − s1 Bi1 − s2 Bi2
and Ni =
=
εi
εi
Nrec,i
with εi =
Ngen,i
(5.1)
(5.2)
(5.3)
where Ndata,i is the number of events in the bin i at the end of the analysis
chain, Bi1 and Bi2 are the background events from simulation, s1 and s2 are
the background scaling factors obtained as described in section 3.4, εi is the
acceptance of bin i (see figure 5.3), Nrec,i is the number of reconstructed signal
events in bin i from MC and Ngen,i the generated number of signal events in
the same bin.
123
Y
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.2 0.4 0.6 0.8
1
0
X
Figure 5.3. The signal acceptance, ε, for each bin of the Dalitz plot distribution.
The error in the acceptance corrected data Ni , σi , includes the error in the
efficiency, data and background:
σi2
=
σNsub,i
2
εi
Nsub,i σεi
+
εi2
2
with σN2sub,i = Ndata,i + s21 · Bi1 + σs21 · B2i1 + s22 · Bi2 + σs22 · B2i2
and σε2i =
εi (1 − εi )
.
Ngen,i
(5.4)
(5.5)
(5.6)
5.3 Comparison with Smearing Matrix Method
To directly compare the acceptance corrected data with the full smearing matrix method, a fit for the Dalitz plot parameters has been performed, similarly
to what is described in section 4.2. In the acceptance corrected data case, the
fit is performed by minimizing
χ =
2
2
Nbins N − N
i
theory,i
∑
i=1
σi
(5.7)
where Ntheory,i = |A(X,Y )|2 dXi dYi and |A(X,Y )|2 N(1+aY +bY 2 +dX 2 +
fY 3 + gX 2Y ) (as before, parameters c, e, h and l could also be included). The
sum is only over bins completely inside the Dalitz plot boundaries and thus the
integral is over the bin, that is, it is an integral over the rectangle Xmin,i < X <
Xmax,i and Ymin,i < Y < Ymax,i . This is calculated with the ROOT [81] function
Integral.
124
The result of this fit is presented below (on the left) together with the full
smearing matrix results (on the right, including systematic errors):
a = −1.092(3)
b = 0.145(3)
a = −1.095(3)(2)
b = 0.145(3)(5)
d = 0.081(3)
d = 0.081(3)(+6
−5 )
f = 0.137(6)
g = −0.044(8)
a = −1.101(3)
a = −1.104(3)(2)
b = 0.142(3)
b = 0.142(3)(+5
−4 )
d = 0.072(3)
f = 0.150(6)
(5.8)
f = 0.141(7)(+7
−8 )
g = −0.044(9)(+12
−13 )
d = 0.073(3)(+4
−3 )
(5.9)
f = 0.154(6)(+4
−5 )
To facilitate comparison, this information is pictorially shown in figure 5.4 for
the parameters a, b, d, f and in figure 5.5 for the parameters a, b, d, f , g, where
only the statistical errors are taken into account for both results. As can be
seen from the numbers or the figures, the results are in agreement, and the
acceptance corrected data is a good approximation.
For completeness, the correlation matrix for the acceptance corrected data
fits is presented in tables 5.1 and 5.2.
Table 5.1. The correlation matrix for the acceptance corrected data, with parameters
a, b, d and f .
b
d
f
a -0.267 -0.359 -0.820
b
0.369 -0.162
d
0.060
Table 5.2. The correlation matrix for the acceptance corrected data, with parameters
a, b, d, f and g.
b
d
f
g
a -0.111 0.010 -0.849 -0.514
b
0.392 -0.214 -0.237
d
-0.134 -0.537
f
0.380
125
Figure 5.4. The results for the Dalitz plot parameters a, b, d, f , comparing the smearing matrix fit (the left point) with the acceptance corrected data fit (right point).
Figure 5.5. The results for the Dalitz plot parameters a, b, d, f , g, comparing the smearing matrix fit with the acceptance corrected data fit.
126
5.4 Results
The background subtracted, acceptance corrected Dalitz plot distribution is
shown in figure 5.6. Tables 5.3 and 5.4 show the 371 bins with the background
subtracted, acceptance corrected, data content. The bin content is normalized
to the bin content of the bin with Xc = 0.0 and Yc = 0.05. These results are
also available as a tab separated ASCII file with one bin per line, available
2
2
in X and 20
in Y , and both these
upon request1 . Note that the bin size is 31
variables are between −1 and 1. As a further test, the fit has been performed
with the values in the table and is very nearly the same as the fit performed on
the unnormalized bin content.
Y
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0
60000
50000
40000
30000
20000
10000
0.2 0.4 0.6 0.8
1
0
X
Figure 5.6. The background subtracted, acceptance corrected data Dalitz plot distribution.
1 please
contact [email protected] or [email protected].
127
Table 5.3. Acceptance corrected data Dalitz plot distribution, normalized to the content in the bin with Xc = 0.0 and Yc = 0.05 (in red).
Xc
Yc
Content
Xc
Yc
Content
Xc
Yc
Content
−0.839
−0.839
−0.774
−0.774
−0.774
−0.710
−0.710
−0.710
−0.645
−0.645
−0.645
−0.645
−0.581
−0.581
−0.581
−0.581
−0.516
−0.516
−0.516
−0.516
−0.516
−0.452
−0.452
−0.452
−0.452
−0.452
−0.387
−0.387
−0.387
−0.387
−0.387
−0.323
−0.323
−0.323
−0.323
−0.323
−0.323
−0.258
−0.258
−0.258
−0.258
−0.258
−0.194
−0.194
−0.194
−0.194
−0.194
−0.194
−0.129
−0.129
−0.129
−0.129
−0.129
−0.129
−0.065
−0.065
−0.065
−0.065
−0.065
0.000
0.000
0.000
−0.050
0.250
−0.150
0.150
0.450
−0.150
0.150
0.450
−0.350
−0.050
0.250
0.550
−0.450
−0.150
0.150
0.450
−0.650
−0.350
−0.050
0.250
0.550
−0.550
−0.250
0.050
0.350
0.650
−0.650
−0.350
−0.050
0.250
0.550
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
1.1713 ± 0.0109
0.8551 ± 0.0093
1.3140 ± 0.0115
0.9275 ± 0.0098
0.6361 ± 0.0080
1.2790 ± 0.0115
0.9369 ± 0.0099
0.6375 ± 0.0081
1.5172 ± 0.0124
1.1498 ± 0.0109
0.8155 ± 0.0093
0.5158 ± 0.0074
1.6404 ± 0.0127
1.2860 ± 0.0114
0.9192 ± 0.0098
0.6060 ± 0.0081
1.8778 ± 0.0136
1.5155 ± 0.0124
1.1585 ± 0.0108
0.7979 ± 0.0092
0.5128 ± 0.0075
1.7575 ± 0.0128
1.3980 ± 0.0118
1.0211 ± 0.0103
0.7223 ± 0.0088
0.4233 ± 0.0069
1.8636 ± 0.0130
1.4912 ± 0.0122
1.1423 ± 0.0107
0.8016 ± 0.0092
0.5014 ± 0.0075
1.9473 ± 0.0131
1.6240 ± 0.0124
1.2573 ± 0.0111
0.9013 ± 0.0097
0.5905 ± 0.0080
0.3447 ± 0.0063
1.8698 ± 0.0129
1.5098 ± 0.0120
1.1315 ± 0.0106
0.7865 ± 0.0091
0.5019 ± 0.0075
2.0928 ± 0.0137
1.7365 ± 0.0126
1.3656 ± 0.0114
1.0031 ± 0.0100
0.6934 ± 0.0086
0.4285 ± 0.0070
1.9726 ± 0.0130
1.6335 ± 0.0124
1.2462 ± 0.0109
0.9088 ± 0.0096
0.5818 ± 0.0080
0.3477 ± 0.0063
1.8267 ± 0.0128
1.4881 ± 0.0120
1.1235 ± 0.0105
0.7864 ± 0.0092
0.4953 ± 0.0074
2.0791 ± 0.0134
1.7267 ± 0.0126
1.3729 ± 0.0115
−0.839
−0.839
−0.774
−0.774
−0.710
−0.710
−0.710
−0.710
−0.645
−0.645
−0.645
−0.645
−0.581
−0.581
−0.581
−0.581
−0.516
−0.516
−0.516
−0.516
−0.516
−0.452
−0.452
−0.452
−0.452
−0.452
−0.387
−0.387
−0.387
−0.387
−0.387
−0.323
−0.323
−0.323
−0.323
−0.323
−0.258
−0.258
−0.258
−0.258
−0.258
−0.258
−0.194
−0.194
−0.194
−0.194
−0.194
−0.194
−0.129
−0.129
−0.129
−0.129
−0.129
−0.065
−0.065
−0.065
−0.065
−0.065
−0.065
0.000
0.000
0.000
0.050
0.350
−0.050
0.250
−0.350
−0.050
0.250
0.550
−0.250
0.050
0.350
0.650
−0.350
−0.050
0.250
0.550
−0.550
−0.250
0.050
0.350
0.650
−0.450
−0.150
0.150
0.450
0.750
−0.550
−0.250
0.050
0.350
0.650
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
1.0704 ± 0.0107
0.7371 ± 0.0086
1.1906 ± 0.0112
0.8198 ± 0.0093
1.5459 ± 0.0125
1.1723 ± 0.0110
0.8347 ± 0.0094
0.5292 ± 0.0074
1.4358 ± 0.0121
1.0498 ± 0.0103
0.7021 ± 0.0086
0.4391 ± 0.0068
1.5137 ± 0.0123
1.1533 ± 0.0108
0.7997 ± 0.0091
0.5009 ± 0.0074
1.7519 ± 0.0129
1.3865 ± 0.0117
1.0288 ± 0.0104
0.6990 ± 0.0087
0.4308 ± 0.0070
1.6416 ± 0.0125
1.2755 ± 0.0113
0.9173 ± 0.0098
0.5728 ± 0.0079
0.3460 ± 0.0063
1.7396 ± 0.0126
1.3906 ± 0.0117
1.0422 ± 0.0103
0.7007 ± 0.0086
0.4203 ± 0.0069
1.8870 ± 0.0130
1.4868 ± 0.0120
1.1348 ± 0.0106
0.7979 ± 0.0093
0.4973 ± 0.0075
2.0905 ± 0.0139
1.7530 ± 0.0127
1.3790 ± 0.0115
1.0008 ± 0.0101
0.6910 ± 0.0086
0.4255 ± 0.0070
1.9664 ± 0.0131
1.6253 ± 0.0124
1.2414 ± 0.0110
0.9129 ± 0.0097
0.5854 ± 0.0080
0.3391 ± 0.0062
1.8522 ± 0.0129
1.4987 ± 0.0119
1.1311 ± 0.0105
0.7865 ± 0.0091
0.4974 ± 0.0075
2.0865 ± 0.0135
1.7469 ± 0.0126
1.3668 ± 0.0115
1.0100 ± 0.0101
0.6823 ± 0.0086
0.4181 ± 0.0069
1.9471 ± 0.0129
1.6045 ± 0.0122
1.2384 ± 0.0109
−0.839
−0.774
−0.774
−0.774
−0.710
−0.710
−0.710
−0.645
−0.645
−0.645
−0.645
−0.581
−0.581
−0.581
−0.581
−0.581
−0.516
−0.516
−0.516
−0.516
−0.452
−0.452
−0.452
−0.452
−0.452
−0.387
−0.387
−0.387
−0.387
−0.387
−0.387
−0.323
−0.323
−0.323
−0.323
−0.323
−0.258
−0.258
−0.258
−0.258
−0.258
−0.258
−0.194
−0.194
−0.194
−0.194
−0.194
−0.129
−0.129
−0.129
−0.129
−0.129
−0.129
−0.065
−0.065
−0.065
−0.065
−0.065
−0.065
0.000
0.000
0.000
0.150
−0.250
0.050
0.350
−0.250
0.050
0.350
−0.450
−0.150
0.150
0.450
−0.550
−0.250
0.050
0.350
0.650
−0.450
−0.150
0.150
0.450
−0.650
−0.350
−0.050
0.250
0.550
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.9580 ± 0.0100
1.4335 ± 0.0123
1.0667 ± 0.0106
0.7209 ± 0.0087
1.4054 ± 0.0122
1.0540 ± 0.0104
0.7303 ± 0.0087
1.6466 ± 0.0128
1.2858 ± 0.0115
0.9270 ± 0.0099
0.6256 ± 0.0082
1.7709 ± 0.0130
1.3919 ± 0.0120
1.0429 ± 0.0104
0.7241 ± 0.0088
0.4347 ± 0.0069
1.6364 ± 0.0127
1.2654 ± 0.0114
0.9261 ± 0.0098
0.6137 ± 0.0081
1.8561 ± 0.0131
1.5093 ± 0.0122
1.1371 ± 0.0108
0.7828 ± 0.0091
0.5181 ± 0.0076
1.9726 ± 0.0135
1.6180 ± 0.0125
1.2656 ± 0.0111
0.9100 ± 0.0097
0.5982 ± 0.0081
0.3421 ± 0.0062
1.7378 ± 0.0128
1.3884 ± 0.0116
1.0177 ± 0.0102
0.6889 ± 0.0086
0.4193 ± 0.0069
1.9730 ± 0.0132
1.6230 ± 0.0124
1.2627 ± 0.0111
0.8922 ± 0.0096
0.5803 ± 0.0080
0.3379 ± 0.0062
1.8632 ± 0.0129
1.4808 ± 0.0119
1.1284 ± 0.0106
0.7802 ± 0.0091
0.5079 ± 0.0076
2.0529 ± 0.0134
1.7228 ± 0.0125
1.3703 ± 0.0115
1.0100 ± 0.0101
0.6877 ± 0.0086
0.3992 ± 0.0067
1.9781 ± 0.0131
1.6206 ± 0.0125
1.2273 ± 0.0109
0.8945 ± 0.0096
0.5973 ± 0.0081
0.3358 ± 0.0062
1.8831 ± 0.0129
1.4697 ± 0.0118
1.1289 ± 0.0105
128
Table 5.4. Acceptance corrected data Dalitz plot distribution, normalized to the content in the bin with Xc = 0.0 and Yc = 0.05 (in red) - continued.
Xc
Yc
Content
Xc
Yc
Content
Xc
Yc
Content
0.000
0.000
0.000
0.065
0.065
0.065
0.065
0.065
0.065
0.129
0.129
0.129
0.129
0.129
0.194
0.194
0.194
0.194
0.194
0.194
0.258
0.258
0.258
0.258
0.258
0.258
0.323
0.323
0.323
0.323
0.323
0.387
0.387
0.387
0.387
0.387
0.452
0.452
0.452
0.452
0.452
0.516
0.516
0.516
0.516
0.516
0.581
0.581
0.581
0.581
0.645
0.645
0.645
0.645
0.710
0.710
0.710
0.710
0.774
0.774
0.839
0.839
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.550
−0.250
0.050
0.350
0.650
−0.650
−0.350
−0.050
0.250
0.550
−0.650
−0.350
−0.050
0.250
0.550
−0.650
−0.350
−0.050
0.250
0.550
−0.450
−0.150
0.150
0.450
−0.450
−0.150
0.150
0.450
−0.350
−0.050
0.250
0.550
−0.050
0.250
−0.050
0.250
1.0000 ± 0.0100
0.6901 ± 0.0087
0.4213 ± 0.0069
1.9851 ± 0.0131
1.5946 ± 0.0122
1.2419 ± 0.0110
0.8973 ± 0.0096
0.5942 ± 0.0081
0.3402 ± 0.0062
1.8539 ± 0.0128
1.4903 ± 0.0119
1.1278 ± 0.0105
0.7985 ± 0.0092
0.5100 ± 0.0075
2.0706 ± 0.0135
1.7274 ± 0.0127
1.3733 ± 0.0115
1.0091 ± 0.0100
0.6855 ± 0.0086
0.4294 ± 0.0070
1.9947 ± 0.0133
1.6226 ± 0.0124
1.2450 ± 0.0109
0.8928 ± 0.0097
0.5935 ± 0.0081
0.3513 ± 0.0064
1.7518 ± 0.0126
1.3641 ± 0.0115
1.0322 ± 0.0102
0.6921 ± 0.0086
0.4182 ± 0.0069
1.8704 ± 0.0130
1.5016 ± 0.0120
1.1250 ± 0.0106
0.7903 ± 0.0091
0.5056 ± 0.0075
1.8835 ± 0.0132
1.5041 ± 0.0120
1.1427 ± 0.0108
0.8238 ± 0.0093
0.5083 ± 0.0075
1.8811 ± 0.0136
1.5174 ± 0.0122
1.1428 ± 0.0107
0.8035 ± 0.0093
0.5081 ± 0.0075
1.6374 ± 0.0125
1.2716 ± 0.0115
0.9095 ± 0.0097
0.5941 ± 0.0080
1.6412 ± 0.0128
1.2861 ± 0.0115
0.9241 ± 0.0098
0.6105 ± 0.0080
1.5493 ± 0.0125
1.1642 ± 0.0108
0.8263 ± 0.0093
0.5178 ± 0.0073
1.1665 ± 0.0109
0.8410 ± 0.0093
1.2007 ± 0.0111
0.8490 ± 0.0093
0.000
0.000
0.000
0.065
0.065
0.065
0.065
0.065
0.129
0.129
0.129
0.129
0.129
0.129
0.194
0.194
0.194
0.194
0.194
0.194
0.258
0.258
0.258
0.258
0.258
0.323
0.323
0.323
0.323
0.323
0.323
0.387
0.387
0.387
0.387
0.387
0.452
0.452
0.452
0.452
0.452
0.516
0.516
0.516
0.516
0.516
0.581
0.581
0.581
0.581
0.645
0.645
0.645
0.645
0.710
0.710
0.710
0.774
0.774
0.774
0.839
0.839
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.550
−0.250
0.050
0.350
0.650
−0.550
−0.250
0.050
0.350
0.650
−0.550
−0.250
0.050
0.350
0.650
−0.350
−0.050
0.250
0.550
−0.350
−0.050
0.250
0.550
−0.250
0.050
0.350
−0.250
0.050
0.350
0.050
0.350
0.9129 ± 0.0097
0.5880 ± 0.0080
0.3495 ± 0.0063
1.8433 ± 0.0128
1.4948 ± 0.0119
1.1258 ± 0.0105
0.7727 ± 0.0090
0.4769 ± 0.0073
2.0506 ± 0.0134
1.7251 ± 0.0125
1.3755 ± 0.0114
0.9951 ± 0.0100
0.6795 ± 0.0086
0.4266 ± 0.0070
1.9784 ± 0.0131
1.5907 ± 0.0123
1.2352 ± 0.0109
0.8919 ± 0.0096
0.5870 ± 0.0080
0.3423 ± 0.0063
1.8482 ± 0.0128
1.4862 ± 0.0120
1.1285 ± 0.0106
0.7877 ± 0.0091
0.4957 ± 0.0074
1.9799 ± 0.0133
1.6229 ± 0.0124
1.2309 ± 0.0110
0.9195 ± 0.0098
0.5861 ± 0.0080
0.3514 ± 0.0063
1.7374 ± 0.0127
1.3593 ± 0.0115
1.0166 ± 0.0102
0.7117 ± 0.0087
0.4165 ± 0.0069
1.7573 ± 0.0128
1.3823 ± 0.0117
1.0171 ± 0.0103
0.7080 ± 0.0087
0.4321 ± 0.0071
1.7314 ± 0.0128
1.4086 ± 0.0118
1.0114 ± 0.0103
0.6988 ± 0.0087
0.4422 ± 0.0071
1.4996 ± 0.0121
1.1489 ± 0.0108
0.8035 ± 0.0092
0.5165 ± 0.0075
1.5339 ± 0.0123
1.1570 ± 0.0109
0.8054 ± 0.0091
0.5236 ± 0.0074
1.4093 ± 0.0119
1.0573 ± 0.0104
0.7258 ± 0.0088
1.4236 ± 0.0122
1.0669 ± 0.0106
0.7171 ± 0.0086
1.0542 ± 0.0105
0.7338 ± 0.0085
0.000
0.000
0.065
0.065
0.065
0.065
0.065
0.065
0.129
0.129
0.129
0.129
0.129
0.129
0.194
0.194
0.194
0.194
0.194
0.258
0.258
0.258
0.258
0.258
0.258
0.323
0.323
0.323
0.323
0.323
0.387
0.387
0.387
0.387
0.387
0.387
0.452
0.452
0.452
0.452
0.452
0.516
0.516
0.516
0.516
0.581
0.581
0.581
0.581
0.581
0.645
0.645
0.645
0.645
0.710
0.710
0.710
0.774
0.774
0.774
0.839
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.650
−0.350
−0.050
0.250
0.550
−0.850
−0.550
−0.250
0.050
0.350
0.650
−0.650
−0.350
−0.050
0.250
0.550
−0.750
−0.450
−0.150
0.150
0.450
0.750
−0.450
−0.150
0.150
0.450
0.750
−0.450
−0.150
0.150
0.450
−0.550
−0.250
0.050
0.350
0.650
−0.250
0.050
0.350
0.650
−0.150
0.150
0.450
−0.150
0.150
0.450
0.150
0.7845 ± 0.0091
0.4988 ± 0.0074
2.0686 ± 0.0134
1.7052 ± 0.0126
1.3625 ± 0.0114
1.0033 ± 0.0101
0.7041 ± 0.0087
0.4188 ± 0.0069
1.9884 ± 0.0131
1.6060 ± 0.0124
1.2338 ± 0.0109
0.8990 ± 0.0097
0.5792 ± 0.0080
0.3470 ± 0.0063
1.8374 ± 0.0128
1.4905 ± 0.0119
1.1161 ± 0.0105
0.7868 ± 0.0091
0.5076 ± 0.0075
2.0853 ± 0.0139
1.7476 ± 0.0126
1.3787 ± 0.0115
1.0184 ± 0.0102
0.6933 ± 0.0086
0.4158 ± 0.0069
1.8781 ± 0.0129
1.4959 ± 0.0121
1.1440 ± 0.0107
0.7792 ± 0.0091
0.4949 ± 0.0074
1.9773 ± 0.0135
1.5947 ± 0.0123
1.2417 ± 0.0111
0.9051 ± 0.0096
0.6075 ± 0.0081
0.3518 ± 0.0063
1.6370 ± 0.0124
1.2599 ± 0.0111
0.9162 ± 0.0098
0.6029 ± 0.0081
0.3473 ± 0.0063
1.6381 ± 0.0126
1.2757 ± 0.0113
0.9282 ± 0.0098
0.5868 ± 0.0079
1.7562 ± 0.0130
1.3986 ± 0.0118
1.0164 ± 0.0103
0.7133 ± 0.0087
0.4301 ± 0.0068
1.3911 ± 0.0117
1.0475 ± 0.0105
0.7340 ± 0.0089
0.4372 ± 0.0068
1.2948 ± 0.0115
0.9256 ± 0.0098
0.6147 ± 0.0079
1.2996 ± 0.0116
0.9417 ± 0.0099
0.6247 ± 0.0079
0.9478 ± 0.0099
129
6. Discussion of Results
This work has produced the Dalitz plot distribution of η → π + π − π 0 with the
highest statistics so far. The distribution is used to extract the most precise
Dalitz plot parameters to date, considering both statistic and systematic uncertainties, resulting in the first determination of the g parameter.
In this chapter, a comparison between the presenteed results and the previous experimental results is done, both in terms of Dalitz plot parameters,
acceptance correced Dalitz plot and charge asymmetries.
6.1 Dalitz Plot Parameters
To facilitate comparison between the results for the Dalitz plot parameters
from this work with previous high statistics results, these are all summarized in
table 6.1. This work results in the first experimental value for the g parameter,
different from zero at a 2.7σ level.
Directly comparing results with different sets of Dalitz plot parameters is
not straightforward, due to the correlations between the parameters. Therefore, we use our result with the a, b, d and f parameters to compare with the
previous experiments which used the same parameter set. It is seen that the
results are not entirely consistent. To further illustrate this, the results for this
parameter set of KLOE(08) [40], WASA [41], BESIII [42] and the present
analysis are plotted in figure 6.1, with the statistical and systematic errors
added in quadrature. The figure also includes the results for the parameter set
with a, b, d, f and g of the present analysis. These are all the experimental
results with high statistics and which include the f parameter. The biggest
tension in the experimental results is for the a and b parameters. The results
from the KLOE(08) [40] and WASA [41] experiments differ for the a parameter by 2σ and for the b parameter by 1.8σ (where σ is the sum in quadrature
of the KLOE and WASA combined errors, considering the asymmetric errors
of KLOE). The BESIII [42] experiment’s value for a is closer to the WASA
result, within 0.7σ , and differs from the KLOE result by 1.5σ , while the result
for the present analysis differs by 0.7σ , 2.2σ and 1.4σ from the KLOE(08),
WASA and BESII results, respectively. The b parameter from BESIII agrees
with the KLOE(08), WASA and present results within 1.4σ or better. With all
results, the previous KLOE value for d seems a bit low and differs by 1.1σ ,
1.4σ and 1.6σ from the WASA, BESIII and present results.
130
Table 6.1. Summary of experimental results with at least the b Dalitz plot parameter,
including the current results. The value of g, only measured in this work, is given in
the last row.
Experiment
−a
Gormley(70)[36] 1.17(2)
Layter(73)[35] 1.080(14)
CBarrel(98)[39] 1.22(7)
KLOE(08)[40] 1.090(5)(+19
−8 )
WASA(14)[41] 1.144(18)
BESIII(15)[42] 1.128(15)(8)
this work (no g) 1.104(3)(2)
this work (g)
1.095(3)(2)
this work (g) g = −0.044(9)(+12
−13 )
b
d
f
0.21(3)
0.03(3)
0.22(11)
0.124(6)(10)
0.219(19)(47)
0.153(17)(4)
0.142(3)(+5
−4 )
0.145(3)(5)
0.06(4)
0.05(3)
0.06(fixed)
0.057(6)(+7
−16 )
0.086(18)(15)
0.085(16)(9)
0.073(3)(+4
−3 )
0.081(3)(+6
−5 )
0.14(1)(2)
0.115(37)
0.173(28)(21)
0.154(6)(+4
−5 )
0.141(7)(+7
−8 )
!
"
#
Figure 6.1. Comparison of the parameters a, b, d and f for the most recent experimental results.
131
Compared to the KLOE(08) result, the present measurement reduces both
statistical and systematic errors. The statistical errors are reduced by about a
factor two, and the systematic errors even more. In the previous analysis the
systematic uncertainties were about two times lager than the statistical ones,
while the present analysis reduces the systematic uncertainties to be of the
same order as the statistical ones.
6.2 Acceptance Corrected Data
The acceptance corrected data provided by the WASA-at-COSY group [41]
allows a direct comparison. By re-binning our result to the WASA bins, the
acceptance corrected Dalitz plot distributions can be compared directly bin by
bin. Figures 6.2-6.4 show the WASA result in red and the result from this
thesis in black, normalized to the same integral as the WASA distribution.
Figure 6.2 shows the 2-dimensional Dalitz plot distribution, while figures 6.3
and 6.4 show one bin wide slices along the Y - and X-axis, respectively. As
can be seen, the values from this thesis are smaller for large negative values
of Y but larger for positive values of Y , specially around X = 0. This seems
consistent with the larger absolute value of a and d for the WASA parameters,
but for a conclusive statement a more detailed analysis taking into account the
correlations between the parameters is needed.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
0.5
X
0
-0.5
-1
1
0.5
0
-0.5
-1
Y
Figure 6.2. Acceptance corrected Dalitz plot distribution, from WASA (red) [41], and
from the current analysis (black), normalized to the same integral as the WASA result.
As can be seen in figure 6.5 and 6.6 , this work generally agrees with the
2
2
+ σKLOE
is the combined staWASA results within 2.5σ , where σ 2 = σWASA
tistical uncertainty.
132
-0.90 < Y < -0.70
-0.70 < Y < -0.50
1.9
2.15
-0.50 < Y < -0.30
1.6
1.85
2.1
1.55
1.8
2.05
2
1.95
1.75
1.5
1.7
1.45
1.65
−1
−0.5
0
0.5
1.4
−1
1
X
-0.30 < Y < -0.10
−0.5
0
0.5
−1
1
X
-0.10 < Y < 0.10
0
0.5
1
X
0.5
1
X
0.5
1
X
0.10 < Y < 0.30
1.4
0.95
1.15
1.35
0.9
1.1
1.3
0.85
1.05
1.25
−0.5
0.8
1
1.2
0.75
0.95
1.15
−1
−0.5
0
0.5
1
X
0.7
0.9 −1
0.30 < Y < 0.50
−0.5
0
0.5
−1
1
X
0.50 < Y < 0.70
−0.5
0
0.70 < Y < 0.90
0.55
0.4
0.7
0.5
0.35
0.65
0.45
0.3
0.6
0.4
0.25
0.55
0.35
0.2
0.5
−1
−0.5
0
0.5
1
X
0.3
−1
−0.5
0
0.5
−1
1
X
−0.5
0
Figure 6.3. The X dependence of the acceptance corrected Dalitz plot distribution, for
each bin in Y , from WASA (red), and from the current analysis (black), normalized to
the same integral as the WASA result .
-0.90 < X < -0.70
-0.70 < X < -0.50
1.1
1
-0.50 < X < -0.30
1.6
1.8
1.4
1.6
1.4
1.2
0.9
1.2
1
1
0.8
0.8
0.8
0.7
0.6
0.6
0.6
−1
−0.5
0
0.5
1
Y
0.4
−1
-0.30 < X < -0.10
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
−0.5
0.4
0
0.5
1
Y
1.2
1
0.8
0.6
0.4
−0.5
0
0.5
−1
1
Y
0.30 < X < 0.50
0
0.5
1
Y
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
−1
0.50 < X < 0.70
1.6
1.8
−0.5
−0.5
0
0.5
1
Y
0.5
1
Y
0.5
1
Y
0.10 < X < 0.30
2
1.8
1.6
1.4
−1
−1
-0.10 < X < 0.10
−0.5
0
0.70 < X < 0.90
1.1
1.6
1.4
1.4
1.2
1
1.2
0.9
1
0.8
1
0.8
0.8
0.6
0.6
0.4
0.4
−1
−0.5
0
0.5
1
Y
0.7
0.6
−1
−0.5
0
0.5
1
Y
−1
−0.5
0
Figure 6.4. The Y dependence of the acceptance corrected Dalitz plot distribution, for
each bin in X, from WASA (red), and from the current analysis (black), normalized to
the same integral as the WASA result .
133
Y
1
2
1
0.5
0
0
−1
−0.5
−2
−3
−1
−1
−0.5
0
0.5
1
X
Figure 6.5. Residuals between the present analysis and the WASA analysis, i.e., the
difference of the acceptance corrected Dalitz plot distributions between this work and
the WASA results, divided by the combined statistical uncertainty σ .
χ2 / ndf
19.48 / 17
Constant
10
Mean
Sigma
8
7.036 ± 1.140
0.1068 ± 0.1551
1.174 ± 0.117
6
4
2
0
−3
−2
−1
0
1
2
3
residuals
Figure 6.6. The distribution of the residuals between the present analysis and the
WASA analysis. The gaussian fit (red curve) is done by the maximum log-likelihood
method.
134
With the statistical uncertainty of the present analysis, the large binning
used in the WASA analysis introduces a bias on the Dalitz plot parameters,
so we do not quote the parameters for this binning. This is the conclusion of
a check performed by comparing the Dalitz plot parameters from a fit to the
signal MC, using the acceptance correction method, with the input MC values,
using the same binning as the WASA result. A shift of the fitted a, b, d and f
parameters of at least three times the statistical uncertainty is found, as well as
a bad χ 2 for the fit.
6.3 Charge Asymmetries
The charge asymmetries ALR , AQ and AS check the charge conjugation invariance symmetry for the η → π + π − π 0 decay. The results to date are summarized in table 6.2 and are all consistent with zero, which implies no violation of
charge conjugation (neglecting the result for ALR from [50], which is believed
to be the result of a systematic bias in the measurement). The same information is presented pictorially in figure 6.7, where the statistic and systematic
uncertainties are added in quadrature if relevant.
Table 6.2. Summary of charge asymmetry results in the η → π + π − π 0 decay. Systematic errors are only explicitly quoted in KLOE(08) and the current results.
Experiment
ALR · 102
AQ · 102
AS · 102
Gormley(68)[50]
Layter(72)[51]
Jane(74)[48]
KLOE(08)[40]
WASA(14)[49]
this work
1.5(5)
−0.05(22)
0.28(26)
0.09(10)(+9
−14 )
0.09(33)
−0.05(4)(+5
−10 )
−0.07(22)
−0.30(25)
−0.05(10)(+3
−5 )
−0.22(33)
0.02(4)(+5
−2 )
0.5(5)
0.10(22)
0.20(25)
0.08(10)(+8
−13 )
−0.06(33)
−0.004(44)(+33
−35 )
Compared to the previous experiments, this result improves the errors: taking the comparison with the previous KLOE result (the result with smallest
errors), this result lowers the statistical errors by a factor 2, and the systematic errors by a similar factor. These results are consistent with the previous
experimental results at better than 2σ level.
6.4 Conclusions
The η → π + π − π 0 isospin violating decay, because of its small electromagnetic contribution, is a sensitive probe of the light quark mass difference. Combining chiral perturbation theory and the partial width of the decay allows to
mu
, through the value
set a constraint on the plane of the quark masses mms vs m
d
d
135
!"#$%
&
&
'
&
&
Figure 6.7. The experimental results for the charge asymmetries: ALR (top), AQ (middle) and AS (bottom). The dotted lines indicate zero.
of Q. To extract this value, a good theoretical description of the decay dynamics is needed, which should be able to describe the experimental Dalitz plot
distribution.
The Dalitz plot distribution of η → π + π − π 0 is usually represented in the
form of the Dalitz plot parameters, a set of coefficients in the polynomial expansion of the distribution, in the dimensionless variables X and Y . The experimental results so far, including the results of this work, show a tension with
the ChPT predictions for the parameters multiplying powers of Y [24]. If this
difference is assumed to be mostly due to pion-pion rescattering in the final
state, then the dispersive calculations can be used to improve the ChPT result
[9, 10]. The dispersive methods (including [11]) can also use the experimental
Dalitz plot distribution as input to extract, together with ChPT, the constraint
on the light quark masses, i.e., the value of Q.
This work provides a precise, high statistics measurement of the Dalitz plot
distribution of η → π + π − π 0 . The Dalitz plot parameters from this work are,
within errors, compatible with almost all previous high statistics experimental
results [36, 35, 39, 40, 41, 42]. A direct comparison of the acceptance corrected Dalitz plot distributin with the WASA result [41] shows deviations in
some regions of the Dalitz plot, but for almost all bins the values are consistent
within 2.5σ . The acceptance corrected Dalitz plot distribution from this work,
presented in section 5.4 and based on ∼ 4.7 · 106 η → π + π − π 0 events, will facilitate comparison with future experiments and can also be used as input for
the dispersive calculations. In fact, in collaboration with the theory groups,
136
one could extract a value of Q which includes the systematic errors related to
the η → π + π − π 0 Dalitz plot distribution input.
In this work we have also calculated the charge asymmetries and found
them consistent with zero, i.e., we find no evidence of violation of charge
conjugation invariance. This is in agreement with the previous experimental
results [50, 51, 48, 40, 49] and our result improves the accuracy.
137
Summary in Swedish - Svensk sammanfattning
Fysik handlar om att förstå och förklara naturlagarna i världen omkring oss,
allt ifrån det stora, såsom universum och galaxer, till väldigt små objekt, såsom
atomer, nukleoner och partiklar. Målet är att kunna beskriva hur alla dessa
ting beter sig och att kunna förutsäga hur de kommer att bete sig utifrån dessa naturlagar. Fysik är uppdelat i flera olika områden som behandlar olika
längdskalor, dvs hur små objekt man tittar på. Den här avhandlingen fokuserar
på området kärn- och partikelfysik.
Som namnet indikerar, handlar kärn- och partikelfysik om att förstå och beskriva atomkärnor och andra partiklar, som till exempel nukleonerna (protoner
och neutroner) som utgör kärnorna. Beskrivningen av atomkärnor är ett eget
delområde inom fysiken, men här fokuserar vi istället på de mindre partiklarna.
Protoner och neutroner, är exempel av en typ av partiklar som kallas hadroner. Hadroner är, till skillnad från t.ex. elektroner, inte elementarpartiklar, utan
de består av andra, mindre, partiklar, som hålls samman av den starka kraften.
Dessa byggstenar kallas kvarkar. Kvarkarna är liksom elektronerna elementarpartiklar, åtminstone så långt vi vet idag. Det finns två välkända typer av
hadroner: baryoner, som består av tre kvarkar, och mesoner, som består av
en kvark och en antikvark. Antikvarkar är kvarkarnas antipartiklar, de är som
kvarkarna i allt förutom att de har motsatta laddningar, t.ex. om en kvark har
en positiv elektrisk laddning så har dess antikvark lika stor men negativ elektrisk laddning. Protonen och neutronen är exempel på baryoner. De lättaste och
vanligaste mesonerna är de tre pionerna (π + , π − , π 0 ) och eta-mesonen (η),
som den här avhandlingen berör.
Vår nuvarande förståelse av elementarpartiklar och dess växelverkan kallas
inom partikelfysiken för standardmodellen. Inom denna modell växelverkar
partiklarna genom tre av naturens fyra fundamentala krafter: den starka kraften, den svaga kraften och den elektromagnetiska kraften. Gravitationskraften
är väldigt svag för elementarpartiklarna och är inte med i standardmodellen.
Kvarkarna är de elementarpartiklar som, tillsammans med kraftpartikeln
gluonen, påverkas av den starka kraften. Det finns sex typer av kvarkar sett till
deras sort: uppkvark (u), nerkvark (d), charmkvark (c), särkvark (s), toppkvark
(t) och bottenkvark (b). Kvarkarna har även färgladdning, vilket är det som gör
att de växelverkar genom den starka kraften. Färgladdningarna kallas för röd,
grön och blå för kvarkar, och antiröd, antigrön och antiblå för antikvarkarna.
Kvarkar förekommer inte fria, utan är alltid ihopbundna i färglösa “vita” partiklar: tre kvarkar med röd, grön och blå färgladdning blir en baryon, en kvark
med en färg och en antikvark med motsvarande antifärg blir en meson.
138
Teorin som beskriver den starka kraften heter kvantkromodynamik (QCD)
och är en del av standardmodellen. Vid låga energier (vilket motsvarar stora avstånd) är denna teori svår att nyttja. Detta är relaterat till att kvarkarna
inte förekommer fria utan är bundna i hadroner. Istället kan man använda en
approximation av QCD, kallad kiral störningsteori. I denna teori räknar man
direkt med växelverkan mellan de lätta mesonerna (som pionerna och etamesonen), utan att behöva använda sig av att det egentligen är kvarkarna som
växelverkar.
Den matematiska formuleringen av standardmodellen beskriver hur elementarpartiklarna, bland annat kvarkarna, växelverkar. I den formuleringen
finns det 19 parametrar som inte kan förutsägas av modellen, utan måste mätas i experiment. Dessa parametrar är t.ex. kvarkarnas massa och krafternas
kopplingskonstanter. Eftersom kvarkarna inte förekommer fria så är det inte
lika lätt att mäta deras massor som för andra elementarpartiklar, t.ex. elektronen. Kvarkarnas massor kan endast mätas indirekt från andra storheter eller
processer. Framförallt massorna hos de lättaste kvarkarna, upp- och nerkvarkarna, är svåra att mäta och är fortfarande föremål för rigorös forskning. Den
här avhandlingen bidrar med en bit till pusslet.
Specifikt handlar denna avhandling om sönderfallet av eta-mesonen till tre
pioner, η → π + π − π 0 . Det här sönderfallet skulle inte kunna ske om upp- och
nerkvarkarna hade samma massa, så det faktum att det sker, och hur sannolikt
det är, ger oss information om dessa kvarkars massa. Genom beräkningar inom
kiral störningsteori kan sönderfallet parametriseras med bl.a. en konstant Q. Q
ger en gräns för vad kvarkarnas massor kan vara: ett värde för Q motsvarar en
ellips i ett plan som utgörs av kvarkmasskvoter och som visas i figur 1. Detta
betyder att en mätning av Q-värdet ger massor hos upp-, ner- och särkvarkarna
på den ellips som motsvarar detta värde. Tillsammans med annan information
om massorna kan man då räkna ut de enskilda kvarkarnas massor, mu , md och
ms .
Figur 1. Ellipsen i planet av kvarkmasskvoter som fås av Q = 24.3.
139
Tidigare experimentella resultat om η → π + π − π 0 sönderfallet tyder på att
kiral störningsteori, med de beräkningar som gjort hittills, inte beskriver sönderfallet särskilt bra. Ett sätt att jämföra teori och experiment är att studera den
så kallade Dalitzfördelningen, en funktion av två variabler som fullt beskriver
sönderfallet. Resultaten från denna avhandling är den mätning av Dalitzfördelningen för η → π + π − π 0 som ses i figur 2, och som likt tidigare experiment visar på skillnader gentemot beräkningarna med kiral störningsteori. Mätningen
har gjorts med data från KLOE-detektorn vid DAΦNE-acceleratorn i Italien,
tagna mellan 2004 och 2005.
25000
20000
15000
10000
5000
0
1 0.8
0.6 0.4
0.2 0
−0.2−0.4
X
−0.2−0.4
0.2 0
0.6 0.4
−0.6−0.8
0.8
−1 1
−0.6−0.8
−1
Y
Figur 2. Dalitzfördelningen för η → π + π − π 0 sönderfallet mätt i den här avhandlingen.
Med olika teoretiska utvidgningar av kiral störningsteori, eller med andra beräkningssätt, försöker man beskriva sönderfallet bättre. Då är det viktigt
att det finns precisa experimentella resultat av Dalitzfördelningen att jämföra
med. De resultat som presenteras i denna avhandling är de mest precisa hittills.
I en viss utvidgning av kiral störningsteori kan man även använda den experimentellt uppmätta Dalitzfördelningen som indata för att beräkna värdet på Q.
På så sätt bidrar resultaten från denna avhandling till att ytterligare begränsa
möjliga värden på massan hos de lätta kvarkarna. I kombination med andra
experiment kan precisa värden för massorna uppnås, vilket ger en djupare förståelse av den starka kraften och standardmodellen.
140
Acknowledgments
These now more than five years as a PhD student have encompassed much:
sometimes really fun and exciting, at other times a struggle. These years have
nonetheless been worthwhile, and now that the end is in sight I can look back
and realize how much I’ve learned. But this journey would not have been
successful without the help and encouragement from a number of people.
First of all, let me thank my supervisors: Andrzej Kupsc and Tord Johansson. Andrzej, thank you for all the physics discussions, your help with the
analysis and for taking the time to learn and explain things. Thanks for believing in this project and not pressing me too much, even when things were a bit
slow. And recently, thanks for the help with making my figures more compact
and understandable! Tord, thank you for always being available even when
you were so busy and for explaining hyperons even though they ended up not
being a part of this thesis. Thanks especially for letting me finish my master
thesis project in the hadron physics group and encouraging me to pursue a
PhD, I certainly would not be here today if not for those events! And thanks
for all the social events that made this time more fun. To both of you, thanks
also for letting me, even encouraging me, to get involved with PhD rights and
representation.
A very big thanks to my internal referees at KLOE: Simona Giovannella
and Antonio di Domenico. Simona, thanks for the endless hours discussing
my analysis and for explaining the oddities of the KLOE reconstruction code.
Antonio, thanks for your attention to detail, both concerning the analysis and
my way of describing it. Thank you both for all the help with the preparation
of the analysis memo.
For helping me get started, thanks to Marek Jacewicz and Camilla di Donato. Your help to start understanding ROOT, the parts of code I got, and your
shared experience from the η → π + π − γ analysis, all greatly helped me in
the beginning. A special thanks to Marek for walking me through the KLOE
reconstruction program and explaining how to make data n-tuples.
Thanks to the members of the Uppsala hadron group for making me feel
welcome and for so many interesting discussions at our Wednesday meetings
and at our seminars. Thanks Stefan Leupold for always having time to explain
the theoretical aspects of what we do, and for making them understandable
to us experimentalists. Thanks Karin Schönning for being a good role model,
easier to relate to than our more senior colleagues. Thank you both for making our “Challenges in hadron physics” seminar series happen, it has been an
amazing learning experience. Thanks to Christoph Redmer (even if you are
141
not in Uppsala anymore), for wonderful physics discussions. You managed to
at the same time be very knowledgeable but also making it ok to have questions and things not understood, it was so easy to talk to you! Thanks to our
current postdocs, Cui Li and Michael Papenbrock, for being a breath of fresh
air in the group, and for being available with questions and answers. Thanks
to our master students over the years, some for the fun times, others for the
physics they taught me. And thanks to the rest of the senior hadron group,
for sharing your knowledge and experience: Bo Höistad, Jozef Zlomanczuk,
Magnus Wolke, Hans Calén, Pawel Marciniewski and Kjell Fransson.
Thank you also to the KLOE and KLOE-2 collaborations, past and present
members, for the data I used and for the discussions about my analysis during our meetings. Thank you Fabio for always being friendly and interested.
Thank you Erika, Antonio de Santis, Eryk and Gianfranco for making the
lunchtimes during my first visits to Frascati so nice. Thanks Michal and Elena
for making my later visits so much fun, and for always finding time to have
coffee and dinner with me, even when you were swamped with work. And my
occasional office mates in Frascati: Daria, Alek, Wojciech and Izabela, thanks
for coffee breaks and dinners!
For the opportunity to visit summer schools and conferences, thank you to
Liljewalchs travel scholarships and to Anna Maria Lundins Travel Grants from
Smålands nation.
Thank you to my PhD colleagues at Uppsala. The ones who to me will
always be senior: Pär-Anders, Erik, Patrik, Carl-Oscar and Glenn, thanks for
sharing your experience. Erik, thanks for sharing the office, for trying to explain hyperon antihyperon production to me and for the loan of a pair of headphones that I never returned. Patrik, I learned so much about our decay from
you, thanks for that and for being such a wonderful person. Thanks for still remembering me and for the postdoc recommendation! Pär-Anders, Carl-Oscar
and Glenn, thanks for sharing your interest and passion in teaching.
To those of you that I feel are “my age”: Lena, Carla, Aila, Daniel, Dominik, Rickard, Henric. For the fun times and also the times of despair, when
you were someone to talk to. For the Swedish-German practice and for all
the games. For the coffee breaks and the discussions about physics or about
ROOT. For all the interest and commitment to improving our labs. My time
here would really not have been the same without you. Maja, you also fit in
here, though senior to us and wiser, I see you as our age at the department.
To the ones who I will always regard as younger: Jim, Bo, Walter, Joachim,
Elisabetta, Hazhar, Lisa, Alex, Max and Mikael, thanks for coffee and lunch
breaks, for at times making me feel experienced and wise, and for letting me
bug you with the importance of PhD representation.
Thanks to all my friends, even if I don’t see you enough. My Möbius
friends, especially Love, Terese, Val, Leia and Linnea, thanks for all the games
and the parties. My gaming group in Portugal, Ariana, Marta and Helena, for
enjoying our sessions even though they are so far apart.
142
A special thanks to my family, for being supportive and believing in me:
mom, dad, Mi, Jan, Peter, Raquel, Gerd, Ingalena and Vôrrogério. Thanks for
being there my whole life (or almost) and for always enjoying to see me.
And lastly, thanks Sebastian. For putting up with me, believing in me,
letting me bore you with hadron physics talk and for all the things we enjoy
doing together. I hope I’ll be as supportive when it’s your turn.
143
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Acta Universitatis Upsaliensis
Uppsala Dissertations from the Faculty of Science
Editor: The Dean of the Faculty of Science
1–11: 1970–1975
12. Lars Thofelt: Studies on leaf temperature recorded by direct measurement and
by thermography. 1975.
13. Monica Henricsson: Nutritional studies on Chara globularis Thuill., Chara zeylanica Willd., and Chara haitensis Turpin. 1976.
14. Göran Kloow: Studies on Regenerated Cellulose by the Fluorescence Depolarization Technique. 1976.
15. Carl-Magnus Backman: A High Pressure Study of the Photolytic Decomposition of Azoethane and Propionyl Peroxide. 1976.
16. Lennart Källströmer: The significance of biotin and certain monosaccharides
for the growth of Aspergillus niger on rhamnose medium at elevated temperature. 1977.
17. Staffan Renlund: Identification of Oxytocin and Vasopressin in the Bovine Adenohypophysis. 1978.
18. Bengt Finnström: Effects of pH, Ionic Strength and Light Intensity on the Flash
Photolysis of L-tryptophan. 1978.
19. Thomas C. Amu: Diffusion in Dilute Solutions: An Experimental Study with
Special Reference to the Effect of Size and Shape of Solute and Solvent Molecules. 1978.
20. Lars Tegnér: A Flash Photolysis Study of the Thermal Cis-Trans Isomerization
of Some Aromatic Schiff Bases in Solution. 1979.
21. Stig Tormod: A High-Speed Stopped Flow Laser Light Scattering Apparatus and
its Application in a Study of Conformational Changes in Bovine Serum Albumin. 1985.
22. Björn Varnestig: Coulomb Excitation of Rotational Nuclei. 1987.
23. Frans Lettenström: A study of nuclear effects in deep inelastic muon scattering.
1988.
24. Göran Ericsson: Production of Heavy Hypernuclei in Antiproton Annihilation.
Study of their decay in the fission channel. 1988.
25. Fang Peng: The Geopotential: Modelling Techniques and Physical Implications
with Case Studies in the South and East China Sea and Fennoscandia. 1989.
26. Md. Anowar Hossain: Seismic Refraction Studies in the Baltic Shield along the
Fennolora Profile. 1989.
27. Lars Erik Svensson: Coulomb Excitation of Vibrational Nuclei. 1989.
28. Bengt Carlsson: Digital differentiating filters and model based fault detection.
1989.
29. Alexander Edgar Kavka: Coulomb Excitation. Analytical Methods and Experimental Results on even Selenium Nuclei. 1989.
30. Christopher Juhlin: Seismic Attenuation, Shear Wave Anisotropy and Some
Aspects of Fracturing in the Crystalline Rock of the Siljan Ring Area, Central
Sweden. 1990.
31. Torbjörn Wigren: Recursive Identification Based on the Nonlinear Wiener Model.
1990.
32. Kjell Janson: Experimental investigations of the proton and deuteron structure
functions. 1991.
33. Suzanne W. Harris: Positive Muons in Crystalline and Amorphous Solids. 1991.
34. Jan Blomgren: Experimental Studies of Giant Resonances in Medium-Weight
Spherical Nuclei. 1991.
35. Jonas Lindgren: Waveform Inversion of Seismic Reflection Data through Local
Optimisation Methods. 1992.
36. Liqi Fang: Dynamic Light Scattering from Polymer Gels and Semidilute Solutions.
1992.
37. Raymond Munier: Segmentation, Fragmentation and Jostling of the Baltic Shield
with Time. 1993.
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52. Erik Larsson: Identification of Stochastic Continuous-time Systems. Algorithms,
Irregular Sampling and Cramér-Rao Bounds. 2004.
53. Per Åhgren: On System Identification and Acoustic Echo Cancellation. 2004.
54. Felix Wehrmann: On Modelling Nonlinear Variation in Discrete Appearances of
Objects. 2004.
55. Peter S. Hammerstein: Stochastic Resonance and Noise-Assisted Signal Transfer.
On Coupling-Effects of Stochastic Resonators and Spectral Optimization of Fluctuations in Random Network Switches. 2004.
56. Esteban Damián Avendaño Soto: Electrochromism in Nickel-based Oxides. Coloration Mechanisms and Optimization of Sputter-deposited Thin Films. 2004.
57. Jenny Öhman Persson: The Obvious & The Essential. Interpreting Software Development & Organizational Change. 2004.
58. Chariklia Rouki: Experimental Studies of the Synthesis and the Survival Probability of Transactinides. 2004.
59. Emad Abd-Elrady: Nonlinear Approaches to Periodic Signal Modeling. 2005.
60. Marcus Nilsson: Regular Model Checking. 2005.
61. Pritha Mahata: Model Checking Parameterized Timed Systems. 2005.
62. Anders Berglund: Learning computer systems in a distributed project course: The
what, why, how and where. 2005.
63. Barbara Piechocinska: Physics from Wholeness. Dynamical Totality as a Conceptual Foundation for Physical Theories. 2005.
64. Pär Samuelsson: Control of Nitrogen Removal in Activated Sludge Processes.
2005.
65. Mats Ekman: Modeling and Control of Bilinear Systems. Application to the Activated Sludge Process. 2005.
66. Milena Ivanova: Scalable Scientific Stream Query Processing. 2005.
67. Zoran Radovic´: Software Techniques for Distributed Shared Memory. 2005.
68. Richard Abrahamsson: Estimation Problems in Array Signal Processing, System
Identification, and Radar Imagery. 2006.
69. Fredrik Robelius: Giant Oil Fields – The Highway to Oil. Giant Oil Fields and their
Importance for Future Oil Production. 2007.
70. Anna Davour: Search for low mass WIMPs with the AMANDA neutrino telescope.
2007.
71. Magnus Ågren: Set Constraints for Local Search. 2007.
72. Ahmed Rezine: Parameterized Systems: Generalizing and Simplifying Automatic
Verification. 2008.
73. Linda Brus: Nonlinear Identification and Control with Solar Energy Applications.
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74. Peter Nauclér: Estimation and Control of Resonant Systems with Stochastic Disturbances. 2008.
75. Johan Petrini: Querying RDF Schema Views of Relational Databases. 2008.
76. Noomene Ben Henda: Infinite-state Stochastic and Parameterized Systems. 2008.
77. Samson Keleta: Double Pion Production in dd→αππ Reaction. 2008.
78. Mei Hong: Analysis of Some Methods for Identifying Dynamic Errors-invariables
Systems. 2008.
79. Robin Strand: Distance Functions and Image Processing on Point-Lattices With
Focus on the 3D Face-and Body-centered Cubic Grids. 2008.
80. Ruslan Fomkin: Optimization and Execution of Complex Scientific Queries. 2009.
81. John Airey: Science, Language and Literacy. Case Studies of Learning in Swedish
University Physics. 2009.
82. Arvid Pohl: Search for Subrelativistic Particles with the AMANDA Neutrino Telescope. 2009.
83. Anna Danielsson: Doing Physics – Doing Gender. An Exploration of Physics Students’ Identity Constitution in the Context of Laboratory Work. 2009.
84. Karin Schönning: Meson Production in pd Collisions. 2009.
85. Henrik Petrén: η Meson Production in Proton-Proton Collisions at Excess Energies
of 40 and 72 MeV. 2009.
86. Jan Henry Nyström: Analysing Fault Tolerance for ERLANG Applications. 2009.
87. John Håkansson: Design and Verification of Component Based Real-Time Systems. 2009.
¯ → Λ̄Λ, Λ̄Σ0 Re88. Sophie Grape: Studies of PWO Crystals and Simulations of the pp
actions for the PANDA Experiment. 2009.
90. Agnes Rensfelt. Viscoelastic Materials. Identification and Experiment Design. 2010.
91. Erik Gudmundson. Signal Processing for Spectroscopic Applications. 2010.
92. Björn Halvarsson. Interaction Analysis in Multivariable Control Systems. Applications to Bioreactors for Nitrogen Removal. 2010.
93. Jesper Bengtson. Formalising process calculi. 2010. 94. Magnus Johansson. Psi-calculi: a Framework for Mobile Process Calculi. Cook
your own correct process calculus – just add data and logic. 2010.
95. Karin Rathsman. Modeling of Electron Cooling. Theory, Data and Applications.
2010.
96. Liselott Dominicus van den Bussche. Getting the Picture of University Physics.
2010.
97. Olle Engdegård. A Search for Dark Matter in the Sun with AMANDA and IceCube.
2011.
98. Matthias Hudl. Magnetic materials with tunable thermal, electrical, and dynamic
properties. An experimental study of magnetocaloric, multiferroic, and spin-glass
materials. 2012.
99. Marcio Costa. First-principles Studies of Local Structure Effects in Magnetic Materials. 2012.