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 Permanent Link:
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Material Information
 Title:
 Decays of the [gamma](1S) into a photon and two charged hadrons
 Creator:
 BrevaNewell, Luis
 Publication Date:
 2004
 Language:
 English
 Physical Description:
 vi, 121 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Angular distribution ( jstor )
Average linear density ( jstor ) Gluons ( jstor ) Hadrons ( jstor ) Luminosity ( jstor ) Momentum ( jstor ) Photons ( jstor ) Pions ( jstor ) Quarks ( jstor ) Signals ( jstor ) Dissertations, Academic  Physics  UF ( lcsh ) Physics thesis, Ph. D ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 2004.
 Bibliography:
 Includes bibliographical references.
 General Note:
 Printout.
 General Note:
 Vita.
 Statement of Responsibility:
 by Luis BrevaNewell.
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 University of Florida
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 University of Florida
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text 
DECAYS OF FIlE T(1S) INTO A PHOTON AND TWO C(tARGED IIADI()NS
By
LUIS BREVANEWELL
A DISSEtITATION PRESENTED TO TttE GRADUATE SCHOOL
OF T1HE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF 'HE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2004
To my Parents.
ACKNOWLEDGMENTS
I owe my gTatitude to many people without whom this work would not have
been possible. First of all, I would like to thank my supervisor, Dr. Yelton. His
office door was always open for me, and he spent long hours of his time listening
to my ideas even though only a few of them ever worked. He gave me freedom to
explore and branch off in different directions while steadily guiding me forward at
the same time. After these years of working together I consider him more than a
mentor, and a think of him as a close friend.
During my stay at the University of Florida I have met many people inside
and outside the physics world. I would like to thank my first year graduate teachers
Dr. Sikivie and Dr. Woodard who were a source of inspiration for me. My thanks
also go to my fellow graduate students Vijay Potlia, Rukshana Patel, Necula
Valentin, Jennifer Sippel, Suzette Atienza, G. Suhas, and many more for all our fun
physics discussions and late homework sessions. I would also like to thank the good
friends ]I made outside the physics building, James Power, Yaseen Afzal (Paki),
Ramji Kamakoti (Ramjizzle), Dan DeKee (Double Down), and Fernando Zamit
(Fernizzle) who always reminded me that there is more to life than Physics.
The most important person I have met during my graduate career is my wife,
Jennifer. I thank her for enduring all those endless days when I would answer her
questions with only grunts and nods while my attention remained fixed on the
computer screen. Her love and support axe invaluable to me.
Finally, I would like to thank my parents Manuel and Charlene, my sister
Teresa, and my brother Gaizka. They have always been there for me.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ..... .......................
ABSTRA CT . . .. .. .. . .
CHAPTER
1 THEO RY . . . .
1.1 Particle Physics ...................
1.1.1 The Standard Model ............
1.1.2 Quantum Chromodynamics .........
1.1.3 Introduction to the Radiative Decays of the
1.2 Radiative Decays of Quarkonia Overview .....
2 EXPERIMENTAL APPARATUS .............
2.1 The Cornell Electron Storage Ring ........
2.2 The CLEO III Detector ...............
2.2.1 Superconducting Coil ............
2.2.2 Tracking System ...............
2.2.3 Crystal Calorimeter ...............
2.2.4 Ring Imaging Cherenkov Detector.....
2.2.5 Muon Detectors ..................
3 ANALYSIS OF THE DATA ...................
3
T(lS) .... 6
. . 14
. . 17
17
20
21
21
24
25
28
.. 31
3.1 Data Sample ..................................... 31
3.1.1 Continuum Subtracted Distributions ................. 32
3.2 Event Selection ....... ............................ 34
3.2.1 Skim Cuts ....... ........................... 35
3.2.2 Analysis Cuts ....... ......................... 37
3.2.3 Cut Summary, Efficiencies, and Fake Rates ............ 43
4 EXCLUSIVE RADIATIVE DECAY T(1S) 'tr,+r .......... 48
4.1 Robustness of The Mass Distribution ..................... 48
4.2 Statistical Fit of the Invariant Mass Distribution: Signal Areas
and Their Significance ..........................
4.3 Angular Distribution of The Signal ................ .
4.3.1 Optimum Mass Interval ......................
4.3.2 Background Subtraction ......................
49
50
51
53
4.3.3 Statistical Fit of the Helicity Angular Distribution: J As
signments and their Probability Distribution ........
5 EXCLUSIVE RADIATIVE DECAY T(1S) "K+K .........
5.1 Statistical Fit of the Invariant Mass Distribution: Signal Areas
and Their Significance ...................
5.2 Angular Distribution of The Signal ............
6 EXCLUSIVE RADIATIVE DECAY T(1S) .ypp ......
7 SYSTEMATIC UNCERTAINTIES .................
7.1 Cuts ................................
7.1.1 Justification of the DPTHMU Cut ........
7.2 Angular Distribution of Signal ................
7.3 Different Hadronic Fake Rates Between IS and 4S .
7.4 Other Systematic Sources ..................
7.5 Overall Systematic Uncertainties ..............
8 RESULTS AND CONCLUSION ..................
* 66
. 67
80
. 86
. 86
. 87
. 88
. 97
. 97
99
.. 100
APPENDIX HELICITY FORMALISM FOR TWO BODY DECAYS. ...
REFERENCES ..........................................
BIOGRAPHICAL SKETCH .................................
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DECAYS OF THE T(1S) INTO A PHOTON AND TWO CHARGED
HADRONS
By
Luis BrevaNewell
December 2004
Chair: John Yelton
Major Department: Physics
Using the CLEO III detector we report on a new study of exclusive radiative
T(1S) decays into the final states yir 7r, 'IK+K and ypp. We present branching
ratios for the decay modes T(1S) + yf2(1270), T(1S) + yf(1525), T(1S) 
yfo(1710) with fo(1710) * K+K and T(1S) > yf4(2050).
CHAPTER 1
THEORY
1.1 Particle Physics
Particle physics is the branch of physics dedicated to the study of matter
and energy at the most fundamental level. This means that the job of a particle
physicist is to identify the smallest constituents of matter and describe how they
interact with each other.
Humankind has been interested in this subject since ancient times. Two of the
first particle physicists in recorded history are the Greek thinkers Empedocles and
Democritus from the fifth century BC. Empedocles stated that our complex world
was made from combining four fundamental elements (earth, air, fire and water) in
different proportions. Democritus on the other hand, believed that the apparently
continuous objects in the natural world were not really continuous, but made from
voids and indivisible particles called atoms.
Over the last half century Particle physics has advanced tremendously and we
now have a beautiful, but incomplete, theory firmly grounded on experiment that
describes the fundamental constituents of matter and how they interact with each
other. This theory is called "The Standard Model."
1.1.1 The Standard Model
According to the Standard Mode], the fundamental building blocks of matter
are point like particles which interact with each other in as many as three different
ways. Each type of interaction, or force, is itself carried by point like particles
called force carriers. The particles which are force carriers are bosons I and are
collectively called gauge bosoms because they are needed for the theory to be gauge
invariant. The nonforce carrying particles are fermions. 2
The three interactions described by the Standard Model are called the
electromagnetic force, the weak force, and the strong force. The force of gravity is
not included in the Standard Model, and this is one of the reasons the Standard
Model is not yet complete. The electromagnetic force carriers are photons, the
weak force carriers are the Z', W+, and W particles, and finally, the strong force
is carried by eight kinds of gluons. Table 11 summarizes the situation.
Table 11: Gauge bosons and the force they carry.
Symbol I Name Force Carried
ly Photon Electromagnetic
Z Z Weak
W+ W+ Weak
W W Weak
g gluon Strong
Particles which interact through a particular force are said to couple to it
and to carry an associated charge. The nomenclature is as follows, particles that
interact through the electromagnetic force have an electromagnetic charge called
electric charge, those which interact through the weak force have a weak charge
called weak isospin, and those which interact strongly have a strong charge called
color.
The rest of the particles in the Standard Model which are not force carriers,
the fermions, are subdivided into different groups depending on their properties
(See table 12) reminiscent of the way chemists organized the elements into
1 Bosons are 'defined as particles with integer spin in quantum mechanics.
2 Fermions are defined as particles with half odd fractional spin.
3
the periodic table during the second half of the nineteenth century. Fermions are
divided into quarks (generically represented by the symbol q) and leptons. The
main difference between these two groups is that quarks interact through the strong
force while leptons do not. There are six types of leptons and six types of quarks
(also called the six quark flavors) which are grouped into three generations. Each
generation consist of two quarks and two leptons. All three generations replicate
the same set of force charges, the main difference between generations is the mass
of the particles (for example, the ratio of the masses of e : 1 : is 1 : 200 : 3500).
For each fermion there is an antifermion with equal mass and spin and opposite
charge.
Table 12: Fermion symbols classified into quarks, leptons, and the three genera
tions along with the generational common charges.
First Second Third if Electric Weak Has
Generation Generation Generationi Charge Charge Color
u c t +2/3 +1/2 Yes
Quarks d s b 1/3 1/2 Yes
Leptons e VT 0 +1/2 No
e [A T 1 1/2 No
1.1.2 Quantum Chromodynamics
Quantum chromodynamics (QCD) is the part of the Standard Model that
describes the strong force. QCD is based on local gauge invariance and color
symmetry. There are three possible color charges for quarks called r (red), b (blue),
and g (green). Antiquarks have opposite colors called f, b, p. The strong force
between quarks only depends on their colors and is independent of their flavor.
A very important characteristic of the strong force is that gluons themselves
carry a color charge and an anticolor charge so they can interact with other quarks
through the strong force and change their color. Since there are three colors and
three anticolors one might think that there are nine different gluons. However,
there is one linear combination of color anticolor states that has no net color and
leaves a quark unchanged. There are therefore 9 1 8 gluons. The 8 individual
gluon color states can be written as follows,
1
(1 > = (rb+br)
V2
i(rb bf)
12 >=1
13>=1 r bb)
1
14 >= (rg+gr)
(1.1)
5> = (r gf)

1
1
8 >z= (rr + g 0 2bb)
and represent the eight different gluons that exist in nature. The single and unique
color state left out is called a color singlet,
j9 >= I (r+b +g )
(1.2)
which is invariant under a redefinition of the color (a rotation in color space). In
group theory this decomposition of the color states into an octet and a singlet is
denoted by 3 o = 8 & 1. It is worth noting here that a colorless sate, such as 13 >
or 18 >, is not necessarily a color singlet state.
This situation is analogous to the perhaps more familiar example of two spin
1/2 particles. Each particle can have their spin up (T) or down (1) corresponding to
four possible combinations, each giving a total spin S = 0 or 1. The S = 1 states
form a triplet,
>LL
12 > = (Tl> +i1 IT>) (1.3)
13 > I 1>
and there is a singlet state with S = 0,
1
14 >= /I( Tl> 1I T>). (1.4)
Since gluons themselves carry color they can interact with each other through
the strong force. This interaction among the force carriers is unique to the strong
force and, when included in perturbative QCD calculations, leads to two important
properties observed in nature called "asymptotic freedom" and "color confinement."
Asymptotic freedom means that the interaction gets weaker at short distances.
Color confinement is the requirement that observed states have neutral color, or in
other words, they must be in a color singlet state.
Confinement explains why free quarks or free gluons, which have a net color
charge, have never been observed. It also explains why no fractional charged
particles made from a qq bound state have never been observed since it is not
possible to construct a color singlet for such a state (in group theory terms
3 9 3 = 6 D 3 where we have a sextet and a triplet, but no singlet). On the other
hand, color singlets can be constructed for a qq or qqq system. The color singlet for
qq is simply the state shown on the right side of Equation 1.2. The color singlet for
qqq can be obtained from the decomposition 3 0 3 0 3 = 10 D 8 (D 8 G) I and is,
qqq > r singlet (rgb grb + brg bgr + gbr rbg). (1.5)
Particles that are bound states of qq are abundant in nature and are called
mesons, those that are bound states of qqq are called baryons and also abound in
nature. Both groups are collectively called hadrons. There have been hundreds
of different hadrons observed, confirming the validity of QCD and the quark
flavors. Nevertheless, QCD leaves room for more possibilities. A bound state of two
gluons gg (sometimes called a glueball) can be in a color singlet and in principle
could be observed. Although glueballs are allowed by QCD there is no convincing
experimental observation of one. Another possibility are bound states that are
a mixture of the previous states, such as aqq + Jgg with arbitrary a, i3. These
states are called "hybrid mesons" or simply "hybrids". It is believed [1], [2], [3] that
hybrids are necessary to explain the spectrum of light mesons between 12 GeV.
As with any quantum mechanical theory, the spectrum of bound states is
a fundamental test. Glueballs are allowed by QCD, yet there is no conclu
sive experimental evidence of their observation, despite intense experimental
searches [4], [5], [6] complemented by lattice QCD calculations [7], [8] and other
theoretical contributions like bag models [9], fluxtube models [10], QCD sum
rules [11], weakly bound boundstate models [12], and QCD factorization formalism
models [13]. Physicists cannot be sure they understand QCD until such states are
observed, or until they can explain why we cannot observe them.
1.1.3 Introduction to the Radiative Decays of the T(1S)
The T(1S) is a meson composed of bb quarks. Mesons of this kind, composed
of a quark and an antiquark of the same flavor, are in general called quarkonia.
There is a convention behind the name of the T(1S) The T symbol is
reserved for particles composed of bb where the combined spin of the quark and
antiquark is 1. The "1S" symbol is borrowed from atomic spectroscopy with the
"1" meaning that the bb pair are in the lowestenergy boundstate, and the "S"
3 A good example of this is the successful description of th6 hydrogen atom's
spectrum by quantum mechanics.
meaning that the bb have a relative angular momentum L 0. A description of
particle naming conventions can be found in [14].
The T(1S) is unstable, existing for only about 1024 seconds after which it
decays into daughter particles (which in turn decay themselves if they are not
stable). The term "radiative decay" is reserved to any T(1S) decay where one of
the stable daughters is a photon.
The different ways the T(1S) can decay must obey the symmetries in nature.
Examples of such symmetries are "parity" and "charge conjugation", both of which
will be described soon. Symmetries are very important in the standard model. The
usefulness of symmetries can be seen in Noethers' theorem, which states that for
every symmetry there is an associated conserved quantity. Examples of Noethers'
theorem are the conservation of momentum connected to translational invariance
and the conservation of angular momentum associated with rotational invariance.
In the next sections we will use the parity and charge conjugation symmetries
to find T(1S) decays which conserve the associated symmetry constants and are
allowed by nature. In particular, we will show that the radiative decay of the
T(1S) through a photon and two gluons (see Figure 11) is allowed. The key
observation is that the two gluons must be in a color singlet since both the T(1S)
and the radiated photon have no color and color must be conserved. This means
that the two gluons satisfy color confinement and could form a glueball, although
more conventional meson states, or hadrons in no bound state at all, are also
possible outcomes. Regardless of what the gluons do, their energy will eventually
manifests itself as hadrons. Sometimes, two charged hadrons of opposite charge will
emerge. This work examines those two hadrons from a radiative T(1S) decay to
experimentally probe the gg spectrum.
Parity
The parity operator, P, reverses the sign of an object's spatial coordinates.
Consider a particle Ja > with a wave function T,a( t). By the definition of the
parity operator,
PXJ(F, t) = Pa Ta(Y, t) (1.6)
where Pa is a constant phase factor. If we consider an eigenfunction of momentum
0~(,t = ei('t (1.7)
then
pqy(Z t) = Pa9(X,t) = PA@pX(, t), (1.8)
so that any particle at rest, with = 0, remains unchanged up to a multiplicative
number, Pa, under the parity operator. States with this property are called
eigenstates with eigenvalue Pa. Pa is also called the intrinsic parity of particle a,
or more usually just the parity of particle a, with the words at rest left implicit.
Since two successive parity transformations leave the system unchanged, Pa = 1,
implying that the possible values for the parity eigenvalue are Pa = 1.
In addition to a particle at rest, a particle with definite orbital angular
momentum is also en eigenstate of parity. The wave function for such a particle in
spherical coordinates is,
IF i (Y, t) = RP1 (r) Y'(0, )(19
where (r, 0, 0) are spherical polar coordinates, R1(r) is a function of the radial
variable r only, and the Y (0, 0) is a spherical harmonic.
The spherical harmonics are well known functions which have the following
property,
Y1m (9, i = (1)lY m(r 0, 7r + ) (1
(1.10)
Hence
Pi nim( ,t) Pailnim(,t) = Pa(1)'Fnim(X, t) (1.11)
proving that a particle with a definite orbital angular momentum I is indeed an
eigenstate of the parity operator with eigenvalue Pa(1)'.
The parities of the fundamental fermions cannot be measured or derived. All
that nature requires is that the parity of a fermion be opposite to that of an anti
fermion. As a matter of convention fermions are assigned P = +1 and antifermions
are assigned P = 1. In contrast, the parities of the photon and gluon can be
derived by applying P to the field equations resulting in P = 1 and P. = 1.
The T(1S) hasP = PbP(1)L = 1 since L = 0.
Parity is a good quantum number because it is a symmetry of the strong and
electromagnetic force. This means, that in any reaction involving these forces,
parity must be conserved.
Charge Conjugation
Charge conjugation is simply the operation which replaces all particles by
their antiparticles. In quantum mechanics the charge conjugation operator is
represented by C. For any particle Ja > we can write,
Cla >= coa > (1.12)
where ca is a phase factor. If we let the C operator act twice to recover the original
state Ja >,
Ia >= 2a >= C(ca, >) = c2aCla >= cacala > (1.13)
which shows that CaCc = 1. If (and only if) a is its own antiparticle, it is an
eigenstate of C. The possible eigenvalues are limited to C = ca ca 1.
All systems composed of a the same fermion and an antifermion pair are
eigenstates of C with eigenvalue C =()(L+s) This factor can be understood
because of the need to exchange both particles' position and spin to recover the
original state after the charge conjugation operator is applied. Exchanging the
particles' position gives a factor of (I)L as was shown in the previous section,
exchanging the particles spin gives a factor of (1)s+l as can be verified by
inspecting Equations 1.3 and 1.4, and a factor of (1) which arises in quantum field
theory whenever fermions and antifermions are interchanged. With this result we
can calculate the charge conjugation eigenvalue for the T(IS) and obtain C = 1
since L + S = 1.
The photon is an eigenstate of C since it is its own antiparticle. The C
eigenvalue for the photon can be derived by inserting C into the field equations and
is C'= 1.
Finally, we consider a system composed of two gluons that are in a color
singlet. The two gluons are bosons and they must have a symmetric wave function,
T1G under a g, i g2 exchange. Under this exchange, the orbital angular momentum
part of the wave function contributes with a factor ( 1)L', the spin part of the
wave function for two spin 1 particles contributes with a factor of (I)S, and the
color singlet part of the wave function contributes with a factor of +1 since it is
symmetric. To ensure that T'G is symmetric we need L + S to be even. This implies
that C = ()L+S = 1. The L = 0 and L = 1 possible gg bound states are shown
in Table 13. The Jpc = 1+ is peculiar because it is impossible for a qq system to
have these quantum numbers. If this state is ever observed, it must be a glueball.
Experimental searches for such a state have been done, for example, in [15].
Table 13: Possible gg bound states with L = 0 or L = 1. The possible quantum
numbers are limited by the condition that J L + S be even, which is needed to
ensure a symmetric wave function for the two gauge bosons.
0 ++
0 2 2++
1 1 0+, 1+, 2k
Charge conjugation is a symmetry of the strong and electromagnetic force. For
those particles that are eigenstates of C, C is a good quantum number because in
any reaction involving these forces C must be conserved.
The P and C values of various particles used in this analysis are shown in
Table 14 along with their quark composition, orbital angular momentum, and
internal spin.
Table 14: Symbol, name, quark composition, angular momentum (L), internal spin
(S), parity (P), and charge conjugation eigenvalues (C) for a few of the particles
used in this analysis.
Symbol[ Name ]Quark Composition ILI S fP IC
T(1S) Upsilon(1S) bb 0 1 1 1
1+ Pion ud 0 0 1 x
Ir Pion dii 0 0 1 x
K+ Kaon us 0 0 1 x
K Kaon sf 0 0 1 x
p Proton uud 0 1/2 +1 x
p Antiproton ia 0 1/2 1 x
Photon x x 1 l 1 1
Possible Decays of the T(1S)
At this point we can understand the different possible ways the T(1S) can
decay. The Possible decays are limited because the strong and electromagnetic
force must conserve color, C and P.
The simplest possibility is for the bb pair to interact electromagnetically and
annihilate into one virtual photon This is allowed by parity, charge conjugation
symmetry and color conservation. The decay of the T(1S) to one gluon is not
allowed by color conservation and is therefore forbidden. T(1S) decays to two
photons are forbidden by charge conjugation. T(1S) decays to two gluons are
4 Such a photon is called virtual because it cannot conserve the 4momentum of
bb and is unstable, only existing for a brief period of time, as allowed by the uncer
tainty principle, after which it decays.
also forbidden by charge conjugation. T(1S) decays to 3 gluons are allowed (3
gluons can form a color singlet). T(1S) decays to 3 photons are also allowed, but
are largely suppressed by decays to one photon since 3 successive electromagnetic
interactions are much less likely to occur than a single one. Finally, T(1S) decays
two one photon and two gluons are also allowed under the condition that the two
gluons be in a color singlet state.
The three different possible T(1S) decays with least amount of interactions
(also called lowest order decays) are shown in Figure 11.
g
(a) 7 Z
b
(b)
9
(c)
1. +
Figure 11: Lowest order decays of the T(1S) allowed by color conservation, charge
conjugation symmetry, and parity. (a) Shows the decay into three gluons, (b) shows
a radiative decay, and (c) shows the electromagnetic decay through a virtual pho
ton that in turn decays electromagnetically into a pair of charged fundamental
particles, such as quarks or charged leptons (the charged leptons are represented by
the symbol 1).
Observable Resonances
In this work we search for a resonance X produced in a radiative T(IS) decay,
T(1S) *X.
(,1.14)
13
Using charge conjugation symmetry on both sides,
1 lCx. (1.15)
Therefore, Cx = +1.
In order for us to observe X it must decay into two charged hadrons,
X * hh, (1.16)
where h = 7r, K, p are the hadrons whose momentum we are going to measure.
Applying charge conjugation to this last decay,
+1 = IL+ S (1.17)
where L and S axe respectively the angular momentum and spin of the h+h
system. This last equation implies that L + S must be even. This has consequences
for the possible X parities we can observe. By parity conservation in 1.16,
Px = ( ) ( ) 1 .. ,1_)L. (1.18)
For h =7r, K, S = 0 and L J must be even, which implies that Px = +1. For
h = p, S can be 0 or 1 and particles with both positive and negative parities can be
detected.
Table 15: Possible S, L, J, P and C values for X from the radiative decay
T(1S) y X reconstructed in different decay modes.
Decay Mode_ P C
Xr + 0 even even +1 +1
X* K+K 0 even even +1 +1
XPP 0 even even + +1
x p I odd even and odd 1 +1
14
1.2 Radiative Decays of Quarkonia Overview
Theoretical models exist for glueball production in quarkonia decay [12]
and for the glueball spectrum. For example, a quenched lattice calculation [71
predicts a jPC = 214 glueball in the 2.2 GeV/c2 mass region (see Figure 12).
According to Table 15, in the charged pion and kaon modes, we are limited to
detect glueballs in the leftmost column where P = C = +1 of Figure 12, while in
the proton mode we are restricted to the two left most columns.
12
10 2+_ 3L
2% ...... .... 1 4
8
3
E 6 2++~
4 04
2
0 0
++ +
PC
Figure 12: Quenched lattice calculation result for the glueball spectrum for differ
ent P and C values. The mass scale is shown in terms of a scaling parameter from
the QCD lattice calculation named r0 on the left and in absolute terms on the right
by taking r1 = 410 MeV.
5 Here J stands for the internal angular momentum (spin) of the glueball,
J= L+S.
For some time bound states not considered to be pure glueballs, such as
f2(1270), f4(2050), r7, and r' have been observed in J/I radiative decays at
the 10 production level [16] 6 In 1996, the BES collaboration claimed the
observation of a resonance they called the fj(2220) particle in the radiative Ji4
system at the 10' level [171 (see Figure 13). A lot of excitement was generated at
the time because it is possible to interpret the fj(2220) as a glueball. However, this
result has not been confirmed.
2.0 2.2 2.4 2.6
Mass (GeV)
Figure 13: The mass spectrum obtained by the BES collaboration in radiative
J/bP decays into different hadronic modes.
Key to identifying a particle as a glueball are (a) suppressed production in
twophoton collisions (unlike quarks, gluons don't carry electric charge and do not
6 The J/ particle is a cI2 bound state. Since the strong force is flavor blind the
situation in J/I decays is in principle similar to that in T(1S) decays.
couple to photons), and (b) flavor symmetric decays, since a pure glueball has no
valence quarks.
CLEO has already done several studies of radiative decays of the T(1S).
Naively, one expects these types of decays to be suppressed by a factor
[(qb/q,)(m/mb)2 0.025 (1.19)
with respect to J'0 radiative decays. This comes from noticing that the quark
photon coupling is proportional to the electric charge and the quark propagator
is approximately 1/m for low momentum quarks. In 1999, CLEO made the first
observation of a twobody T(IS) radiative decay [18]. The spin of the observed
resonance could not be measured, ut its mass and width where consistent with the
f2(1270) particle. Under this assumption, comparing the measured branching ratio
of T(IS)  f2(2220) to the measured branching fraction of the J1,0 7 f2(1270),
a suppression factor of 0.06 0.03 was obtained. After the BES result for the
fj(2220) in radiative .JA)v decays, a corresponding search was performed by CLEO
in the radiative T(IS) system 1191. This analysis put limits on the fj(2220)
production in radiative T(IS) decays.
In this work we are privileged to have available the largest collection of
radiative T(1S) decays in the world. With it, we can study the structure of color
singlet gg hadronization, and shed more light on the fj(2220) result from BES.
CHAPTER 2
EXPERIMENTAL APPARATUS
To carry out, our study of the di.hadron spectrum we need to first produce
the T(1S) resonance and secondly observe its daughter particles flying away at
relativistic speeds. These two tasks are respectively accomplished by the Cornell
Electron Storage Ring and the CLEO III detector.
2.1 The Cornell Electron Storage Ring
The Cornell Electron Storage Ring (CESR), located at Cornell University. is a
circular particle accelerator that produces e+e collisions.
In order to produce such collisions electrons and positrons need to be created,
accelerated and stored. CESR's different components, shown in Figure 21, have
been carrying out this task since 1979.
A typical CESR run begins at the linear accelerator (LINAC) where electrons
and positrons are produced. To create positrons, electrons are evaporated off' a
filament and linearly accelerated by electromagnetic fields towards a tungsten
target. The collision creates a spray of electrons, positrons and photons. The
electrons are cleared away with magnetic fields and the positrons are introduced
into the synchrotron. The filling procedure is identical, except that the tungsten
target is removed.
Once the electron and positron beams are introduced into the synchrotron,
they are accelerated to the operating energy. in hour case they are accelerated to
the point where their combined energy is the T(1S) mass, 9.46 GeV.
Once the beams are at the desired energy they are transfered to the storage
ring, where they will remain for about an our. At one point in the storage ring the
beams are forced to cross paths. This is the point where e+e collisions occur 1
and where the center of the CLEO III detector is located. If the accelerator is
0 BUNCH OF POSITRONS
o BUNCH OF ELECTRONS
Figure 21: The Wilson Laboratory accelerator located about 40 feet beneath
Cornell University's Alumni Fields.
performing well a high collision rate results. A high collision rate is crucial for the
success of an accelerator and the experiments it serves. The important figure is the
number of possible collisions per second per unit area; this is called the luminosity.
In order to maximize the luminosity, the beams are focused as small as possible at
This point is not fixed in space, but varies from event to event inside a small
volume of space called the interaction region (IR).
19
the IR. During the CLEO III installation several magnetic quadrupoles were added
to CESR to improve the beam focus. CESR has consistently outdone itself while
collecting luminosity over the years (see Figure 22).
223100100
8000
7000
6000
.E
00.
14000
01
00
'C 0
Figure 22: CESR yearly luminosity. The gaps in 1995 and 1999 correspond to
down times when the CLEO II.V and CLEO III detectors where being installed.
The CLEO III detector measures the time integrated luminosity over a period
of time by counting how many times a benchmark process occurs during a certain
time interval at the IR. For redundancy, there are two benchmark processes that
are used, one where the e e particles interact to produce a new e+e pair,
and the other one where the e+e annihilate and produce two photons. Using
the known crosssection for each process, the number of events is converted to a
luminosity called the Bhabha integrated luminosity for the first process, and the y',
integrated luminosity for the second one. The term "integrated" is sometimes left
out and the total luminosity is referredto as simply the Bhabha or y luminosity,
with the time integration left implicit.
2.2 The CLEO III Detector
When the e+e collision occurs, the two particles are annihilated we enter
the world of particle physics. Nature decides what to do with the energy from
the annihilation. We have no chance of directly observing what is happening at
the annihilation point, but eventually long lived semistable particles are created
that fly off at relativistic speeds. These particles carry information about what
happened after the e~e collision, and can tell. us what nature did. Ehe CLEO III
detector has the important mission of detecting and measuring such particles.
As one can see in Figure 23, the CLEO III detector is a composite of many
detector elements. These subdetectors are typically arranged as concentric cylin
ders. The entire detector is approximately cube shaped, with one side measuring
about 6 meters, and weighs over 500 thousand kilograms.
C L E O IIIS l no d C l
Barrel
RiCH Drift
Silicon I
SC Ouadrupole
Pylon
CEndcap
SC Calorimeter
Iron Chambers
Figure 23: The CLEO III detector.
As implied by the name. CLEO III is not the only CLEO detector. CLEO
III was preceded by CLEO II.V, CLEQ.I1, CLEO I.V and the original CLEO
detectors. Fhe CLEO III detector was a major upgrade compared with the
previous version of CLEO [201, ,21], and has an improved particle identification
system together with a new drift chamber and a new silicon vertex detector.
2.2.1 Superconducting Coil
All the detector subsystems except, for the muon chambers are located inside
a superconduction coil. The coil remains unchanged since CLEO II. It is kept in
a superconducting state by liquid helium. The purpose of the coil is to provide a
1.5 Tesla magnetic field, which is mforin to 0.2%, to bend the paths of charged
particles in the detector. By measuring how much much a charged particle bends,
experimenters can measure the moment am of the particle.
The coil inner radius is 1.45m and its outer radius is 1.55 m, with a radial
thickness of 0.10 in. The total length of the coil in z is 3.50 m. It is wound from
a 5mm x 16 tnmi superconducting cable (Al surrounding CuNbTi strands). It
is wound in 2 layers, with 650 turns per layer, on an aluminum shell. When in
operation a current of 3300 amps flows through the coil.
2.2.2 TlYacking System
After particles from the interaction point pass through the beam pipe, they
begin to encounter the active detector elements of the tracking system. There are
two subdetectors responsible for tracking the curving path of charged particles.
The first one encountered by particle, is the silicon vertex detector, and the second
one is the central drift chamber. The CLEO li tracking system is responsible
for tracking a charged particle's path and measuring its momentum. Typical
momentum resolution is 0.3% for 1 GeV tracks. The tracking system also measures
ionization energy losses with an accuracy of about 6%.
Silicon Vertex Detector
The silicon vertex detector in CLEO III [22], also called SVD III, is a four
layer barrelonly structure with no endcaps that surrounds the beam pipe. This
detector (see Figure 24) provides four 0 and four z measurements covering over
over 93% of the solid angle. The average radius of inner surface of the four layers is
25 mm, 37.5 mn, 72 mm, 102 mm. The detector is constructed from 447 identical
doublesided silicon wafers, each 27.0 mm in 52.6 mm in z and 0.3 mnm thick.
The wafers are instrumented and read out on both sides. The instrumentation on
each side consists of an array of aluminum strips oil the wafer surface. These strips
are connected to preamplifiers at the end of the detector. The inner side has 512
strips in thc z direction and the outer side has 512 in the 6 direction. Therefore
each wafer contains 512+512 sensors. The 447 wafers are arranged in the 4 layers,
as follows: 7 sections in 0, each with 3 wafers in z, total = 21 wafers in the first
layer; 10 sections in each with 4 wafers in z, total = 40 wafers in the second
layer; 18 sections in each with 7 wafers in z, total = 126 wafers in the third
layer; 26 sections in phi, each with 10 in Z, total = 260 wafers in the fourth layer.
Charged particles traversing the wafer lose energy and create electron hole
pairs. Approximately 3.6 eVis required to create a single electronhole pair. The
electrons and holes then travel in opposite directions in the electric field applied
to the surfaces of the wafers until they end up on the aluminum strips, and the
detector registers a "hit'". When combined together, the hit on the inner side of a
wafer and the hit onl the outer side give a measurement of the (z, Q@). Tho wafer
position itself determines r.
The Central Drift Chamber
The CLEO III central drift chamber (DR 111) is full of a gas mixture with 60%
Helium and 40% propane held at about 270 K and at a pressure slightly above one
atmosphere. The drift chamber is strung with array of anode wires of goldplated
23
1 cm
tLJ
Figure 24: View of the SVD III along the beampipe.
tungsten of 20 rn in diameter and cathode wires of goldplated aluminum tubes
of 130 ,m in diameter. The anodes are kept at a positive voltage (about 2000 V),
and the cathodes are kept grounded, which provides an electric field between the
anode and the cathode wires. Anode and cathode wires are often called "sense"
and "field" wires respectively.
As a charged particle passes through the DR 1II, it interacts electromagneti
cally with the gas molecules giving energy to the outer electrons which become free
in a process called ionization. The free electrons from the ionized gas molecules
drift in the electric field toward the nearest anode wire. As the electrons get close
to the anode, the electric field becomes very strong which causes an avalanche
as further ionization is induced. The result of the avalanche is a large number of
electrons collapsing upon the sense wire in a very short amount of time (less than
one nanosecond). When this happens to a sense wire, we say that there is a "hit".
The current on the anode wire from the avalanche is amplified and collected at
the end of the anode wire. Both the amount of charge and the time it takes it to
move to the end of the detector are measured. A calibration of the drift chamber
is used to convert the amount of charge to a specific ionization measurement of the
incident particle. A calibrated drift chamber can also convert the time to roughly
measure the position along the sense wire where the charge was deposited.
The wires are strung along the z direction. About 2/3 of the outer part of
the drift chamber (the farthest part from the interaction point) is strung in with a
slight angle (about 25 miliradians) with respect to the z direction to help with the
z measurement. Wires strung in the z direction are called "axial" wires, while those
that are strung at a slight angle are called "stereo" wires.
The DR III consists of an inner stepped section with 16 axial layers, and an
outer part with conical endplates and 31 small angle stereo layers. There are 3
field wires per sense wire and they approximately form a 1.4 cm side square. The
drift resolution is around 150 Im in r and about 6 mm in z. All wires are held
at sufficient tension to have only a 50 /im gravitational sag at the center (z = 0).
There are 1696 axial sense wires and 8100 stereo sense wires, a 9796 total.
2.2.3 Crystal Calorimeter
The CLEO Crystal Calorimeter (CC) is composed of 7784 thalliumdoped
CsI crystals. Each crystal is 30 cm long (16.2 radiation lengths) with 5cm x 5cm
square front face. The crystals absorb any incoming electron or photon which
cascades into a series of electromagnetic showers. The electronic system composed
of 4 photodiodes present at the back of each crystal are calibrated to measure the
energy deposited by the incoming particle. Other incoming particles other than
photons and electrons are partially, and sometimes fully, absorbed by the crystal
giving an energy reading.
The CC is arranged into a barrel section and two endcaps, together covering
95% of the solid angle. The CC barrel section is unchanged since CLEO II; the
endcaps have been rebuilt for CLEO II to accommodate the new CESR interaction
region quadrupoles. The barrel detector consists of an array of 6144 crystals, 128
in 0 and 48 in z, arranged in an almostprojective barrel geometry. That is, the
crystals are tilted in z to point to a few cm away from the interaction point, and
there is also a small tilt in 0. The CC barrel inner radius is 1.02 m, outer radius
is 1.32 m, and the length in z at the inner radius is 3.26 m. It covers the polar
angle range from 32 to 148 degrees. The barrel crystals are tapered towards the
front face (there are 24 slightly different tapered shapes), the endcap crystals are
rectangular, but shaved near the outer.radius to fit in the container. The CC
endcaps consist of two identical end plugs, each containing 820 crystals of square
crosssection, aligned parallel to the beam line (not projective). There are 60
crystals in the "fixed" portion of the "keystone" piece of the endcap. 760 in the
part that slides. The keystone is made up of two parts, one on top that has 12
crystals that for mechanical removal reasons is separate from a container holding 48
crystals. The endcap extends from 0.434 m to 0.958 m in r. The front faces are z =
1.308 m from the interaction point. It covers the polar angle region from 18 to 34
degrees in +z, and 146 to 162 in z.
The photon energy resolution in the barrel (endcap) is 1.5% (2.5%) for 5
GeVphotons, and goes down to 3.8% (5.0%) for 0.1 GeVphotons.
2.2.4 Ring Imaging Cherenkov Detector
The Ring Imaging Cherenkov (RICH) detector [23] is a new detector subsys
tem for CLEO III. It replaces the CLEO I.V time of flight system designed to
measure particles' velocities.
Cherenkov radiation occurs when a particle travels faster than the speed of
light in a certain medium,
v > c/n. (2.1)
Where v is the velocity of the particle, c is the speed of light in vacuum, and n
is the index of refraction of the medium the particle is traveling in. The charged
particle polarizes the molecules of the medium, which then turn back rapidly
to their ground state, emitting radiation. The emitted light forms a coherent
wavefront if v > c/n and Cherenkov light is emitted under a constant Cherenkov
angle, J, with the particle trajectory forming a cone of light. The cone halfangle is
given by the Cherenkov angle which is,
cos 6 = 1. (2.2)
vn On*
If the radiation angle, 6, is measured, the speed of the incident particle is known.
This measurement, combined with the momentum measurement from the tracking
system, gives a measurement of the particles mass, and can be used in particle
identification.
The threshold velocity at which Cherenkov radiation is emitted is vmin n
When a particle traveling at the threshold velocity transverses the medium a very
small cone with 6 0 is produced. The maximum emission angle occurs when
v,,. = c and is given by
1
cos 6max (2.3)
n
The RICH (see Figure 2 5) consists of 30 modules in phi, 0.192 m wide and
2.5 m long. The detector starts at a radius of 0.80 m and extends to 0.90 m. Each
module has 14 panes of solid crystal LiF radiator at approximately 0.82 m radius,
0.192 m wide, 0.17 m long, 1 cm thick. Inner separation between radiators is
typically 50/um. The LiF index of refraction is n = 1.5. The radiators closest to z
= 0 in each module have a 45 degree sawtooth outer face, to reduce total internal
reflection of the Cherenkov light for normal incident particles (see Figure 26). The
radiators axe followed by a 15.7 cm (radial) drift space filled with pure N2. The
drift space is followed by the photodetector, a thingap multiwire photosensitive
proportional chamber.
Figure 25: The RICH detector subsystem.
track
I
10 MM
track
Y7
10m MM
Figure 26: The two kinds of RICH LiF radiators. For normal incidence particles
(z 0) a sawtooth radiator is necessary to avoid internal reflection.
With this index, particles in the LiF radiator with beta = 1 produce
Cherenkov cones of halfangle cos'(1/n) = 0.84 radians. With a 16 cm drift
space, this produces a circle of radius 13 cm. The RICH is capable of measuring
the Cherenkov angle with a resolution of a few miliradians (see Figure 27). This
great resolution allows for good separation between pions and kaons up to about 3
GeV as Figure 28 shows.
tz
L
2
0
1 2 3 4 5
Radiator row
6 7
Figure 27: Cherenkov angle resolutions per track as a function of radiator row for
Bhabha events. Row 1 corresponds to the two rows closest to z = 0, etc.
2.2.5 Muon Detectors
The muon detectors (MU) are the most external subsystem of the CLEO III
detector. They remain unchanged from CLEO II, and are composed of plastic
proportional tubes embedded in the magnet iron return yoke. They cover 85%
of the 47r solid angle (roughly 30150 degrees in polar angle). If a series of hits is
detected in the muon chamber layers they most likely correspond to muons because
other particles are blocked by the iron. Besides detecting muons, the heavy iron
I I I I I *
*Data
o Monte Carlo
0
I I I o I I I ,
0.4
: : :
" 0 .3 ...................... ..................... ...................... ..... ................... ..............: ........7 ''
0 0.2 ........
0 .1 ................. ........................ .... ..................... ..................... r.... ...........
0.5 1.0 1.5 2.0 2.5 3.0
Momentum (GeV/c)
Figure 28: Pion fake rate as a function of particle momentum for kaon efficiency of
80% (circles), 85% (squares) and 90% (triangles).
of the return yoke protects the inner subsystems of the CLEO III detector from
cosmic ray background (except for cosmic ray muons of course).
There are three planes of chambers in the barrel section, arranged in 8 octants
in 0. The plastic barrel planes lie at depths of 36, 72, and 108 cm of iron (at
normal incidence), corresponding to roughly 3, 5, and 7 hadronic interaction
lengths (t6.8 cm in iron) referred to as DPTHMU. There is one plane of chambers
in each of the two endcap regions, arranged in 4 rough quadrants in 6. They lie
at z = 2.7 m, roughly covering the region 0.80 < I cos(O)I < 0.85. The planar
tracking chambers use plastic proportional counters at about 2500 V with drift gas
of 60% He, 40% propane, identical to (and supplied by the same system as) the
drift chamber gas. Individual counters are 5 m long and 8.3 m wide, with a space
30
resolution (along the wire, using charge division) of 2.4 cm. The tracking chambers
are made of extruded plastic, 8cm wide by 1 cm thick by 5 m long, containing eight
tubes, coated on 3 sides with graphite to form a cathode, with 50 Pm silverplated
CuBe anode wires held at 2500 V. The orthogonal coordinate is provided by
8cm copper strips running perpendicular to the tubes on the side not covered by
graphite.
CHAPTER 3
ANALYSIS OF THE DATA
This work builds on the techniques developed by previous CLEO radiative
T(1S) analyses [18], [19], modified to be used with CLEO III data. A new
technique based on kinematic fitting is developed to, together with the new
RICH detector, improve efficiency and particle identification.
We search for radiative T(1S) decays in the modes T(1S) . yr+Tr, yK+K,
and ypp. The e+e collision data, has both resonant events, where the e+e
annihilate to give a T(1S) and continuum events, where the e+e collision does
not give a T(1S) To be sure we are observing T(1S) and not a continuum process,
the continuum must be subtracted by using a pure source of correctly scaled
continuum events. Pure continuum data can be obtained by operating CESR at an
energy different form the T(1S) mass. After subtracting the underlying continuum,
we examine the dihadron invaxriant mass spectrum in search of resonances. We
determine the spin and production helicity (the projection of the spin on the
momentum vector at production time) of any found resonances by examining the
photon and hadron angular distributions.
3.1 Data Sample
The analysis presented here is based on CLEO III data. Throughout this
document, and unless otherwise stated we use 7y luminosity (the luminosity types
used by CLEO are defined at the end of Section 2.1). We prefer to use the 7y over
the Bhabha luminosity because the resonant process T(1S) e+e artificially
increases the reported Bhabha luminosity by about 3% in T(1S) data. This extra
contribution would need to be accounted for when doing a continuum subtraction.
Choosing the yy luminosity avoids this complication.
The CLEO III data is divided into numbered sets. Sets 18 and 19 have a
luminosity of 1.13 0.02fb1 in the beam energy range 4.7274.734 GeV. This
data, which we call the T(1S) data (or simply the IS data), has both resonance
e+e , T(1S) and continuum events. We take the number of resonant events
from [24], NT(1s) = (2.1 0.1) x 107. This number contrasts with the previous
generation measurement of CLEO II, where Nr(ls) 0.15 x 10' were available.
In datasets and 18 and 19 there are also 0.192 0.004fb1 taken below the
T(1S) beam energy (4.7144.724 GeV). This data, which we call the T(1S)off data
(or simply the 1Soff data), has relatively low statistics and corresponds to purely
continuum events.
'To improve our continuum statistics we use 3.49 0.07f b' from datasets
9, 10, 12, 13, and 14 of data taken near the centerofmass energy of the T(4S),
which for our purposes is defined as data with beam energy in the 5.2705.300 GeV
range. This set of data, which we call the T(4S) data (or simply the 4S data), is a
source of pure continuum because no T(4S) * BB resonant event can survive our
"cuts" (the cuts are presented in Section 3.3).
3.1.1 Continuum Subtracted Distributions
We use the continuum data taken at the T(4S) energy to subtract the under
lying continuum present in the T(1S) data. This is important because continuum
background processes like e+e  'yp with p , 7r+7., e4e * y/ with
S K+K, and direct e+e + .yh (we will use the convention h = 7, K, p
from now on), look like the signal events we are searching for. To first order, the
cross section of these continuum process scales like 11s, where s is the square of
1 Cuts are simply conditions that an event must satisfy to be considered in the
,analysis. Cuts are necessary to eliminate background that would otherwise make a
measurement difficult or even impossible.
the center of mass energy of the c+e system. Taking the luminosityweighted
average beamenergies of each interval,, and the y'f luminosities (see Table 31), we
calculate that the T(4S) data scaled down by a factor of 0.404 2 represents the un
derlying continuum in the T (IS) data. This is true up to differences in momentum
distributions and phase space. The error in the continuum scale factor is unknown
because the luminosity ratio is expected to have a small but undetermined sys
tematic error. We make the somewhat arbitrary decision to retain three significant
digits in the continuum scale factor because it is sufficient for our purposes and
there are 0.5% effects from second order terms in the cross section formulae.
To eliminate the contribution of continuum events from a T(1S) data variable
distribution (e.g., the invariant mass of two tracks, the photon angular distribution)
we proceed as follows,
Obtain the T(4S) data distribution for the same variable.
Efficiency correct both the T(1S) and T(4S) distributions, using a
GEANT [25] based Monte Carlo (MC) simulation of the detector. Examples of
MC efficiency distributions are shown in Figures 33 and 73.
2 If we are to be mathematically strict we should calculate the scale factor as
E lSrun / I 4S run
1S runs S lun 4S runs S4S run
with obvious notation. This is equivalent to redefining the average energy as
1 _1 ELrun
E2 L EE2
runs run
However, our energy intervals are sufficiently narrow and this calculation does 4
not change the last significant digit of the original scale factor. Similarly, taking
into account second order terms in the energy dependence of the cross section, like
the one that appears as mn/s in the explicit formula for the e+e + 7p cross sec
tion, also has an insignificant effect on the scale factor.
Subtract the T(4S) distribution from the T(1S) distribution using a T(4S)
scale factor of 0.404.
We call this set of steps "continuum subtraction" by definition. For com
pactness, we call such a distribution the "continuum subtracted
distribution/plot" or " continuum subtracted distribution/plot". Except
for statistical fluctuations and phase space effects, the resulting 1S distribution
should not have any contribution from continuum processes that scale as 1/s.
It is important to notice that any continuum subtracted distribution is
efficiency corrected. This means that a fit to a continuum subtracted distribution
(for example, the continuum subtracted invariant mass distribution) gives the
efficiency corrected number of events directly. Strictly speaking, this number of
events is only correct if all the other variables we cut on have the same initial
distribution in data and MC, or if the efficiency does not depend oil them. In this
note we use "flat MC", defined as MC that is generated with a flat distribution in
the mass and the helicity angles 0, 0, (these angles are defined in the appendix).
For example, the number of events obtained from the continuum subtracted
invariant mass fit needs to be corrected to account for the fact that the helicity
angle distributions are not flat in data (see for example Figure 49, and the
efficiency is highly dependent on these variables (see Figure 73). This correction is
done in Section 7.2.
A summary of the results from this section is shown in Table 31.
3.2 Event Selection
Event, selection for T(1S) * yh+h is straightforward and can be thought of
in terms of three major stages.
First we skim the data, keeping only those events that contain exactly one
highenergy photon and two tracks. Next, we require that the total 4momentum
of these three elements be consistent with the colliding e+e 4momentum. Finally,
35
Table 31: Summary of the data used in this analysis. The continuum scale factor
is obtained using yy luminosities because the Bhabha luminosity is artificially high
during T(lS) running due to the process T(1S) ee.
T(1S) T(4S) T(1S)off
Dataset 18, 19 9, 10, 12, 13, 14 18, 19
Average Ebeam (GeV) 4,730 5.286 4.717
Range of Ebam(GeV) 4.7274.734 5.270 5.300 4.714 4.724
Z(re)(fb') 1.20 0.02 3.56 : 0.07 0.201 0.004
L(y7)(fb1) 1.13 0.02 3.49 0.07 0.192 0.004
T(1S) continuum scale factor 1 0.404 5.84
in the third stage, we project the surviving data onto the three different hadronic
modes via hadron separation and QED suppression cuts.
However, checking the 4momentuml involves using the tracks masses. This
means that the information from stage 2 should somehow be useful in stage 3. This
is indeed the case, and the details of how we do it are revealed in this section.
3.2.1 Skim Cuts
We skim over the data in the "hardGam" subcollection. The hardGam
subcollection was developed with this type of analysis in mind. For an event that
passes the triggers 3 to be classified as hardGam it must pass the following cuts,
eGaml > 0.5
eSh2 < 0.7
eOverP1 < 0.85
eVis > 0.4
aCosTh < 0.95 where "eGaml" is the highest isolated shower energy
relative to the beam energy, "eSh2" is the energy of the second highest shower
relative to the beam energy, "eOverPl" is the matched shower energy relative to
3 Triggers are basic criteria that an event must satisfy to recorded during the
data collection processes. Triggers are designed to get rid of trash and noise and
reduce the size of the data sample while keeping all of the important information.
36
the momentum of the track with highest momentum, "eVis" is the total energy
detected (charged tracks are assumed to be pions) relative to the center of mass
energy, and "aCosTh" is the z component of the unitary total momentum vector.
MonteCarlo predicts that about 75% of the generated T(1S) 4 yh+h signal
passes the hardware and software triggers and gets classified as hardGam.
As mtrentioned above, we use this data to make our skim. To write an event
from the hardGam subcollection into our skim we require the following topological
cuts,
There are exactly two "good tracks"; there can be any number of tracks that
are not "good tracks" but these are not used in the analysis. We define a "good
track" as a track that satisfies the following cuts; drift chamber track ionization
energy loss (dE/dX) information is available, the ratio of number of wire hits to
those expected is between 0.5 and 1.5, the pion fit has x2/d.o.f. < 20 (here d.o.f.
stands for degrees of freedom), and the distance of closest approach to the beam
spot in the xy plane (called DBCD) is less than 5 3.8P (nun) if P < 1 GeV/c
(where P is the tracks momentum in GeV/c) and less than 1.2 mm for tracks with
P > 1 GeV/c. This DBCD cut is common in the more sophisticated CLEO 11/11.5
analyses. It performs better than a simple DBCD < 5 mm cut, because it takes
into account the fact that traks with higher momentum have a better measurement
of DBCD since they scatter less.
There is exactly one "good shower", there can be any number of showers
that are not "good showers" but these are not used in the analysis. We define a
"good shower" as an unmatched shower with energy > 4 GeV.
These topological cuts are about 85% efficient for generated signal events that
have passed the triggers and have been classified as hardGam.
The overall skim efficiency is between 60,65%, depending on the mode (see
Table 34).
3.2.2 Nnalysis Cuts
After our skim we call any cuts we make "analysis cuts". These cuts are done
at analysis time and are mode dependent. As a convention, and unless otherwise
stated, efficiencies for individual analysis cuts are reported relative to the events in
the skim (not relative to the events generated).
4momentum Cut
All fully reconstructed events should have the 4momentum of the 6 e
system. This constraint is usually implemented with a simple twodimensional
AEp box cut, where AE is the difference between the reconstructed energy for
the event and the colliding e c energy (EcMA). and p is the magnituiide of the
reconstructed total momentum for the event. Typical values for these cuts are
0.03 < AEiEcM < 0.02 and p < 150 MeV/Ic (taken from [19)).
The traditional AEp box cut is somewhat useful. However, it does not take
into account the correlation between the measured energy and momentum. Indeed,
the signal lies in diagonal bands in the AEp plane, making a boxsbaped cut not
optimal (see Figures 31a and 31b).
We use an alternative approach to the 4momentum cut. After a simple
substitution, Ey = Ay = ECM Eh+h (where E,. is the photon's energy, p, is
the magnitude of the photons momentum, ECM is the energy of the e6 system.
and Eh+h is the energy of the hadron pair), we can write the Ep conservation
equations as:
Ph +h + (EC  Eh+h )P, =P AM (3.1)
where, Pgh+h is the dihadron momentum, fr is the photon's momentum unit
vector, and i5CAM is the momentum of the &~c' systen (which is a few MeV because
of the crossing angle). Equation 3.1 is a 3constraint subset of the 4momentum
constraint and has the convenient property of avoiding the use of the measured
photon's energy, which has nonGaussian asymmetric errors. It is important to
notice that Equation 3.1 contains the dihadron energy, therefore, it can help
discriminate between the various particle hypotheses.
We proceed as follows. After vertexing the hadron pair using the beam spot
with its error matrix, we calculate the photon's direction from the hadron pair's
vertex and the shower position. We then fit the event to the 3 constraints expressed
in Equation (1) using the techniques outlined in [26] and cut on the X2 of the
3constraint fit, yX p(h) < 100. To complete the 4momentum requirement, we
calculate AE(h) = Eh+h + E,  ECM, where Eh+h is the updated dihadron
energy after the constraint, and E, is the measured photon's energy, and require
0.050 ECM < AE(h) < 0.025 FrM. Furthermore, we now have available
x2_P/h) differences between different particle hypotheses, which help in particle
identification (ID). This is discussed in more detail in the next section. Figure 31
compares the performance of the old and new approaches to the Ep cut.
At this point it is a good idea to check that the 4momentum cut, rejects
background events that make it through our skim cuts (see Section 3.1). These
events typically have one high energy shower, two tracks, and (an) additional
element(s). One such background is T(IS) , ,yr+r7r0. Out of 25000 T(1S)
with 71 y 7r+r MC events 4 only 4 survive our 4momentum cut. We conclude
that our 4momentum cut is good at rejecting background events that pass the
skim cuts but have additional elements such as an extra photon, 7r, pair of tracks,
etc.
Hadron Separation
We define AX D(h,  h2) for the particle hypotheses h, and h2 of our charged
track pair (e.g. AX ( K)) as follows,
4 Thanks to Vijay Potlia for generating these events
0.
0.
0.'
0.1
0.50 0.40
0.30 0.20 0.10 0.00
AE (GeV)
Figure 31: Distributions for different 4momentum cuts. Signal MC events
(T(1 S) , y7r+r) are represented by the black dots, and "background" M( events
(e+e   K+K) appear as (red) triangles. Plot a) has no cuts. Plot b)
has the old 4momentum boxcut. Plot c) has the new 4momentum cuts. Plot d)
has the new 4momentum cuts, and also a cut defined as X9 __(7r) X 2_K) < 0.
The particle ID potential of the newly available X 2_P(h) is evident.
ID(h, h2) adE/dX(h+) E/dX(h2) + dX(h ,dx(h
I d 2 dd; 1 d,
2 log(LRIcH (h)) + 2 log(LRIcH(h4))
(3.2)
2 tog(LRIC1 (h)) 4 2 log(C R CH(h2)).
where the idea is to combine the dE/dX and RICH information into one number.
Pairs of tracks with AXID(hi h2) < 0 are more likely to be of type hb than of typc
h2,
In practice, we only add a tracks RICH information if its momentum is above
the Cherenkov radiation threshold for both mass hypotheses and there are at least
3 photons within 3a of the Cherenkov angle for at least one of the mass hypothesis.
50Q I ... f ... I i. i
25 a)
25 b)
00 AA 1
25 0
A
25 d) .
Jil 1 1>f ALL:
0.
0.
0.
n3
0.10 0.20
We also require that both hypotheses were actually analyzed by RICH. during
pass2.
In addition to RICH and dE/dX, and as hinted in the previous section, the
difference in XEP(h) from the constraint expressed in Equation 3.1 can help the
particle ID (see Figures 31c and 31d). We define,
A(E_ hj  h2) X,, P(hi) (3.3)
Events with AXEP(hl h2) < 0 are more likely to be of type h., than of type h2.
In this analysis, to select h, and reject h2 the default cut is simply AXID(hl 
h2) < 0. This simple cut is highly efficient, has low fake rates., and is sometimes
sufficient. However, out of the six possible cases when one hadron fakes another,
there are three important cases where it pays off to also use AX 2(h h2)
together with AXID(h h2) in an optimal way:
1. 7r background to K. This background comes trom the continuum process
e+W +ip, p 7"f.
2. r background to p. Again, this background comes fiom the continuum
process ee ',p, p  iri
3. K background to p. This background comes from the continuum process
e yp K K
In other words, the important cases occur when the lighter mass hypothesis
fakes the heavier mass hypothesis.
Mathematically, one would expect that simply adding both AXD and AXE
together (like we just did when combining RICH and dE/dX), and cutting on the
grand AX 2 is the way to go. Unfortunately. this simple approach fails because of
large nonmathematical tails in the individual XEP distributions.
Instead, for each of these three cases we define the best cut values (cl. c2) in
A D < cl and AX _P < c2 as those that maximize
T'CC2 R (34)
where Re (Rf) is the efficiency (fake rate) of the particle ID cuts and W is the
rough ratio of the background to signal in the data sample for each case. Each W
can't be known a prwri, but a rough idea of its value can be obtained by doing
a first iteration of the analysis with, for example, c1 = c2 = 0. We use W = 20,
W = 60, and W = 30 for cases 13 respectively. Figure 32 shows F(c, c2) for each
case. Table 32 shows the optimized cut values and their effect on particle ID.
10 =
10
10 r
10 5 0 5 10
Cl
Figure 32: Contour plots of F(cl, c2) as defined in Equation 3.4 for different sig
nals and backgrounds from flat MC. Top has a K signal and a 7r background. Cen
ter has a p signal and a ir background, Bottom has a p signal and a K background.
Quantum Electrodynamic Suppression
Quantum Electrodynamic (QED) background in this analysis comes from the
abundant processes e+e* Ie+e and e+e'  y .
42
Table 32: Cut values, efficiencies, and fake rates for AXAD < cl and AX2_ < c2 in
fiat MC. The different c1, c2 values axe chosen so F from Equation 3.4 is maximun.
Efficiency and fake rates of each cut are reported relative to events in the skim.
Statistical errors in the efficiencies are 0.1% or less. Errors in the fake rates are
statisticall and are shown for completeness only.
CutValue Signal efficiency (%) Fake rate (%)
cl c2 AXeD AX Both AXeD AXEZP Both
w faking K 3 2 94.2 95.2 89.9 1.73 0.09 14.5 + 0.3 0.31 fr 0.04
7r faking p 1 2 99.0 99.8 98.9 0.89 0.06 1.03 + 0.07 0.03 0.01
K faking p 1 3 98.8 99.3 98.1 1.89 0.09 1.09 + 0.07 0.10 0.02
To reject e+e + e+e we require both tracks to have a matched shower with
energy E such that IE/p(7r)  0.951 > 0.1, and to have 'ZWD(h  e) < 0.
To reject e e ",/lA+p we simply require DPTIIMU < 5 (DPTHA LU
was defined in Section 2.2.5) for both tracks in the K and p modes because particle
ID cuts make the pion (and therefore the muon) fake rate small (see Tables 32
and 36). For the 7r mode we cannot use particle ID in a practical way because
muons and pions have similar masses. Instead, to separate pions and muons we
use a much stronger cut requiring that both tracks be within the barrel part of
the muon chamber (I cos(O)I < 0.7), both have P > 1 GeV/c and both have
DPTHMU < 5
To improve the overall muon suppression cut efficiency with virtually no
increase in muon fakes, we flag an event as "not muonic" if any of the tracks
deposit more than 600 MeV in the CC., This increases the cut efficiency by about
90% in the ir mode and makes the detector more hermetic.
5 Other analyses (for example [18]) typically use DPTHMU < 3, our CLEO III
MC has too many pion tracks with 3 < DPTHIMU < 5 and two few tracks with
DPTHMU = 0.
43
3.2.3 Cut Summary, Efficiencies, and Fake Rates
Table 33 summarizes the cuts used in this analysis. Figure 33 shows the
overall Monte Carlo efficiency after all cuts. Figure 34 shows the fake rates
according to the MC and the data for different particle ID cuts. The data fake
rates and limit fake rates, which are measured for pions and kaons faking other
hadrons, are calculated from the p and peaks in the continuum (see Figures 42
and 51). Tables 34 through 36 sununarize the results of Figures 33 and 34.
Table 33: Cuts used in this analysis.
Motivation Definition for T(1S)  h+h (where h = 7r, K, or p)
Skim cuts
Data acquisition Event must pass hardware (Levell) and software triggers
t (pass2) and be of type hardGarn
Topological There are only two good tracks and only one isolated shower
with E > 4 GeV
Analysis cuts
Reconstructed event must X2_p(h) < 100.0 and 0.050 < E(h)/EcM <0.025
'have 4momentum of the
centerofmass system
Hadron separation Default is, AXID < 0. The three cases with a large fake pop
ulation because of continuum processes use a simultaneously
optimized cut on AXID and AXEp, and are summarized in
Table 32
QED background Both tracks have a matched shower energy E that satisfies
e+e 7e+e IE/p(r) 0.951 > 0.1, and AxeD(h e) < 0
QED background For h K, p both tracks have DPTHMU < 5. For
e e yy / h = 7r (At least one track has a matched shower energy >
600 MeV) or (( Both tracks have cos(O) < 0.7 and P >
1 GeV/c) and (both tracks have DPTHMU < 5))
Table 34: Efficiencies in % for cuts (as outlined in Table 33) for flat signal MC.
Efficiencies in the second group are reported relative to the number of candidates
that make it to the skim. The third part of the table shows the overall reconstruc
tion efficiency. Statisticall errors are 0.1% or less.
Skim cuts 7r 1K _
Hardware Trigger 89.7 89.0 90.9
Software Trigger 96.5 96.2 96.6
hardGam 78.4 79.9 76.5
Topological (ntrarck, = 2 & Er > 4GeV) 73.3 67.6 72.3
Overall Skim efficiency 64.3 60.362.4
Analysis cuts 7r K p
4momentum 98.6 98.5 99.0
QED e+e  suppresion 93.9 87.4 93.1
QED p+p A+p suppresion 74.7 93.0 98.3
Hadron separation 97.1 89.0 97.8
Overall analysis efficiency 66;.9 79.1 89.1
Overall reconstruction efficiency 43 4 7.6] 55.6
'Table 35: Efficiencies in % for cuts (as outlined in Table 33) for flat 4S MC. Effi
ciencies in the second group are reported relative to the number of candidates that
make it to the skim. The third part of the table shows the overall reconstruction
efficiency. Statisticall errors are 0.1% or less.
Skim cuts 7I K[ p
Hardware Trigger 88.4 88.0 89.0
Software Trigger 94.6 94.6 94.8
hardGam 79.2 80.5 77.9
Topological (ntraks = 2 & E > 4GeV) 75.2 69.5 75.8
Overall Skim efficiency 66.9_ 6.7 66.6
Analysis cuts 7 r K p
4momentum { 97.1 97.0 97.4
QED e+e + ye e suppresion 93.4 89.1 .93.2
QED p~t 4 y p+p suppresion 75.8 92.5 98.1
Hadron separation 91.4 96.6
Overall analysis efficiency ]67.4] 80.2 87.3
Overall reconstruction efficiency i45.0] 5.3 58.2,
Table 36: Final efficiencies and fake rates after all cuts in %. Statistical errors
for efficiencies are 0.1% or less. Statistical errors for fake MC rates are shown for
completeness only. The IS DATA corresponds to the IS off resonance data sample.
The MC was generated flat.
cuts= K cuts p cuts
IS 7r MC 43.0 0.14 0.01 < 0.007
4S 7r MC 45.0 0.29 0.02 < 0.01
1S p DATA  < 1.2 < 0.2
4S p DATA 0.20 0.06 < 0.06
IS K MC 1,27 0.03 47.6 < 0.02
4S K MC 0.92 0.04 50.3 < 0.06
1S DATA < 3.8 < 2.0
4S DATA 4.14 0.69 < 0.4
IS p MC 0.08 0.02 0.34 0.04 55.6
4S p MC 0.05 0.01 0.37 0.01 58.2
1 2
Invariant mass (GeV/c2)
Figure 33: Final efficiencies for each mode as a function of invariant mass for the
iS (solid) and 4S (dashed) Monte Carlo data. The MC was generated with a flat
angular distribution.
0.50
0.25
0.00
W
o 0.025
0.000
0.050
0.000
. .. I
1TIT f kin K'K
I I
n'nt faking pop
K K fokin pop
II I
Invariant Mass of Fakes (GeV/c2)
Figure 34: Fake rates for IS (hollow circles) and 4S (hollow squares) according to
flat MC. The (red) downward pointing triangle is obtained using the p peak in 4S
data.
CHAPTER 4
EXCLUSIVE RADIATIVE DECAY T(1S) ,T+7r
In Figure 42 the 7r+r invariant mass plot is shown for both 1S and 4S data.
Figure 43 shows the continuum subtracted 7r+ r invariant mass distribution (as
defined in Section 3.2.1) with the most likely statistical fit overlayed (which is
described in the next section). The number of events within 11 of the p region
(0.62 0.92 GeV/c2) left after the continuum subtraction is 200 300, and 50 of
these belong to the fo(980) lowmass tail.
4.1. Robustness of The Mass Distribution
In [27] the decay T(1S)  77r' is analyzed, and it is shown how the analysis
stream warps the shape of the reconstructed resonance. This effect, which arises
because of the particular 7r' behavior, raises problems when fitting the invariant
mass distribution.
In Section 3.2.1 we claimed that if the data and MC had the same O.y and
0, distributions, the fit to the continuum subtracted invariant mass distribution
automatically gives the correct efficiency corrected number of events.
Here we test this claim. To this end, we generate 10000 T(1S) y f2(1270),
with f2(1270)  7r+ with flat 0, and 0,. distributions. We treat this MC
as data and carry out the first two steps of the continuum subtraction process.
The resulting peak has a mass of 1.278 0.002 GeV/c2 and a width of 0.193
0.006 GeV/c2, consistent with the generated mass and width of 1.275 GeV/c2 and
0.185 GeV/c2. More importantly, the number of reconstructed events from the fit is
10040 180, which is consistent with the number of generated events.
Figure 41 shows the reconstructed and efficiency corrected events, a fit to
them, and the generated events.
49
We conclude that there is no warping of the mass distribution, and that the
analysis stream behaves like we expect when obtaining the efficiency corrected
number of events from data.
N Reconstructed and Elf. Corrected
MC Generated
2400
2000 "
"u 1600 1
>
~12O
N
> 800
W
400
1.00 1.10 1.20 1.30 1.40 1.50
i*7T Invariant Mass
Figure 41: Reconstructed events and efficiency corrected events, a fit to them, and
the generated events for T(1S)  'yf2(1270), with f2(1270) 4 7r+r with flat 01,
and 0, distributions.
4.2 Statistical Fit of the Invariant Mass Distribution: Signal Areas and
Their Significance
The results of this section are summarized in Table 41.
Figure 43 shows possible signals for for T(1S) * yfo(980), T(1S)
yf2(1270) and T(1S) 'yf4(2050). To determine the number of events in each
signal we fit the invariant i+Tr mass continuum subtracted distribution with three
spindependent, relativistic BreitWigner line shapes. The masses and widths are
allowed to float, except for the width of the f4(2050) which has a very large error
if allowed to float and is set to its PDG value. The PDG values[16] for the mass
and width of the three resonances are mtfo(980) = 980 10 MeV/c2, fo(980)
70 + 15 MeV/c2, mf2(127,) = 1275.4 + 1.2 MeV/c2. rF2(1270) = 185.1+3.4 MeV/c2,
mf4(2050) = 2034 11 MeV/c2, and Pf4(2050) = 222 + 19 MeV/c2.
To measure the statistical significance of each signal we do multiple X2 fits
fixing the signal area to different values while letting the mass and width of
the signal whose significance is being measured to float within 2a of the PDG
values. At the same time the rest of the fit parameters are fixed to the values that
originally minimized the X2. We assign each of these multiple fits a probability
proportional to eX/2 and then normalize. We calculate the chance of the signal
being due to a random fluctuation by adding the normalized probabilities for the
fits with a negative or 0 signal. This method fails for the highly significant f2(1270)
signal because the e2/2 value of fits with negative or 0 f2(1270) signal is of the
order of 1054 and our computing software can only handle numbers as small as
1041. For completeness we state that the significance of this signal is < 10'5.
To measure the upper limit for the fj(2220) we also do multiple X2 fits
for different fixed signal values, while keeping its mass and width constant at
myfj(2220) = 2.234 GeV/c2 and Ffj(2220) 17 MeV/c2 as in [17]. The resulting
probability plot is shown in Figure 45.
We find clear evidence for the f2(1270), evidence for the fo(980) and weak
evidence for the f4(2050). We also put a 90% confidence level upper limit on
fj(2220) production. Fit results are shown in Table 41.
Table 41: Results for T(1S) * rtr. The branching fractions of f2(1270) and
f4(2050) are taken from the PDG [16]. Errors shown are satistical only.
Mode Area B.F. (105) Significance
yfo(980), fo(980) 7r+r 340, T6 8.3 x 106 (4.30
,yf2(1270) 1230 100 10.4 0.8 < 1045 (> 14a)
f4(2050) 85 30 3.6 1.3 5.2 x 101 (2.6u)
yfj(2220), fj(2220) * 7r+7r < 13 < 6.2 x 102 _
4.3 Angular Distribution of The Signal
The helicity angle distributions of 0, and Oy are defined and described in the
appendix. In this section we first obtain the helicity angular distributions of the
51
fo(980), f2(980). and f4(2050) and then fit them to the predictions of the helicity
formalism.
In practical terms, obtaining the helicity angle distribution of a particular
resonance from data consists of two steps. First, we choose an invariant mass
interval around the resonance peak to select events from the resonance and obtain
a helicity angular distribution which has both signal and background events.
Second, we subtract the contribution to the helicity angular distribution of the
background events in the chosen mass interval to obtain what we want; the helicity
angle distribution of the resonance.
Choosing the mass interval is not a trivial thing. If its too wide there will
be too much background, and if its to narrow there will not be enough signal.
To choose the optimum mass interval we need to know how much signal and
background we are selecting. Therefore, the two steps described in the previous
paragraph are related. How we deal with this is revealed in the next two sections.
4.3.1 Optimum Mass Interval
As described above, the first step in obtaining the helicity angle distribution
for a resonance is choosing an invariant mass interval to select events from such a
resonance. A standard 11' (which corresponds to 1.6a for a spin 0 BreitWigner)
cut around the mean mass of the resonance can be chosen as a "standard" interval.
We could proceed this way, but in our case because of the large subtractions
involved when obtaining the angular distribution, a considerable increase statistical
significance of each bin in the helicity angular distribution can be achieved by
choosing the mass interval carefully (see the last column of Table 42).
Let's consider for example the f0(980) angular distribution. We begin with
the T(1S) and the T(4S) efficiency corrected distributions. Before the continuum
subtraction each bin in the fo(980) angular distribution has contributions from
the high end mass tail of the p and the low mass tail of the f2(1270). In order to
52
get the final angular distribution, both of these contributions are taken away by
first doing a continuum subtraction using the scale factor a = 0.404, and then
by subtracting the f2(1270) distribution outside the mass interval scaled by an
appropriate scale factor /3 equal to the ratio of f2(1270) inside the mass interval
and outside of it 1 After all this, the contributions to the bins in the final angular
distribution are,
Nfo(980) := nfo(98o) + np(T(1S)) + 'ff2(1270) anp(T(4S)) /On'2(1270) hfo (98o) (4.1)
where nfo(980) (nf2(1270)) is the number of fo(980)(f2(1270)) obtained by integrating
the fitted spindependent, relativistic BreitWigner function inside the mass
interval, np(T(lS))(np(T(4S))) is the number of p's inside the mass interval from
continuum events at the T(1S)(T(4S)) energy, and n'f2 (270) is the number of
f2(1270) outside the mass interval being used to subtract the contribution of
f2(1270) to the fo(980) angular distribution. Following this last definition, =
nf2(1270)/nf2(1270).
Each of these terms has an associated error. Assuming that An v, that
the efficiency correction has infinite statistics, and ignoring the errors on the
continuum scale factor a, the overall error is,
ANf0(980) = ./nfo(980) + (a + 2)np(T(4S)) + (2 + 0 + /32)nf2(1270)
We arrive to the conclusion that the mass interval (Fn Am, fn + Am) which
produces the helicity angle distribution with smallest relative binerrors is the one
1 Strictly speaking this is correct only to first order. The f2(1270) distribution
itself has a small contamination from fo(980) and f4(2050). This effect, which we
call crosscontamination, is ignored in Equation 4.1. Later, in section 4.2.2, we will
show how to eliminate this small crosscontamination using all the resonances. The
difference in how we actually get the heicity distributions and Equation 4.1 has an
insignificant effect when calculating the optimal mass interval
that maximizes,
Ff(90)(, A) nf (980) (4.2)
F/')fo(980) + (a + a2)np(T(4S)) + (2 + 3 T f2)n (127o)
The plot of Ff0(980) (in, Am) is shown in figure 46.
This same technique can be applied to the f2(1270) and f4(2050) resonances.
Results are shown in Table 42.
Table 42: Mean masses, widths in GeV/ic2 and inverse of the average bin relative
error (F) from background subtractions for the angular distribution of different
resonances. Standard mean masses and widths, corresponding to 1f, are taken
from the fit in Figure 43, and are labeled with the subscript "s", while those that
maximize F are labeled with the subscript "m". The last column shows the factor
by which the effective statistics increase.
Resonance (f, Am,) F(m5, Am5) (fn., Amm) F(m, F m:A
fo(980) (0.970,0.070) 5.7 (0.985,0.060) 6.0 1.11
f2 (1270) (1.270,0.120) 18.3. (1.590,0.420) 20.6 1.27
f4(2050) (2.120, 0.220) 3.2 (2.240, 0.250) 3.4 1.13
4.3.2 Background Subtraction
The continuum subtraction of the helicity angle distribution is defined in
Section 2.1. The subtraction of the tails from other resonances requires a closer
look. Let us call the continuum subtracted helicity angle distribution of the events
in the fo(980), f2(1270), and f4(2050) mass intervals cf0(9O), Cf2(127o), and cf4(2o50)
respectively. Let us call the felicity angle distribution of the events that come
exclusively from the resonance we are trying to select in the same mass intervals
(that is, the true helicity angle distribution of the resonance) tfo(980), tf2(127o), and
tf4(205o) respectively. The continuum subtracted helicity angle distribution of a
resonance is being contaminated by the tails of the other resonances. Keeping this
in mind we write 2
Cfo(980) tfo(980) + Otf2(1270)
Cf2(1270) tfo(1270) + ytf0(980) + btf4(2050) (4.3)
Cf4(2050) ifo(2050) + dtf2(1270)
Where the small numbers /3, y 6, and E are the ratios of the number of events
from a resonance in the mass interval where the contamination is taking place to
the number of events from the same resonance in the mass interval used to select
it. Using the (mm, Am,,) values in Table 42 and the fit in Figure 43 we obtain
/3 8.5 x 103, = 6.0 x 102, 6 = 9.7 x 102, and e = 0.13.
To obtain the background subtracted helicity angular distributions we simply
invert the system of equations expressed in Equation 4.3. The solution can be
conveniently expressed as,
tf2(l27O) = (Cfo(127o) "YCfo(980) 6Cf4(2050))
tfo(980) = Cfo(980) Otf2(1270) (4.4)
tf4(2050) = Cfo(2050) Ftf2(1270)
As a check, if we ignore the second order terms the previous solution becomes,
2 Here the cross contamination between the fo(980) and the f4(2050) is ignored.
There is no mathematical problem in including these contamination terms, but the
system of equations would be more complicated than it needs to be since this type
of cross contamination is negligible.
tf2(1270) Cfo(1270) 'YCfo(980) bCf4(2050)
tf0(980) CfO(980) OCf2(1270)
tf4(2O50) =Cf0(2o50)  f(20
which is indeed is the solution when crosscontamination is ignored.
4.3.3 Statistical Fit of the Helicity Angular Distribution: J Assign
ments and their Probability Distribution
For each resonance, we fit the data to the simultaneous cos 01, and cos 0,
helicity angle distribution obtained from data using Equation 4.4 to the helicity
formalism prediction, Equations 2327. projected on each angle and folded in
opposite directions around their syrmnetry axis in order to show both distributions
on the same plot. The corresponding J value of the best fit for each resonance,
shown in Figures 48 through 410, is defined as the J assignment (J,). We obtain
J = 1 for the fo(980) which is inconsistent with the known spin of the fo(980)
which is J = 0. For the f2(1270) and the f4(2050) we obtain J = 2 and Ja = 4
respectively, which is consistent with. their known spins.
To have an idea of how well the angular distribution determines J among the
hypotheses J = 0, 1, 2, 3, 4, we do a statistical fit for each hypothesis and assign
each one a probability proportional to e(x'++2d.f)12 where d.o.f. are the degrees
of freedom in the fit. The resulting normalized probability distributions give an
idea of the assigned J significance and are shown in Figure 411. In particular,
this figure shows that the J, = I for the fo(980) inconsistency can be due to a
statistical fluctuation.
File r/userluiuLdeo3/ntp/data_50.ntp
ID IDB Symb Datefflme
1 1 1 040807/0140
1 2 1 040807/0140
98765 981 1 040807/0203
Area
9112.
2.1195E+04
569.1
Meau RM.S
0.8080 0.2895
0.7647 0.2072
1.227 0.7650
0 1 2 3
Mass (GeV/c )
Figure 42: Invariant mass of 7r+r for IS (top) and 4S (bottom) data. For the 4S
data we show a blowup of the mass region 1.5 3 GeV/c2 where the p* can be
seen.
57
MINUIT X' Fit to Plot 9&3
UIS*Gamma h h. masspipi axis
File: Generated internally
Plot Area Total/Fit 1544.1 / 1542.1
Func Area TotaliFit 1359.7 / 1359.7
2
x = 44.7 for 54 8 d.o.f.,
Errors Par.
Function 1:f0
MOD 435.70 1l
MEAN 0.96672 1.5
WIDTH 7.05219E02 3.2
* SPIN 0.00000E+00 .0
* THRESHOLD 0.28000 0
Function 2: f2
MOD 998.59 77
MEAN 1.2678 7,5
WIDTH 0.12353 1.4
*SPIN 2.0000 00
* THRESHOLD 0,28000 0.0
Function 3:A_4
MOD 10437 38
MEAN 2.1191 5.4
* WIDTH 022200 0.0
* SPIN 4,0000 0.0
* THRESHOLD 0.28000 00
abolic
5.4
714E02
912E 02
DOOEi 00
OOE+00
'96
905E .03
273E02
000E+00
OOOE400
.84
51 5E.02
)00E 00
OOOE+00
O00E00
Minos
167.0
1.2374E02
4.3455E02
0.0000E+00
0.0000E+00
78.12
7.1319E03
1.3459E02
0.0000E+00
0 0000E+00
38.76
5.8898E02
00000E+00
0.0000E+00
0.0000E+00
22JUL2004 11:51
Fit Status 3
E.D.M. 8.848E06
C,L.= 52.7%
177.3
1.81 60E02
4.4879E02
0.3000E+00
0.OOOOE+00
77.92
8.0978E03
1 5382E02
0,0000.E00
0.OOOOE+00
38.91
4 9096E02
0.OOOOE+00
0.0000E+00
0.OOOOE+00
300 1 1 1 1 1 1 1 f. f. I
0 1 2 3
Mass (GeV/c2)
Figure 43: Continuum subtracted (as defined in Section 3.2.1) invariant mass of
7+7r from T(1S)  '+.r.
I
cite: Gen rolt iyntenally
ID iD8 Symb
9 3 32
98765 981 1
967 5 982 2
DOeFTime
040806/1839
040806/1845
040806/18A5
mean RM S.
1.408 0.4996
1.416 0.4800
1.422 0.4836
1.5 2.0 2.5 3.0
Mass (GeVic )
Figure 44: Blow up of the invariant mass of 7r+7w in the f4(2050) mass region,
including the upper limit on the fj(2220) shown as a dashed line.
File: tmp mntbohrhserlluicleo3mnJitulsgammahthtemp.m"p
go IoB Symb DateTime Area
1 29 1 04080811845 1.000
1 25 1 04086/1845 o0,M01
Mean RMS
6. 94 5008
5,132 3650
0 25
Area
Figure 45: Normalized probability distribution for different fj(2220)  +7 r sig
nal areas. The shaded area spans 90% of the probability.
1010 Fr
50 Lk W 1, L 1i., .
950 975
600 F
400'
200
1290 1490 1690
1000 1025 1050
1890
1990 2090 2190 2290 2390
Mean Mass (MeV/c)
Figure 46: Contour plot of the inverse of the average relative binerror from back
ground subtractions in the fo(980) (top), f2(1270) (middle), and f4(2050) (bottom)
angular distribution.
MINUIT )2 Fit to Plot 9&82
Ul S*Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 606.18/60618
Func Area Tota/Fit 436.31 / 436.31
X 2= 25.6 for 20 1 d.o.f.,
Errors Parabolic
Function 1: JO
AREA 218.15 42.27
300 ,
200
100
z
Minos
 42.27
5AUG2004 23:06
Fit Status 3
E.O.M. 6.301E13
C.L.= 14.1%
+ 42.27
1.0 0.5 0.0 0.5 1.0
Icosel IcoseI
Figure 47: Angular distribution for the excess events in the fo(980) mass region.
The fit corresponds to J = 0.
MINUIT X2 Fit to Plot 9&82
UlS4Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 606.18/606.18
Func Area Total/Fit 478.89 / 478.89
2
6=,rs 15.2 for 20 2 d.o.f.,
rmrs Parabolic
Function 1:J1
AREA 239.45 39.31
e 1.32483E03 0,3852
Minos
39.33
0.OOOOE+00
5AUG2004 23:06
Fit Status 3
E.D.M 1.115E05
C.L.= 65.1%
+ 39.29
+ 0.3537
1.0 0.5 0.0 0.5 1.0
ICOS0t IcosO, l
Figure 48: Angular distribution for the excess events in the fo(980) mass region.
The fit corresponds to J = 1.
MINUIT X2 Fit to Plot 9&72
Ul S*Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 1673.4/ 1673.4
Func Area Total/Fit 17338/1733.8
X2= 27.3 for 20 3 d.o.f.,
Errors Parabolic
Function 1:J2
AREA 866.97 43.10
E 4.25267E04 + 7.7843E02
0.29838 0.1098
Minos
45.85
0.OOOOE+00
0.1478
5AUG2004 23:06
Fit Status 3
E.D.M. 9.701 E06
C.L.= 5.4%
+ 46.07
+ 0.1238
+ 9.9862E02
, 200
0
0
o
_%
0
1.0 0.5 0.0 0.5 1.0
IcosO J IcoseI
Figure 49: Angular distribution for the excess events in the f2(1270) mass region.
The fit corresponds to J = 2.
MINUIT X,2 Fit to Plot 9&92
U1 SGamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 15993 / 159.93
Func Area Total/Fit 127.08 / 127.08
2
X2= 19.6 for 20 3 d.o.f.,
Errors Parabolic
Function 1:J4
AREA 63.528 1.408
0 1.5697 5.8174E02
4) 0.16919 0.6887.
Minos
14.91
0.3398
0,OOOOE+00
5AUG2004 23:06
Fit Status 3
E.D.M. 5.300E07
C.L.= 29.3%
+ 0.OOOOE+00
+ 0.0000E+00
+ 0.O00E+00
. I i I I I I I I I I 1 6i
1* TTiF
0.5
Icos0eI
0.5
Icosl tI
Figure 410: Angular distribution for the excess events in the f4(2250) mass region.
The fit corresponds to J = 4.
40 F
dI
40
1.0
.... I
0
0.8
20.4
I 0.0
0.8
0.4
0.0
0 2 4
Figure 411.: J, probability distribution for resonances in the fo (980) (top),
f2(1270) middlele, and f4(2050) (bottom) invariant mass region when only the
hypotheses Ja = 0, 1, 2, 3, 4 are considered.
CHAPTER 5
EXCLUSIVE RADIATIVE DECAY T(1S) K+K
In figure 51 the K+K invariant mass plot is shown for both resonance and
continuum running. This figure also has an inset showing the UdE/dX(K) for both
tracks for events in the 1.1 3 GeV/c2 region. This inset indicates that most
of the events in this mass region have indeed two kaons and that the amount of
p reflection is small. Furthermore, when the 7r+7 invariant mass is plotted for
these events only 40 out of 700 can be fit under a p peak (this fit is what we use to
calculate the pion faking kaon rate in Table 36).
Figure 52 shows our fit to the K+K invariant mass continuum subtracted
plot as defined is Section 3.2.1 with the most likely statistical fit overlayed (which
is described in the next section). The number of events near the 0 region (1.01 
1.03 GeV/c2) left after the continuum subtraction is 50 70.
5.1 Statistical Fit of the Invariant Mass Distribution: Signal Areas and
Their Significance
The results of this section are summarized in Table 51.
From the measurement of the previous section we expect a small contribution
( 50 events assuming no interference) from f2(1270) + K+K. We do find
some evidence for f2(1270)  K+K events (110 40) in the fit. We also find
strong evidence for the resonance f2(1525) and weak evidence for the fo(1710)
resonance. The f2(1270), and fo(1710) are fitted with their widths fixed to their
PDG values[16] because they have large errors if allowed to float. The rest of
the resonances parameters are consistent with their PDG values[16], which are
mf2(1525) = 1525 5 MeV/c,, Ff2(1525) = 76 110 MeV/c2, nif,(1710) = 1715 +
67
6 MeV/c2, and Ff(1710) = 125 10 MeV/c2. We also observe an excess of events in
the 2 3 GeV/c2 region which we can't attribute to any known resonances.
Significances of the signals of the identified resonances in the fit are calculated
as described in the previous section. The T(1S)  'fj(2220), fj(2220) K+K
upper limit is also calculated using multiple fits, except that this time the events
under the fj(2220) are of unknown origin, so we use a first order polynomial
allowed to float. The significance of the excess of events in the 2 3 GeV/c2
invariant mass region is calculated assuming a normal distribution; we simply add
up the number of events in each bin along with its error. Results are shown in
Table 51.
Table 51: Results for T(1S) + K+K
Mode Area Branching Fraction (105) Significance
7f2(1270) 110 40 23 8 5.4 x 104(3.3y)
f(1525) 360+8o7 3'.9 +09 < 1045(> 1a
yfo(1710), f0(1710) K+K 75 30 0.35 0.14 7.5 X 104(3.2,7)
,K+K (2 3 GeV/c2) 220 20 1.03 0.12 8.8a
yfj(2220), fj(2220) K+K < 10 < 5 x 102
5.2 Angular Distribution of The Signal
In this section we adapt the ideas presented in Section 3.4.3 to the T(1S) 
yK+K situation.
The derived statistical errors from the signal and background subtractions can
be used to calculate the mass interval which best represents the helicity angular
distribution. The inverse of the expected average relative bin error as a function of
the mass interval is shown in Figure 55, and the mass interval that maximizes it
are tabulated in Table 52.
The tails from the resonances contribute to the continuum subtracted helicity
distributions,
68
Table 52: Mean masses, widths inmass and inverse of the average bin relative
error (F) from background subtractions for the angular distribution of different
resonances. Standard mean masses and widths, corresponding to if, are taken
from the fit in Figure 52, and are labeled with the subscript "s", while those that
maximize F are labeled with the subscript "m". The last column shows the factor
by which the effective statistics increase.
Resonance (in,, Am,) F(m,, Am,) I (rnh, Amm) F(mm, Amm) F(m'ZmL
f2(1270) (1.276,0.185) 3.5 (1.300,0.100) 4.1 1.37
f2(1525) (1.540,0.085) 9.6 (1.565,0.100) 9.8 1.04
fo(1710) (1.760,0. 125) 3.1 (1.780,0.095) 3.3 1.13
Cf2(1270) tf2(1270) + Otf2(1525)
Cf2(1525) tf2'(1525) d Ytf2(1270) + btfo(1710) (5.1)
Cfo(1710) tf]o(110)  tfI(i525)
Where again the small numbers 3, y, 3, and c are the ratios of the number
of events from a resonance in the mass interval where the contamination is taking
place to the number of events from the same resonance in the mass interval used
to select it. Using the (mm, Am,,) values in Table 52 and the fit in Figure 52 we
obtain /3 = 1.3 x 102, y = 0.14, 3 = 0.11, and E = 0.19.
The background subtracted helicity angular distributions are,
tf2(1525) yb_(cf2(1525) 'YCf2(1270) 6Cfo(1710))
tf"(1270) =C(1270) Otf2'(1525) (5.2)
tf(=7(0) C,(17) t(1525)
The the best fit for each resonance and the excess of events in the 23 GeV/c2
are shown in Figures 56 through 59. The best spin assignment for the f2(1270) is
Ja = 2, for the f (1525) it is Ja = 2, for the it is fo(1710), and it is J = 1 for the
excess of events in the 2 3 GeV/c2 mass region. The J = 2 value for the fo(1710)
69
is inconsistent with its known spin. Also, examination of the normalized helicity
amplitudes for the f2(1270) reveals that they are inconsistent with those obtained
for the f2(1270) in the 7r+7r mode.
The assigned J probability distributions are shown in Figure 511. They reveal
that the inconsistencies in the f2(1270) and fo(1710) are not significant and can be
attributed to the statistical uncertainty.
File: "/bohr/userl fluis/de3/ntp/data_50.ntp
ID IDB Symb Date/Time
1 1 1 040807/0219
1 2 1 040807/02 19
1 10 1 040807/0235
Area Mean R.M S.
1274. 1.320 04283
2031. 1.156 02585
705.0 0.2244 1.208
01480 1.170
600 I
400
200
1200 _
6
800 .2
6 2 2 6
a~d(K )
400
0
0.9 1.4 1.9 2.4 2.9
Mass (GeV/c )
Figure 51: Invariant mass of K+K for IS (top) and 4S (bottom) data. For
the 4S data the inset shows the O'dE/dX (K) for both tracks for events in the
1.1 3 GeV/c2 mass region. This inset indicates that most of the events are con
stant with having two kaons.
MINUIT )? Fit to Plot 9&3
U1S4Garnmah h. masskk axis
File: Generated internally
Plot Area Total/Fit 794.78 / 575.02
Func Area Total/Fit 494.50 / 429.38
2
x = 37.5 for 40 7 d.o.f.,
Errors Para
Function 1: f_2(1270)
AREA 197.37 64.
MEAN 1.2757 3.31
*WIDTH 0.18510 + 00(
* SPIN 2.0000 + 0.
* THRESHOLD 0.98000 0.0
Function 2: f2'(1510)
MOD 318.76 64.
MEAN 1.5384 7.8
WIDTH 8.54208E02 2.7
* SPIN 2.0000 + 0.04
*THRESHOLD 0.98800 + 00
Function 3: f0(1710)
MOD 96.117 + 38.
MEAN 1.7622 2.2
* WIDTH 0.12500 0.04
* SPIN 0.00000E+00 004
* THRESHOLD 0.98800 0.04
15
bolic
92
I89EQ2
)00E00
)OOE+00
61
)67E 03
982E02
)OOE+00
00E+00
27
476E02
000E+00
)OOE+00
)OOE+00
Minos
64.93
8.9556E02
0.OOOOE+00
0.0000E+00
0.0000E+00
59.53
7.5447E03
2.2981 E02
0.0000E+00
0.0000E+00
38.70
2.0985E02
0.0000E+00
0.0000E+00
0.0000E+00
7AUG2004 14:23
Fit Status 3
E.D.M. 1.627E06
C.L.= 26.9%
64.91
3.0045E02
0.OOOOE+00
0.0000E+00
0.0000E+00
7290
8.8799E03
3.6400E02
0.0000E+00
0.0000E+00
37.77
2.6333E02
0.0000E+00
0.OOOOE+00
0.OOOOE+00
0.9 1.4 1.9 2.4 2.9
Mass (GeVic2)
Figure 52: Continuum subtracted invariant mass of K+K.
01
10
0 El
T
File: Generated internally
ID IDB Symb
9 3 32
98765 981 1
98765 982 2
1.80
Date/Time
040807/1422
040807/1426
040807/1426
2.00 2.20 2.40
Mass (GeV/c2)
Figure 53: Blow up of the fj(2220) region, with the 90% CL upper limit over
laid. The mass and width are taken to be mfj(2220) = 2.234 GeV/c2 and
Ffj'(2220) = 17 MeV/c2 as in [17].
Mean R.MS.
1.739 0.4861
2.086 0.5922
2.091 0.5835
2.60
File: /tmp mntvbohr/userlAuis/cdeo mn ftiulsgammahh'tern'.tmu
ID IDB Symb DatefTme Area
1 29 1 040807/1426 1.000
1 25 1 040807/1426 0.9054
R.M.S,
3.582
2.723
0 25
Area
Figure 54: Normalized probability distribution for different fj(2220) K+K
signal areas. The shaded area spans 90% of the probability.
1200 1250 1300 1350 1400
150 1
100 lo
1500 1550 1600 1650 1700
100
50 L 1 i i
1700 1750 1800 1850 1900
Mean Mass (MeV/c2)
Figure 55: Contour plot of the inverse of the average relative binerror from back
ground subtractions in the f2(1270) (top), f2(1525) (middle), and fo(1710) (bot
tom) angular distributions.
MINUIT X2 Fit to Plot 9&82
UlSGamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area TotaVFit 517.49 / 517.49
Func Area Total/Fit 129 01/129.01
X2= 17.8 for 20 3 d.o.f.,
Errors Parabo!lic
Function 1:J2
AREA 64505 17.78
e 7.95385E03 1.347
D 1.4108 0.4602
Minos
17.78
0.OOE+00
 0.2689
5AUG2004 23.16
Fit Status 3
E.D.M 5.475E05
C.L.= 40.4%
17.78
0.5745
0.0000E400
. I  I I I I I I I I
II ~
I I I, ,
I I I I I I
0.5
IcosO)
. I
. I I I
0.5
IcosOOeI
Figure 56: Background subtracted K+K angular distribution in the f2(1270)
mass region as defined in the text. The fit corresponds to J = 2.
. I
I
1IT
I I
MINUIT X2 Fit to Plot 9&72
U1S4Garnma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area TotaV/Fit 284.44 284.44
Func Area Total/Fit 367.87 / 36787
X2= 20.9 for 20 3 o.o.f.,
Errors Parabolic
Function 1:J2
AREA 183.94 t 18.70
0 6.76443E04 + 0.1885
0.50343 + 0.1298
o 50
"0
Minos
19.29
0.OOOOE+00
0.1593
5AUG2004 23:16
Fit Status 3
E.D.M. 3.771 E06
C.L.= 23.3%
+ 19.25
+ 0.2757
+ 0.1248
1.0 0.5 0.0 0.5
IcosO) tcosO"t
Figure 57: Background subtracted K K angular distribution
mass region as defined in the text. The fit corresponds to J = 2.
a>
I
1.0
in the .f (1525)
MINUIT X2 Fit to Plot 9&92
UlS4Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 129.47 / 129.47
Func Area Total/Fit 79.031 /79.031
X2= 28.0 for 20 3 d.o.f.,
Errors Parabolic
Function I: J2
AREA 39.516 10.80
8 2.99860E04 0.3847
1.5696 0.2633
Minos
10.81
0.0000E+00
0 3060
5AUG2004 23:16
Fit Status 3
E.D.M. 1 721E05
C.L.= 4.5%
10.80
+ 0.4645
+ .O.0E+00
25
0
1.0 0.5 0.0 0.5 1.0
IcoseO4 Icosel
Figure 58: Background subtracted K+K angular distribution in the fo(1710)
mass region as defined in the text. The fit corresponds to J = 2.
MINUIT X' Fit to Plot 9& 102
U1SGamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area TotaVFit 411.73 i 411.73
Func Area 7otal/Fit 370.86 / 370.86
2
x = 18.9 for 20 2 d.o.f.,
Errors Parabolic
Function 1:J1
AREA 165b43 17.19
G 015542 + 0.3003
100
0
Figure 59: K+K
Minos
17.19
0.0000E00
5AUG2004 23:16
Fit Status 3
E.D.M. 4.735E09
C.L.= 40.0%
+ 17.18
+ 0.1921
0.5 0.0 0.5 1.0
Icosel lcosej
angular distribution for events within the 2 3 GeV/c2.
MINUIT X2 Fit to Plot 9&92
U1 S*Garnma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 129.47 / 129.47
Func Area Total/Fit 62.194 / 62.194
2
m, 33.3 for 20 1 d.o.f., Parabolic
Function 1: JO
AREA 31.097 + 0.92
50 ,
5AUG2004 23:16
Fit Status 3
E.D.M. 3.484E15
C.L.= 2.2%
Minos
10.92 + 10.92
1.0 0.5 0.0 0.5 1.0
IcosO7 lcosOI
Figure 510: Background subtracted K+K angular distribution in the fo(1710)
mass region as defined in the text. The fit corresponds to J = 0.
79
1.2 I
0.8
0.4 
1 0.0
0.8
0.4
1 0.0 _
0.8
0.4I
1 0.0
0.8
0.4
0.0 
0 2 4
Ja
Figure 511: Ja probability distribution for resonances in the f2(1270)(top),
f2(1525) (middletop), fo(1710) (middlebottom) invariant mass region and the
excess of events in the 2 3 GeV/c2 (bottom). region when only the hypotheses Ja
= 0, 1, 2, 3, 4 are considered.
CHAPTER 6
EXCLUSIVE RADIATIVE DECAY T(1S) pp
In figure 61 the pp5 invariant mass plot is shown for both IS and 4S data, with
an inset showing that most of the events indeed have a proton and an antiproton
and that the p and 6 reflections are small. The enhancement at 3.1 GeV/c2 in
the 4S plot corresponds to the process e+e  7yJ/0 with J/ * pp. This
enhancement is not as pronounced in the IS plot because the IS data only has
32.4% of the luminosity of the 4S data. The number of events in the J/ invariant
mass region after continuum subtraction is 6 3 and is consistent with 0.
Figure 62 shows the continuum subtracted invariant ppi mass distribution
(as defined in Section 3.2.1) with a 90% confidence level upper limit for fj(2220)
overlaid. A direct fit to the fj(2220) yields 12 5 events.
There is an excess of events in the continuum subtracted invariant mass plot.
We measure this excess and the upper limit of e+e 4 yfj(2220), fj(2220)  pp
the same way we measured the excess of events inside 2 GeV/c2 < m(K+K) <
3 GeV/c2 region and the fj(2220) upper limit in Section 5.1. Results are shown in
Table 61.
The pp angular distribution for the mass range 2 GeV/c2< m(pp) < 3 GeV/c2
is shown in Figure 64 with the most likely J assignment, J = 1, fit overlaved.
The probability distribution for Ja is shown in Figure 65.
Table 61: Results for T(1S) 'p!.
Mode Area B.F. or 90 % U.L. (105) Significance
ypp (2 3 GeV/c2) 85 18 0.41 0.08 4.85u
yfj(2220), fj(2220) + pp < 20 < 9 X 102
Rle: *Abohr/user1Auis/deo3/ntp/data_50ntp
ID IDB Symb Date,'Time
1 1 1 040807/0303
1 2 1 040807/0303
1 10 1 040807/0304
2.0 2.2 2.4 2.6
Mass (GeV/c2)
Area Mean R.M.S.
104.0 2.334 0.3628
141.0 2.187 0.3419
139.0 0.2583 1.259
0.1224 1.119
2.8 3.0 3.2
Figure 61: Invariant mass of p3 for IS (top) and 4S (bottom) running. For the 4S
running the inset is consistent with the events having a proton and an antiproton.
Fle: Generated internally
ID IDB Symb
9 3 32
98765 981 2
98765 982 1
Date/Time
040807/1446
040807/1446
040807/1446
1.8 2.0 2.2 2.4 2.6
Mass (GeV/c2)
Mean R.M.S.
2.521 0.3093
2.685 0.3291
2.813 0.2553
2.8 3.0 3.2
Figure 62: Invariant mass of pp. The plot is continuum subtracted and efficiency
corrected. An overlay with the 90% confidence level upper limit for fj(2220)
is shown. The mass and width are taken to be mrf,(2220) 2.234 GeV/c2 and
Ffj(222o) = 17 MeV/c2 as in [17].
File: tmpmntroohr/userlAuis/ldeo3/mnjitlulsgammahh/temp.tmp
ID IDB Symb Date/Time Area
1 29 1 040807/1446 1.000
1 25 1 040807/1446 0.9110
Mean R.M.S.
12.78 5.430
11 80 4.590
0 25
Area
Figure 63: Normalized probability distribution for different fj(2220) * pp signal
areas. The shaded area spans 90% of the probability.
MINUIT X2 Fit to Plot 9&22
UlS,Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 256.69 / 256.69
Func Area Total/Fit 134.57 / 134.57
X2= 19.5 for 20 2 d.o.f.,
Errors Parabolic
Function 1:J1
AREA 67.287 10.91
0 1.20611E03 0.4148
25
0
C,
Minos
10.93
0,0000E+00
2AUG2004 18:12
Fit Status 3
E.D.M. 1.845E05
C.L.= 36.4%
+ 10.88
+ 0.3544
1.0 0.5 0.0 0.5
IcoseY Icos0"I
Figure 64: pp angular distribution
2 GeV/c2< m(pp) < 3 GeV/c2.
for the excess of events in the mass range
1.0
0.8
0.6
0.4
0.2
1 0.0
0
2 4
Assigned J
Figure 65: J, probability distribution for the excess of events in the mass range
2 GeV/c2< m(pp) < 3 GeV/c2 when only the hypotheses Ja = 0, 1. 2, 3, 4 are
considered.
CHAPTER 7
SYSTEMATIC UNCERTAINTIES
Systematic uncertainties are any sources of experimental uncertainty other
than the statistical ones. Limits on the accuracy of our detector simulation and any
physical processes that interferes with the experimental measurement are typical
examples of systematic uncertainties. These uncertainties need to be identified,
quantified, and if possible, corrected.
7.1 Cuts
From now on we report individual cut efficiencies relative to the events that
survive all the other cuts.
According to our MC, most of the skim cuts, except for the hardGam re
quirement, are nearly 100% efficient. Therefore, such cuts should not be a source
of systematic uncertainty. The only skim cut worth taking a closer look at is the
hardGam cut, which is about 93% efficient in MC. Measuring this efficiency in our
data is not possible because events not classified as hardGam are not in the data
to begin with. Closer examination of our MC reveals that that most of the 7%
inefficiency in hardGam comes from the eOverP1 cut (4%), some from the Sh2 cut
(2%), and the rest (1%) from the other cuts present in hardGam. We can measure
the efficiency of eOverP1 in data by looking at the eOverP2 distribution of data
tracks from p and 0 decay. We are satisfied by this check on the MC modeling (see
Tables 71 and 72), and we won't measure how well MC models the rest of the
hardGam cuts, which are 97% efficient in MC.
To quantify the quality in the MC modeling of our analysis cuts in the 7r
and K modes we use the p and signals present in our data' In this case the
process is straightforward. We measure the p and 0 signal signals in data and MC
over a floating background function with all cuts in place, and with all in place
cuts except the one under consideration. From these numbers we calculate the
effective efficiency of the cut. The differences between data and MC are taken as
the systematic errors which are added in quadrature. Results are shown in Tables.
71 72. The 4momentum cut does not appear because its efficiency is close to
100%.
Cuts MC eff. Data eff. Systematic Error
eOverP2 96.2 96.6 0.4
QED e+e  ye+e suppresion 93.1 94.2 1.2
QED M+p + ,p+p suppresion 75.3 77.7 3.2
Hadron separation 96.9 96.8 0.1
Overall analysis cut systematic error [ I 3
Table 71: Efficiencies for T(1S)  yr+r for flat signal MC, efficiencies from
data (p), and the derived systematic error in %. Efficiencies are reported as the
number of signal events after all cuts divided by the number of signal events with
all cuts except the one under consideration. Statistically errors are 0.1% or less.
For the proton case we don't have a clean sample with high statistics of
e+e + yop5 events in data. By extension we take the systematic error in this mode
to be 10%.
7.1.1 Justification of the DPTHMU Cut
The reason we prefer to use DPTHMU < 5, instead of the more traditional
(see [18]) DPTHMU < 3 used in CLEO II, is that, for some unknown reason, our
CLEO III MC has too many 7r tracks with 3 < DPTHMU < 5.
1 A study using K or A signals from hadronic environments would have larger
statistics, but is problematic because the large number of tracks and showers artifi
cially decrease the cut efficiency. See [28] Appendix A.3 for an example.
Cuts MC Eff. Data Eff. Systematic Error
eOverP2 98.5 100 1.5
QED e+e  e+e suppresion 98.1 99.3 1.3
QED p+ly * / suppresion 93.5 96.4 3.1
Hadron separation 88.4 82.2 7.0
Overall analysis cut systematic error 8
Table 72: Efficiencies for T(1S) * yK+K for flat signal MC, efficiencies from
data (0), and the derived systematic error in %. Efficiencies are reported as the
number of signal events after all cuts divided by the number of signal events with
all cuts except the one under consideration. Statisticall errors are 0.1% or less.
To observe this fact we first select a relatively clean sample of pion tracks
by requiring the event to have a 7r+ invariant mass consistent with the p mass,
and to pass all our analysis cuts except for the cut on DPTHMU on one track.
For such events we plot the rate as a function of momentum, at which the 7r track
whose DPTHMU cut we released has DPTHMU > 3 and DPTHMU > 5. To
increase our statistics we do this procedure twice, once for each track, and average
the fake rate. Figure 71 shows the results. Clearly the CLEO III MC we are using
has some problem modeling the DPTHMU < 3 cut.
To keep the systematic error low we choose a cut at DPTHMU < 5
(DPTHMU < 3 gives a systematic error of about 20%). This does not change the
efficiency in data very much, but it increases the efficiency r,, ted by MC, bring
ing it closer to reality. The increase in IL fakes after loosening the cut is estimated
to be low using QED MC (see Figure 72).
7.2 Angular Distribution of Signal
The photon and tracks from the process T(1S)  'X with X  h h have
a different angular distribution than that of flat MC. Examples of possible angular
distributions are shown in the appendix.
89
File: /bohr/userlAuis/deo3Mtp/data50.ntp
ID IDB Symb Date/Time Area Mean R.M.S.
1 4 31 040525/1408 0.2350 2.758 0.8547
1 3 31 040525/1408 0.4317 2.684 0.8088
1 6 31 040525/1408 9.679E02 2.996 0.7400
1 5 31 040525/1408 6.0516E02 2.792 0.6995
0.20
0.10
0.00
0.040
0.020 
0.000
I I I
0 1.6 3.2
Track Momenta (GeV)
Figure 71: Pion faking muon fake rates for a cut on DPTHMU < 3 (top) and
DPTHMU < 5 (bottom). MC is shown as solid circles while p from data is shown
as hollow circles. Fake rates are reported relative to all events that pass all cuts,
except for the DPTHMU cut for one of the tracks.
Figure 73 shows the efficiency in flat MC as a function of cos 0. and
cos Oh+' Note that the K and p modes are nearly insensitive to the track an
gular distribution, while the 7 mode is more sensitive to Oh+. This happens because
of the stronger muon rejection cut in the 7r mode.
We measure the systematic effects of flat MC efficiency by convoluting each
plot in Figure 73 with different possible angular distributions calculated in the
2 0, and Oh+ are the helicity angles of the sequential decay, defined in the ap
pendix (see Figure 2).
File: r/u erluisde 3 tp/mc s2 mumu ga ma new. n p
ID IDB Symb Date/Time Area
1 3 31 040518/1402 3.676
1 30 32 040518/1402 3.022
Mean R.M.S.
2.361 0.9088
2.549 0.8590
Track Momenta (GeV)
Figure 72: Muon faking pion fake rate (in %) for DPTHMU
and DPTHMU < 5 (solid circles). There is a complete overlap
< 3 (solid squares)
in the last bin.
appendix. Tables 7374 show the necessary correction factors relative to flat MC
efficiency for decays with definite 7 and X helicities A. and Ax, due to the nonflat
photon and hadron distributions. We call these factors E'X~x(,) and Ex (h)
respectively.
The efficiency of a decay with definite A,, Ax can be obtained using the flat
MC efficiency corrected by a factor JX X= E 'x (h). In general, the
=\'A~X (7) X E,\Yx () ngnrl h
final state is a mixture of all possible Aly, Ax pairs, and the efficiency correction
factor is,
iJx = cos2 9 cos2 (DE J + sin2 E)EJ< + cos2 E) sin2 (D(.1
I I I
(7.1)
U U9
0 0.40
LL]
0.0001
0.00 L
1 0 1
cosOe
0.40 hi i 0
0.80
UOS L 1 7
0.60 r
0.40K
0.20 !
0.00 0 1
cosO,"
0.60
0.40
0.20
1 0 1
cosO.
Figure 73: Flat MC efficiency as a function of cos9, (left column) and as a func
tion of CoSOh+ (right column). Top row corresponds to the pion mode, middle row
to the kaon mode and bottom row to the proton mode.
The fits in Figures 48 and 410 and 5658 measure the pair ((3, 4)). These
values are summarized in Table 75, where they are used to obtain Ex for each
mode.
The pair ((, 4)) carry an error which is a source of systematic uncertainty.
We calculate this systematic uncertainty by inspecting the differences in efficiency
when (6, 45) move away from the value which gives the minimum chisquared, 2
alo or a12
E=0
all
7 t 0.910 1.18
K 0.925 1.15
p 0.922 1.16
Table 73: ExJX (7), efficienciy correction with respect to flat MC factors due to
the nonfiat angular distribution of the photon in T(1S) + yX for different X spin
values (Jx) and y, X helicities (Ay, Ax).
Mode Jx alo al a12
0=0,4)=0 19=1 0
0 1.00 
1 0.785 1.11 
X > 7+7 2 0.829 0.934 1.15
3 0.898 0.853 1.03
4 0.899 0.863 0.922
0 1.00 
1 0.944 1.03 
X 4 K+K 2 0.950 0.988 1.04
3 0.957 0.971 1.01
4 0.965 0.962 0.992
Table 74: & xx (h), efficienciy correction factors with respect to nonfiat MC due
to the track angular distribution in T(1S) * yX, with X * h+h for different X
spin values (Jx) and y, X helicities (Ay, Ax).
under the condition X 2 < X2 + 1 For Jx > 1 both (, (D) are free to move,
defining a surface in the (E, 4) plane. These surfaces are shown in Figure 74 for
the f2(1270), f4(2050), and f2(1525).
At this point we can check whether (0, 1) depend on the mass of the de
cay. We split the f2(1270) and the f2(1525) into a high mass and a low mass
region. The plots of the error surfaces of the measured (), 1') show no significant
separation for the different mass regions (see Figures 75 and 76).
3 In a two dimensional linear problem such a set of points defines the surface of
the standard error ellipse.
Results for the correction factor and its systematic error are shown in Table 7
5.
Upper limits on T(1S) * yf2(2220), f2(2220) * h+h are changed to
include the angular distribution's effect on efficiency. Since we can't measure
the helicity amplitudes in this case, we choose the worst possible case where the
corrected efficiency is lowest. This always corresponds to 8 =D 0. Results are in
Table 76.
1.5
0E f2(1270)
E f4(2050)
[O f2'(1525)
1.0 
0.5
0.0 I I
0.0 0.5 1.0 1.5
9 (radians)
Figure 74: Surfaces in the (8, D) plane used to determine the systematic uncer
tainty in the efficiency correction factor for the.modes with f2(1270), f4(2050),
and f2(1525). The f4(2050) surface may seem large, but when drawn in spherical
coordinates it is a small "north pole cap".
0.0 0.5 1.0 1.5
0 (radians)
Figure 75: Measurecd (9, D) surfaces for the f2(1270) high and low mass regions.

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/' 2
DECAYS OF THE T(1S) INTO A PHOTON AND TWO CHARGED HADRONS
By
LUIS BREVANEWELL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2004
To my Parents.
ACKNOWLEDGMENTS
I owe my gratitude to many people without whom this work would not have
been possible. First of all, 1 would like to thank my supervisor, Dr. Yelton. His
office door was always open for me, and he spent long hours of his time listening
to my ideas even though only a few of them ever worked. He gave me freedom to
explore and branch off in different directions while steadily guiding me forward at
the same time. After these years of working together I consider him more than a
mentor, and a think of him as a close friend.
During my stay at the University of Florida I have met many people inside
and outside the physics world. I would like to thank my first year graduate teachers
Dr. Sikivie and Dr. Woodard who were a source of inspiration for me. My thanks
also go to my fellow graduate students Vijay Potlia, Rukshana Patel, Necula
Valentin, Jennifer Sippel, Suzette Atienza, G. Suhas, and many more for all our fun
physics discussions and late homework sessions. I would also like to thank the good
friends 1 made outside the physics building, James Power, Yaseen Afzal (Paki),
Ramji Kamakoti (Ramjizzle), Dan DeKee (Double Down), and Fernando Zamit
(Fernizzle) who always reminded me that there is more to life than Physics.
The most important person I have met during my graduate career is my wife,
Jennifer. I thank her for enduring all those endless days when I would answer her
questions with only grunts and nods while my attention remained fixed on the
computer screen. Her love and support are invaluable to me.
Finally, I would like to thank my parents Manuel and Charlene, my sister
Teresa, and my brother Gaizka. They have always been there for me.
iii
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS Ã¼i
ABSTRACT vi
CHAPTER
1 THEORY 1
1.1 Particle Physics 1
1.1.1 The Standard Model 1
1.1.2 Quantum Chromodynamics 3
1.1.3 Introduction to the Radiative Decays of the T(1S) .... 6
1.2 Radiative Decays of Quarkonia Overview 14
2 EXPERIMENTAL APPARATUS 17
2.1 The Cornell Electron Storage Ring 17
2.2 The CLEO III Detector 20
2.2.1 Superconducting Coil 21
2.2.2 Tracking System 21
2.2.3 Crystal Calorimeter 24
2.2.4 Ring Imaging Cherenkov Detector 25
2.2.5 Muon Detectors 28
3 ANALYSIS OF THE DATA 31
3.1 Data Sample 31
3.1.1 Continuum Subtracted Distributions 32
3.2 Event Selection 34
3.2.1 Skim Cuts 35
3.2.2 Analysis Cuts 37
3.2.3 Cut Summary, Efficiencies, and Fake Rates 43
4 EXCLUSIVE RADIATIVE DECAY T(1S) > ttt+tT 48
4.1 Robustness of The Mass Distribution 48
4.2 Statistical Fit of the Invariant Mass Distribution: Signal Areas
and Their Significance 49
4.3 Angular Distribution of The Signal .' . . . 50
4.3.1 Optimum Mass Interval 51
4.3.2 Background Subtraction 53
IV
4.3.3Statistical Fit of the Helicity Angular Distribution: J AsÂ¬
signments and their Probability Distribution 55
5 EXCLUSIVE RADIATIVE DECAY T(1S) > 7K+K~ 66
5.1 Statistical Fit of the Invariant Mass Distribution: Signal Areas
and Their Significance 66
5.2 Angular Distribution of The Signal 67
6 EXCLUSIVE RADIATIVE DECAY T(1S) 7PP 80
7 SYSTEMATIC UNCERTAINTIES 86
7.1 Cuts 86
7.1.1 Justification of the DPTHMU Cut 87
7.2 Angular Distribution of Signal 88
7.3 Different Hadronic Fake Rates Between IS and 4S 97
7.4 Other Systematic Sources 97
7.5 Overall Systematic Uncertainties 99
8 RESULTS AND CONCLUSION 100
APPENDIX HELICITY FORMALISM FOR TWO BODY DECAYS .... 103
REFERENCES 119
BIOGRAPHICAL SKETCH 121
v
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DECAYS OF THE T(1S) INTO A PHOTON AND TWO CHARGED HADRONS
By
Luis BrevaNewell
December 2004
Chair: John Yelton
Major Department: Physics
Using the CLEO III detector we report on a new study of exclusive radiative
T(15) decays into the final states 77r+/T~, 7K+K~ and 7pp. We present branching
ratios for the decay modes T(1S) â€”> 7/2(1270), T(lS') â€”> 7/3(1525), T(IS') â€”â–º
7/o(1710) with /o(1710) K+K~ and T(15) > 7/4(2050).
vi
CHAPTER 1
THEORY
1.1 Particle Physics
Particle physics is the branch of physics dedicated to the study of matter
and energy at the most fundamental level. This means that the job of a particle
physicist is to identify the smallest constituents of matter and describe how they
interact with each other.
Humankind has been interested in this subject since ancient times. Two of the
first particle physicists in recorded history are the Greek thinkers Empedocles and
Democritus from the fifth century BC. Empedocles stated that our complex world
was made from combining four fundamental elements (earth, air, fire and water) in
different proportions. Democritus on the other hand, believed that the apparently
continuous objects in the natural world were not really continuous, but made from
voids and indivisible particles called atoms.
Over the last half century Particle physics has advanced tremendously and we
now have a beautiful, but incomplete, theory firmly grounded on experiment that
describes the fundamental constituents of matter and how they interact with each
other. This theory is called â€œThe Standard Model.â€
1.1.1 The Standard Model
According to the Standard Model, the fundamental building blocks of matter
are point like particles which interact with each other in as many as three different
ways. Each type of interaction, or force, is itself carried by point like particles
1
2
called force carriers. The particles which are force carriers are bosons 1 and are
collectively called gauge bosons because they are needed for the theory to be gauge
invariant. The nonforce carrying particles are fermions. 2
The three interactions described by the Standard Model are called the
electromagnetic force, the weak force, and the strong force. The force of gravity is
not included in the Standard Model, and this is one of the reasons the Standard
Model is not yet complete. The electromagnetic force carriers are photons, the
weak force carriers are the ZÂ°, IT+, and W~ particles, and finally, the strong force
is carried by eight kinds of gluons. Table 11 summarizes the situation.
Table 11: Gauge bosons and the force they carry.
Symbol
Name
Force Carried
7
Photon
Electromagnetic
ZÂ°
ZÂ°
Weak
w+
W+
Weak
w~
W~
Weak
9
gluon
Strong
Particles which interact through a particular force are said to couple to it
and to carry an associated charge. The nomenclature is as follows, particles that
interact through the electromagnetic force have an electromagnetic charge called
electric charge, those which interact through the weak force have a weak charge
called weak isospin, and those which interact strongly have a strong charge called
color.
The rest of the particles in the Standard Model which are not force carriers,
the fermions, are subdivided into different groups depending on their properties
(See table 12) , reminiscent of the way chemists organized the elements into
1 Bosons are defined as particles with integer spin in quantum mechanics.
2 Fermions are defined as particles with half odd fractional spin.
3
the periodic table during the second half of the nineteenth century. Fermions are
divided into quarks (generically represented by the symbol q) and leptons. The
main difference between these two groups is that quarks interact through the strong
force while leptons do not. There are six types of leptons and six types of quarks
(also called the six quark flavors) which are grouped into three generations. Each
generation consist of two quarks and two leptons. All three generations replicate
the same set of force charges, the main difference between generations is the mass
of the particles (for example, the ratio of the masses of e : /x : t is 1 : 200 : 3500).
For each fermion there is an antifermion with equal mass and spin and opposite
charge.
Table 12: Fermion symbols classified into quarks, leptons, and the three generaÂ¬
tions along with the generational common charges.
First
Generation
Second
Generation
Third
Generation
Electric Weak Has
Charge Charge Color
Quarks
u
d
c
s
t
b
(2/3 +1/2 Yes
1/3 1/2 Yes
Leptons
Ve
e
vn
V
vr
T
0 +1/2 No
1 1/2 No
1.1.2 Quantum Chromodynamics
Quantum chromodynamics (QCD) is the part of the Standard Model that
describes the strong force. QCD is based on local gauge invariance and color
symmetry. There are three possible color charges for quarks called r (red), b (blue),
and g (green). Antiquarks have opposite colors called f, b, g. The strong force
between quarks only depends on their colors and is independent of their flavor.
A very important characteristic of the strong force is that gluons themselves
carry a color charge and an anticolor charge so they can interact with other quarks
through the strong force and change their color. Since there axe three colors and
three anticolors one might think that there are nine different gluons. However,
there is one linear combination of color anticolor states that has no net color and
4
leaves a quark unchanged. There are therefore 9 â€” 1 = 8 gluons. The 8 individual
gluon color states can be written as follows,
Â° *â€¢
1 > =
2 > =
3 > =
4 > =
72
(rb + br)
(â– rb â€” br)
(rf â€” bb)
72
J_
75
j=(rg + gf)
(1.1)
5 > = f={rggr)
6 > = \{bg + gb)
V2
7 > â€¢= 7/=(&Â£ 
8 > = =(rf + gg  266)
V6
and represent the eight different gluons that exist in nature. The single and unique
color state left out is called a color singlet,
9 >=~(rr+ bb +gg) (1.2)
which is invariant under a redefinition of the color (a rotation in color space). In
group theory this decomposition of the color states into an octet and a singlet is
denoted by 3 0 3 = 8 Â® 1. It is worth noting here that a colorless sate, such as 3 >
or Â¡8 >, is not necessarily a color singlet state.
This situation is analogous to the perhaps more familiar example of two spin
1/2 particles. Each particle can have their spin up (T) or down (f) corresponding to
four possible combinations, each giving a total spin S = 0 or 1. The S = 1 states
5
form a triplet,
1 > =  T'T>
2 > = J~( tl> +1 U>) (1.3)
3> =  U>
and there is a singlet state with S = 0,
4 >= b( Tl> I lt>). (14)
Since gluons themselves carry color they can interact with each other through
the strong force. This interaction among the force carriers is unique to the strong
force and, when included in perturbative QCD calculations, leads to two important
properties observed in nature called â€œasymptotic freedomâ€ and â€œcolor confinement.â€
Asymptotic freedom means that the interaction gets weaker at short distances.
Color confinement is the requirement that observed states have neutral color, or in
other words, they must be in a color singlet state.
Confinement explains why free quarks or free gluons, which have a net color
charge, have never been observed. It also explains why no fractional charged
particles made from a qq bound state have never been observed since it is not
possible to construct a color singlet for such a state (in group theory terms
3 3 = 6 Â® 3 where we have a sextet and a triplet, but no singlet). On the other
hand, color singlets can be constructed for a qq or qqq system. The color singlet for
qq is simply the state shown on the right side of Equation 1.2. The color singlet for
qqq can be obtained from the decomposition 3<8>3<8>3 = 10Â©8Â®8Â©1 and is,
\qqq >coior singlet= A={rgb  grb + brg bgr + gbr  rbg). (1.5)
V6
Particles that are bound states of qq are abundant in nature and are called
mesons, those that are bound states of qqq are called baryons and also abound in
6
ft
nature. Both groups are collectively called hadrons. There have been hundreds
of different hadrons observed, confirming the validity of QCD and the quark
flavors. Nevertheless, QCD leaves room for more possibilities. A bound state of two
gluons gg (sometimes called a glueball) can be in a color singlet and in principle
could be observed. Although glueballs are allowed by QCD there is no convincing
experimental observation of one. Another possibility are bound states that are
a mixture of the previous states, such as aqq + /3gg with arbitrary a, (3. These
states are called â€œhybrid mesonsâ€ or simply â€œhybridsâ€. It is believed [1], [2], [3] that
hybrids are necessary to explain the spectrum of light mesons between 12 GeV.
As with any quantum mechanical theory, the spectrum of bound states is
a fundamental test. 3 Glueballs are allowed by QCD, yet there is no concluÂ¬
sive experimental evidence of their observation, despite intense experimental
searches [4], [5], [6] complemented by lattice QCD calculations [7], [8] and other
theoretical contributions like bag models [9], fluxtube models [10], QCD sum
rules [11], weakly bound boundstate models [12], and QCD factorization formalism
models [13]. Physicists cannot be sure they understand QCD until such states are
observed, or until they can explain why we cannot observe them.
1.1.3 Introduction to the Radiative Decays of the Y(1S)
The Y(1S) is a meson composed of bb quarks. Mesons of this kind, composed
of a quark and an antiquark of the same flavor, are in general called quarkonia.
There is a convention behind the name of the T(1S) . The T symbol is
reserved for particles composed of bb where the combined spin of the quark and
antiquark is 1. The â€œISâ€ symbol is borrowed from atomic spectroscopy with the
â€œ1â€ meaning that the bb pair are in the lowestenergy boundstate, and the â€œSâ€
3 A good example of this is the successful description of the hydrogen atomâ€™s
spectrum by quantum mechanics.
7
meaning that the bb have a relative angular momentum L â€” 0. A description of
particle naming conventions can be found in [14].
The T(1S) is unstable, existing for only about 10â€œ24 seconds after which it
decays into daughter particles (which in turn decay themselves if they are not
stable). The term â€œradiative decayâ€ is reserved to any T(1S) decay where one of
the stable daughters is a photon.
The different ways the Y(1S) can decay must obey the symmetries in nature.
Examples of such symmetries are â€œparityâ€ and â€œcharge conjugationâ€, both of which
will be described soon. Symmetries are very important in the standard model. The
usefulness of symmetries can be seen in Noethersâ€™ theorem, which states that for
every symmetry there is an associated conserved quantity. Examples of Noethersâ€™
theorem are the conservation of momentum connected to translational invariance
and the conservation of angular momentum associated with rotational invariance.
In the next sections we will use the parity and charge conjugation symmetries
to find T(1S) decays which conserve the associated symmetry constants and are
allowed by nature. In particular, we will show that the radiative decay of the
Y(1S) through a photon and two gluons (see Figure 11) is allowed. The key
observation is that the two gluons must be in a color singlet since both the Y(1S)
and the radiated photon have no color and color must be conserved. This means
that the two gluons satisfy color confinement and could form a glueball, although
more conventional meson states, or hadrons in no bound state at all, are also
possible outcomes. Regardless of what the gluons do, their energy will eventually
manifests itself as hadrons. Sometimes, two charged hadrons of opposite charge will
emerge. This work examines those two hadrons from a radiative Y(1S) decay to
experimentally probe the gg spectrum.
8
Parity
The parity operator, P, reverses the sign of an objectâ€™s spatial coordinates.
Consider a particle a > with a wave function \ka(x, t). By the definition of the
parity operator,
P*{x,t)=PaVa(X,t) (16)
where Pa is a constant phase factor. If we consider an eigenfunction of momentum
'S>f{x,t) = ei&2Et) (1.7)
then
P$p{x,t) = Pa%i~x,t) = Pa^p{x,t), (1.8)
so that any particle at rest, with p = 0, remains unchanged up to a multiplicative
number, Pa, under the parity operator. States with this property are called
eigenstates with eigenvalue Pa. Pa is also called the intrinsic parity of particle a,
or more usually just the parity of particle a, with the words at rest left implicit.
Since two successive parity transformations leave the system unchanged, P% = 1,
implying that the possible values for the parity eigenvalue are Pa = Â±1.
In addition to a particle at rest, a particle with definite orbital angular
momentum is also en eigenstate of parity. The wave function for such a particle in
spherical coordinates is,
W2.0 = Mr)17"(M. (19)
where (r, 0, ) are spherical polar coordinates, Rni(r) is a function of the radial
variable r only, and the Yjm(0, (Â¡>) is a spherical harmonic.
The spherical harmonics are well known functions which have the following
property,
>7(0, Â«Hi/vn*Â«,* + <Â»â€¢
(1.10)
9
Hence
PVnlm&t) == Pa^mm(X,t) = Pa(l)l^nlm(x,t) (1.11)
proving that a particle with a definite orbital angular momentum l is indeed an
eigenstate of the parity operator with eigenvalue Pa(â€”1).
The parities of the fundamental fermions cannot be measured or derived. All
that nature requires is that the parity of a fermion be opposite to that of an antiÂ¬
fermion. As a matter of convention fermions are assigned P = +1 and antifermions
are assigned P = â€”1. In contrast, the parities of the photon and gluon can be
derived by applying P to the field equations resulting in P7 = â€” 1 and Pg = â€”1.
The T(1S) has P = PbPb{ 1)L = 1 since L = 0.
Parity is a good quantum number because it is a symmetry of the strong and
electromagnetic force. This means, that in any reaction involving these forces,
parity must be conserved.
Charge Conjugation
Charge conjugation is simply the operation which replaces all particles by
their antiparticles. In quantum mechanics the charge conjugation operator is
represented by C. For any particle a > we can write,
C\a>=ca\a> (1.12)
where ca is a phase factor. If we let the C operator act twice to recover the original
state  a >,
a >= C2\a >= C(ca\a >) = caCja >= cacsa > (1.13)
which shows that caca = 1. If (and only if) a is its own antiparticle, it is an
eigenstate of C. The possible eigenvalues are limited to C = ca = c5 = Â±1.
All systems composed of a the same fermion and an antifermion pair are
A >
eigenstates of C with eigenvalue C = (â€”l)(i+sb This factor can be understood
because of the need to exchange both particlesâ€™ position and spin to recover the
10
original state after the charge conjugation operator is applied. Exchanging the
particlesâ€™ position gives a factor of (â€”1)L as was shown in the previous section,
exchanging the particles spin gives a factor of (â€”l)s+1 as can be verified by
inspecting Equations 1.3 and 1.4, and a factor of (1) which arises in quantum field
theory whenever fermions and antifermions are interchanged. With this result we
can calculate the charge conjugation eigenvalue for the T(IS) and obtain C â€” â€”l
since L + S = 1.
The photon is an eigenstate of C since it is its own antiparticle. The C
eigenvalue for the photon can be derived by inserting C into the field equations and
is C1 = â€”1.
Finally, we consider a system composed of two gluons that are in a color
singlet. The two gluons are bosons and they must have a symmetric wave function,
Tg under a gx exchange. Under this exchange, the orbital angular momentum
part of the wave function contributes with a factor ( 1)L, the spin part of the
wave function for two spin 1 particles contributes with a factor of (â€”l)s, and the
color singlet part of the wave function contributes with a factor of +1 since it is
symmetric. To ensure that 'I'g is symmetric we need L + S to be even. This implies
that C = (â€”l)i+s = +1. The L â€” 0 and Lâ€” 1 possible gg bound states are shown
in Table 13. The JPC = 1â€œ+ is peculiar because it is impossible for a qq system to
have these quantum numbers. If this state is ever observed, it must be a glueball.
Experimental searches for such a state have been done, for example, in [15].
Table 13: Possible gg bound states with L â€” 0 or L = 1. The possible quantum
numbers are limited by the condition that J = L + S be even, which is needed to
ensure a symmetric wave function for the two gauge bosons.
L
S
JFC
0
0
0++
0
2
2++
1
1
0+,l~+,2~+
11
Charge conjugation is a symmetry of the strong and electromagnetic force. For
those particles that are eigenstates of O, C is a good quantum number because in
any reaction involving these forces C must be conserved.
The P and C values of various particles used in this analysis are shown in
Table 14 along with their quark composition., orbital angular momentum, and
internal spin.
Table 14: Symbol, name, quark composition, angular momentum (L), internal spin
(S), parity (P), and charge conjugation eigenvalues (C) for a few of the particles
used in this analysis.
Symbol
Name
Quark Composition
L
S
P
c
T(1S)
Upsilon(lS)
bb
0
1
1
1
7r+
Pion
ud
0
0
1
X
7râ€œ
Pion
dÃ¼
0
0
1
X
K+
Kaon
us
0
0
1
X
K~
Kaon
su
0
0
1
X
P
Proton
uud
0
1/2
+1
X
P
Antiproton
Ã¼Ã±d
0
1/2
1
X
7
Photon
X
X
1
1
1
Possible Decays of the T(1S)
At this point we can understand the different possible ways the T(1S) can
decay. The Possible decays are limited because the strong and electromagnetic
force must conserve color, C and P.
The simplest possibility is for the bb pair to interact electromagnetically and
annihilate into one virtual photon 4 . This is allowed by parity, charge conjugation
symmetry and color conservation. The decay of the Y(lS) to one gluon is not
allowed by color conservation and is therefore forbidden. Y(1S) decays to two
photons are forbidden by charge conjugation. T(1S) decays to two gluons are
4 Such a photon is called virtual because it cannot conserve the 4momentum of
bb and is unstable, only existing for a brief period of time, as allowed by the uncerÂ¬
tainty principle, after which it decays.
12
also forbidden by charge conjugation. Y(1S) decays to 3 gluons are allowed (3
gluons can form a color singlet). Y(1S) decays to 3 photons are also allowed, but
are largely suppressed by decays to one photon since 3 successive electromagnetic
interactions are much less likely to occur than a single one. Finally, Y(1S) decays
two one photon and two gluons are also allowed under the condition that the two
gluons be in a color singlet state.
The three different possible T(1S) decays with least amount of interactions
(also called lowest order decays) are shown in Figure 11.
(a)
(b)
(C)
Figure 11: Lowest order decays of the T(1S) allowed by color conservation, charge
conjugation symmetry, and parity, (a) Shows the decay into three gluons, (b) shows
a radiative decay, and (c) shows the electromagnetic decay through a virtual phoÂ¬
ton that in turn decays electromagnetically into a pair of charged fundamental
particles, such as quarks or charged leptons (the charged leptons are represented by
the symbol l).
Observable Resonances
In this work we search for a resonance X produced in a radiative T(1S) decay,
T(1S) Â» 7X.
(.1.14)
Using chaige conjugation symmetry on both sides,
â€” 1 = â€”1 Cx
(1.15)
Therefore, Cx = +1.
In order for us to observe X it must decay into two charged hadrons,
X 4 h+h~, (1.16)
where h = tt, K, p are the hadrons whose momentum we are going to measure.
Applying charge conjugation to this last decay,
+1 = (1)^+*, (1.17)
where L and S are respectively the angular momentum and spin of the h+h~
system. This last equation implies that L + S must be even. This has consequences
for the possible X parities we can observe. By parity conservation in 1.16,
Px = (1)L(1)(1) = (l)i. (1.18)
For h = 7T, K, S â€” 0 and L â€” J must be even, which implies that Px = +1. For
h â€” p, S can be 0 or 1 and particles with both positive and negative parities can be
detected.
Table 15: Possible 5, L, J, P and C values for X from the radiative decay
T(1S) â€”> 7X reconstructed in different, decay modes.
Decay Mode
s
L
J
P
C
X 4 7T + 7T~
0
even
even
+1
+1
X 4 K+K~
0
even
even
+1
+1
X >pp
0
even
even
+1
+1
x > PP
1
odd
even and odd
1
+1
14
1.2 Radiative Decays of Quarkonia Overview
Theoretical models exist for glueball production in quarkonia decay [12]
and for the glueball spectrum. For example, a quenched lattice calculation [7]
predicts a Jpc = 2++ glueball 5 in the 2.2 GeV/c2 mass region (see Figure 12).
According to Table 15, in the charged pion and kaon modes, we are limited to
detect glueballs in the leftmost column where P = C = +1 of Figure 12, while in
the proton mode we are restricted to the two left most columns.
4
3
>
CD
o
C!
2 E
1
0
Figure 12: Quenched lattice calculation result for the glueball spectrum for differÂ¬
ent P and C values. The mass scale is shown in terms of a scaling parameter from
the QCD lattice calculation named r0 on the left and in absolute terms on the right
by taking rp1 = 410 MeV.
0 Here J stands for the internal angular momentum (spin) of the glueball,
J = L + 5.
%
15
For some time bound states not considered to be pure glueballs, such as
/2(1270), /4(2050), 77, and rf have been observed in J/ip radiative decays at
the 10~3 production level [16] 6 . In 1996, the BES collaboration claimed the
observation of a resotiance they called the /./(2220) particle in the radiative J/ip
system at the 10~5 level [17] (see Figure 13). A lot of excitement was generated at
the time because it is possible to interpret the /j(2220) as a glueball. However, this
result has not been confirmed.
27 70301 *004
Figure 13: The mass spectrum obtained by the BES collaboration in radiative
JÂ¡ip decays into different hadronic modes.
Key to identifying a particle as a glueball are (a) suppressed production in
twophoton collisions (unlike quarks, gluons donâ€™t carry electric charge and do not
6 The J/ip particle is a cc bound state. Since the strong force is flavor blind the
situation in J/ip decays is in principle similar to that in T(1S) decays.
couple to photons), and (b) flavor symmetric decays, since a pure glueball has no
valence quarks.
V;/*
CLEO has already done several studies of radiative decays of the T(lS').
Naively, one expects these types of decays to be suppressed by a factor
[(Qb/qc)(mc/mb)}2 Â» 0.025 (1.19)
with respect to J/xp radiative decays. This comes from noticing that the quark
photon coupling is proportional to the electric charge and the quark propagator
is approximately 1/m for low momentum quarks. In 1999, CLEO made the first
observation of a twobody T(15) radiative decay [18]. The spin of the observed
resonance could not be measured, but its mass and width where consistent with the
/2(1270) particle. Under this assumption, comparing the measured branching ratio
of T(1S) â€”> 7/2(2220) to the measured branching fraction of the J/xp â€”> 7/2(1270),
a suppression factor of 0.06 Â± 0.03 was obtained. After the BES result for the
fj (2220) in radiative J/xp decay's, a corresponding search was performed by CLEO
in the radiative T(1S) system [19]. This analysis put limits on the fj(2220)
production in radiative Y(1S) decays.
In this work we are privileged to have available the largest collection of
radiative T(1S) decays in the world. With it, we can study the structure of color
singlet gg hadronization, and shed more light on the fj(2220) result from BES.
CHAPTER. 2
EXPERIMENTAL APPARATUS
To carry out our study of the dihadron spectrum we need to first produce
the T(1S) resonance and secondly observe its daughter particles flying away at
relativistic speeds. These two tasks are respectively accomplished by the Cornell
Electron Storage Ring and the CLEO III detector.
2.1 The Cornell Electron Storage Ring
The Cornell Electron Storage Ring (CESR), located at Cornell University, is a
circular particle accelerator that produces e+e~ collisions.
In order to produce such collisions electrons and positrons need to be created,
accelerated and stored. CESRâ€™s different components, shown in Figure 21, have
been carrying out this task since 1979.
A typical CESR run begins at the linear accelerator (LINAC) where electrons
and positrons are produced. To create positrons, electrons are evaporated off a
filament and linearly accelerated by electromagnetic fields towards a tungsten
target. The collision creates a spray of electrons, positrons and photons. The
electrons are cleared away with magnetic fields and the positrons are introduced
into the synchrotron. The filling procedure is identical, except that the tungsten
target is removed.
Once the electron and positron beams are introduced into the synchrotron,
they are accelerated to the operating energy. In hour case they are accelerated to
the point where their combined energy is the Y(1S) mass, 9.46 GeV.
Once the beams are at the desired energy they are transfered to the storage
ring, where they will remain for about an our. At one point in the storage ring the
17
18
SYNCHROTRON
WEST
TRANSFER
LINE
EAST
TRANSFER
LINE
LINAC e
beams are forced to cross paths. This is the point where e+e collisions occur 1
and where the center of the CLEO III detector is located. If the accelerator is
â€¢ BUNCH OF POSITRONS
O BUNCH OF ELECTRONS
Figure 21: The Wilson Laboratory accelerator located about 40 feet beneath
Cornell Universityâ€™s Alumni Fields.
performing well a high collision rate results. A high collision rate is crucial for the
success of an accelerator and the experiments it serves. The important figure is the
number of possible collisions per second per unit area; this is called the luminosity.
In order to maximize the luminosity, the beams are focused as small as possible at
1 This point is not fixed in space, but varies from event to event inside a small
volume of space called the interaction region (IR).
19
the IR. During the CLEO III installation several magnetic quadrupÃ³les were added
to CESR to improve the beam focus. CESR has consistently outdone itself while
collecting luminosity over the years (see Figure 22).
9000
8000
7000
5000
5000
13
a
~ 4000
CA
o
c
E
3
300G
2000
1000
Figure 2 2: CESR yearly luminosity. The gaps in 1995 and 1999 correspond to
down times when the CLEO II.V and CLEO III detectors where being installed.
The CLEO III detector measures the time integrated luminosity over a period
of time by counting how many times a benchmark process occurs during a certain
time interval at the IR. For redundancy, there are two benchmark processes that
are used, one where the e+e~ particles interact to produce a new e+eâ€” pair,
and the other one where the e+e~ annihilate and produce two photons. Using
the known crosssection for each process, the number of events is converted to a
luminosity called the Bhabha integrated luminosity for the first process, and the 77
integrated luminosity for the second one. The term â€œintegrated" is sometimes left
out and the total luminosity is referred to as simply the Bhabha or 77 luminosity,
with the time integration left implicit.
2.2 The CLEO III Detector
When the e+e~ collision occurs, the two particles are annihilated we enter
the world of particle physics. Nature decides what to do with the energy from
the annihilation. We have no chance of directly observing what is happening at
the annihilation point, but eventually long lived semistable particles are created
that fly off at relativistic speeds. These particles carry information about what
happened after the e+e~ collision, and can tell us what nature did. The CLEO III
detector has the important mission of detecting and measuring such par ticles.
As one can see in Figure 23, the CLEO III detector is a composite of many
detector elements. These subdetectors are typically arranged as concentric cylinÂ¬
ders. The entire detector is approximately cube shaped, with one side measuring
about 6 meters, and weighs over 500 thousand kilograms.
CLEO III
SC Quadrupole
Pylon
SC
Quadrupoles
Rare Earth
Quadrupole
Solenoid Coil
Barrel
Calorimeter
RICH
Drift
Silicon /
oeampipe
Magnet
Iron
Endcap
Calorimeter
Iron
Polepioce
Barrel Muon
Chambers
Figure 23: The CLEO III detector.
21
As implied by the name. CLEO ill is not the only CLEO detector. CLEO
III was preceded by CLEO II.V, CLEQ^JI, CLEO I.V and the original CLEO
>
detectors. The CLEO III detector was a major upgrade compared with the
previous version of CLEO [20], [21], and. has an improved particle identification
system together with a new drift, chamber and a new silicon vertex detector.
2.2.1 Superconducting Coil
All the detector subsystems except for the muon chambers are located inside
a superconduction coil. The coil remains unchanged since CLEO II. It is kept in
a superconducting state by liquid helium. The purpose of the coil is to provide a
1.5 Tesla magnetic field, which is uniform to 0.2%, to bend the paths of charged
particles in the detector. By measuring how much much a charged particle bends,
experimenters can measure the momentum of the particle.
The coil inner radius is 1.45m and its outer radius is 1.55 m, with a radial
thickness of 0.10 m. The total length of the coil in z is 3.50 m. It is wound from
a 5mm x 16 mm superconducting cable (A1 surrounding CuNbTi strands). It
is wound in 2 layers, with 650 turns per layer, on an aluminum shell. When in
operation a current of 3300 amps flows through the coil.
2.2.2 Tracking System
After particles from the interaction point pass through the beam pipe, they
begin to encounter the active detector elements of the tracking system. There are
twro subdetectors responsible for tracking the curving path of charged particles.
The first one encountered by particles is the silicon vertex detector, and the second
one is the central drift chamber. The CLEO III tracking system is responsible
for tracking a charged particleâ€™s path and measuring its momentum. Typical
momentum resolution is 0.3% for 1 GeV tracks. The tracking system also measures
ionization energy losses with an accuracy of about 6%.
22
Silicon Vertex Detector
The silicon vertex detector in CLEO III [22], also called SVD III, is a four
layer barrelonly structure with no endcaps that surrounds the beam pipe. This
detector (see Figure 24) provides four 0 and four z measurements covering over
over 93% of the solid angle. The average radius of inner surface of the four layers is
25 mm, 37.5 mm, 72 mm, 102 mm. The detector is constructed from 447 identical
doublesided silicon wafers, each 27.0 mm in 0, 52.6 mm in z and 0.3 nun thick.
The wafers are instrumented and read out on both sides. The instrumentation on
each side consists of an array of aluminum strips on the wafer surface. These strips
are connected to preamplifiers at the end of the detector. The inner side has 512
strips in the z direction and the outer side has 512 in the 0 direction. Therefore
each wafer contains 512+512 sensors. The 447 wafers are arranged in the 4 layers,
as follows: 7 sections in 0, each with 3 wafers in z, total = 21 wafers in the first
layer; 10 sections in 0, each with 4 wafers in z, total = 40 wafers in the second
layer; 18 sections in 0, each with 7 wafers in z, total = 126 wafers in the third
layer; 26 sections in phi, each with 10 in Z, total = 260 wafers in the fourth layer.
Charged particles traversing the wafer lose energy and create electron hole
pairs. Approximately 3.6 eVis required to create a single electronhole pair. The
electrons and holes then travel in opposite directions in the electric field applied
to the surfaces of the wafers until they end up on the aluminum strips, and the
detector registers a â€œhitâ€. When combined together, the hit on the inner side of a
wafer and the hit on the outer side give a measurement of the (z, 0). The wafer
position itself determines r.
The Central Drift Chamber
The CLEO III central drift chamber (DR III) is full of a gas mixture with 60%
Helium and 40% propane held at about 270 K and at a pressure slightly above one
atmosphere. The drift chamber is strung with array of anode wires of goldplated
ff
23
Figure 24: View of the SVD III along the beampipe.
tungsten of 20 Â¿Â¿m in diameter and cathode wires of goldplated aluminum tubes
of 130 /iin in diameter. The anodes are kept at a positive voltage (about 2000 V),
and the cathodes are kept grounded, which provides an electric field between the
anode and the cathode wires. Anode and cathode wires are often called â€œsenseâ€
and â€œfieldâ€ wires respectively.
As a charged particle passes through the DR III, it interacts electromagneti
cally with the gas molecules giving energy to the outer electrons which become free
in a process called ionization. The free electrons from the ionized gas molecules
drift in the electric field toward the nearest anode wire. As the electrons get close
to the anode, the electric field becomes very strong which causes an avalanche
as further ionization is induced. The result of the avalanche is a large number of
electrons collapsing upon the sense wire in a very short amount of time (less than
one nanosecond). When this happens to a sense wire, we say that there is a â€œhitâ€.
24
The current on the anode wire from the avalanche is amplified and collected at
the end of the anode wire. Both the amount of charge and the time it takes it to
move to the end of the detector are measured. A calibration of the drift chamber
is used to convert the amount of charge to a specific ionization measurement of the
incident particle. A calibrated drift chamber can also convert the time to roughly
measure the position along the sense wire where the charge was deposited.
The wires are strung along the 2: direction. About 2/3 of the outer part of
the drift chamber (the farthest part from the interaction point) is strung in with a
slight angle (about 25 miliradians) with respect to the 2 direction to help with the
z measurement. Wires strung in the z direction are called â€œaxialâ€ wires, while those
that are strung at a slight angle are called â€œstereoâ€ wires.
The DR III consists of an inner stepped section with 16 axial layers, and an
outer part with conical endplates and 31 small angle stereo layers. There are 3
field wires per sense wire and they approximately form a 1.4 cm side square. The
drift resolution is around 150 pm in r â€” 4> and about 6 mm in 2. All wires are held
at sufficient tension to have only a 50 pm gravitational sag at the center (2 = 0).
There are 1696 axial sense wires and 8100 stereo sense wires, a 9796 total.
2.2.3 Crystal Calorimeter
The CLEO Crystal Calorimeter (CC) is composed of 7784 thalliumdoped
Csl crystals. Each crystal is 30 cm long (16.2 radiation lengths) with 5cm x 5cm
square front face. The crystals absorb any incoming electron or photon which
cascades into a series of electromagnetic showers. The electronic system composed
of 4 photodiodes present at the back of each crystal are calibrated to measure the
energy deposited by the incoming particle. Other incoming particles other than
photons and electrons are partially, and sometimes fully, absorbed by the crystal
giving an energy reading.
25
'
i
The CC is arranged into a barrel section and two endcaps, together covering
95% of the solid angle. The CC barrel section is unchanged since CLEO II; the
endcaps have been rebuilt for CLEO III to accommodate the new CESR interaction
region quadrupoles. The barrel detector consists of an array of 6144 crystals, 128
in cf> and 48 in 2, arranged in an almostprojective barrel geometry. That is, the
crystals are tilted in 2 to point to a few cm away from the interaction point, and
there is also a small tilt in . The CC barrel inner radius is 1.02 m, outer radius
is 1.32 m, and the length in 2 at the inner radius is 3.26 m. It covers the polar
angle range from 32 to 148 degrees. The barrel crystals are tapered towards the
front face (there are 24 slightly different tapered shapes), the endcap crystals are
rectangular, but shaved near the outer radius to fit in the container. The CC
endcaps consist of two identical end plugs, each containing 820 crystals of square
crosssection, aligned parallel to the beam line (not projective). There are 60
crystals in the â€™â€™fixedâ€ portion of the â€™â€™keystoneâ€ piece of the endcap. 760 in the
part that slides. The keystone is made up of two parts, one on top that has 12
crystals that for mechanical removal reasons is separate from a container holding 48
crystals. The endcap extends from 0.434 m to 0.958 m in r. The front faces are 2 =
Â±1.308 m from the interaction point. It covers the polar angle region from 18 to 34
degrees in +z, and 146 to 162 in 2.
The photon energy resolution in the barrel (endcap) is 1.5% (2.5%) for 5
GeVphotons, and goes down to 3.8% (5.0%) for 0.1 GeVphotons.
2.2.4 Ring Imaging Cherenkov Detector
The Ring Imaging Cherenkov (RICH) detector [23] is a newr detector subsysÂ¬
tem for CLEO III. It replaces the CLEO II.V time of flight system designed to
measure particlesâ€™ velocities.
26
Cherenkov radiation occurs when a particle travels faster than the speed of
light in a certain medium,
v > c/n. (2.1)
Where v is the velocity of the particle, c is the speed of light in vacuum, and n
is the index of refraction of the medium the particle is traveling in. The charged
particle polarizes the molecules of the medium, which then turn back rapidly
to their ground state, emitting radiation. The emitted light forms a coherent
wavefront if v > c/n and Cherenkov light is emitted under a constant Cherenkov
angle, 5, with the particle trajectory forming a cone of light. The cone halfangle is
given by the Cherenkov angle which is,
cos 5 = â€”
vn
1
/3n
(2.2)
If the radiation angle, 5, is measured, the speed of the incident particle is known.
This measurement, combined with the momentum measurement from the tracking
system, gives a measurement of the particles mass, and can be used in particle
identification.
The threshold velocity at which Cherenkov radiation is emitted is vmin =
When a. particle traveling at the threshold velocity transverses the medium a very
small cone with Ã³ Â« 0 is produced. The maximum emission angle occurs when
Vmax = c and is given by
COS Â¿max = â€¢ (2.3)
n
The RICH (see Figure 25) consists of 30 modules in phi, 0.192 m wide and
2.5 m long. The detector starts at a radius of 0.80 m and extends to 0.90 m. Each
module has 14 panes of solid crystal LiF radiator at approximately 0.82 m radius,
0.192 m wide, 0.17 m long, 1 cm thick. Inner separation between radiators is
typically 50/im. The LiF index of refraction is n = 1.5. The radiators closest to z
= 0 in each module have a 45 degree sawtooth outer face, to reduce total internal
27
reflection of the Cherenkov light for normal incident particles (see Figure 26). The
radiators are followed by a 15.7 cm (radial) drift space filled with pure N2. The
drift space is followed by the photodetector, a thingap multiwire photosensitive
proportional chamber.
Figure 25: The RICH detector subsystem.
track
X
/
10 mm \
TV? 1
t Lâ€”
170
mm ^
track
Figure 26: The two kinds of RICH LiF radiators. For normal incidence particles
(z ~ 0) a sawtooth radiator is necessary to avoid internal reflection.
With this index, particles in the LiF radiator with beta = 1 produce
Cherenkov cones of halfangle cos_1(l/n) = 0.84 radians. With a 16 cm drift
28
space, this produces a circle of radius 13 cm. The RICH is capable of measuring
the Cherenkov angle with a resolution of a few miliradians (see Figure 27). This
great resolution allows for good separation between pions and kaons up to about 3
GeV as Figure 28 shows.
TD
cu
E
â€”1â€”'â€”
tâ€”Tâ€”iâ€”'â€”iâ€”Tâ€”iâ€”'â€”r
1



â€¢ Data
â–
o Monte Carlo
1 
â–
T â–
â–

o
"
â€œ

lili
â–
â€¢
t I T T
â–
" 1
* 'ii

â€¢ t
â–
0

â–
â–


â€¢
â–
â–
â–
I
J . 1 . 1 . 1 i L
â– 1
1 2 3 4 5 6 7
Radiator row
Figure 27: Cherenkov angle resolutions per track as a function of radiator row for
Bhabha events. Row 1 corresponds to the two rows closest to z = 0, etc.
2.2.5 Muon Detectors
The muon detectors (MU) are the most external subsystem of the CLEO III
detector. They remain unchanged from CLEO II, and are composed of plastic
proportional tubes embedded in the magnet iron return yoke. They cover 85%
of the 47t solid angle (roughly 30150 degrees in polar angle). If a series of hits is
detected in the muon chamber layers they most likely correspond to muons because
other particles are blocked by the iron. Besides detecting muons, the heavy iron
29
Figure 28: PiÃ³n fake rate as a function of particle momentum for kaon efficiency of
80% (circles), 85% (squares) and 90% (triangles).
of the return yoke protects the inner subsystems of the CLEO III detector from
cosmic ray background (except for cosmic ray muons of course).
There are three planes of chambers in the barrel section, arranged in 8 octants
in (f>. The plastic barrel planes lie at depths of 36, 72, and 108 cm of iron (at
normal incidence), corresponding to roughly 3, 5, and 7 hadronic interaction
lengths (16.8 cm in iron) referred to as DPTHMU. There is one plane of chambers
in each of the two endcap regions, arranged in 4 rough quadrants in d>. They lie
at z â€”Â± 2.7 m, roughly covering the region 0.80 < cos(0) < 0.85. The planar
tracking chambers use plastic proportional counters at about 2500 V with drift gas
of 60% He, 40% propane, identical to (and supplied by the same system as) the
drift chamber gas. Individual counters are 5 m long and 8.3 m wide, with a space
30
resolution (along the wire, using charge division) of 2.4 cm. The tracking chambers
arc made of extruded plastic, 8cm wide by 1 cm thick by 5 m long, containing eight
tubes, coated on 3 sides with graphite to form a cathode, with 50 gm silverplated
CuBe anode wires held at 2500 V. The orthogonal coordinate is provided by
8cm copper strips running perpendicular to the tubes on the side not covered by
graphite.
CHAPTER 3
ANALYSIS OF THE DATA
This work builds on the techniques developed by previous CLEO radiative
Y(1S) analyses [18], [19], modified to be used with CLEO III data. A new
technique based on kinematic fitting is developed to, together with the new
RICH detector, improve efficiency and particle identification.
We search for radiative T(1S) decays in the modes T(1S) â€”â– > 77r+7r~,7K+K~,
and 7pp. The e+e~ collision data has both resonant events, where the e+e~
annihilate to give a T(1S) , and continuum events, where the e+e~ collision does
not give a Y(1S) . To be sure we are observing Y(1S) and not a continuum process,
the continuum must be subtracted by using a pure source of correctly scaled
continuum events. Pure continuum data can be obtained by operating CESR at an
energy different form the Y(1S) mass. After subtracting the underlying continuum,
we examine the dihadron invariant mass spectrum in search of resonances. We
determine the spin and production helicity (the projection of the spin on the
momentum vector at production time) of any found resonances by examining the
photon and hadron angular distributions.
3.1 Data Sample
The analysis presented here is based on CLEO III data. Throughout this
document, and unless otherwise stated we use 77 luminosity (the luminosity types
used by CLEO are defined at the end of Section 2.1). We prefer to use the 77 over
the Bhabha luminosity because the resonant process T(1S) â€”> e+e~ artificially
increases the reported Bhabha luminosity by about 3% in T(1S) data. This extra
contribution would need to be accounted for when doing a continuum subtraction.
Choosing the 77 luminosity avoids this complication.
31
32
The CLEO III data is divided into numbered sets. Sets 18 and 19 have a
luminosity of 1.13 Â± 0.02/fr1 in the beam energy range 4.7274.734 GeV. This
data, which we call the T(1S) data (or simply the IS data), has both resonance
e+e~ â€”> Y(1S) and continuum events. We take the number of resonant events
from [24], Nr(is) = (21 Â± 0.1) x 107. This number contrasts with the previous
generation measurement of CLEO II, where Nr(is) ~ 0.15 x 10' were available.
In datasets and 18 and 19 there are also 0.192 Â± 0.004/6â€œ1 taken below the
Y(1S) beam energy (4.7144.724 GeV). This data, which we call the T(lS)ofF data
(or simply the lSoff data), has relatively low statistics and corresponds to purely
continuum events.
To improve our continuum statistics we use 3.49 Â± 0.07fb~l from datasets
9, 10, 12, 13, and 14 of data taken near the centerofmass energy of the Y(4S),
which for our purposes is defined as data with beam energy in the 5.2705.300 GeV
range. This set of data, which we call the Y(4S) data (or simply the 4S data), is a
source of pine continuum because no T(4S) â€”> BB resonant event can survive our
â€œcutsâ€ 1 (the cuts are presented in Section 3.3).
3.1.1 Continuum Subtracted Distributions
We use the continuum data taken at the T(4S) energy to subtract the underÂ¬
lying continuum present in the T(1S) data. This is important because continuum
background processes like e+eâ€œ â€”* yp with p â€”* 7r+7râ€œ, e+eâ€œ â€”> 74> with
4> â€”> K+K~, and direct e+e~ â€”> 7h+h~ (we will use the convention h = 7r, K, p
from now on), look like the signal events we are searching for. To first order, the
cross section of these continuum process scales like 1/s, where s is the square of
1 Cuts are simply conditions that an event must satisfy to be considered in the
Â«analysis. Cuts are necessary to eliminate background that would otherwise make a
measurement difficult or even impossible.
33
the center of mass energy of the e+e~ system. Taking the luminosityweighted
average beamenergies of each interval,, and the 77 luminosities (see Table 31), we
calculate that the T(4S) data scaled down by a factor of 0.404 2 represents the unÂ¬
derlying continuum in the T(1S) data. This is true up to differences in momentum
distributions and phase space. The error in the continuum scale factor is unknown
because the luminosity ratio is expected to have a small but undetermined sysÂ¬
tematic error. We make the somewhat arbitrary decision to retain three significant
digits in the continuum scale factor because it is sufficient for our purposes and
there are 0.5% effects from second order terms in the cross section formulae.
To eliminate the contribution of continuum events from a T(1S) data variable
distribution (e.g., the invariant mass of two tracks, the photon angular distribution)
we proceed as follows,
â€¢ Obtain the T(4S) data distribution for the same variable.
â€¢ Efficiency correct both the Y(1S) and T(4S) distributions, using a
GEANT [25] based Monte Carlo (MC) simulation of the detector. Examples of
MC efficiency distributions are shown in Figures 33 and 73.
2 If we are to be mathematically strict we should calculate the scale factor as
with obvious notation. This is equivalent to redefining the average energy as
However, our energy intervals are sufficiently narrow and this calculation does
not change the last significant digit of the original scale factor. Similarly, taking
into account second order terms in the energy dependence of the cross section, like
the one that appears as m2/s in the explicit formula for the e+e~ â€”> 7p cross sec
tion, also has an insignificant effect on the scale factor.
34
â€¢ Subtract the T(4S) distribution from the Y(1S) distribution using a Y(4S)
scale factor of 0.404.
We call this set of steps â€œcontinuum subtractionâ€ by definition. For comÂ¬
pactness, we call such a distribution the â€œcontinuum subtracted
distribution/plotâ€ or â€œ continuum subtracted distribution/plotâ€. Except
for statistical fluctuations and phase space effects, the resulting IS distribution
should not have any contribution from continuum processes that scale as 1/s.
It is important to notice that any continuum subtracted distribution is
efficiency corrected. This means that a fit to a continuum subtracted distribution
(for example, the continuum subtracted invariant mass distribution) gives the
efficiency corrected number of events directly. Strictly speaking, this number of
events is only correct if all the other variables we cut on have the same initial
distribution in data and MC, or if the efficiency does not depend on them. In this
note we use â€œflat MCâ€, defined as MC that is generated with a flat distribution in
the mass and the helicity angles #7, 8n (these angles are defined in the appendix).
For example, the number of events obtained from the continuum subtracted
invariant mass fit needs to be corrected to account for the fact that the helicity
angle distributions are not flat in data (see for example Figure 49, and the
efficiency is highly dependent on these variables (see Figure 73). This correction is
done in Section 7.2.
A summary of the results from this section is shown in Table 31.
3.2 Event Selection
Event selection for Y(1S) â€”> ~fh+h~ is straightforward and can be thought of
in terms of three major stages.
First we skim the data, keeping only those events that contain exactly one
highenergy photon and two tracks. Next, we require that the total 4momentum
of these three elements be consistent with the colliding e+e~ 4momentum. Finally,
35
Table 31: Summary of the data used in this analysis. The continuum scale factor
is obtained using 77 luminosities because the Bhabha luminosity is artificially high
during T(1S) running due to the process T(1S) â€”> e+e~.
T(1S)
T(4S)
T(lS)off
Dataset
18, 19
9, 10, 12, 13, 14
18, 19
Average Â¿Â£wi(GeV)
4.730
5.286
4.717
Range of Ffceam(GeV)
4.7274.734
5.270  5.300
4.7144.724
Â¿(e+eK/ir1)
1.20 Â± 0.02
3.56 Â± 0.07
0.201 Â± 0.004
â– C(77)(fbâ€˜)
1.13 Â±0.02
3.49 Â± 0.07
0.192 Â±0.004
T(1S) continuum scale factor
1
0.404
5.84
in the third stage, we project the surviving data onto the three different hadronic
modes via hadron separation and QED suppression cuts.
However, checking the 4momentum involves using the tracks masses. This
means that the information from stage 2 should somehow be useful in stage 3. This
is indeed the case, and the details of how we do it are revealed in this section.
3.2.1 Skim Cuts
We skim over the data in the â€œhardGamâ€ subcollection. The hardGam
subcollection was developed with this type of analysis in mind. For an event that
passes the triggers 3 to be classified as hardGam it must pass the following cuts,
â€¢ eGaml > 0.5
â€¢ eSh2 < 0.7
â€¢ eOverPl < 0.85
â€¢ eVis >0.4
â€¢ aCosTh < 0.95 where â€œeGamlâ€ is the highest isolated shower energy
relative to the beam energy, â€œeSh2â€ is the energy of the second highest shower
relative to the beam energy, â€œeOverPlâ€ is the matched shower energy relative to
3 Triggers are basic criteria that an event must satisfy to recorded during the
data collection processes. Triggers are designed to get rid of trash and noise and
reduce the size of the data sample while keeping all of the important information.
36
if
the momentum of the track with highest momentum, â€œeVisâ€ is the total energy
detected (charged tracks are assumed to be pions) relative to the center of mass
energy, and â€œaCosThâ€ is the z component of the unitary total momentum vector.
MonteCarlo predicts that about 75% of the generated T(1 S) â€”â–º 'yh+h~ signal
passes the hardware and software triggers and gets classified as hardGam.
As mentioned above, we use this data to malee our skim. To write an event
from the hardGam subcollection into our skim we require the following topological
cuts,
â€¢ There are exactly two â€œgood tracksâ€; there can be any number of tracks that
are not â€œgood tracksâ€ but these are not used in the analysis. We define a â€œgood
trackâ€ as a track that satisfies the following cuts; drift chamber track ionization
energy loss (dE/dX) information is available, the ratio of number of wire hits to
those expected is between 0.5 and 1.5, the pion fit has x2/do.f. < 20 (here d.o.f.
stands for degrees of freedom), and the distance of closest approach to the beam
spot in the xy plane (called DBCD) is less than 5 â€” 3.8P (mm) if P < 1 GeV/c
(where P is the tracks momentum in GeV/c) and less than 1.2 mm for tracks with
P > 1 GeV/c. This DBCD cut is common in the more sophisticated CLEO II/II.5
analyses. It performs better than a simple DBCD < 5 mm cut, because it takes
into account the fact that traks with higher momentum have a better measurement
of DBCD since they scatter less.
â€¢ There is exactly one â€œgood showerâ€, there can be any number of showers
that are not â€œgood showrersâ€ but these are not used in the analysis. We define a
â€œgood showerâ€ as an unmatched shower with energy > 4 GeV.
These topological cuts are about 85% efficient for generated signal events that
have passed the triggers and have been classified as hardGam.
The overall skim efficiency is between 6065%, depending on the mode (see
Table 34).
37
3.2.2 Analysis Cuts
After our skim we call any cuts we make â€œanalysis cutsâ€. These cuts are done
at analysis time and are mode dependent. As a convention, and unless otherwise
stated, efficiencies for individual analysis cuts are reported relative to the events in
the skim (not relative to the events generated).
4momentum Cut
All fully reconstructed events should have the 4momentum of the eTe~
system. This constraint is usually implemented with a simple twodimensional
AEp box cut, where AE is the difference between the reconstructed energy for
the event and the colliding e4e~ energy (Ecm), &nd P is the magnitude of the
reconstructed total momentum for the event. Typical values for these cuts are
â€”0.03 < AE/Ecm < 0.02 and p < 150 MeV/c (taken from [19]).
The traditional AEp box cut is somewhat useful. However, it does not take
into account the correlation between the measured energy and momentum. Indeed,
the signal lies in diagonal bands in the AEp plane, making a boxshaped cut not
optimal (see Figures 3la and 3lb).
We use an alternative approach to the 4momentum cut. After a. simple
substitution, = p^ = Ecm ~ Ek+h (where is the photonâ€™s energy, p7 is
the magnitude of the photons momentum, Ecm is the energy of the e+e~ system,
and Eh+h is the energy of the hadron pair), we can mite the Ep conservation
equations as:
Vh+h T {Ecm  Eh+h )Pi â€” Pcm (3.1)
where, Ph+h is the dihadron momentum, p7 is the photonâ€™s momentum unit
vector, and pcm is the momentum of the e 're~ system (which is a few MeV because
of the crossing angle). Equation 3.1 is a 3constra.int subset of the 4momentum
constraint and has the convenient property of avoiding the use of the measured
38
photonâ€™s energy, which has nonGaussian asymmetric errors. It is important to
notice that Equation 3.1 contains the dihadron energy, therefore, it can help
discriminate between the various particle hypotheses.
We proceed as follows. After vertexing the hadron pair using the beam spot
with its error matrix, we calculate the photonâ€™s direction from the hadron pairâ€™s
vertex and the shower position. We then fit the event to the 3 constraints expressed
in Equation (1) using the techniques outlined in [26] and cut on the x2 of the
3constraint fit, XepW <100. To complete the 4momentum requirement, we
calculate AE(h) = Eh+h + E1  ECm, where Eh+h is the updated dihadron
energy after the constraint, and Ey is the measured photonâ€™s energy, and require
0.050 * Ecm < AE(h) < 0.025 * Ecm Furthermore, we now have available
X2Ep[h) differences between different particle hypotheses, which help in particle
identification (ID). This is discussed in more detail in the next section. Figure 31
compares the performance of the old and new approaches to the Ep cut.
At this point it is a good idea to check that the 4raomentum cut rejects
background events that make it through our skim cuts (see Section 3.1). These
events typically have one high energy shower, two tracks, and (an) additional
element(s). One such background is T(1S) â€”> 77r+7r~7rÂ°. Out of 25000 T(lS') â€”Â» 77,
with i] â€”> 7r+7r_7rÂ° MC events 4 only 4 survive our 4momentum cut. We conclude
that our 4momentum cut is good at rejecting background events that pass the
skim cuts but have additional elements such as an extra photon, 7rÂ°, pair of tracks,
etc.
Hadron Separation
We define Ax2Â£>(/q â€” h2) for the particle hypotheses h\ and h2 of our charged
track pair (e.g. AxjD(^ â€” K)) as follows,
4 Thanks to Vijay Potlia for generating these events
39
0.50
0.25
0.00
0.25
TT
\
0 0.00
o
CL
0.25
C.00
0.25
0.00
0.50 0.40 0.30 0.20 0.10 0.00 0.10 0.20
AE (GeV)
Figure 31: Distributions for different 4momentum cuts. Signal MC events
(T(L9) â€”Â» 77T+7râ€œ) are represented by the black dots, and â€œbackgroundâ€ MC events
(e+e~ 70, 0 â€”> K+ K ~) appear as (red) triangles. Plot a) has no cuts. Plot b)
has the old 4momentum boxcut. Plot c) has the new 4momentum cuts. Plot d)
has the new 4momentum cuts, and also a cut defined as XepM ~ Xep(K) < 0.
The particle ID potential of the newly available X2E~p(h) is evident.
1 ~ ^2) â€” adE/dx(hi ) ~ adE/dx(^2 ) + adE/dx(h 1 ) ~ adE/dx(^2 )
2 \og(C rich (ht)) +2 log (Crich^z)) fi.2)
2 log(C,rich(K)) P 2log(Â£ Ricnih'i)).
where the idea is to combine the dE/dX and RICH information into one number.
Pairs of tracks with AxjD(hi â€” h<Â¿) <0 axe more likely to be of type hi than of type
h2.
In practice, we only add a tracks RICH information if its momentum is above
the Cherenkov radiation threshold for both mass hypotheses and there are at least
3 photons within 3
We also require that both hypotheses were actually analyzed by RICH during
pass2.
In addition to RICH and dE/dX, and as hinted in the previous section, the
difference in XepW from the constraint expressed in Equation 3.1 can help the
particle ID (see Figures 3lc and 3ld). We define,
Ax_p(/ii â€œ M = xlP(^i)  (33)
Events with Ax\^p{hi  h^) < 0 are more likely to be of type h\ than oÃ type hi.
In this analysis, to select hi and reject hi the default cut is simply AxjD{hi â€”
hi) < 0. This simple cut is highly efficient, has low fake rates, and is sometimes
sufficient. However, out of the six possible cases when one hadron fakes another,
there are three important cases where it pays off to also use Ay_p(hi â€” hi)
together with ~ ^2) in an optimal way;
1. 7T background to K. This background comes from the continuum process
e+e_ â€”â–º 7p, p â€”â–º 7r+7r~.
2. 7r background to p. Again, this background comes from the continuum
process e+eâ€œ â€”> yp, p â€”> 7r+7r~.
3. K background to p. This background comes from the continuum process
e+e_ â€”> 70, 0 â€”> K+K~.
In other words, the important cases occur when the lighter mass hypothesis
fakes the heavier mass hypothesis.
Mathematically, one would expect that simply adding both Ax]D and Axep
together (like we just did when combining RICH and dE/dX), and cutting on the
grand Ay2 is the way to go. Unfortunately, this simple approach fails because of
large nonmathematical tails in the individual Xep distributions.
Instead, for each of these three cases we define the best cut values (C1.C2) in
Ax2d < ci and AXep < c2 as those that maximize
41
F(cl5c2) *=â€¢
1L
(3.4)
/Re + W * Rfâ€™
where Re (Rf) is the efficiency (fake rate) of the particle ID cuts and IT is the
rough ratio of the background to signal in the data sample for each case. Each W
canâ€™t be known a priori, but a rough idea of its value can be obtained by doing
a first iteration of the analysis with, for example, c\ â€” c2 = 0. We use W = 20,
W â€” 60, and W â€” 30 for cases 13 respectively. Figure 32 shows F(ci,c2) for each
case. Table 32 shows the optimized cut values and their effect on particle ID.
10 5 0 5 10
c,
Figure 32: Contour plots of F(ci, c2) as defined in Equation 3.4 for different sigÂ¬
nals and backgrounds from flat MC. flop has a K signal and a tt background. CenÂ¬
ter has a p signal and a vr background. Bottom has a p signal and a K background.
Quantum Electrodynamic Suppression
Quantum Electrodynamic (QED) background in this analysis comes from the
abundant processes e+e~ â€”> 7e+e'â€œ and e+eâ€œ â€”â–º 7p^pT.
42
Table 32: Cut values, efficiencies, and fake rates for Ax}D < C\ and AXep < c2 in
flat MC. The different ci, C2 values are chosen so F from Equation 3.4 is maximun.
Efficiency and fake rates of each cut are reported relative to events in the skim.
Statistical errors in the efficiencies are 0.1% or less. Errors in the fake rates are
statistical! and are shown for completeness only.
Cut Value
Cl c2
Signal efficiency (%)
aX/d A Xev Both
Fake rate (%)
Axiâ€ž
Both
7r faking K
3
2
94.2
95.2
89.9
1.73 Â±0.09
14.5 Â± 0.3
0.31 Â± 0.04
7r faking p
1
2
99.0
99.8
98.9
0.89 Â± 0.06
1.03 Â± 0.07
0.03 Â±0.01
K faking p
1
3
98.8
99.3
98.1
1.89 Â±0.09
1.09 Â± 0.07
0.10 Â±0.02
To reject e+e â€”* ^e+e~ we require both tracks to have a matched shower with
energy E such that \E/p(n) â€” 0.95 > 0.1, and to have AXw(h â€” e) < 0.
To reject e+e~ â€”> ryp+p~ we simply require D PTH MU < 5 (D PTH MU
was defined in Section 2.2.5) for both tracks in the K and p modes because particle
ID cuts make the pion (and therefore the muon) fake rate small (see Tables 32
and 36). For the 7r mode we cannot use particle ID in a practical way because
muons and pions have similar masses. Instead, to separate pions and muons we
use a much stronger cut requiring that both tracks be within the. barrel part of
the muon chamber ( cos(0) < 0.7), both have P > 1 GeV/c and both have
D PTH MU < 55 .
To improve the overall muon suppression cut efficiency with virtually no
increase in muon fakes, we flag an event as â€œnot muonicâ€ if any of the tracks
deposit more than 600 MeV in the CC, This increases the cut efficiency by about
90% in the n mode and makes the detector more hermetic.
5 Other analyses (for example [18]) typically use D PTH MU < 3, our CLEO III
MC has too many pion tracks with 3 < I) PTH MU < 5 and two few tracks with
DPTHMU = 0.
43
3.2.3 Cut Summary, Efficiencies, and Fake Rates
Table 33 summarizes the cuts used in this analysis. Figure 33 shows the
overall Monte Carlo efficiency after all cuts. Figure 34 shows the fake rates
according to the MC and the data for different particle ID cuts. The data fake
rates and limit fake rates, which are measured for pions and kaons faking other
hadrons, are calculated from the p and Ã© peaks in the continuum (see Figures 42
and 51). Tables 34 through 36 summarize the results of Figures 33 and 34.
Table 33: Cuts used in this analysis.
Motivation
Definition for T(1S) â€”* 7h+h (where h = tt, K, or p)
Skim cuts
Data acquisition
Event must pass hardware (Levell) and software triggers
(pass2) and be of type hardGam
Topological
There are only two good tracks and only one isolated shower
with E > 4 GeV
Analysis cuts
Reconstructed event must
have 4momentum of the
centerofmass system
XEp(h) < 100.0 and 0.050 < AE{h)/ECM < 0.025
Hadron separation
Default is, AxjD < 0 The three cases with a large fake popÂ¬
ulation because of continuum processes use a simultaneously
optimized cut on AxjD an(l AXepi aiid are summarized in
Table 32
QED background
e+e~ â€”* je+e~
Both tracks have a matched shower energy E that satisfies
\E/p(tt) â€” 0.95 > 0.1, and AXidÃ^1 ~ e) < 0
QED background
e+e â€”> 7p.~
For h = K, p both tracks have D PTH MU < 5. For
h = n (At least one track has a matched shower energy >
600 MeV) or (( Both tracks have cos(0) < 0.7 and P >
1 GeV/c) and (both tracks have DPTHMU < 5))
45
r
Table 3 4: Efficiencies in % for cuts (as outlined in Table 33) for flat signal MC.
Efficiencies in the second group are reported relative to the number of candidates
that make it to the skim. The third part of the table shows the overall reconstrucÂ¬
tion efficiency. Statistical! errors are 0.1% or less.
. .a... ..... .
Skim cuts
7T
K
P
Hardware Trigger
89.7
89.0
90.9
Software Trigger
96.5
96.2
96.6
hardGam
78.4
79.9
76.5
Topological (ntracks = 2 &. E1 > 4GeV)
73.3
67.6
72.3
Overall Skim efficiency
64.3
60.3
62.4
Analysis cuts
7T
K
V
4momentum
98.6
98.5
99.0
QED e+e~ â€”> ye+e" suppresion
93.9
87.4
93.1
QED /D"/x~ â€”> 7/x+/i~ suppresion
74.7
93.0
98.3
Hadron separation
97.1
89.0
97.8
Overall analysis efficiency
66.9
79.1
89.1
Overall reconstruction efficiency
43.0
47.6
55.6
Table 35: Efficiencies in % for cuts (as outlined in Table 33) for flat 4S MC. EffiÂ¬
ciencies in the second group are reported relative to the number of candidates that
make it to the skim. The third part of the table shows the overall reconstruction
efficiency. Statistical! errors are 0.1% or less.
Skim cuts
7T
K
V
Hardware Trigger
88.4
88.0
89.0
Software Trigger
94.6
94.6
94.8
hardGam
79.2
80.5
77.9
Topological (ntracks â€” 2 & E1 > 4GeV)
75.2
69.5
75.8
Overall Skim efficiency
66.9
62.7
66.6
Analysis cuts
7T
K
V
4momentum
97.1
97.0
97.4
QED e+e_ â€”* 7e+e~ suppresion
93.4
89.1
.93.2
QED /â€”> 7^'l'/i_ suppresion
75.8
92.5
98.1
Hadron separation
98.0
91.4
96.6
Overall analysis efficiency
67.4
80.2
87.3
Overall reconstruction efficiency
45.0
50.3
58.2
46
Table 36: Final efficiencies and fake rates after all cuts in %. Statistical errors
for efficiencies are 0.1% or less. Statistical errors for fake MC rates are shown for
completeness only. The IS DATA corresponds to the IS off resonance data sample.
The MC was generated flat.
7T cuts.
K cuts
p cuts
IS 7r MC
43.0
0.14 Â±0.01
< 0.007
4S 7T MC
45.0
0.29 Â±0.02
<0.01
IS p DATA

< 1.2
< 0.2
4S p DATA
 â–
0.20 Â± 0.06
< 0.06
IS K MC
1.27 Â± 0.03
47.6
< 0.02
4S K MC
0.92 Â± 0.04
50.3
< 0.06
IS <Â¡> DATA
< 3.8

< 2.0
4S
4.14 Â±0.69

< 0.4
IS p MC
0.08 Â± 0.02
0.34 Â± 0.04
55.6
4S p MC
0.05 Â±0.01
0.37 Â±0.01
58.2
0 12 3
Invariant mass (GeV/c2)
Figure 33: Final efficiencies for each mode as a function of invariant mass for the
IS (solid) and 4S (dashed) Monte Carlo data. The MC was generated with a flat
angular distribution.
47
Figure 34: Fake rates for IS (hollow circles) and 4S (hollow squares) according to
flat MC. The (red) downward pointing triangle is obtained using the p peak in 4S
data.
CHAPTER 4
EXCLUSIVE RADIATIVE DECAY T(1S) * ^+'n~
In Figure 42 the 7r+7râ€œ invariant mass plot is shown for both IS and 4S data.
Figure 43 shows the continuum subtracted 7r+7r~ invariant mass distribution (as
defined in Section 3.2.1) with the most likely statistical fit overlayed (which is
described in the next section). The number of events within ir of the p region
(0.62 â€” 0.92 GeV/c2) left after the continuum subtraction is 200 Â± 300, and 50 of
these belong to the /o(980) lowmass tail.
4.1 Robustness of The Mass Distribution
In [27] the decay T(1S) 77rÂ°7rÂ° is analyzed, and it is shown how the analysis
stream warps the shape of the reconstructed resonance. This effect, which arises
because of the particular 7rÂ° behavior, raises problems when fitting the invariant
mass distribution.
In Section 3.2.1 we claimed that if the data and MC had the same 07 and
6k distributions, the fit to the continuum subtracted invariant mass distribution
automatically gives the correct efficiency corrected number of events.
Here we test this claim. To this end, we generate 10000 T(1S) â€”Â» 7/2(1270),
with ^(1270) â€”* 7r+7r~ with flat 07 and 0* distributions. We treat this MC
as data and carry out the first two steps of the continuum subtraction process.
The resulting peak has a mass of 1.278 Â± 0.002 GeV/c2 and a width of 0.193 Â±
0.006 GeV/c2, consistent with the generated mass and width of 1.275 GeV/c2 and
0.185 GeV/c2. More importantly, the number of reconstructed events from the fit is
10040 Â± 180, which is consistent with the number of generated events.
Figure 41 shows the reconstructed and efficiency corrected events, a fit to
them, and the generated events.
.1 ;
48
49
We conclude that there is no warping of the mass distribution, and that the
analysis stream behaves like we expect when obtaining the efficiency corrected
number of events from data.
â– Reconstructed and Eff. Corrected
râ€”i MC Generated
Figure 41: Reconstructed events and efficiency corrected events, a fit to them, and
the generated events for Y(1S) â€”Â» 7/2(1270), with /2(1270) â€”> 7r+7r~ with flat 67
and On distributions.
4.2 Statistical Fit of the Invariant Mass Distribution: Signal Areas and
Their Significance
The results of this section are summarized in Table 41.
Figure 43 shows possible signals for for T(15) â€”> 7/o(980), T(IS') â€”â–º
7/2(1270) and T(15) â€”> 7/4(2050). To determine the number of events in each
signal we fit the invariant n+n~ mass continuum subtracted distribution with three
spindependent, relativistic BreitWigner line shapes. The masses and widths are
allowed to float, except for the width of the /4(2050) which has a very large error
if allowed to float and is set to its PDG value. The PDG values[16] for the mass
and width of the three resonances are nif0^8o) â€” 980 Â± 10 MeV/c2, F/0(98o) â€”
70 Â± 15 MeV/c2, mh{1270) = 1275.4 Â± 1.2 MeV/c2, r/2(127o) = 185.ll; MeV/c2,
ra/4(2050) = 2034 Â± 11 MeV/c2, and T/42050) = 222 Â± 19 MeV/c2.
50
To measure the statistical significance of each signal we do multiple x2 fits
fixing the signal area to different values while letting the mass and width of
the signal whose significance is being measured to float within 2a of the PDG
values. At the same time the rest of the fit parameters are fixed to the values that
originally minimized the x2 We assign each of these multiple fits a probability
proportional to e*2/2 and then normalize. We calculate the chance of the signal
being due to a random fluctuation by adding the normalized probabilities for the
fits with a negative or 0 signal. This method fails for the highly significant /2(1270)
signal because the e~x2/2 value of fits with negative or 0 /2(1270) signal is of the
order of 1054 and our computing software can only handle numbers as small as
1CT45. For completeness we state that the significance of this signal is < 1045.
To measure the upper limit for the /j(2220) we also do multiple x2 fits
for different fixed signal values, while keeping its mass and width constant at
mfj(2220) = 2.234 GeV/c2 and ^(2220) = 17 MeV/c2 as in [17]. The resulting
probability plot is shown in Figure 45.
We find clear evidence for the /2(1270), evidence for the /o(980) and weak
evidence for the /4(2050). We also put a 90% confidence level upper limit on
fj (2220) production. Fit results are shown in Table 41.
Table 41: Results for T(15) â€”> 77r+7r~. The branching fractions of /2(1270) and
/4(2050) are taken from the PDG [16]. Errors shown are satistical only.
Mode
Area
B.F.(IO5)
Significance
7/o(980), /o(980) â€”> 7r+7r
340ÃS
1 fi+u'2
7/2(1270)
1230 Â± 100
10.4 Â±0.8
< lO"45 (> 14a)
7/4(2050)
85 Â±30
3.6 Â±1.3
5.2 x 10~3 (2.6a)
7//(2220),/j(2220) â€”> 7r+7r
< 13
< 6.2 x IQ2

4.3 Angular Distribution of The Signal
The helicity angle distributions of 6n and #7 are defined and described in the
N
appendix. In this section we first obtain the helicity angular distributions of the
51
/o(980), /2(980), and /4(2050) and then fit them to the predictions of the helicity
formalism.
In practical terms, obtaining the helicity angle distribution of a particular
resonance from data consists of two steps. First, we choose an invariant mass
interval around the resonance peak to select events from the resonance and obtain
a helicity angular distribution which has both signal and background events.
Second, we subtract the contribution to the helicity angular distribution of the
background events in the chosen mass interval to obtain what we want; the helicity
angle distribution of the resonance.
Choosing the mass interval is not a trivial thing. If its too wide there will
be too much background, and if its to narrow there will not be enough signal.
To choose the optimum mass interval we need to know how much signal and
background we are selecting. Therefore, the two steps described in the previous
paragraph are related. How we deal with this is revealed in the next two sections.
4.3.1 Optimum Mass Interval
As described above, the first step in obtaining the helicity angle distribution
for a resonance is choosing an invariant mass interval to select events from such a
resonance. A standard ir (which corresponds to 1.6tr for a spin 0 BreitWigner)
cut around the mean mass of the resonance can be chosen as a â€œstandardâ€ interval.
We could proceed this way, but in our case because of the large subtractions
involved when obtaining the angular distribution, a considerable increase statistical
significance of each bin in the helicity angular distribution can be achieved by
choosing the mass interval carefully (see the last column of Table 42).
Letâ€™s consider for example the /o(980) angular distribution. We begin with
the T(15) and the T(4S) efficiency corrected distributions. Before the continuum
subtraction each bin in the /o(980) angular distribution has contributions from
the high end mass tail of the p and the low mass tail of the /2(1270). In order to
52
get the final angular distribution, both of these contributions are taken away by
first doing a continuum subtraction using the scale factor a = 0.404, and then
by subtracting the /2(1270) distribution outside the mass interval scaled by an
appropriate scale factor 8 equal to the ratio of /2(1270) inside the mass interval
and outside of it 1 . After all this, the contributions to the bins in the final angular
distribution are,
Nf0(980) â€” ?Vo(980) + Â«p(T(lS)) + Uf2(i270) ~
where TC/o(980)(n/2(i270)) is the number of /o(980)(/2(1270)) obtained by integrating
the fitted spindependent, relativistic BreitWigner function inside the mass
interval, np^[(is))(^p(r(4S))) is the number of pâ€™s inside the mass interval from
continuum events at the T(15)(T(4S)) energy, and n^(1270j is the number of
/2(1270) outside the mass interval being used to subtract the contribution of
/2(1270) to the /o(980) angular distribution. Following this last definition, (3 =
nMl270)/n'f2(1270)
Each of these terms has an associated error. Assuming that An = \fn, that
the efficiency correction has infinite statistics, and ignoring the errors on the
continuum scale factor a, the overall error is,
AN/o(980) = \J Tifo (980) + (Q + 22)ftp(T(4S)) + (2 + (3 + (32)rif2( i270)
We arrive to the conclusion that the mass interval (m â€” Am, m + Am) which
produces the helicity angle distribution with smallest relative binerrors is the one
1 Strictly speaking this is correct only to first order. The /2(1270) distribution
itself has a small contamination from /0(980) and /4(2050). This effect, which we
call crosscontamination, is ignored in Equation 4.1. Later, in section 4.2.2, we will
show how to eliminate this small crosscontamination using all the resonances. The
difference in how we actually get the helicity distributions and Equation 4.1 has an
insignificant effect when calculating the optimal mass interval
53
y
that maximizes,
Ff0(9so){m, Am)
n/o(980)
\A/o(980) + (<* + Â«2)Â«p(T(4S)) + (2 + (5 + /?2)n/2(i270)
(4.2)
The plot of F/O(980)(m, Am) is shown in figure 46.
This same technique can be applied to the /2(1270) and /4(2050) resonances.
Results are shown in Table 42.
Table 42: Mean masses, widths in GeV/c2 and inverse of the average bin relative
error (F) from background subtractions for the angular distribution of different
resonances. Standard mean masses and widths, corresponding to IT, are taken
from the fit in Figure 43, and are labeled with the subscript â€œsâ€, while those that
maximize F are labeled with the subscript â€œmâ€. The last column shows the factor
by which the effective statistics increase.
Resonance
(ms, Ams)
F(ms, Ams)
(Tnm, Amm)
F(mm, Amm)
F'Â¿(mm,Amm)
F2(ms,Arris)
/o(980)
(0.970,0.070)
5.7
(0.985,0.060)
6.0
1.11
/2(1270)
(1.270,0.120)
18.3 .
(1.590,0.420)
20.6
1.27
/4(2050)
(2.120,0.220)
3.2
(2.240,0.250)
3.4
1.13
4.3.2 Background Subtraction
The continuum subtraction of the helicity angle distribution is defined in
Section 2.1. The subtraction of the tails from other resonances requires a closer
look. Let us call the continuum subtracted helicity angle distribution of the events
in the /o(980), /2(1270), and /4(2050) mass intervals c/o(980), C/2(mo), and c/4(2050)
respectively. Let us call the helicity angle distribution of the events that come
exclusively from the resonance we are trying to select in the same mass intervals
(that is, the true helicity angle distribution of the resonance) Â¿/O(980), Â¿/2(i270)> and
2050) respectively. The continuum subtracted helicity angle distribution of a
54
resonance is being contaminated by the tails of the other resonances. Keeping this
in mind we write 2
c/o(980) = i/o(980) + fit f2(1270)
< C/2( 1270) â€” Ã/o(1270) + 7Ã/o(980) + 5tfA(2050)
C/4(2050) = Ã/O(2050) + ^/2( 1270)
Where the small numbers /?, 7, <5, and e are the ratios of the number of events
from a resonance in the mass interval where the contamination is taking place to
the number of events from the same resonance in the mass interval used to select
it. Using the (mm, Amm) values in Table 42 and the fit in Figure 43 we obtain
/3 == 8.5 x 1CT3, 7 = 6.0 x 1(T2, 5 = 9.7 x 1(T2, and e = 0.13.
To obtain the background subtracted helicity angular distributions we simply
invert the system of equations expressed in Equation 4.3. The solution can be
conveniently expressed as,
i/2(1270) = (C/o( 1270)  7^/o(980) ~ Â¿C/4(2050))
Â£/o(980) = C/0(980) ~ /%a(1270)
(4.4)
^/4(2050) = C/0(2050) â€” ei/2( 1270)
As a check, if we ignore the second order terms the previous solution becomes,
2 Here the cross contamination between the /o(980) and the /4(2050) is ignored.
There is no mathematical problem in including these contamination terms, but the
system of equations would be more complicated than it needs to be since this type
of cross contamination is negligible.
4
i y ' .
* t â–
55
Ã/2(1270) = C/â€ž(1270) ~ 7C/o(980) â€œ Â¿C/4(2050)
<
Ã/4(2050) â€” C/0(2050) ~ ec/2(l270)
which is indeed is the solution when crosscontamination is ignored.
4.3.3 Statistical Fit of the Helicity Angular Distribution: J AssignÂ¬
ments and their Probability Distribution
For each resonance, we fit the data to the simultaneous cos 81 and cos 9n
helicity angle distribution obtained from data using Equation 4.4 to the helicity
formalism prediction, Equations 2327. projected on each angle and folded in
opposite directions around their symmetry axis in order to show both distributions
on the same plot. The corresponding J value of the best fit for each resonance,
shown in Figures 48 through 410, is defined as the J assignment (Ju). We obtain
Ja = 1 for the /o(980) which is inconsistent with the known spin of the /o(980)
which is J â€” 0. For the /2(1270) and the /4(2050) we obtain Ja â€” 2 and Ja = 4
respectively, which is consistent with their known spins.
To have an idea of how well the angular distribution determines J among the
hypotheses J = 0, 1, 2, 3, 4, we do a statistical fit for each hypothesis and assign
each one a probability proportional to e_^x2+do ^^2 where d.o.f. are the degrees
of freedom in the fit. The resulting normalized probability distributions give an
idea of the assigned J significance and are shown in Figure 411. In particular,
this figure shows that the Ja â€” 1 for the /o(980) inconsistency can be due to a
statistical fluctuation.
56
RÃe: â€¢/bohr/usen/luis/deo3/ntp/data_50.ntp
ID
IDB
Symb
Date/Time
Area
Mean
R.M.S
1
1
1
040807/0140
9112
0.8080
0.2895
1
2
1
040807/0140
2.1195E+04
0.7647
0.2072
98765
981
1
040807/0203
569.1
1.227
0.7650
Mass (GeV/c2)
Figure 42: Invariant mass of 7r+7r for IS (top) and 4S (bottom) data. For the 4S
data we show a blowup of the mass region 1.5 â€” 3 GeV/c2 where the p* can be
seen.
y â€¢
i
i
/'* *
MINUIT x2 Fit to Plot 9&3 '
UlS>Gamma h h. masspipi axis
File: Generated internally
Plot Area Total/Fit 1544.1 /1542.1
Func Area Total/Fit 1359.7 /1359.7
22JUL2004 11:51
Fit Status 3
E.D.M. 8.843E06
x2= 44.7 for 54 8d.o.fâ€ž
Errors
Function 1: f_0
Parabolic
Minos
C.L.= 52.7%
MOD
435.70
t
165.4
 167.0
+
177.3
MEAN
0.96672
t
1.5714E02
1.2374E02
1.8160E02
WIDTH
7.05219E02
Â±
3.2912E 02
4.3455E02
+ â–
4.4879E02
* SPIN
O.OOOOOE+OO
Â±
O.OOOOErOO
 O.OOOOE+OO
+
O.OOOOE+OO
* THRESHOLD
Function 2:1 2
028000
Â±
0 00006+00
 O.OOOOE+OO
+
O.OOOOE+OO
MOD
998.59
Â±
77.96
 7812
+
77.92
MEAN
1.2678
Â±
7 59056 03
 7.1319E03
+
8.0978E03
WIDTH
0.12353
Â±
1.4273E02
 1.3459602
1 5382E02
* SPIN
2.0000
Â±
0 OOOOEâ€™OO
 O.OOOOE+OO
+
O.OOOOE+OO
* THRESHOLD
Function 3: f 4
028000
Â±
O.OOOOE+OO
 0 0000E+00
+
O.OOOOE+OO
MOD
104.37
+
38.84
 38.76
+
38.91
MEAN
2.1191
Â±
5.4515E02
5.8893E02
+
4 9096E02
* WIDTH
0.22200
Â±
0 0000E t00
 0 OOOOE+OO
+
O.OOOOE+OO
* SPIN
4 0000
Â±
O.OOOOE+OO
 O.OOOOE+OO
+
O.OOOOE+OO
* THRESHOLD
0.28000
Â±
0 OOOOE+OO
O.OOOOE+OO
+
O.OOOOE+OO
Figure 43: Continuum subtracted (as defined in Section 3.2.1) invariant mass of
7r+7râ€œ from T(15l) â€”â™¦ 77r+7r~.
Events / ( 50.0 MeV/c )
f
58
1.
File: Generated internally
ID
ID8 Syrr.b
Qete/Ti'rie
Area
Mean
R.M.S.
9
3 32
040806/1839
1544.
1.408
0.4996
98765
981 1
040806/1845
1360.
1.416
0.4800
98765
982 2
040806/18*5
1371.
1.422
0.4836
1.5 2.0 2.5 3.0
Mass (GeV/c2)
Figure 44: Blow up of the invariant mass of 7r+7r in the /i(2050) mass region,
including the upper limit on the fj(2220) shown as a dashed line.
Probability distribution
59
File: Amp_mnt/boMuser1.1uis/deo3/mnJit/u1sgammahh/temp.tmp
ID
IDB Symb
Date/Time
Area
Mean
R.M.S.
1
29 1
040806/1845
1.000
6.294
5008
1
25 1
040806/1845
0 9001
5.132
3.650
0.080
0.060
0.040
0.020
0.000
0
25
50
Area
Figure 45: Normalized probability distribution for different fj(2220)
nal areas. The shaded area spans 90% of the probability.
7T+7T' sig
60
>u tz_j iimimiiii .Mb i l i i , i j l=
1990 2090 2190 2290 2390
Mean Mass (MeV/c )
Figure 46: Contour plot of the inverse of the average relative binerror from backÂ¬
ground subtractions in the /â€ž(980) (top), /2(1270) (middle), and /4(2050) (bottom)
angular distribution.
J>
61
MINUIT x2 Fit to Plot 9&82
U1S>Gamma h h. ABS(COS(PHOTHFTA'i) axis
File: Generated internally
Plot Area Total/Fit 606.18 / 606.18
Func Area Total/Fit 436.31 / 436.31
x2= 25.6 for 20  1 d.o.f.,
Errors Parabolic
Function 1: JO
AREA 218.15 + 42.27 .
5AUG2004 23:06
Fit Status 3
E.D.M. 6.301 E13
C.L.= 14.1%
Minos
42.27 + 42.27
IcosB^I cos6n
Figure 47: Angular distribution for the excess events in the /o(980) mass region.
The fit corresponds to J = 0.
dN/( 0.1 dcos0re)
62
MINUIT x2 Fit to Plot 9&82
U1S>Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 606.13/606.18
Func Area Total/Fit 478.89 / 478.89
x2= 15.2for 20 2d.o.f.,
Errors Parabolic
Function 1: J1
AREA 239.45 Â± 39.31
Â© 1.32483E03 Â± 0 3852
5AUG2004 23:06
Fit Status 3
E.D.M. 1.115E05
C.L.= 65.1%
Minos
39.33 + 39.29
O.OOOOE+OO + 0.3537
cos0Tl cos0n
Figure 48: Angular distribution for the excess events in the /o(980) mass region.
The fit corresponds to J = 1.
dN/( 0.1 dcos0J)
63
MINUIT x2 Fit to Plot 9&72
U1S>Gamma h h. A3S(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 1673.4 /1673.4
Func Area Total/Fit 1733.8 /1733.8
X2= 27.3 for 20  3 d.o.f.,
Errors Parabolic
Function 1: J2
AREA
0
866.97 Â± 43.10
4.25267E04 Â± 7.7843E02
0.29838 Â± 0.1098
5AUG2004 23:06
Fit Status 3
E.D.M 9.701 E06
C.L.= 5.4%
Minos
45.85 + 46.07
O.OOOOE+OO + 0.1238
0.1478 + 9.9862E02
1.0
0.5 0.0 0.5 1.0
cos6y IcosOJ
Figure 49: Angular distribution for the excess events in the /2(1270) mass region.
The fit corresponds to J = 2.
dN/( 0.1 dcos0J)
64
MINUiT %2 Fit to Plot 9&92
U1SÂ»Gamma h h. ABS(COS(PHOTHErA)) axis
File: Generated internally
Plot Area Total/Fit 159 93 /159.93
Func Area Total/Fit 127.08 /127.08
Xz= 19.6 for 20 3d.o.f.,
Errors Parabolic
Function 1: J4
AREA
0
<Â¡>
63.528 Â± 1.408
1.5697 Â± 5.8174E02
0.16919 Â± 0.6887
5AUG2004 23:06
Fit Status 3
E.D.M. 5.300E07
C.L.= 29.3%
Minos
14.91 + 0.0000E+00
0.3398 + 0.0000E+00
O.OOOOE+OO + O.OOOOE+OO
1.0 0.5 0.0 0.5 1.0
IcosG^I IcosGJ
Figure 410: Angular distribution for the excess events in the /4(2250) mass region.
The fit corresponds to J = 4.
dN/( 0.1 dcos6J)
65
<*
Figure 411: Ja probability distribution for resonances in the /o(980) (top),
/2(1270) (middle), and /4(2050) (bottom) invariant mass region when only the
hypotheses Ja = 0, 1, 2, 3, 4 are considered.
CHAPTER 5
EXCLUSIVE RADIATIVE DECAY T(15) â–º 7K+K~
In figure 51 the K+K ~ invariant mass plot is shown for both resonance and
continuum running. This figure also has an inset showing the
tracks for events in the 1.1  3 GeV/c2 region. This inset indicates that most
of the events in this mass region have indeed two kaons and that the amount of
p reflection is small. Furthermore, when the 7r+7r_ invariant mass is plotted for
these events only 40 out of 700 can be fit under a p peak (this fit is what we use to
calculate the pion faking kaon rate in Table 36).
Figure 52 shows our fit to the K+ K~ invariant mass continuum subtracted
plot as defined is Section 3.2.1 with the most likely statistical fit overlayed (which
is described in the next section). The number of events near the region (1.01 â€”
1.03 GeV/c2) left after the continuum subtraction is 50 Â± 70.
5.1 Statistical Fit of the Invariant Mass Distribution: Signal Areas and
Their Significance
The results of this section are summarized in Table 51.
From the measurement of the previous section we expect a small contribution
(Â« 50 events assuming no interference) from /2(1270) â€”> K+K~. We do find
some evidence for /2(1270) â€”> K+K~ events (110 Â± 40) in the fit. We also find
strong evidence for the resonance /2(1525) and weak evidence for the /o(1710)
resonance. The /2(1270), and /o(1710) are fitted with their widths fixed to their
PDG values[16] because they have large errors if allowed to float. The rest of
the resonances parameters are consistent with their PDG values[16], which are
"V'(1525) = 1525 Â± 5 MeV/c2, r/'(1525) = 76 Â± 10 MeV/c2, m/o(mo) = 1715 Â±
66
67
6 MeV/c2, and T^mo) = 125 Â± 10 MeV/c2. We also observe an excess of events in
the 2 â€” 3 GeV/c2 region which we canâ€™t attribute to any known resonances.
Significances of the signals of the identified resonances in the fit are calculated
as described in the previous section. The Y(IS') â€”â–º 7/j(2220), /j(2220) â€”> K+K~
upper limit is also calculated using multiple fits, except that this time the events
under the fj{2220) are of unknown origin, so we use a first order polynomial
allowed to float. The significance of the excess of events in the 2 â€” 3 GeV/c2
invariant mass region is calculated assuming a normal distribution; we simply add
up the number of events in each bin along with its error. Results are shown in
Table 51.
Table 51: Results for T(IS') â€”> ^K+K~
Mode
Area
Branching Fraction (10 5)
Significance
7/2(1270)
7/2(1525)
110 Â±40
23 Â±8
5.4 x 10 4(3.3(j)
360/7Â°
39 ij?
< 1045(> 14cr)
7/o(1710), /o(1710) â€”> K+K~
75 Â±30
0.35 Â±0.14
7.5 x 10~4(3.2cr)
^K+K~(2 â€” 3 GeV/c2)
220 Â± 20
1.03 Â±0.12
8.8u
7/7(2220), fj(2220)*K+K
< 10
< 5 x 10~2

5.2 Angular Distribution of The Signal
In this section we adapt the ideas presented in Section 3.4.3 to the T(IS') â€”*
7K+K~ situation.
The derived statistical errors from the signal and background subtractions can
be used to calculate the mass interval which best represents the helicity angular
distribution. The inverse of the expected average relative bin error as a function of
the mass interval is shown in Figure 55, and the mass interval that maximizes it
are tabulated in Table 52.
The tails from the resonances contribute to the continuum subtracted helicity
distributions,
68
Table 52: Mean masses, widths inmass and inverse of the average bin relative
error (F) from background subtractions for the angular distribution of different
resonances. Standard mean masses and widths, corresponding to IF, are taken
from the fit in Figure 52, and are labeled with the subscript â€œsâ€, while those that
maximize F axe labeled with the subscript â€œmâ€. The last column shows the factor
by which the effective statistics increase.
Resonance
(ms, A ms)
F(ms, Ams)
(mm, A mm)
F{mm,Amm)
F2(ms,Arris)
/2(1270)
(1.276,0.185)
3.5
(1.300,0.100)
4.1
1.37
/i(1525)
(1.540,0.085)
9.6
(1.565,0.100)
9.8
1.04
/o(1710)
(1.760,0.125)
3.1
(1.780,0.095)
3.3
1.13
C/a (1270) = 1270) + /3t/'( 1525)
< C/'(]525) = Â£/'(1525) + 7^/2(1270) + <%>(1710)
C/0(1710) = Â¿/0(1710) + eÂ£/'( 1525)
Where again the small numbers 8, 7, 5, and e are the ratios of the number
of events from a resonance in the mass interval where the contamination is taking
place to the number of events from the same resonance in the mass interval used
to select it. Using the (mm, Amm) values in Table 52 and the fit in Figure 52 we
obtain /3 = 1.3 x 102, 7 = 0.14, 8 = 0.11, and e = 0.19.
The background subtracted helicity angular distributions are,
Ã/'(1525) = (C/2(1525) ~ 7C/2(1270) â€œ Â¿C/o(m0))
i Â¿/2(1270) = c/2(1270) ~ /%Â£( 1525) (^2)
^/o(1710) = c/o(1710) ~ 1525)
The the best fit for each resonance and the excess of events in the 2â€”3 GeV/c2
are shown in Figures 56 through 59. The best spin assignment for the /2(1270) is
Ja â€” 2, for the /2(1525) it is Ja = 2, for the it is /o(1710), and it is Ja = 1 for the
excess of events in the 2 â€” 3 GeV/c2 mass region. The Ja = 2 value for the /o(1710)
69
is inconsistent with its known spin. Also, examination of the normalized helicity
amplitudes for the /2(1270) reveals that they are inconsistent with those obtained
for the /2(1270) in the 7r+7T_ mode.
The assigned J probability distributions are shown in Figure 511. They reveal
that the inconsistencies in the /2(1270) and /o(1710) are not significant and can be
attributed to the statistical uncertainty.
File: 7bohr/user1 /Iuis/deo3/ntp/data_50.ntp
ID IDB Symb Date/Time
1 1 1 040807/0219
1 2 1 040807/0219
1 10 1 040807/0235
Area
Mean
R.MS.
1274.
1.320
0.4283
2031.
1.156
0.2585
705.0
0.2246
1.208
0.1480
1.170
Figure 51: Invariant mass of K+K~ for IS (top) and 4S (bottom) data. For
the 4S data the inset shows the (JdE/dxi.^) fÂ°r both tracks for events in the
1.1 â€” 3 GeV/c2 mass region. This inset indicates that most of the events are conÂ¬
stant with having two kaons.
Events / ( 25.0 MeV/c )
70
MINUIT x Fit to Plot
U1S>Gamma h h. masskk axis
9&3
File: Generated internally
7AUG2004 14:23
Plot Area Total/Fit
794.78/575.02
Fit Status 3
Func Area Total/Fit
494.50/429.38
E.D.M. 1.627E06
y2= 37.5 for 40 
7 d.o.f.
Â»
C.L.= 26.9%
Errors
Parabolic
Minos
Function 1: f_2(1270)
AREA
197.37
Â±
64.92
 64.93
+
64.91
MEAN
1.2757
Â±
3.3189E02
 8.9556E02
+
3.0045E02
* WIDTH
0.18510
Â±
0 OOO0E+0O
 O.OOOOE+OO
+
O.OOOOE+OO
* SPIN
2.0000
Â±
0.0000E+00
 O.OOOOE+OO
+
O.OOOOE+OO
* THRESHOLD
0.98000
Â±
0 0000E+00
 O.OOOOE+OO
+
O.OOOOE+OO
Function 2:f_2'(1510)
MOD
318.76
Â±
64.61
 59.53
+
72.90
MEAN
1.5384
Â±
7 8867E03
 7.5447E03
+
8.8799E03
WIDTH
8.54208E02
Â±
2.7982E02
 2.2981 E02
+
3.6400E02
* SPIN
2.0000
Â±
O.OOOOE+OO
 O.OOOOE+OO
+
O.OOOOE+OO
* THRESHOLD
0.98800
+
0 OOOOE+OO
 O.OOOOE+OO
+
O.OOOOE+OO
Function 3: f_0(1710)
MOD
96.117
Â±
38.27
 38.70
+
37.77
MEAN
1.7622
Â±
2.2476E02
 2.0985E02
+
2.6333E02
* WIDTH
0.12500
Â±
O.OOOOE+OO
 O.OOOOE+OO
+
O.OOOOE+OO
* SPIN
O.OOOOOE+OO
Â±
O.OOOOE+OO
 O.OOOOE+OO
+
O.OOOOE+OO
Â»THRESHOLD
0.98800
Â±
O.OOOOE+OO
 O.OOOOE+OO
+
O.OOOOE+OO
0.9 1.4 1.9 2.4 2.9
Mass (GeV/c2)
Figure 52: Continuum subtracted invariant mass of K+K .
Events / ( 25.0 MeV/c'1)
71
File: Generated internally
ID
IDB
Symb
Date/Time
Area
Mean
R.M.S.
9
3
32
040807/1422
794.8
1.739
0.4861
98765
981
1
040807/1426
308.2
2.086
0.5922
98765
982
2
040807/1426
318.3
2.091
0.5835
1.80 2.00 2.20 2.40 2.60
Mass (GeV/c2)
Figure 5â€”3: Blow up of the /j(2220) region, with the 90% CL upper limit overÂ¬
laid. The mass and width are taken to be m/j(2220) = 2.234 GeV/c2 and
F/j(2220) = 17 MeV/c2 as in [17].
Probability distribution
72
File: Amp_mnt/bohr/user1/luis/deo3/mnJit/u1sgammahh/temp.tmp
ID IDB Symb Date/Time Area Mean
1 29 1 040807/1426 1.000 4.746
1 25 1 040807/1426 0.9054 3.971
fi.M.S.
3.582
2.723
0.20
0.15
0.10
0.05
0.00
l I L.
50
Figure 54: Normalized probability distribution for different fj(2220)
signal areas. The shaded area spans 90% of the probability.
K+K
150
100
50
1700
1750 1800 1850 1900
2
Mean Mass (MeV/c )
Figure 55: Contour plot of the inverse of the average relative binerror from backÂ¬
ground subtractions in the /2(1270) (top), /2(1525) (middle), and /o(1710) (botÂ¬
tom) angular distributions.
tf
74
â™¦
MI NU IT %2 Fit to Plot 9&B2
U1S>Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 517.49/517.49
Func Area Total/Fit 129 01 /129.01
X2= 17.8 for 20 3d.o.fâ€ž
Errors Parabolic
Function 1:J2
AREA 64.505 Â± 17.78
0 7.95385E03 Â± 1.347
5 AUG2004 23:16
Fit Status 3
E.D.M. 5.475E05
C.L. 40.4%
Minos
17.78 + 17.78
O.OOOOE+OO + 0.5745
0.2689 + Q.OOOOEtOO
1.0
0.5 0.0 0.5 1.0
IcosG^J cos0n
Figure 5 6: Background subtracted K+K angular distribution in the /2(1270)
mass regionas defined in the text. The fit corresponds to J = 2.
dN/( 0.1 dlcosBJ)
75
MINUIT x2 Fit to Plot 9&72
U1 S>Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 284.44 / 284.44
Func Area Total/Fit 367.87 / 367.87
x2= 20.9 for 20  3 d.o.f.,
Errors Parabolic
Function 1:J2
AREA 183.94 + 18.70
0 6.76443E04 Â± 0.1885
5AUG2004 23:16
Fit Status 3
E.D.M. 3.771 E06
C.L.= 23.3%
Minos
19.29 + 19.25
O.OOOOE+OO + 0.2757
0.1593 + 0.1248
1.0 0.5 0.0 0.5 1.0
ICOS0.J ICOS0J
Figure 57: Background subtracted K* K~ angular distribution in the /2(1525)
mass region as defined in the text. The fit corresponds to J = 2.
dN/( 0.1 dcos0rt)
76
MINUIT x2 Fit to Plot 9&92
U1S>Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 129.47 /129.47
Func Area Total/Fit 79.031 / 79.031
5AUG2004 23:16
Fit Status 3
E.D.M 1 721E05
%'= 28.0 for 20  3 d.o.f.,
Errors
Function 1: J2
Parabolic
AREA 39.516 Â± 10.80
9 2.99860E04 Â± 0.3847
<}> 1.5696 Â± 0.2633
C.L.â€” 4.5%
Minos
10.81 ^ 10.80
O.OOOOE+OO + 0.4645
0.3060 + O.OOOOE+OO
1.0
0.5 0.0 0.5 1.0
IcosQ^J IcosOJ
Figure 58: Background subtracted K+K angular distribution in the /o(1710)
mass region as defined in the text. The fit corresponds to J â€” 2.
dN/( 0. f djcosBJ )
MINUIT x2 Fit to Plot 9&102
U1S>Gammah h. ABS(CCS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 411.73 / 411.73
Fane Area Total/Fit 370.86 / 370.86
%2= 18.9 for 20 2d.o.f.,
Errors Parabolic
Function 1: J1
AREA 185.43 Jt 17.19
0 015542 + 0.3003
5AUG2004 23:16
Fit Status 3
E.O.M. 4.735E09
C.L.= 40.0%
Minos
 17.19 + 17.18
â€¢ O.OOOOEtOO + 0.1921
1.0 0.5 0.0 0.5 1.0
IcosQ^J IcosBJ
Figure 59: K+K~ angular distribution for events within the 2 â€” 3 GeV/c
( TqsooIp 10 )/NP
78
MINUIT x2 Fit to Plot 9&92
U1S~>Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 129.47 /129.47
Func Area Total/Fit 62.194 / 62.194
y2= 33.3 for 201 d.o.f.,
Errors Parabolic
Function 1: JO
5AUG2004 23:16
Fit Status 3
E.D.M. 3.484E15
C.L.= 2.2%
Minos
1.0 0.5 0.0 0.5 1.0
Icose^l cos0rt
Figure 510: Background subtracted K+K~ angular distribution in the /o(1710)
mass region as defined in the text. The fit corresponds to J = 0.
dN/( 0.1 dcos0TC)
79
83
â– C
O
Cm
Cm
I
0 2
4
J
a
Figure 511: Ja probability distribution for resonances in the /%{1270)(top),
/2(1525) (middletop), /o(1710) (middlebottom) invariant mass region and the
excess of events in the 2 â€” 3 GeV/c2 (bottom) region when only the hypotheses Ja
â€” 0, 1, 2, 3, 4 are considered.
CHAPTER 6
EXCLUSIVE RADIATIVE DECAY T(1S) > 7pp
In figure 61 the pp invariant mass plot is shown for both IS and 4S data, with
an inset showing that most of the events indeed have a proton and an antiproton
and that the p and
the 4S plot corresponds to the process e+e_ â€”â–º 7 J/ip with J /ip â€”> pp. This
enhancement is not as pronounced in the IS plot because the IS data only has
32.4% of the luminosity of the 4S data. The number of events in the J/ip invariant
mass region after continuum subtraction is 6 Â± 3 and is consistent with 0.
Figure 62 shows the continuum subtracted invariant pp mass distribution
(as defined in Section 3.2.1) with a 90% confidence level upper limit for fj{2220)
overlaid. A direct fit to the fj(2220) yields 12 Â± 5 events.
There is an excess of events in the continuum subtracted invariant mass plot.
We measure this excess and the upper limit of e+e_ â€”> 7/j(2220), fj(2220) â€”â–º pp
the same way we measured the excess of events inside 2 GeV/c2 < m(K+K~) <
3 GeV/c2 region and the fj(2220) upper limit in Section 5.1. Results are shown in
Table 61.
The pp angular distribution for the mass range 2 GeV/c2 < m(pp) < 3 GeV/c2
is shown in Figure 64 with the most likely J assignment, Ja â€” 1, fit overlayed.
The probability distribution for Ja is shown in Figure 65.
Table 61: Results for Y(IS') â€”+ 7pp.
Mode
Area
B.F. or 90 % U.L. (10~5)
Significance
7pp (2 â€” 3 GeV/c2)
85 Â± 18
0.41 Â± 0.08
4.85cr
7/7(2220), /j(2220) â€”â™¦ pp
<20
< 9 x 102

80
Events / ( 25.0 MeV/c )
81
File: 7bohr/user1 /Iuis/cleo3/ntp/data_50.ntp
ID
IDB
Symb
1
Date/Time
Area
Mean
R.M.S.
1
1
040807/0303
104.0
2.334
0.3628
1
2
1
040807/0303
141.0
2.187
0.3419
1
10
1
040807/0304
139.0
0.2583
0.1224
1.259
1.119
1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2
Mass (GeV/c2)
Figure 61: Invariant mass of pp for IS (top) and 4S (bottom) running. For the 4S
running the inset is consistent with the events having a proton and an antiproton.
82
File: Generated internally
ID
IDB
Symb
Date/Time
Area
Mean
R.M.S.
9
3
32
040807/1446
85.43
2.521
0.3093
98765
981
2
040807/1446
72.17
2.685
0.3291
98765
982
1
040807/1446
55.91
2.813
0.2553
Mass (GeV/c2)
Figure 62: Invariant mass of pp. The plot is continuum subtracted and efficiency
corrected. An overlay with the 90% confidence level upper limit for /j(2220)
is shown. The mass and width are taken to be 771^(2220) â€” 2.234 GeV/c2 and
T/j(2220) = 17 MeV/c2 as in [17].
83
RÃe: Amp_mnt/bohr/user1/luis/deo3/mn Jit/u1 sgammahh/temp.tmp
ID
IDB
Symb
Date/Time
Area
Mean
R.M.S.
1
29
1
040807/1446
1.000
12.78
5.430
1
25
1
040807/1446
0.9110
11.80
4.590
S3
O
3
r2
H
43
XA
'S
C3
2
0,
0.060
0.040
0.020
0.000
Area
Figure 63: Normalized probability distribution for different /j(2220)
areas. The shaded area spans 90% of the probability.
pp signal
84
MINUIT x2 Fit to Plot 9&22
U1S>Gamma h h. ABS(COS(PHOTHETA)) axis
File: Generated internally
Plot Area Total/Fit 256.69/256.69
Func Area Total/Fit 134.57 /134.57
X2= 19.5 for 20  2 d.o.f.,
Errors Parabolic
Function 1:J1
AREA 67.287 Â± 10.91
0 1.20611E03 Â± 0.4148
2AUG2004 18:12
Fit Status 3
E.D.M. 1.845E05
C.L.= 36.4%
Minos
10.93 + 10.88
O.OOOOE+OO + 0.3544
1.0 0.5 0.0 0.5 1.0
IcosG^j cos6n
Figure 64: pp angular distribution for the excess of events in the mass range
2 GeV/c2< m(pp) < 3 GeV/c2.
(lVÂ°oP ro )/NP
85
Figure 65: Ja probability distribution for the excess of events in the mass range
2 GeV/c2< rn(pp) < 3 GeV/c2 when only the hypotheses Ja â€” 0, 1, 2, 3, 4 are
considered.
CHAPTER 7
SYSTEMATIC UNCERTAINTIES
Systematic uncertainties are any sources of experimental uncertainty other
than the statistical ones. Limits on the accuracy of our detector simulation and any
physical processes that interferes with the experimental measurement are typical
examples of systematic uncertainties. These uncertainties need to be identified,
quantified, and if possible, corrected.
7.1 Cuts
From now on we report individual cut efficiencies relative to the events that
survive all the other cuts.
According to our MC, most of the skim cuts, except for the hardGam reÂ¬
quirement, axe nearly 100% efficient. Therefore, such cuts should not be a source
of systematic uncertainty. The only skim cut worth taking a closer look at is the
hardGam cut, which is about 93% efficient in MC. Measuring this efficiency in our
data is not possible because events not classified as hardGam are not in the data
to begin with. Closer examination of our MC reveals that that most of the 7%
inefficiency in hardGam comes from the eOverPl cut (4%), some from the Sh2 cut
(2%), and the rest (1%) from the other cuts present in hardGam. We can measure
the efficiency of eOverPl in data by looking at the eOverP2 distribution of data
tracks from p and
Tables 71 and 72), and we wonâ€™t measure how well MC models the rest of the
hardGam cuts, which are 97% efficient in MC.
86
87
To quantify the quality in the MC modeling of our analysis cuts in the 7r
and K modes we use the p and 0 signals present in our data1 . In this case the
process is straightforward. We measure the p and
over a floating background function with all cuts in place, and with all in place
cuts except the one under consideration. Prom these numbers we calculate the
effective efficiency of the cut. The differences between data and MC are taken as
the systematic errors which are added in quadrature. Results axe shown in Tables.
71 72. The 4rnomentum cut does not appear because its efficiency is close to
100%.
Cuts
MC eff.
Data eff.
Systematic Error
eOverP2
96.2
96.6
0.4
QED e+e~ â€”â–º ye+eâ€ suppresion
93.1
94.2
1.2
QED pPpT â€”> 7p+p~ suppresion
75.3
77.7
3.2
Hadron separation
96.9
96.8
0.1
Overall analysis cut systematic error
Â±3
MC, efficiencies from
Table 71: Efficiencies for T(15) â€”> 'yn+n~ for flat signa
data (p), and the derived systematic error in %. Efficiencies are reported as the
number of signal events after all cuts divided by the number of signal events with
all cuts except the one under consideration. Statistical! errors axe 0.1% or less.
For the proton case we donâ€™t have a clean sample with high statistics of
e+e_ â€”> 7pp events in data. By extension we take the systematic error in this mode
to be 10%.
7.1.1 Justification of the DPTHMU Cut
The reason we prefer to use DPTHMU < 5, instead of the more traditional
(see [18]) DPTHMU < 3 used in CLEO II, is that, for some unknown reason, our
CLEO III MC has too many 7r tracks with 3 < DPTHMU < 5.
1 A study using Kg or A signals from hadronic environments would have larger
statistics, but is problematic because the large number of tracks and showers artifiÂ¬
cially decrease the cut efficiency. See [28] Appendix A.3 for an example.
88
Cuts
MC Eff.
Data Eff.
Systematic Error
eOverP2
98.5
100
1.5
QED e+e~ â€”â–º qe+e suppresion
98.1
99.3
1.3
QED p+p~ â€”> 7P+/r suppresion
93.5
96.4
3.1
Hadron separation
88.4
82.2
7.0
Overall analysis cut systematic error
Â±8
Table 72: Efficiencies for T(lS') â€”â– > 7K+K~ for flat signal MC, efficiencies from
data (
number of signal events after all cuts divided by the number of signal events with
all cuts except the one under consideration. Statistical! errors are 0.1% or less.
To observe this fact we first select a relatively clean sample of pion tracks
by requiring the event to have a n+n~ invariant mass consistent with the p mass,
and to pass all our analysis cuts except for the cut on D PTH MU on one track.
For such events we plot the rate as a function of momentum, at which the 7r track
whose DPTHMU cut we released has D PTH MU > 3 and DPT H MU > 5. To
increase our statistics we do this procedure twice, once for each track, and average
the fake rate. Figure 71 shows the results. Clearly the CLEO III MC we are using
has some problem modeling the DPTHMU < 3 cut.
To keep the systematic error low we choose a cut at DPTHMU < 5
(.DPTHMU < 3 gives a systematic error of about 20%). This does not change the
efficiency in data very much, but it increases the efficiency rQEfted by MC, bringÂ¬
ing it closer to reality. The increase in p fakes after loosening the cut is estimated
to be low using QED MC (see Figure 72).
7.2 Angular Distribution of Signal
The photon and tracks from the process T(IS') â€”â–º 7X with X â€”â–º h+h~ have
a different angular distribution than that of flat MC. Examples of possible angular
distributions are shown in the appendix.
89
File: â€¢/bohr/user1/luis/deo3/ntp/data_50.ntp
ID
IDB
Symb
Date/Time
Area
Mean
R.M.S.
1
4
31
040525/1408
0.2350
2.758
0.8547
1
3
31
040525/1408
0.4317
2.684
0.8088
1
6
31
040525/1408
9.6769E02
2.996
0.7400
1
5
31
040525/1408
6.0516E02
2.792
0.6995
0 1.6 3.2
Track Momenta (GeV)
Figure 71: Pion faking muon fake rates for a cut on D PTH MU < 3 (top) and
D PTH MU < 5 (bottom). MC is shown as solid circles while p from data is shown
as hollow circles. Fake rates are reported relative to all events that pass all cuts,
except for the D PTH MU cut for one of the tracks.
Figure 73 shows the efficiency in flat MC as a function of cos 61 and
cos dh+'1 . Note that the K and p modes are nearly insensitive to the track anÂ¬
gular distribution, while the tt mode is more sensitive to 6h+. This happens because
of the stronger muon rejection cut in the 7r mode.
We measure the systematic effects of flat MC efficiency by convoluting each
plot in Figure 73 with different possible angular distributions calculated in the
2 6L and 6h+ are the helicity angles of the sequential decay, defined in the apÂ¬
pendix (see Figure 2).
90
Rle:'/bohr/user1/luis/deo3/ntp/mcpass2_mumu_gamma_new.ntp
ID IDB Symb Date/Time Area Mean R.M.S.
1 3 31 040518/1402 3.676 2.361 0.9088
1 30 32 040518/1402 3.022 2.549 0.8590
ID
4â€”*
c3
cs
pH
Track Momenta (GeV)
Figure 72: Muon faking pion fake rate (in %) for D PTH MU < 3 (solid squares)
and D PTH MU < 5 (solid circles). There is a complete overlap in the last bin.
appendix. Tables 7374 show the necessary correction factors relative to flat MC
efficiency for decays with definite 7 and X helicities A7 and A^, due to the nonflat
photon and hadron distributions. We call these factors eAxAx (7) and ey)Ax (h)
respectively.
The efficiency of a decay with definite A7, A^ can be obtained using the flat
MC efficiency corrected by a factor eA*Ax â€” â‚¬x^xx (7) x eAxAx(h). In general, the
final state is a mixture of all possible A7; \x pairs, and the efficiency correction
factor is,
eJx = cos2 0 cos2 Te/o + sin2 0qf + cos2 0 sin2 .
(7.1)
91
o
tÃ
o
o
Ui
ttÃ
>>
o
tÃ
O
*5
E
tu
cos0y
COS0JJ
Figure 73: Flat MC efficiency as a function of cosO7 (left column) and as a funcÂ¬
tion of cos6h+ (right column). Top row corresponds to the pion mode, middle row
to the kaon mode and bottom row to the proton mode.
The fits in Figures 48 and 410 and 5â€”65â€”8 measure the pair (0, ). These
values are summarized in Table 75, where they are used to obtain eJx for each
mode.
The pair (0.4>) carry an error which is a source of systematic uncertainty.
We calculate this systematic uncertainty by inspecting the differences in efficiency
when (0.4>) move away from the value which gives the minimum chisquared, xLn>
92
Mode
aio or ai2
0 = 0
II Â§
S5=i
7T
0.910
1.18
K
0.925
1.15
P
0.922
1.16
Table 73: Â£\x\x (7), eÃÃ±cienciy correction with respect to flat MC factors due to
the nonflat angular distribution of the photon in Y(15) â€”â–º 7X for different X spin
values (Jx) and 7, X helicities (A7, Ax)
Mode
Jx
Qio
0 = 0, $ = 0
an
0 = i
w 2
G12
0 = 0, $ = f
0
1.00


1
0.785
1.11

X â€”> 7T+7r_
2
0.829
0.934
1.15
3
0.898
0.853
1.03
4
0.899
0.863
0.922
0
1.00


1
0.944
1.03

X > K+K
2
0.950
0.988
1.04
3
0.957
0.971
1.01
4
0.965
0.962
0.992
Table 74: ex*Xx(h), eÃÃ±cienciy correction factors with respect to nonflat MC due
to the track angular distribution in T(lS') â€”> 7X, with X â€”> h+h~ for different X
spin values (Jx) and 7, X helicities (A7, A*).
under the condition x2 < Xmin + 1 3 â€¢ For Jx > 1 both (Â©, $) are free to move,
defining a surface in the (0,
the /2(1270), /4(2050), and /Ã¡(1525).
At this point we can check whether (0, ) depend on the mass of the deÂ¬
cay. We split the /2(1270) and the /2(1525) into a high mass and a low mass
region. The plots of the error surfaces of the measured (0, ) show no significant
separation for the different mass regions (see Figures 75 and 76).
3 In a two dimensional linear problem such a set of points defines the surface of
the standard error ellipse.
93
Results for the correction factor and its systematic error axe shown in Table 7
5.
Upper limits on Y(1S) â€”* 7/2(2220), ^(2220) â€”> h+h~ are changed to
include the angular distributionâ€™s effect on efficiency. Since we canâ€™t measure
the helicity amplitudes in this case, we choose the worst possible case where the
corrected efficiency is lowest. This always corresponds to 0 = $ = 0. Results are in
Table 76.
0 (radians)
Figure 74: Surfaces in the (Â©,
tainty in the efficiency correction factor for the.modes with /2(1270), /i(2050),
and /2(1525). The /i(2050) surface may seem large, but when drawn in spherical
coordinates it is a small â€œnorth pole capâ€.
(radians)
94
Figure 75: Measurer. (0, $) surfaces for the /2(1270) high and low mass regions.
(radians)
95
Figure 76: Measurer (0, ) surfaces for the f2(1525) high and low mass regions.
Table 75: Measured (0, 4>), calculated eJx, efficiency correction factor interval when (0, 4>) move away from their measured
value under the condition that \2 stay within one unit of its minimum value, and the systematic uncertainty on the correcÂ¬
tion factor.
Mode
Jx
0
â‚¬Jx
Interval
Sys. Err. (%)
7/o(980)
7/0 (980)
7/2(1270)
7/4(2050)
7/^(1525)
7/o(1710)
ryK+K~, 2 GeV/c2 < m(K+K") < 3 GeV/c2
7pp, 2 GeV/c2 < m(pp) < 3 GeV/c2
0
1
2
4
2
0
1
1
0
o.oÂ±8;g
O.OO/oioo
1 C7+0.00
10'0.34
0.00/Â°;2'
0
016Â±8;ii
900/o!oo
0
0
0.30/â€; 10
0.2 Â±0.7
OSOÃaÃe
0
0
0
0.91
0.71
0.78
1.02
0.90
0.93
0.88
0.92
0.91  0.91
0.71  0.80
0.76  0.80
1.00  1.02
0.89  0.91
0.930.93
0.87  0.91
0.92  0.95
0
+ 13
Â±3
2
Â±1
0
+3
1
+3
Table 76: Worst .case efficiency correction factors and their effect on upper limits of Y(l5) â€”Â» 7/j(2220), fj(2220) â€”Â» h+h
decays.
Mode
Worst case correction
Upper limit increase (%)
fj{2220) > 7T+7T
0.75
+33
fj(2220)  K+K
0.88
+14
/H2220)  pp
0.92
+9
97
7.3 Different Hadronic Fake Rates Between IS and 4S
Differences in the fake rates of hadronic events between the 15 and 45
data are another potential source of systematic errors when doing a continuum
subtraction.
In principle this is a second order effect since the contamination is proportional
to the difference of two small numbers. However since we have such a large number
of p and 4> events in the underlying IS continuum, we should quantify any possible
signal contamination. Section 3.2.2 outlines the three cases we need to worry
about; p events contaminating the K+K~ invariant mass plot, p contaminating the
pp invariant mass plot, and 0 events contaminating the pp invariant mass plot.
It is reasonable to assume that the IS and 4S fake rates stay within 50%
of each other (see Section 3.3.2 for some examples from MC measurements).
With this assumption, taking np Â« 20000 and n$ ~ 2000 in the IS data, and
the fake rates measured in data (see Table 36) we can calculate the systematic
uncertainties. We expect Â±27 events from p to contaminate the K+K~ invariant
mass plot. However, these events can be ignored as source of systemtic uncertnty
because 90% of them fall in the 1 â€” 1.5 GeV/c2 region, below the 7^(1525) peak.
We also expect Â±7 from p and Â±3 events from 0 contaminating the pp invariant
mass plot. This represents a Â±8% systematic error for the T(15) â€”> 'ypp mode.
7.4 Other Systematic Sources
Besides the systematic uncertainties described above, we also add a 2%
systematic effect from track finding (1% per track) and 5% from the number of
T(15).
We find no evidence for nonresonant hadron pairs that could interfere with
the resonances. There is a possible interference between /2(1275) and /2(1525)
because they both are in the J = 2 state. From the ^(1275) measurement in the
7r+7r_ mode we expect 3 events from this source in the /2(1525) mass region 1.45 â€”
98
1.6 GeV/c2. In the most extreme cases we have, N oc 7ly'(1525)2 + I IAf2(i270) 112 ^
2^4(1525) 1111^4/2(1270) 11> where Ay'i,1525j and Af^mo) are the amplitudes associated
with /2(1525) and /2(1270) respectively. From the data we have N Â« 300, and
using the fact that 1/2(i270) 112 contributes with 3 events, the contribution from
P/; '(1525) 112 can range from 400 to 200 events. This represents a 30% variation from
the central value. Since this is the extreme interference case, we take half of this
value, 15%, as a reasonable systematic uncertainty from possible interference in the
Â¿(1525).
The possible decay T(IS') â€”> pir could contaminate our 7r+ 7r~ invariant mass
plot, since a 7rÂ° with a momentum greater than 4 GeV/c2 looks very much like a
photon. Using phase space MG and isospin symmetry, we estimate that about 40%
of possible Y(IS') â€”> pit have such a 7TÂ°. The latest measured upper limit [24]
for this decay mode is 1.9 x 10 "6. This gives an upper limit of 16 events in the
efficiency corrected ir+ n~ invariant mass plot, of which 10 events fall in the p mass
region, behind the /o(980) mass region. The remaining potential 6 events could
contaminate the rest of the invariant mass region. None of the signal regions is
affected by more than 1%.
To test the robustness of the statistical fits, we redo them with all masses and
widths of possible resonances required to stay within 1
values [16]. The results from fitting the data in this way are within the statistical
errors when compared to the fit results summarized in Table 41 and 51. We do
not observe a systematic difference between these two fitting techniques.
The branching ratios for /2(1270) â€”> 7r+7r~, /4(2050) â€”â–º 7T+7T, and /Ã¡(1525) â€”>
K+K~ are taken from the PDG [16] and contribute to the systematic uncertainty
with jj, Â±9, and Â±3 percent respectively.
99
7.5 Overall Systematic Uncertainties
We combine the systematic errors of each section in quadrature. Tables 77
78 summarize the systematic errors.
Table 77: Systematic errors expressed as % from the different sources described
in the text. Not shown in the table, but included in the total, are a systematic
uncertainty of 5% from the number of Y(1S), a 2% systematic uncertanty from
MC tracking in all the modes, a 15% systematic uncertainty in the 7/2(1525) from
possible interference with 7/2 (1270), an 8% systematic uncertainty in the 7pp
mode from possible hadronic contamination, and the systematic uncertanties in the
/2(1270), /4(2050), and ^(1525) hadronic branching fractions.
Mode
Analysis
Cuts
Angular
Distribution
Total
7/o(980)
Â±3
0
Â±6
7/2(1270)
Â±3
Â±3
Â±7
7/4(2050)
Â±3
+2
Â±7
7/5(1525)
Â±8
Â±1
Â±20
7/o(1710)
Â±8
0
Â±10
7if+A'(23 GeV/c2)
Â±8
+1
3
Â±10
7pp(2 â€” 3 GeV/c2)
Â±10
3
Â±14
Table 78: Increase in the upper limits of T(lS') â€”â–º 7/j(2220) for different fj(2220)
decay modes due to the possible systematic effects described in the text added in
quadrature. Not shown in the table, but included in the total, are a 5% contribuÂ¬
tion from the number of Y(IS') and of 2% contribution from MC tracking.
Mode
Analysis
Cuts
Angular
Distribution
Total
fj(2220) + 7T+7T
Â±3%
Â±33%
Â±34%
/j(2220) â€”> K+K~
Â±8%
Â±14%
Â±17%
fj(2220)  pp
Â±10%
Â±9%
Â±14%
CHAPTER 8
RESULTS AND CONCLUSION
We report on a new search for twobody radiative T(1S) decays. We place
stringent limits on the production of the /j(2220) particle in the pion and kaon
modes, and a less stringent limit in the proton mode where some excess of events is
observed in the region of interest.
In the decay channel T(IS') â€”â–º 77r+7r_ we find clear evidence for the resonance
/2(1270) and measure a branching fraction of (13.3 Â± 1.4) x 10~5, which is
consistent with the earlier CLEO measurement of (7.4/^) x 1(U5 [18]. The angular
distributions of the photon and tracks strongly indicate that the hadron pairs we
assign to the /2(1270) are indeed in a J = 2 state, and that nature prefers to
produce the /2(1270) with 0 helicity. In contrast, for J/ij) â€”> 7/2(1270) it was
found [29] that the /2(1270) is produced at equal rates with both helicity 0 and
helicity 1, and at a rate consistent with 0 for helicity 2, but more recently [30]
measured helicity 0 dominance for this same mode.
There is a barely significant (4.3a) excess of events in the /o(980) â€”> 7r+7r~
invariant mass region. The angular distributions of the photon and tracks indicate
that this excess of events is in a J = 1 state, rather than J = 0 which would be the
case if the excess were due to /o(980) decays. It may be that this excess is due to
nonresonant â€¢y7c+n~ production. We conclude that more data is needed to resolve
this situation.
We find weak evidence for the production of the resonance /4(2050) in the
T(1S) 4 77T+7r~ decay channel.
In the decay channel T(15l) â€”> 7K+K~ we find strong evidence for the
production of the resonance /[(1525) and weak evidence for the production of
100
101
/o(1710). The photon and track angular distributions show that the two tracks
attributed to the (1525) are indeed in a J = 2 state and that the /2(1525) is
produced mostly with helicity 0.
There is also evidence of an excess of events in the 2 â€” 3 GeV/c2 K+K~
invariant mass region, which we cannot attribute to any known resonances.
Finally, we also find evidence of an excess of events in the channel Y(IS') â€”>
7pp, which we cannot attribute to any known resonances. We find no evidence
of an enhancement near pp threshold. Such an enhancement was recently reÂ¬
ported by [31] in the J/t/j system, and is currently being interpreted (See
e.g. [32], [33], [34]).
When comparing the radiative T(1S) decays studied in this analysis to raÂ¬
diative J/i/> decays we observe a suppression ratio of 0.09 that is in reasonable
agreement with with the naive expectation of 0.04 obtained from scaling arguÂ¬
ments.
Table 81 summarizes our results.
While we find hadronic resonances at a level consistent with the estimate
of Section 1.2, we do not observe any glueball states. This is not the result we
would expect from a naive interpretation of QCD. as the two gluons from a T(1S)
radiative decay can form a bound state directly and need to go through at least
two strong interaction vertexes to form a hadron.
Internal consistency in QCD predicts the existence of glueballs. Even though
theoretical calculations of the glueball mass spectrum exist, a clear glueball
observation has not been made yet. We conclude that more glueball searches, such
as those proposed in the GlueX project, are necessary to clarify this situation.
Table 81: Final measured branching ratios, measured branching ratios reported relative to J/tp branching ratios, and staÂ¬
tistical significance for each decay channel. The measured branching ratios have been corrected by the factors calculated in
Appendix A2. For the branching ratios the first uncertainty is statistical and the second is systematic.
Channel
Branching Fraction x(10 5)
Y(1 S)
J/+
Significance
7/o(980), /0(980) >
7T+7T
i8io:?Â±o.i

4.3(7
7/2(1270)
13.3 Â±1.0 Â±0.9
0.09 Â± 0.01
> 14 a
7/4(2050)
3.5 Â± 1.3 Â±0.2
0.019 Â± 0.006
2.6cr
7/2(1525)
4.3/Â¿ g Â± 0.9
0.09ÃÂ£$
> 14(7
7/o(1710), /o(1710)
K+K~
0.38 Â±0.16 Â±0.04
0.007/^005
3.2a
1k+k~
1.14 Â± 0.14 Â±0.11

9.1(7
7PP
0.41 Â±0.08 Â±0.10
0.011 Â±0.005
4.8(7
7//(2220), fj(2220) 
Â» 7r+7T
< 0.08


7/j(2220), fj(2220) >
K+K
< 0.06


7//(2220), fj(2220)
PP
< 0.11


APPENDIX
HELICITY FORMALISM FOR TWO BODY DECAYS
Helicity formalism can be used to obtain the angular distribution of a decay
process. In this appendix we restrict ourselves to the case of two body decays.
The situation is as follows: particle A with spin J4, and zaxis spin projection
decays to two particles, A â€”â™¦ B + C with definite helicity A# and Ac. Let the
direction of the of the decay be n(6, ) = n(0s, 4>b) = Ps â€” â€”Pc (see Figure 1).
The amplitude for this decay is,
AZ% = < 0^b\c\H\JaMa > . (1)
We have little chance of directly calculating 1. But we can exploit the fact that
H conserves angular momentum by considering the basis {< jmAsAcI}, where j
is the total angular momentum of B + C and in its projection along the z axis1 .
Inserting this basis into Equation 1,
c ~ < ^â€˜Mb'Mj^iAbAc >< jm\B\c\H\ JaMa >
jm
â€” X/ < ^^bAcIj'^AbAc > 5jjASmMAA\Bxc
jm
= < 9(})\b\c\JaM\\b\c >
1 The rotational invariance of helicities is a necessary condition to construct
{< jmXB\c\} as a basis for the twoparticle state B + C. Rotational invariance
is one of two key ideas in helicity formalism. The other key idea, which is boost
invariance, is>useful in sequential decays.
103
104
The above equation contains two factorized terms. The first term is indepenÂ¬
dent of the interaction H and contains the angular distribution information we are
interested in. The second term. A\g\c. is usually called the â€œhelicity coupling amÂ¬
plitudeâ€ . It is independent of the angular distribution, and contains the physics of
the decay (in particular it depends on Ag and Ac, but not on Ma). In the helicity
formalism, the helicity coupling amplitudes are unknown parameters.
For strong and electromagnetic decays there is another invariance of H we
can exploit; parity. For these kind of interactions it can be shown by inserting the
parity operator that,
^AsAc = (~ 1) â€œ Jc nAr]B r]CAxB \c (2)
where tia^bVc is the product of the parity of particles A, B and C. This useful
property of A\B\c reduces the number of unknown parameters in the angular
distribution of strong and electromagnetic decays.
The next step is to derive explicit formulae for < 9(Â¡)XbXc\JaMaAbAc > for
the B + C system. Such derivation, which is out of the scope of this appendix, will
a. J 
105
be omitted here2 . The result can be expressed in terms of the Wigner functions,
< 64,\b\c\JaMa\b\c >= ÃMt) (3)
where X = Xb â€” Xc and,
Â». 4>) = W
The functions dJ^A x(0) have real values, and some of them are given in the particle
data book.
We have finally arrived to the following important relation,
All helicity distribution calculations are reduced to handling these basic ampliÂ¬
tudes. Here are some of their very important properties,
4^=0 if Ab  Ac > Ja,
(6)
A'"v 11=11 <%
(7)
/â– 27T
Jo
dd>AMA A*Ma â€” 2ttS('Ma~M^AMa A*Ma
a(P'A\BXc'Axâ€™x'  Z7r0(XX') ^XbXc^x'x'
B^Câ€™
(8)
/
dnAâ€œ\â€žXf?, = ^ H!
XbXc^x' x'c (AA')
(9)
2 The interested reader can find this derivation in [35]
Property 6 is a reflection of the physical fact that the projection of the total
angular momentum along a particular axis cannot be greater than the total angular
momentum itself.
 A^Xr 2 is the angular distribution of a final state where MA and the
helicities of B and C are known (see Equation 1). However, it is usually the case
that Ma^b^c are not known. In such cases we can only measure the overall
angular distribution, which is simply the average over possible initial states of the
incoherent sum of final states,
dP_
dÃœB
Ma\b^c
d
dCln
E Ãvj ii2
Ma^b^c
(10)
where Pma is the weight of the initial state Ma and dBts â€” d(cos(0B))d
Next, consider a sequential decay, A â€”> 13 4 C with C * D + E (see Figure
2). The rotational and boost invariance of helicities makes it easy to extend the
helicity formalism to this case. To do so we consider a new reference frame, Oâ€™,
at rest with C, such that z' â€” n{6B, 4>b)Z â– Ail particle helicities remain the same
in Oâ€™, and in particular MZ{C)  Ac (the negative sign arises from the fact
that n(#B, (j)B) is the direction in which B is traveling). It is now straightforward
to calculate the intermediate amplitude at Oâ€™. Of course is still
calculated at A's original rest frame. Since Ac is an intermediate state, the helicity
amplitude for the final state Xb\d^e is,
ACMa
EaMa nâ€”\C
Ãa
(11)
3 This condition does not mxlikely determine Oâ€™. We can be more precise by
defining O' â€” L(pc)R(B, #b> 0)0, where L is the Lorenz Boost operator, R is the
rotation operator in'terms of the Euler angles, and O is Aâ€™s original rest frame.
107
Figure 2: A â€”> B +C and C â–º D + E. The drawing is peculiar in that Cs daughÂ¬
ters are drawn as seen in Câ€™s rest frame (backtoback).
where we have coherently summed over the intermediate state C. If Ma^b^d^e
are not known.
dP
E
Ma^b^d^b
P
Mi iiladik,
= E II
MaXbXd^e
(12)
The sequential decay amplitudes have the following important properties,
ac~Ma
XbXdXe
1=11
(13)
p2n r2n
/ #Â«/ #D^VJ2=4*2Ell
Jo Jo >c
This kind of logic can be extended to a general twobody decay chain.
Example: T(1 S) > 7/2(1270)(/2(1270) > tt+ttâ€™)
From our previous discussion we can immediately write (see Equation 11),
^Afr(is) _ V"' ^Atr(is) r~xM1270)
^r\r + \â€” ZJ \+\
A/a(1270)
108
In this case, A* = A^+ = 0, A7 = Â±1. In CLEO Mx(is) = Â±1, there is no
Mr (is) â€” 0 since at the T(l,9) energy electrons couple only to positrons of the
opposite helicity, this way â€” P_i = ~ and Pq â€” 0 is an excellent approximation..
There are therefore four amplitudes we must consider, AClm. AC^, AC l 100 and
4Â£ioo
The overall angular distribution is (see Equation 12),
dP = \ II AC
dÂ£l1dÂ£llt+
loo IP +7
2 II
n i
I2 +2 II 4Â£ioo II2 +2 II ACtfjQ Â¡I"
(15)
=11 ^Cloo Â¡Â¡2 +  AC1
100
where we have used property 13. The two relevant amplitudes are,
a:1
100
â€” ^io^oo T A\iCfu( Al^Cm
n''oo
l12'~'00
*^ioo ~ ^io^oo + â€œ^Aii^oo + v41_1_2CÂ§0
where we have used the fact that A7 â€” A/2(i270) I < Jr(iS) = 1 to sum over all
possible A/2(i270)
All that remains is to substitute,
^loo ll2~â€˜ II ^Ão^oo I!2 T ll ^n^oo1 II" + II A\2C0q Â¡1â€œ +
{^â€¢Ão^oo^lÃboo 1 + ^10^00^12^00 2 + Ail^oo1 ^12^00 + cc}
II AC1 II2
II HUioo II "
A] C
0
10''00
+ II AlA
00
+ l AL^Cqo Â¡i? +
{410^00â€¢'^ll^'OO + ^lO^OO*^1! 2^'00 + ^â– 11^'00â– ^12^'00 + ccâ€¢}
(16)
into Equation 15.
Doing so will give us the full angular distribution of the decay chain. However,
using property 8, all the cross terms inside {...} vanish if we integrate over (pn+ or
7. We will return to the
109
47T2[( A\0 !2 + II ALW f) II cÂ°0 II2 +
(IMli II2+ Â¡1 4111II2) II Cll2 +
(ÃI 412 l2 + II Â«4Ã.12 l2) I! ^00 lH
= fill Al0 2 Cm II2 (dln(ej2 + Â¿Ã_1(Â«1)2)dSo(Â«,+)2 +
II An 2 Coo 2 (Â¿Â¡â€ž((
II a12 2 Coo II2 (Â¿U(Â«,)2 + <Â¡}_1(ft,)2)4)(Mil]
= ~ II Coo II2 [II Aio II2 i(l + cos2 #.,)(? cos2 0Â»+  l)2 +
II An II2 sin2 sin 0â€ž+ cos0x+)2 +
II Ai2 !2 i(l + cos2 0,)( sin2Â«Â»+)].
At this point it is convenient to define the normalized helicity amplitudes,
Aio
â€œI0 = v4l ATolF+ifln II2 + II Alo II2
__ 4n
â€œâ€œ " '7prFTri7p+ii a12 ip
4x2
â€œ'2 â€œ /liATo II2 + II An II2 + II A ,2 II2
terms inside {...} later.
[2* Â¡in dp
which satisfy,
Â¡I aio 2 + II an 2 + Â¡I ai2 2= 1. (17)
The normalized probability distribution, Â¿^g^cose +). can be obtained by
dividing the probability distribution by the branching fraction of the sequential
decay,
110
P = 2  Cqo 2 (II A 10 2 +  An 2 +  Ayi 2)
= II Coo 2 A10 2 + II Coo 2 An 2 +  Coo 2 A\i Â¡2 +
II Coo 2 <4io II2 +  Coo 2 Ann 2 +  Cqo 2l A_i_2 II"
This way,
dU
d(cos07)d(cos07r+)
aio 2 (1 + cos207)(3cos29t+  l)2 +
O O
an [2 7 sin2 61^(sm0n+ cos^t)2 +
aw 2 ^(1 + eos2 07)~sin40,r+
It is useful to project the angular distribution on 07 and 0V+,
dU~ = aw II2 ^(3cos207r+â€”1)2+  an 2 ^(sin^ cos07r+)2+  an 2 ^sin40ff+
d cos 9.
8
16
(18)
JcosFâ€ž = ^ a"â€˜ ^ + H â€œ1! (+'1 + cos2 II 011 II2 i S1IIâ€˜ F (19)
Let us now return to the terms inside {...} in Equationl6. It turns out that for
these terms a substantial simplification occurs if we integrate over cos 9K+. After
some algebra, we arrive at the following expression,
dfa  (t>n+) = 2^(1 â€œ 7{aioa*2 + aiÂ°Gl2} coÂ®[2(^7 ~ (2Â°)
Example: T(L9) Â» 7/4(2050)(/4(2050) + tt+tT)
For the case /4(2050) the calculation is very similar to the /Â¿(1270) case
discussed above. The allowed helicities for /4(2050) are the same as the allowed
helicities for /2(1270). This makes the angular distribution calculation very similar
I
111
to the /2(1270) case, the only difference being that the functions ^a/220270)o are
replaced by di/4(a050,o '
The photonâ€™s angular distribution comes out to be the same. The result for
the pionâ€™s normalized angular distribution is.
=  aio 2 (35 cos4 0*+  30 cos2 6n+ + 3)2 +
45
II an IP 'rx(sin07T+ cos07r+)2(7cos20n+ â€” 3)2 f
45
II 0,12 II2 rj sin4 0w+(7 cos2 0*+  l)2.
04
(21)
Fit of the Helicity Angular Distribution Obtained from Data
In practice, a particular bin in the helicity angular distribution obtained from
data has bins with dN â€” NdU events, where N is the total number of events
(which is allowed to float during the fit), and dU is the portion of the normalized
probability assigned to the bin by the helicity formalism prediction.
To incorporate Equation 17 directly into the fit we make the following change
of variables,
it flio â€” cos0cos$;  ai2 jâ€” cosÂ©sinÃÂ»;  an = sinÂ©. (22)
Because of this definition, both 0 and i> should be required to be within the
interval (0, ) during the fit. The new variables have the convenient property that
for J 0, i> = 0 â€” 0, and for J â€” 1, Tâ€”0. In general there are 1, 2 and 3 degrees
of freedom for the fits with J â€” 0, J â€” 1, and J > 1 respectively. One degree
of freedom always corresponds to the number of events, N. and the other two
112
correspond to 0 and T, which determine the magnitude of the normalized helicity
amplitudes through Equation 22 4 .
When fitting, it is preferable to choose the full 2D helicity angular distribution
which is a function of 0f and 6W+ instead of its one dimensional projections. The
2D choice is better not only because the 2D distribution carries the maximum
amount of information, but because in general the efficiency depends on both the
pion and photon helicity angles, which can lead to complications when efficiency
correcting the onedimensional projections. Building on the two previous examples
the 2D helicity angular distributions for J = 0, 1, 2, 3, 4 are (for compactness we
define x = cos 6n+ and y â€” cos 07),
dNJ=o
dxdy
= Wj=o(l + y2)
(23)
dNJ=1
dxdy
= NJ=i(cos2 0 5(1 + y2)^x2 +
sin2 0 5(1y2)5(lx2))
(24)
Obtaining the error of the normalized helicity amplitudes requires some matrix
Ã flio
multiplication. For compactness let us define a â€” O12 ) and u
\ant
matrix for the normalized helicity amplitudes is,
'0N
. The error
< 5'dSa1 >â€” < 8Ã¼SÃ¼j >
Ouj
:..T ^
da/
dCo
where < 8Â¿oÃ³ujr > is the error matrix obtained from the fit.
113
*
= NJ=2( cos2 0cos2 $ ~(1 1 y2)(3x2  1)2+
dxdy 8 8
sin20 "(1  y2)â€”i2(l  x2)+ (25)
cos2(Wsf(l+y2))f(l x2)2)
8 ' lo
Nj=3(cos2 0cos2 $ ?(1 + y2)^(5x3  3x)2+
sin2 0 5(i _ y2)â€˜^{ 1  x2)(5x2  1)2+ (26)
cos2 0sin2 $ (1 + y2)^.T2(l  x2)2)
8 lo
= NJt=4( cos2 Â© cos2 5(1 + r)7^(35^  30x2 + 3)2+
axa?/ 8 Lz8
sin2 Â© 5(i _ j/2)~x2( 1  x2)(7x2  3)2+ (27)
t: O Z
cos2 Â©sin2 5(1 + y2)^5(i _ x2)2(7x2 â€” l)2).
In this analysis, because of our low statistics (and by low statistics we mean
that there axe numerous bins with < 1 events in the 2D plots), we project EquaÂ¬
tions 2327 on x = cos 6^, and y = cos 07. We also fold each distribution around its
symmetry axis x = 0 and y = 0 to increase each bin's statistics. Finally, we do a
simultaneous fit to both distributions and obtain N, 0, and $.
Admittedly, this procedure has the problems described previously when 2D fits
where advocated, which can give rise to systematic effects in the measured helicity
amplitudes. In essence, we need to use a unitary weight function proportional
to the efficiency distribution (see Figure 73) in the integrals used to make the
projections. This affects the relative strength of each helicity amplitude and the
overall efficiency, but does not change each projected shape. However, the helicity
amplitudes themselves are only used to calculate systematic effects on the efficiency
dNj^
dxdy
114
of resonances, so by simultaneously fitting the unweighted ID projections we are
just ignoring the systematics on the systematics.
Finally, let us consider the 7 â€” 4>n+ distribution for J = 2. Equation 20
becomes,
jrjâ€”â€”r = ~~~(1 Â¡={cos2 0 sin 24Â» cos A} cos[2(d>7 â€” (f>n+)]) (28)
d\(p7 *p7T^') V6
where the new degree of freedom A is the relative phase between aio and 0,20 The
deviation from the flat distribution is characterized by R â€” T?{cos2 0 sin 2T cos A} <
The bigger this factor is, the easier it is to observe in data. Figure 3 shows the
continuum subtracted â€” â€žâ– + distribution for events near the /,2(1270) resonance.
The factor R obtained from fitting the distribution is nonsignificant. It is a bit
tedious but straightforward to calculate the 7 â€” (j)n+ distribution for other J values.
We will not present those results here. They are not very interesting because we
need large statistics to observe deviations from the flat distribution, and there is no
effect on the efficiency from nonfiat â€” n+ distributions.
115
MINUIT x* Fit to Plot 9&3
U1Sâ€”>Gamma h h. pipiphi axis
File: Generated internally
Plot Area Total/Fit 779.39 / 779.39
Func Area Total/Fit 769.52 / 769.52
yz= 5.2 for 10  2 d.o.f.,
Errors Parabolic
Function 1: COMIS Function XMNCMJ
N 769.49 Â± 56.78
R 6.87430E02 t 0.1062
7AUG2004 00.36
Fit Status 3
E D.M. 1 509E06
C.L.= 74.0%
Minos
O.OOOOE+OO + O.OOOOE+OO
O.OOOOE+OO + 0 OOOOE+OO
M*
Figure 3: Continuum subtracted â€” T+ distribution for events near the /2(1270)
resonance with a fit to Equation 28 over fayed.
116
Example: Continuum Events.
Our previous examples tackled radiative decays of the T(1S). Now let us
consider the continuum â€œradiative beamâ€ process e+e" â€”Â» 77r+7r (see Figure 4).
At first glance it might seem that the helicity formalism cannot be applied in this
case. But a closer look at the diagram of this process, shown in Figure 4, indicates
otherwise. The key point is that, regardless of the particular coupling, helicity
formalism can still be applied at the CM of the 7r+7râ€œ vertex, where the initial state
has Jp â€” 1~.
7T+
Figure 4: e+e â€”> 77r+7râ€˜ in continuum. 7r+7r pair production usually occurs
through p â€” 7 coupling.
We only need to know that we are dealing with 1 â€”Â» 0 0 at the 7r+7r
vertex. There is only one helicity amplitude we need to consider,
K>= VS23'Â»'4â„¢ (29)
which after a couple simple steps leads to,
dN . t 3 . a.
â€” = N sm2i4+.
a cos 0n+ 4
This result is also valid for the kaon case.
.. (30)
On the other hand, the photonâ€™s angular distribution cannot be calculated
using twobody helicity formalism. A good approximation is (see [36] Appendix B),
117
dP l + cos2#7 /01,
j T ^ 1 27Tâ€™ b31)
d cos p7 1 â€” cos21/7
which is not valid at low 6 angles (6 â€”> 0, 9 â€”â–º 7r) because of the effects on the
approximation of the neglected electron mass,
V
Figure 5 shows these angular distributions in 4S â€œradiative beamâ€™ events. The
agreement between theory and experiment is excellent. This is reassuring, because
in the main analysis we rely on our angular distribution measurements to get the
correct efficiency in data and to identify the J value of each resonance.
118
Figure 5: Helicity angle distribution for â€œradiative beamâ€ continuum events
e+eâ€œ â€”â–º 7p and e+eâ€œ â€”> 7 in the 4S data. Top left (right) plot shows in gray
the events selected as p â€”> 7r+7r (0 â€”> K+K~). Middle left (right) plot shows
the 7T+7r_ (K+K~) helicity angle distribution for the selected events, and has the
helicity formalism prediction, Equation 30, overlÃ¡id. Bottom left (right) plot shows
the 7 angular distribution for the photon accompanying the p (0), and has the first
order QED prediction, Equation 31, overlayed.
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BIOGRAPHICAL SKETCH
Luis BrevaNewell was born in Boulder, Colorado on December 11, 1975.
When he was two years old his family moved to the Basque Country region of
Spain. He graduated High School from GETXO III and moved to Madrid where he
obtained a 5year degree in Physics from the Universidad Complutense de Madrid.
He was accepted to attend graduate School at the University of Florida in 1999.
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
John Yelton
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
David banner
Distinguished Professor of Physics
1 certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Pierre Sikivie
Professor of Phvsics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adeqK^te, in scope and
quality, as a dissertation for the degree of Doctor of rhv
Charle:
PpfysÃ¡br of Civil and
Coastal Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
This dissertation was submitted to the Graduate Faculty of the
Department of Physics in the College of Liberal Arts and Sciences and
to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
December 2004
Dean, Graduate School
LD
20<2it

