A Study of Hadronic Final States from e e- Annihilation
as a Function of Center of Mass Energy
by
Michael H. Capell
B.S. (Physics), University of California, Davis
(1979)
Submitted to the Department of Physics
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
at the
Massachusetts Institute of Technology
January 1986
Massachusetts Institute of Technology 1986
Signature of Author
Signature redacted
SigrSjgnature redacted
Certified by
Department of Physics
January 6, 1986
grProfessor Ulrich J. Becker
Thesis Supervisor
Signature redacted
Accepted by
Professor George F. Koster
MASSACHUSETTS INSTITUTE
OF TECHNOtOGY
Chairman, Departmental Graduate Committee
FEB 14 1986
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1
Abstract
A Study of Hadronic Final States from e+e- Annihilation
as a Function of Center of Mass Energy
by
Michael H. Capell
Submitted to the Department of Physics in partial fulfillment of the
the requirements for the degree of Doctor of Philosophy, January 1986
The multihadron data from e+e- annihilation collected with the MARK-J detector at
PETRA over the center of mass energy range 22 < Vs- < 47 GeV have been examined. Several
measures of gluon bremsstrahlung and fragmentation models were used. The strong coupling
constant has been determined from the entire energy range to be:
a,(Vw/= 34.6 GeV) = 0.119
0.005 (statistical)
0.020 (systematic)
by comparing the predictions of perturbative QCD through complete second order to the data.
Including lower energy measurements the strong coupling constant has been shown to
run with center of mass energy, the functional dependence being that predicted by QCD.
The measured dependence of event topologies on
F has been directly compared with the
perturbative calculation via the Energy-Energy Correlation Asymmetry. Use of the data over
a restricted range allowed the determination of an
a, value consistent with those obtained with
current fragmentation models over the full range. An improved description was obtained over
almost the full range with a simple, quantitative, parameterization of the non-perturbative
effects which yielded a QCD scale parameter of
A = 114 +I4 (statistical) MeV
in the MS renormalization scheme.
Thesis Supervisor: Ulrich J. Becker
Title: Professor of Physics
2
Acknowledgements
My first acknowledgement is made with pleasure to the people who have constructed and
run the MARK-J experiment.
My thesis supervisor, Ulrich Becker, has a clear insight into what it is useful to measure
and how to measure it, which is the root of experimental physics. I thank him for teaching
me some parts of this and for sharing his experience in the field.
The groundwork for the analysis presented herein was laid by Prof. J. G. Branson, H. Newman and R. B. Clare. This study and I have profited from the attentions of Prof. M. Chen,
G. M. Swider, R.-Y. Zhu and Ll. Garrido.
I thank Prof. Samuel C. C. Ting for his leadership in building the MARK-J detector and
collaboration, and for the efficient way he has helped to solve the resulting physics questions.
I thank A. Dworak, J. Hudson, R. Meisel, P. Slade and Dr. S. M. Ting for their administrative support.
A debt of thanks is owed to the DESY directorate and the PETRA Machine Group for
proving a good place to do high energy physics, and to the people of America and Germany who
have helped to support MARK-J, PETRA, and this study. Part of this work was supported
under DOE contract DE-AC02-76ER03069.
Other, but no less necessary, support has been provided by my friends and family, especially Susan.
3
Table of Contents
A bstract .................................................................................
A cknow ledgem ents .......................................................................
2
3
Chapter 1
Introduction ...............................................................
5
Chapter 2
2.1
2.2
2.2.1
2.2.2
Experim ent ................................................................
7
The MARK J Detector at PETRA ...........................................
7
Hadronic Event Selection and Reconstruction.................................9
Topological Considerations.................................................9
Selection and Analysis ....................................................
10
Chapter 3
3.0
3.1
3.2
3.3
3.4
Theories, Models and Their Implementation ..........................
12
Present Assumptions Used to Describe the Data e+e-- Hadrons............12
12
T he Initial State ............................................................
Perturbative Q CD ..........................................................
13
H adronization ..............................................................
17
Detector Simulation.........................................................18
Chapter 4
Determination of the Strong Coupling Strength
as a Function of Center of Mass Energy ...............
4.0
4.1
4.2
4.3
4.4
4.5
Methods in Measuring Gluon Bremsstrahlung ...............................
a, from the Energy Flow Variables..........................................22
a. from the Jet M ultiplicity.................................................25
a, from the Energy-Energy Correlation Asymmetry .........................
C om parison .................................................................
Measurement of the Running of the Strong Coupling Strength ...............
Chapter 5
A Stringent Test of QCD Versus Energy...............................34
Chapter 6
C onclusions...............................................................37
R eferences ..............................................................................
Tables ..................................................................................
Figure C aptions.........................................................................51
F igures .................................................................................
4
20
20
26
29
31
38
43
55
Chapter 1
Introduction
The strong or nuclear interaction is long known but not well understood from basic principles. The strong force manifests itself in the interaction of hadrons. This study investigates
multihadron final states produced by e+e- annihilation in order to gain insight into the strong
interaction.
Quarks were proposed by Gell-Mann [1-1] and by Zweig [1-2] as the mathematical building
blocks of strongly interacting particles. Just three different "flavors" of fractionally charged
spin I quarks and the symmetry of the group SU(3) [1-3] were required to construct the
dozens of known hadrons: mesons were "built" from quark-antiquark pairs and baryons from
triplets of quarks. A success of the model was the prediction of the existence and properties
of a new particle [1-4], the 11-, which was subsequently discovered [1-5].
However, the 1- pointed out the disagreement of the simple quark model with spin
statistics. As a ground state baryon of spin 1 the l- was postulated to be made up of
three
identical spin i quarks, violating Fermi-Dirac statistics. Greenberg solved this by introducing
an additional quantum number for the quarks, called color, which could take three values [16].
The f-
was explained as being composed of 3 quarks in a color antisymmetric state.
Antiquarks were assigned anticolor.
Color tripled the number of "fundamental" particles, solving two other puzzles. Calculations of the w* decay rate [1-7a] and of the total hadronic cross section in e+e- annihilation
[1-7b] both needed a factor of three more channels to agree with the data, and color provided
this factor.
The idea that quarks are more than a tool for building hadrons was advanced by the
deep inelastic lepton-nucleon scattering experiments at SLAC [1-8]. These revealed pointlike
constituents - partons - within nucleons, with all the properties of quarks, including fractional
charge [1-9]. The lepton-parton scattering results also showed that, within the nucleon, partons behaved as quasi-freely moving particles. This property, that at short distances or large
momentum transfers the partons are unbound, has become known as asymptotic freedom.
Later experiments revealed roughly collinear fluxes of hadrons moving in the expected directions of the struck parton and the remaining target partons [1-10]. This led to the concept
of confinement: partons do not appear as free particles at large distances, rather, they dress
themselves into hadrons. If a parton has sufficient momentum in the Lab frame, this dressing
appears as a "jet", i.e., a collimated, or limited PL, flux of hadronic particles or energy. To
observe jets, they must be isolated by regions of low flux, such that the angle between jets
is larger than the angular width of the jets. The simple quark-parton model with color did
not explain a further result from deep inelastic scattering, that only one half of the nucleon
momentum was carried by the charged quarks [1-11].
Direct evidence that quarks are the physical building blocks of hadrons came with the
discovery of the J/0' [1-12]. This was interpreted as the bound state of a quark-antiquark
5
pair with a new quark flavor, charm, in the same way that analysis of the hydrogen spectrum
revealed the component electron and proton [1-13]. Futhermore the binding potential could
be derived from a simple model of confinement.
Quantum ChromoDynamics (QCD) is the theory of the strong interactions which has
grown out of these observations [1-14]. Similar to Quantum ElectoDynamics (QED), it is a
renormalizable gauge theory [1-15]. The interaction of charges in QED is replaced by the
interaction of colors in QCD. The Abelian U(1) group symmetry of QED is replaced by the
non-Abelian SU(3) group symmetry of color. Massless spin 1 quanta called gluons carry the
strong force between colors, analogous to photons in QED. In contrast to electrically neutral
photons, gluons carry one unit of color and one unit of anticolor. With three colors (and three
anticolors) there are nine gluons. The group symmetry arranges these nine into an octet and
a singlet (3 0 3 = 8 E 1). The coupling of the color neutral singlet gluon is set to zero. The
colored octet gluons interact with both quarks and with other gluons. Gluons account for the
missing momentum in the nucleons.
The direct evidence for gluons came with the observation that about 10% of the multihadron events in e+e- annihilation at Fs > 25 GeV appeared as three jets [1-16].
The
agreement of both the rate of these events and the angular distribution of the jets within the
events with the expected process of gluon bremsstrahlung from the quark or antiquark [1-17],
was a major success of QCD.
In QCD the strength of the quark-gluon and gluon-gluon interactions is determined by
the strong coupling constant,
a,, in the same way as the fine structure constant, a, determines
the strength of the electric charge-photon interaction. To account for both asymptotic freedom
and confinement the coupling in QCD "runs". It is small at short distances and large at long
distances (fractions of a fermi). In momentum space this means that the coupling is small
when gluons are transferring large amounts of momenta (a few GeV). The perturbation series
expansion of QED in terms of a is paralleled in the expansion of QCD in terms of a, to provide
a calculable theory. Perturbative QED is useful over a wide range of energies, but, because
a
is large for processes involving low momenta, the expansion in QCD is only expected
to be valid for reactions involving large momenta.
Even in this region a, is larger than a
>> e = ) and the effects
(a,of ~higher to
orders are worrisome.
a
This study uses the data collected with the MARK-J detector to precisely determine the
strong coupling constant, a., over the wide energy range available at PETRA, 22 < v/ < 47
GeV. The functional dependence of the running of the coupling constant is tested over this
range and with the inclusion of lower energy data. The energy dependence of the data is
used to probe the validity of the present theoretical understanding and to quantify the as yet
uncalculated effects.
6
Chapter 2
2.1
Experiment
The MARK-J Detector at PETRA
The e+e- storage ring PETRA [2-11 and its injection apparatus are diagrammed in
Fig 2.1. Electrons are injected from Linac I into DESY, accelerated to 7 GeV, and injected
into the PETRA ring to form two bunches. Positrons, produced in Linac II and accumulated
in PIA, are likewise accelerated in DESY and injected into PETRA to form bunches. Each
bunch contains (1 to 2)-1011 particles. As the ring circumference is 2.3 km this yields a current
of 2 to 4 mA/bunch. The two pairs of counter-circulating bunches are further accelerated
and focused to collide inside the four detectors, with a crossing frequency of 250 kHz. After
the first physics runs in November 1978, the ring elements were modified to provide higher
luminosities and higher beam energies. The peak instantaneous luminosity obtained with the
MARK-J detector was 1.6 -10
31
/cm 2 /sec, with up to 650 nb-
1
collected in one day. The
integrated luminosity collected with the MARK-J detector over the resulting large range of
center of mass energy, ,F, from 12 GeV up to the world's present highest e+e- energy of
46.78 GeV, is shown in Fig 2.2.
The MARK-J detector, Fig 2.3a and 2.3b, is composed of a vertex detector, electromagnetic and hadron calorimeters, and a muon spectrometer.
Particles leaving the e+e-
interaction region traverse the detector layers shown schematically in Fig 2.3c. Outside of the
beampipe is the vertex detector (labelled DT or drift tubes in Fig 2.3). The next layer is the
electromagnetic or inner calorimeter (A,B,C). Surrounding this calorimeter are the inner drift
chambers of the muon spectrometer (S,T). Proceeding radially outward the magnet toroids of
the spectrometer form the absorber for the hadronic or outer calorimeter (K). The outermost
part of the detector completes the muon spectrometer with trigger counters (D), more magnetized iron, and drift chambers (P,R). Hadronic events are selected using the vertex detector
and the calorimeters, the analysis of these events utilizes the reconstructed energy depositions
in the calorimeters.
The remainder of this section provides a description of these detector
elements (more detail can be found in [2-2]). The following section describes event selection
and reconstruction.
The vertex detector measures the charged multiplicity and the vertex position along the
beamline. It is composed of 2616 cylindrical drift tubes arrayed perpendicular to the beams
in four rectangular layers. Each tube is 1 cm in diameter and 30 cm long. The array actively
covers the polar angle from 0 = 10' to 170* and the entire azimuth less 70 in each corner.
The resolution per tube is 0.03 cm. Tracks are fit to hits in each layer. The best overall fit
for an event is determined by constraining the tracks to a common vertex along the beam
direction, z, and minimizing the X 2 per track. The number of tracks pointing to this common
vertex yields a measure of the charged multiplicity. The r.m.s. widi of the best fit vertices
for hadronic events is just that expected from the PETRA bunch length, Az = 1.3 cm.
7
The inner calorimeter provides the bulk of the information about hadronic events. It is
subdivided into three azimuthally segmented layers of shower counters (A,B,C in Fig 2.3).
Each counter is made up of 0.5 cm thick pieces of scintillator alternated with 0.5 cm thick
lead plates.
At normal incidence this yields 3, 3 and 12 radiation lengths or a total of 1
absorption length. The 20 A counters are arrayed parallel to the beamline outside of an iron
box which surrounds the vertex detector. They cover the polar angle 0 = 120 to 1680 with
no azimuthal holes.
The 24 B and 16 thicker C counters are arranged similarly to the A
counters, but are offset in 0 and shorter, the B counters covering 0 = 160 to 164' and the C
counters 0 = 260 to 1540. Each of these sixty counters is instrumented with a phototube at
each end
[2-3]. The time and magnitude of both phototube pulses are digitized with TDCs
and ADCs and recorded.
The longitudinal hit position is measured by a weighted average
of the positions determined from the time difference between the two pulses and from their
relative magnitude. Comparing to the the positions extrapolated from tracks fit in the vertex
detector this method yields a resolution for single hits of A0 = 50 per counter. The azimuthal
segmentation combined with shower sharing between counters yields a resolution of AO
= 7*.
The energy deposited in each counter is determined from the two pulse heights corrected for
attenuation. From the direction with respect to the interaction region and magnitude of the
energy deposited in a counter an "energy vector", $4, is formed. This is not a proper vector.
Under addition the resultant direction is taken as the vector sum and the resultant magnitude
as the algebraic sum. The leakage of electromagnetic showers from electrons and gamma rays
into the outer calorimeter is less than 4%. The resultant resolution is AEIE = 7% at E = 17
GeV. On average, a hadronic event deposits 35% of its energy in the A counter layer, 15% in
B and 25% in C. Distributions of the energy deposited in the inner calorimeter are shown in
Fig 2.4 and 2.5.
The outer calorimeter absorbs the remaining 25% of the energy from a hadronic event.
192 scintillation counters are arranged in four layers interleaved with 2.5 to 10 cm of iron for a
total of 2 absorption lengths at normal incidence. The two inner (outer) layers cover the polar
range 0 = 430 to 137' (260 to 1540) with azimuthal holes of 100 (40). In this calorimeter the
4
resolution is ~ 2* better than in the inner calorimeter because of the finer segmentation. The
longitudinal resolution is worse because only one end of each counter is viewed by a phototube.
To improve this resolution clusters of hits in the same (
90
in 4 and t
~ 0.1 in cos 0) solid
angle are assigned to a "counter track". The longitudinal hit positions in the outer calorimeter
are adjusted on to these counter tracks. The resulting angular distribution is shown in Fig 2.6.
Using the position information from the inner and outer calorimeters results in the energy
weighted 4 distribution of counter tracks shown in Fig 2.7. The angular resolution for the
axis of a hadron jet, determined from Monte Carlo studies, is A0 =Z\ cos 0 - A0 = 70 for the
entire detector. The active solid angle is 95% of 47r.
8
Hadronic Event Selection and Reconstruction
2.2
2.2.1
Topological Considerations
The "signal" events for this study are multihadron final states resulting from e+e- annihilation into a virtual photon.
The procedure for selecting these events from the other
channels and backgrounds which constitute the "noise" is best prefaced by descriptions of
their respective appearances in MARK-J. Fig 2.8a is a schematic representation of a typical
one photon hadron event; the five major sources of background are diagrammed in Fig 2.8b-f.
Fig 2.9 shows a computer reconstruction of a typical "signal" event, taken at a beam energy of
22.05 GeV. The characteristic high multiplicity of penetrating particles appears in MARK-J
as several tracks in the vertex detector (11 for this event, Fig 2.9b) originating from the e+einteraction region and significant energy deposition in the outer calorimeter (9.2 GeV). This
is distinct from electromagnetic final states which are contained in the inner calorimeter. Two
other distinctive features visible in Fig 2.9 are that the energy depositions in the calorimeter
are balanced across the origin and the sum of the energy depositions, EV, is close to the avail2
E~eam. Specifically, this event has net energy imbalances
able center of mass energy, Fs
of 6.0, 4.2 and -0.4 GeV in the x, y and z directions and a total reconstructed energy of 41.5
GeV out of 44.1 GeV available in the center of mass.
Tau pair production, Fig 2.8b, where both taus decay semi-leptonically into hadrons,
is characterized by the missing energy carried off by the neutrinos and by two narrow jets
of hadrons. This is observed as an event with a net energy imbalance, reduced Ey and low
multiplicity, though a significant fraction of energy may penetrate to the outer calorimeter.
Tau pairs can also contaminate the sample by one tau decaying leptonically into an electron
and the other into hadrons. This appears as a narrow, low multiplicity hadronic jet back-toback with an electromagnetic shower.
The appearance of two photon multihadron events, Fig 2.8c, depends on the number of
tagged electrons. If both electrons are scattered sufficiently to enter the detector they are
observed as nonpenetrating showers in the inner calorimeter with a single matching track in
the vertex detector. An electron escaping down the beampipe leaves behind an event which
is reconstructed with a net energy imbalance along the beam direction and with reduced
Ey. The hadronic system may be roughly balanced but its energy is substantially less than
2EBeam, hence it leaves a reduced fractional energy deposition in the outer calorimeter.
Bhabha scattering (ee- _- e+e- including radiated photons), Fig 2.8d, leads to < 2
tracks in the vertex detector with 2 or 3 electromagnetic showers in the inner calorimeter and
no energy in the outer calorimeter. The bulk of these events occur at low scattering angles.
Electron-beam gas scattering, Fig 2.8e, appears as a highly unbalanced event with E
~
EBeam and usually a reconstructed vertex away from the e+e- interaction region.
High energy cosmic rays, Fig 2.8f, interacting in the outer parts of the detector can
deposit large amounts of energy in the calorimeters. These events are distinguished by their
9
external origin, as indicated by the muon spectrometer and by time-of-flight between the
muon trigger counters (D in Fig 2.3).
Cosmic rays which sneak through the spectrometer
only rarely produce tracks in the vertex detector which point to the e+e- interaction region,
and the energy depositions are typically concentrated in one area of the detector.
2.2.2
Selection and Analysis
The selection of signal events from the noise is done stepwise. The first step in the event
selection is implemented by the online data acquisition system [2-2,4]. Of the eight triggers
employed to initiate data collection, two are significant for multihadron events.
The first
relies on the coincidence of hits in several calorimeter layers, the second on the coincidence
of energy depositions in groups of counters opposite each other in 0. On receipt of either of
these triggers a fast sum of the total energy deposition is calculated. This approximate sum
is conservatively required to exceed
fI/6, suppressing beam gas, cosmic ray, and two photon
events where neither electron is tagged. Combined with the other triggers the typical trigger
rate is 5 Hz and the accepted trigger rate is 2 Hz, with a deadtime of less than 9%. Monte
Carlo studies indicate the trigger inefficiency for the signal events is less than 1%.
The energy depositions from the accepted triggers are then reanalyzed offline by a fast, but
approximate, algorithm. At this step the total energy requirement is increased to EV > 0.30fi
and a loose balance cut of energy depositions greater than 0.10of in opposing # quadrants
is applied.
These two cuts reduce the sample 75-90% (depending on beam conditions) by
further rejecting beam gas and low energy two photon events. Monte Carlo studies indicate
95% of the signal events pass this step.
The remaining candidates are accurately analyzed.
vertex detector (e.g., see Fig 2.9b).
Tracks and a vertex are fit in the
The energy vectors, fi, of each hit in the calorime-
try are precisely determined from the combination of ADC and TDC data with calibration values.
Counter tracks are formed as described above.
The visible energy require-
ment, EV =_ E|J4|, is increased to 0.35N/s and the net energy imbalances perpendicular,
AE.1 =
/(
Ei
) + (, Ej
, and parallel, AE,, =1 - . + - [ , to the beamline are re-
quired to be less than 0.60EV. The remaining cosmic ray events are rejected via the difference
in the hit times between the outer muon trigger counters. After this step the sample contains
one and two photon hadronic events, Bhabha events, and tau pair events.
A complicated pattern recognition algorithm then attempts to sort each event into one of
these categories. Each counter track is assigned probabilities of being hadronic, electromagnetic, or the result of a tau decaying into hadrons. This assignment is based on: the charged
multiplicity of the counter track, estimated from the number of matching vertex detector
tracks; the energy fraction of the track in the outer calorimeter, this fraction being a function
of the direction of the track; and the total energy of the track, this also being a fraction of
the track direction. The event is then viewed as belonging to each of the categories and the
assigned probabilities folded with the possible sources. Events which do not strongly agree
10
with exactly one hypothesis or have an unusual feature are classified by physicists scanning
detailed event pictures such as Fig 2.9. (Examples of unusual features are a vertex fit > 3away from the interaction region or large energy hits in the inner calorimeter.) About 20% of
the sample at this stage requires this manual classification, half of which is accepted. Of the
other 80%, 5-10% are scanned as a cross check.
The data used in this study were collected over the last six years and over a wide range of
energies, so systematic variations in the detector response are a concern. These can be induced
by scintillator aging, by decreasing phototube gain, by ADC pedestal shifts, etc.
During
data taking and in the subsequent analysis chain these variations are largely eliminated by
recalibration. The calibration values for each counter and its associated electronics are derived
from studies using Bhabha events, cosmic rays taken between beam crossings, and multihadron
events [2-5]. They are readjusted every 2-4 weeks. For hadronic events this has maintained
the total energy resolution at A(Ev/v/)
~ 18% independent of -
as shown in Fig 2.10.
V
The dark circles in Fig 2.11 show the mean values of detector related quantities versus
measured using hadron events, and display the consistency of the detector response versus
both time and
V/-.
Studies using both data and Monte Carlo indicate that events with Ev
0.50fi are
predominately multihadron final states from the one photon annihilation channel, with an
acceptance of 83% at
F > 32 GeV, and with a contamination from two photon processes of
1.3% and from tau pairs of 4.5%. To reduce these two remaining backgrounds to negligible
levels and to insure accurate reconstruction of the event shape by the calorimetry, the events
0.70v/-, as marked in Fig 2.10, and AE 1 <
used in this study are required to have Ev
0.50Ev, AE, < 0.50Ev, as marked in Fig 2.12. The data are distributed in
F as shown in
Fig 2.13. For the bulk of the analysis presented below, these data have been combined into
the bins summarized in Table 2.1. The data at 14 GeV is not used in this study because it is
near to the b-meson threshold and both b-production and b-decay are not well understood
in this region.
11
Theories, Models and Their Implementation
Chapter 3
Present Assumptions Used to Describe the Data e+e--+ Hadrons
3.0
Only a partial understanding of the process of e+e- annihilation into hadrons is existing.
This study aims to increase this understanding. This chapter sketches the theories and models
used to describe the data, and how they have been implemented for the analysis presented.
At present e+e- annihilation into hadrons is postulated to proceed as sketched in Fig 3.1,
[3-1]. From left to right the abscissa can be taken as increasing time, decreasing particle
momentum, or, roughly, decreasing understanding. The three subprocesses, (1) e+e- _-
y*
(2) -y* -+ qq, -+ qqg and (3) q, g -+ hadrons, are assumed to have independent probabilities
The first step is well understood using QED with Electroweak
which can be multiplied.
The next step, involving the high momentum quarks and gluons known as
modifications.
partons, is described by perturbative QCD. The final step, the evolution and association of
partons into real hadrons, is called hadronization or fragmentation. This step has no solid
theoretical basis and resort is made to models. At PETRA energies the association during
fragmentation results in the hadrons appearing collimated in jets. The jets of hadrons are
the observables and they reflect the original parton energies and directions. The utility of the
factorization is that it allows the calculable, perturbative steps (1) and (2) to be examined
without requiring a full understanding of the non-perturbative step (3).
Because of the multistep and complex nature of the predictions, the comparison of data
and theory relies heavily on Monte Carlo (MC) techniques.
"Events" are generated with
different parton configurations, distributed according to the matrix elements of steps (1) and
(2). Step (3) is then implemented on an event by event basis using different models.
3.1
The Initial State
A beauty of e+e- annihilation physics is that the initial state is governed primarily
by QED. This gives the cross section for e+e-
-+ Y* -+ ff
(massless, pointlike, unit charge,
2
a is the fine structure constant and q is the momentum
transferred by the -*. Hence, for e+e- - y*, q is equal to the center of mass energy: q = N.
spin
I pairs) as aPt
ra /q , where
=
For hadron production the total cross section is:
ao(e+e
-
*
-
Q2
hadrons) = N,
(3
2
pt
-+1
32
,
(3-1)
f
where N, = 3 is the number of colors, the sum runs over the active flavors (at PETRA
energies, ./
> 14 GeV, f=u,d,s,c,b), Qf is the quark charge (+ for u,c and -.
for d,s,b),
#2 = 1 - 4m2 /s and mf is the quark mass (taken to be, in GeV: ~ 0.01 for u, d; ~ 0.15 for s;
~ 1.2 for c; and ~ 5.0 for b). These masses are not well determined. They are inferred from
the bound states and mesons displaying flavor associated features. The initial state radiative
corrections (RC) are given by QED and were included in step (1) to order cO [3-2]. Should a
12
hard photon be emitted, steps (2) and (3) are carried out in the boosted hadronic center-ofmass frame with a decreased
\Fs.
The detector response to the hadrons and the photon is, of
course, simulated in the Lab frame.
At PETRA energies the interference of virtual Z 0 's with the annihilation photon is imElectroweak theory
~
[3-3] predicts the total hadronic
c to !Mzo- The standard
portant,
cross section is modified to:
(
OEw(e e -
Z*,ZO) -+ hadrons)
= Ne
S [Q2 + 2Qf~wsge gl
2
+ X2 s 2 (g
+ g 2 )(gf
2
+ gi 2 )],
(3-2)
f
where
Xw = [4sin 2
OW
gky =
T'3L-
2Qj s sw2
gA
TL
=
TiL =
0
cos 2
Ow(M2
-
-26 at
)
F
= 35 GeV,
is the weak vector coupling,
is the weak axial coupling,
+1/2 for i =
c
-1/2 for i = d, s, b, e-
is the third component of the weak isospin,
w is the weak mixing angle, Mzo is the Z 0 mass, and the width of the Z0 and the quark
masses have been neglected. The bulk of the modification comes from the interference term,
oc Qfgf. In this study variables normalized to the total cross section are used exclusively.
For this case the modification to the fractional contribution of the heavy flavors is PL gf
/Qf.
Within the statistics, this is not observable.
3.2
Perturbative QCD
QCD, the theory of the strong interactions, was introduced in Chapter 1. In this section
the calculations of the perturbation series expansion of QCD in terms of the strong coupling
constant, a,, is outlined up to second order in a, in the form used for the analysis in this
study. Many excellent descriptions of these calculations exist (e.g. see [3-1]).
The calculation to first order requires the diagrams of Fig 3.2. (The interference of the
simple quark pair production diagram of Fig 3.2a and the virtual correction diagrams of
Fig 3.2c is first order.) The quarks are taken as massless to facilitate the calculations. For
qqg final states evaluation of the gluon bremsstrahlung diagrams of Fig 3.2b leads to the
differential cross section [3-4]:
where CF = (N2
-
,
QC2
CF
27r
+X
q
(1 - Xq)(1-
2P
~ -(3-3)
_-
)'
X'"
r
,
d 20
o0 dxq dxq
1
1)/2N, = 4/3 is the Casmir operator from the evaluation of the color
SU(3) transformation of the quark-gluon coupling. Eqn (3-3) displays the infrared or soft
(Xq
and x- -+ 1; Pg -
0) divergence, and the collinear (xq or
13
xq -
1; Pg |
or Pg || P1)
divergences. It also shows the cross section is low for "Mercedes-Benz" events, where all three
partons have equal energy (xq
= X-
= Xg =
2/3), and are separated by 1200.
The soft and collinear singularities are familiar from QED, where a similar divergence
arises in the calculation of e+e-
-
P+/-ry when the -y is not resolved. Evaluation of the
diagrams is facilitated via a resolution cut which separates the distinguishable 3-parton events
from 3-parton events where the gluon is not resolved.
Events where three partons can be
distinguished are called "resolvable 3-parton" events. For events from the 3-parton diagrams
which fail the resolution cut a pair of partons is recombined, these events are called "quasi-2parton". That is, a "resolvable-n-parton" event is an event where an n-parton configuration
has passed the resolution cut, or an (n+ 1)-parton configuration has failed the resolution cut.
When the (n + 1)-parton configuration has failed the cut it is also know as a quasi-n-parton
configuration.
In the calculations the divergences in the quasi-2-parton amplitudes cancel
those from the interference between the diagrams in Fig 3.2a and 3.2c [3-5]. Summing the
finite remainder and the zeroth order diagram, Fig 3.2a, yields the total "resolvable 2-parton"
cross section. In agreement with the "Kinoshita-Lee-Nauenburg"
(KLN) theorem [3-6] the
resulting resolvable 2- and 3-parton cross sections are finite. Summing them gives the total
cross section to first order [3-7]:
01= (1+ 3CF
)uo
4
?r
(3-4)
This defines the strong coupling constant to be:
Swith
bo log(s/A 2 )'
11N-2Nf
bo
6
- 23(3-5)
6'
where Nf = 5 is the number of flavors and where A is the QCD scale parameter. A is the free
parameter of the theory.
The resolution cut deserves some consideration [3-7a]. In the pyuy case the functional form
of the cut depends on the quantity observed, e.g. the muon acollinearity or the photon energy.
The cut for qqg is selected similarly. The resolution cut must be less than the resolution for the
variable being measured, taking into account the smearing from hadronization. Conversely,
the cut must be hard enough to avoid double counting of the soft gluons which are also
involved in hadronization. More importantly, to fixed order in the perturbation expansion,
too soft a cut can lead to negative probabilities, which is incompatible with the MC technique.
In addition, the observables of an event which moves from one category to another as the cut
is varied should change smoothly, as the combining of partons in configurations that fail the
resolution criteria is more in the nature of bookkeeping than physics.
Three reasons to undertake the extension to second order, which involves the diagrams in
Fig 3.3, are: (1) a, measured in first order is large,
-
0.17 at I = 35 GeV, which necessitates
at least next order calculations. (2) The non-Abelian nature of QCD, e.g. the triple gluon
vertices in Fig 3.3, only appears in greater or equal to second order diagrams. (3) The relation
14
of the scale parameter A to c, depends on the renormalization scheme. In first order this is
poorly specified, e.g. a rescaling of A(-+ A' oc A) yields a constant which can be absorbed or
not during the renormalization. Equivalently "N/s" is not well specified, so that neither the
same measurements at different center of mass energies nor measurements involving different
reactions can be compared. In second order the scale is uniquely defined for e+e- annihilation
into hadrons.
Here the modified minimal subtraction (MS) renormalization scheme [3-8] is
used exclusively, where the scale, i.e.
fi,
is exactly that of the virtual photon.
The total cross section can not yet be evaluated by summing the partial dressed 2-, 3and 4-parton cross sections. Instead, much as in the optical theorem, the relation between
the total cross section and the imaginary part of the inverse photon propagator is exploited
and the latter is evaluated via the diagrams in Fig 3.4. This yields the total cross section to
complete second order [3-9]:
3
a
2)2),o ,
CF- + Kjs(
?rX
3
Kis - CF(0.538Nc - - CF - 0.08645Nf) = 1.405,
a2 =
with
(1 +
4
(3-6)
32
where the coupling strength is given by:
2x'
as =
2r(3-7)
S
bo log(s/A 2 ) + (bi/bo) - log(log(s/A 2 ))
wit
with
b1
bi -
The presence of both the log(
17N2 - 5NeNf
wv
6
-
3CFNf
_
68
6
) and the log(log() terms indicates the unique relation between
a, and A in second order.
The full contribution of the interference of the fourth order, 2-parton diagrams, Fig 3.3c,
with the zeroth order diagram, Fig 3.2a, is not yet finished [3-10]. Instead the resolvable
2-parton cross section, finite by the KLN theorem, is taken as the total cross section less the
resolvable 3- and 4-parton cross sections. The 4-parton tree level diagrams, Fig 3.3a, are
evaluated above a suitable resolution cut yielding the resolvable 4-parton cross section [3-11].
As in first order the assorted divergences below the resolution cut have been shown to cancel
with interference terms [3-12a].
The quasi-3-parton cross section has been calculated by several groups and the results
have not agreed. The earliest calculation [3-12a] was precise but difficult for experimentalists
to apply because the results were expressed in terms of a single variable, thrust (this variable
is defined in
4.1).
This variable, it turned out, is very dependent on the resolution cut
and the calculation was done with a cut orders of magnitude smaller than that dictated
by the requirement of non-negative cross sections and the desire to avoid double counting
soft gluons. A subsequent attempt [3-12b] obtained an analytic expression in a general form
with reasonable resolution cuts, but achieved this by approximating a few terms and by an
approximate scheme of parton recombination. These approximations turned out to be not a
15
good idea and the calculation was found to be accurate only to the order of the resolution
cut. These two calculations were eventually shown to be equivalent in the limit of vanishing
resolution cut [3-13]. This study uses a third calculation [3-12c], where the precise methods
of [3-12a] were extended to relevant values of the resolution cut by a MC integration of the
4-body phase space, and the results were expressed in terms of the x qand x- defined in Eqn
(3-3), which completely specify a 3-parton event. The accuracy of this calculation is limited
by the Monte Carlo statistics. Ten million events were used and the theoretical uncertainties
exceed the statistical error. Work continues on this topic, giving some hope of a useful analytic
evaluation [3-12d].
The calculation of [3-12c] implemented two different forms of resolution cuts over a range
of values, the scaled pseudo-invariant mass cut, defined by
Y = 2 minPP
and the Sterman-Weinberg cuts
C = min
i
S (>Z Pi)
where
[3-7a],
P-P
Ecut
-
P'
2
(Zs P )
i~
' > Ycut,
(3-8)
defined by:
and
cos(26)
#- -PN
max
i+1 PPi
< cos(26cut),
(3-9)
P, P are the momentum vector and magnitudes of the partons i, j,k. These momentum
based definitions allow the QCD calculations for massless partons to be extended to massive
quarks.
The Sterman-Weinberg cuts have simple interpretations: if a parton has too low
an energy it cannot be resolved and if two partons are too close together they cannot be
distinguished from one parton. The invariant mass cut achieves the same effect by excluding
large parton energies but can be calculationally more convenient. For example in the qqg case
the cut excludes large parton momenta: Y > Ycut
-- >
xqq,g <
1 - Ycut, where x is from
Eqn (3-3).
To give a concrete example MC events can be placed in one of four categories, labeled 2,
3, 3' and 4. "2" refers to all events which contain 2 resolved partons. "3" refers to the first
order resolvable 3-parton events, i.e. those events from Fig 3.2b which pass the resolution cut.
"3'" refers to the second order resolvable 3-parton events. Contributions to this category come
from: the finite terms in the evaluation of the diagrams of Fig 3.3a which fail the resolution
cut once and the interference between the third order virtual correction diagrams of Fig 3.3b
with the simple first order single gluon bremsstrahlung diagrams of Fig 3.2b. "4" refers to the
second order resolvable 4-parton events, i.e. configurations passing the cut from the tree-level
diagrams, Fig 3.3a. At Fi = 35 GeV for A = 0.10 GeV(=> a, = 0.12), with ecut = 0.1 and
cos(2Scut) = 0.9 the fractional partial cross sections are calculated to be:
1 - (F 3 + Fs' + F4),
)/Utot = 0.57
F2 = u (resolvable 2-parton
F
= u(resolvable 3-parton, O(a))/uott = 0.31
F-' = u(resolvable 3-parton, O(a2))/otot = 0.06,
16
= o-(resolvable 4-parton, 0(a))/utot = 0.06
.
F
For the analysis presented below F' and F 4 were calculated from the generation, including
radiative corrections, of one million events of both parton types at each center of mass energy.
The resulting statistical error is negligible (- 3 parts in 104).
3.3
Hadronization
Because the evolution of partons into hadrons involves the non-perturbative or "confine-
ment" regime of QCD and is not yet calculable, the comparison of data and theory requires
the use of fragmentation models. While these models are partially motivated by theory, they
are not useful if they cannot be adjusted to agree with the data. Two models are used in
this study, the model due to Ali et al., an independent jet model [3-14], which is an extension
of the earlier Feynman-Field qq fragmentation parameterization [3-15], and the Lund color
string model [3-16]. Of the models which can be "tuned" to reproduce the data, these two
represent extreme viewpoints with respect to the effects of the initial parton topology on
the direction and energy of the hadrons produced. The model dependence of the results is
taken as a conservative estimate of the uncertainty owing to the ignorance of the details of
fragmentation.
The models are well described in the references given (see also [3-17]), but a few points
need to be reiterated. Each model produces hadrons by popping a series of sea quark pairs out
of the vacuum. Several free parameters describe the generation of these pairs, these parameters
are fixed using inclusive and exclusive particle spectra [3-18]. Of these parameters, the mean
transverse momentum, o-q, has the largest effect on the resulting event shape.
The Ali model fragments quarks one at a time into hadrons through a chain of decays:
quark -+ meson + sea quark, where the meson contains the initial quark and the sea antiquark.
Gluons are treated as a collinear quark pair, each of which is fragmented in turn, so gluons
produce a broader jet (i.e. two overlapping jets).
A problem with this approach is that
the production of massive jets from massless partons explicitly fails to conserve energy and
momentum.
This is fixed post facto by a suitable Lorentz boost of the jets of hadrons.
which introduces a mild global dependence of the final jet directions on the initial parton
configuration. In a qqg event this results in the boost usually being directed antiparallel to
the gluon direction, which moves quark jets off of their original parton directions and towards
the gluon jet. Hence the event appears somewhat more like a 2-jet event. Another problem
with the model is its discontinuous behavior with parton pairs just passing or just failing
the resolution cut. Because the number of links in the fragmentation chain varies about as
the square root of the initial parton energy, a (qg) subsystem just passing the resolution cut
produces about twice as many hadrons as when it fails the resolution cut and is recombined
into a single quark.
The Lund model fragments strings, which are narrow tube-like fluxes of the color field,
that run between the partons. This is a Lorentz invariant approach a n d avoids discontinuities
17
across the resolution cut. Formally the Lund model does not produce "quark" jets or "gluon"
jets and the intuitive association of jets with partons present in the Ali model is lost. The
fragmentation of a qqg event is depicted for the two schemes in Fig 3.5, the Lund model
producing particles along hyperbolas connecting the quark and gluon and connecting the
gluon and antiquark. Functionally, the Lund model produces jets shifted from the original
parton directions, the "quark" jets swinging towards the "gluon" jet. This is a larger effect
than the imposition of energy-momentum conservation via a Lorentz boost in the Ali model.
Consequently a 3-parton event appears less 3-jet like when fragmented with the Lund model
than with the Ali model.
This implies that to predict a measured number of 3-jet events
the Lund model requires a systematically larger partial cross section for 3-parton events than
the Ali model, that is, a larger a.. The magnitude of this shift is seen to be 14% in chapter
4, it would be nice to eliminate this shift beforehand by excluding one of the models. The
MARK-J data does not rule out either model. Investigations which have claimed to favor the
Lund model have been based on the soft particle fluxes between jets (e.g. see [3-19]), a region
neither model was designed to simulate.
There are several other models proposed to mimic fragmentation.
Hoyer independent jet model,
These include the
[3-201, which, in the usual implementation, achieves energy-
momentum conservation by rescaling the hadron energies while maintaining their directions,
and numerous "gluon-shower-cascade" models [3-21]. To date they do not yield satisfactory
descriptions of our data and remain under development both at MARK-J and elsewhere.
3.4
Detector Simulation
After event selection the analysis of hadronic events uses primarily the energy vectors, Ej,
of the hits in the various calorimeter layers. For this reason a complete detector simulation is
more important for MARK-J than for detectors which measure, for example, the momentum
of charged tracks. The bulk of this study uses a fast, but complete, simulation. Particles are
tracked through the detector and their intersections with active detector elements calculated.
The energy deposited in each counter is determined from tables that give the dependence on
penetration depth, angle, incident energy, and particle. Energy resolution and longitudinal
shower fluctuations are also simulated using tabulated information. These tables were generated using test beam data for 0.5 to 10 GeV electrons and pions, experimental calorimeter
studies [3-22], and a shower Monte Carlo program [3-23], but have been much improved by
studies using the data collected at PETRA from multihadron events and the very collinear
hadron jets from r decays. After correction for time of flight, attenuation and propagation
delays in the scintillator, time slewing due to varying pulse heights. and multiple hits, the
counter ADC and TDC information is digitized.
The drift chambers are also simulated in
detail, and the result is written out and analyzed using the offline programs, including the
same cuts, described in 2.2.
18
As a cross check another, more accurate, detector simulation called GHEISHA is used [324]. This program carefully follows not only each particle but also its associated secondaries
through the complete detector. Multiple and nuclear scattering are included, as are many
other effects. Again, the output is digitized and analyzed with the programs used for data.
For technical reasons, only a fraction of the events simulated with the fast shower program
are available.
The results of the simulations are compared with the data in Fig 2.4-2.7, 2.10 and 2.11.
Both simulations reproduce the response of the inner and outer calorimeters reasonably well.
The effect of the inconsistencies, e.g., the shift in the inner calorimeter energy distribution
seen in Fig 2.4b, are investigated in the analysis presented below.
19
Chapter 4
4.0
Determination of the Strong Coupling Strength
as a Function of Center of Mass Energy
Methods in Measuring Gluon Bremsstrahlung
The fundamental process of QCD is the interaction of gluons. The strength of this interaction is given by the strong coupling constant, a.. This chapter presents the determination
of a. over a wide energy range using the rate of hard gluon bremsstrahlung. The presence of
a single radiated gluon can be tagged because it modifies the topology of the event. Qualitatively, a two parton event is observed as two narrow back-to-back "quark" jets. An event with
a gluon emitted at low energy or small angle to one of the quarks appears as a narrow "quark"
jet opposed by a broader "quark + gluon" jet. Events with a higher energy gluon emitted
at larger angles show three distinct coplanar jets. Four parton events appear in a variety of
topologies: two broad jets, a very broad jet opposing a narrow jet, and more isotropic configurations. Hence three parton, qqg events are identified by analysis of the spatial distributions
of observed hadron momentum vectors.
Intrinsic to QCD is the variation of the strong coupling with energy, accounting for confinement at low energies and asymptotic freedom at high energies. Eqn (3-7) relates the
coupling constant, a,, at a given center of mass energy, fi, to the QCD scale parameter, A.
The perturbation expansion leading to this equation is expected to be valid for 5>
A, and
well above quark flavor production thresholds, \s > 2Mf. Measurements of a, over the PETRA energy range which meet these conditions (i.e., excluding the data \ < 14GeV ~ 2Mb)
can then be related, provided a renormalization scheme is specified and a full second order
calculation is used. This study uses the MS renormalization scheme and such a calculation,
as related in 3.2. In the MS scheme, for e+e- -- hadrons, the scale of the reaction is unambiguously defined to be the energy of the annihilation photon, so that the rate of gluon
bremsstrahlung yields a, immediately, i.e. a, is the physical quantity and A is the theoretical
parameter. For this reason the determinations of the strong coupling strength in 4.1, 4.2
and 4.3 which combine several center of mass energies are given in terms of a, evaluated
at an V'8, with the expected variation of a, with F from Eqn (3-7) taken into account.
For this study the mean \/F of the high statistics data 33 < \Is < 36 GeV is taken as \F:
o = 34.63 GeV, and the conversion to AT is delayed to 4.4.
The primary difficulty in determining the strong coupling strength is to disentangle the
effects of hadronization from the phenomena described by perturbative QCD. This is where
the use of a wide range of center of mass energies becomes advantageous. The perturbative effects have a slight, logarithmic, dependence over the energy range: for A = 100 MeV , a,(\/, =
22GeV) = 0.154 and a,(x/5 = 46GeV) = 0.118. The fragmentation effects are expected to
1/ 8 b, where the value of b ranges from } to 2 depending
vary much more quickly, roughly
on the variable considered [4-1]. Hence, using variables which show only a small, ~ logarithmic, dependence on 5F to determine the strong coupling strength has the advantage that
20
the uncertainty due to the incalculable power corrections from non-perturbative effects are
minimized.
A wealth of variables exist to quantify the observed topologies of hadronic events. Their
suitability for determining a, can be examined with respect to the related criteria of: insensitivity to the details of fragmentation; infrared stability; independence from the parton
resolution cut; and small higher order corrections.
The particle multiplicity of a jet is an
examplary detail of fragmentation. Variables which depend quadratically or more on the particle momenta or energy are sensitive to the multiplicity. These variables are unstable against
particle decays, for example p -+ 7rxr and against the emission of soft or collinear gluons. In
addition there is no manner to connect such variables with parton level calculations. Parton
resolution cuts cannot then be related to the experimental resolution and their functional
form cannot be chosen to minimize their effect on the measurement. Variables which depend
linearly on the particle momenta avoid these problems and are called "infrared safe".
The higher order (0 (a
3
)) calculations do not exist and can be roughly estimated in two
fashions: the relative size of the first and second order contributions, and the dependence on
the parton resolution cut. The reasoning behind the former guess is clear, the latter guess is
motivated by the parton level calculations of these variables, where the expansion in powers of
the resolution cut parallels the perturbation expansion. Stability as the resolution cut varies
indicates that next order corrections may be stable, stability as the resolution cut vanishes
indicates some hope of small higher order corrections.
In this study two additional criteria need to be imposed: adaptability to the calorimetric
measurements of the MARK-J detector and good discrimination of "3-jet-like" events as
opposed to the easier discrimination of "not-2-jet-like" events. The MARK-J calorimetry
does not measure the individual particle momenta in a jet, but a series of energy depositions in
the different shower counter layers, resulting from the interaction of an undetermined number
of charged and neutral particles. This dovetails with independence from the exact particle
multiplicity and imposes no further requirements. The discrimination of 3-jet events fits in
equally well with the criteria of small higher order corrections, because the next order is
expected to have a larger relative effect on the > 4-parton distributions than on the 3-parton
distributions.
Assuming a suitable variable has been chosen, the point is to compare the prediction,
which depends on a,, with the data. To do this for some variable, call it z, the prediction is
given by:
XThy(a)=
ZzAF(as)/
AF(es)
(4-1)
where the sum runs over the parton types defined at the end of 3.2, i= 2,3,3',4. A' is the
acceptance per parton type, and xi is the value of the variable per part on type evaluated using
accepted MC events. The denominator on the right side of Eqn (4
1) is the acceptance, as
Ai is 5 for all parton types, the acceptance is practically independent of ci. The dependence
21
of
XThy
on
a, enters solely through the fractional partial cross sections, Fi, and the quality
of a variable in picking out single gluon bremsstrahlung is then the difference between x 3 , X'
and x 2 , x 4 . The coupling strength is determined by the condition XThy (a,)
-
extension to obtaining the prediction for a distribution is straight forward.
xData =
0. The
To determine
ci (Vs-o) from measurements at several energies, Eqn (3-7) is used to evaluate a, at each
v's.
The determination of o, using three suitable but different variables is presented in
4.1, 4.2 and 4.3. The three methods are compared in 4.4. The "running" of the strong
coupling "constant" is discussed in 4.5, non-perturbative effects are quantitatively examined
in chapter 5.
4.1
a, from the Energy Flow Variables
In this section the Energy Flow variables [4-2] are used to quantify event topologies,
resulting in a measure of the rate of single gluon bremsstrahlung. Having this rate, the strong
coupling strength is determined. As depicted in Fig 4.1, the variables are determined stepwise
for each event. First, the thrust axis, , and thrust value, T, are found by maximizing T while
varying t:
i
ZE,
= [0.5,1]
(4-2a)
,
Thrust = T = max
t is also known as the event axis. Next, the major axis, mh, and value, M, are found by
maximizing M while varying m, keeping r^ I t:
- = 0,0.5] ,
Major = M = max
l
_
t,
(4-2b)
(t, rA)
defines the event plane. The normal to this plane is the minor axis,
n, the projected energy flow along n^ is known as minor, m:
for planar events
minor = m =
E ini -Al
= [0, M) , n^ = t X M^,
(4-2c)
minor is very close to the minimum projected energy flow.
At the parton level 2-parton events have T = 1, M = 0, m = 0. A 3-parton event has
T = 1 - Y,
M
=
2 pguon/,F/,
m = 0, where pflun is the gluon momentum perpendicular
to t and Y is from Eqn (3-8). Consequently the "Y-cuts" are used with the Energy Flow
variables. A "Mercedes-Benz" configuration, 3 equal energy partons separated by 1200, has
T = 2/3, M = 1/-V, m = 0. Fragmentation smears these directions. Including the detector
resolution, MC studies indicate a resolution of 5* for the event axis determined in qq events,
and 100 for the event plane (i.e. A) determined in qqg events. The values are also smeared,
but the coplanar events from gluon emission still have a significantly larger Major than minor,
i.e. they are oblate [4-3]:
Oblateness = 0 = Major - minor
22
(4-2d)
Again at the parton level, O(2-parton) = 0 and O(3-parton)= M =
F. An advan-
2Plu"n/
tage of Oblateness over Major and Thrust is the subtraction which effects a partial cancellation
of the hadronization smearing. Other advantages of Oblateness over Thrust as a variable to
discern single gluon emission are that 0 specifically tags these events whereas T yields the
"2-jettiness" of an event and that 0 is much less sensitive to the parton resolution cut [3-12c].
More detail is obtained by dividing the event into hemispheres using the (tm, n) plane.
For q-qg events the hemisphere with less energy flow perpendicular to C, the narrow (N) side,
usually (70% at \/F = 35 GeV) contains the "quark" jet. The opposite, or broad (B), side
contains either the "quark + gluon" jet or two distinct jets. Recalculating the variables on
each side yields TN, TB, MN,...,ON, OBThe data and MC predictions using both models with
a.(fi 0 ) = 0.144 are plotted
versus -/F for several of the Energy Flow variables in Fig 4.2a-e. Reassuringly, the agreement
is good. The dot-dashed curves are the predictions for qq events alone. For the mean values
(1 - T), (M) and (m), Fig 4.2a-c, the q-q effect, which is purely from fragmentation, is large.
These are not good 3-jet measures. Better separation comes using the fraction of events with
3-jet like topologies. The fraction of events with large
0
., Fig 4.2e, is the preferred measure
of single gluon emission, having a small contribution from fragmentation and only a slight
dependence.
VS
Qualitatively, requiring a large value of OB selects events with energy flow confined in
a half plane on the broad side. Using this variable the strong coupling is determined. For
example at 35 GeV with Yet = .04 QCD predicts (using the Ali model):
X2
= (fraction of accepted 2-parton events
(
(
3-parton events ,0q(a),
"
) = 0.25,
3-parton events ,Oi(ai),
"
=(
4-parton events ,O(ce.),
"
)=
)=
S=
X
X
=
4
with 0B > 0.3) = 0.04,
0.24,
0.35.
Folding in the acceptances and fractional partial cross sections and comparing these fractions
to the fraction measured, XData = 0.13, & la Eqn (4-1), determines the strong coupling
constant: a. = 0.14. This implies 76% of the events with 0B
>
0.3 are from first and second
order qqg events. Of course this percentage depends on the Ycut used to define qqg events.
For each of the five center of mass energy points in Table 2.1 ten to forty thousand MC
events were generated, fragmented, and put through the complete detector simulation for
each of the fragmentation models. The strong coupling constant determined at each
fi
is
plotted in Fig 4.3 (see also Table 4.1). Also plotted are the results from simultaneously fitting
all the data, (22 GeV < Vs
47 GeV), which yielded a,(fO;OB) = 0.134
0.012(syst.) using the Ali Model and o,(xo; OB) = 0.154
the Lund Model.
23
0.004 (stat.)
0.004 (stat.)
0.011(syst.) using
The systematic error estimation is summarized in Table 4.1, the following sources were
considered:
(1) Parton level resolution cut in the QCD calculation: This was varied from Ycut = 0.03
to 0.05. At 35 GeV this results in a variation F3 from 0.37 to 0.23 and of F3' from 0.14 to
0.09 but the variation in
a, determined at each of the five energies and over the full range is
less than a 5%. The measurement is stable over the experimentally relevant region. However,
from [3-12c] the parton level calculation indicates a&,(Ycut -+ 0) ~ 20%.
(2)
0
B
0
cut: Fig 4.4a shows the slight dependence of the measurement on the exact
value. The ordinate is a,(,F0 ; 0),
fitting the data 22 <
fv
. cut
i.e. the strong coupling strength determined from
< 47 GeV.
(3) Acceptance cuts: The cuts on EV, E1 and E, which define the sample of accepted events
(see 2.2.2 and Fig 2.10, 2.12) were varied. In addition, cuts on the maximum fractional
Ey and on cos 0j, the angle between the event axis and the beam axis, were imposed. The
motivation is two-fold: First, background events such as 2-photon hadron events or tau
pairs decaying into hadrons are characteristically more imbalanced and have either too
little visible energy (electrons down the beampipe or neutrinos) or too much (the energy
of an electro-magnetic shower is over estimated when evaluated as a hadronic shower) as
related in 2.2.1. Second, an event topology would be misconstrued if a large fraction of
the event vanished into one of the holes in the detector response. The largest holes are
around the beampipe ( 2.1), and the cut on cos 0j ensures the event is well into the active
solid angle. The eight sets of cuts examined and the resulting estimated acceptances are
given in Table 4.2. The acceptance varies by a factor of two but the result is stable for
each of the five energies and, as shown in Fig 4.5, for the fit over the entire
V/-
range.
At the two points with higher statistics the following sources could also examined:
(4) Oq: Fig 4.6 demonstrates the stability of the determination over a wide range of
The leftmost point was determined using the Ali model with
Uq
Uq
values.
~ 0, and gives a feeling
for the size of the corrections due to fragmentation. As mentioned, [3-18], the uncertainty
on
Uq
is ;
12%, i.e. 270 < oq(Ali) < 330MeV and 380 < Uq(Lund) < 470MeV.
(5) Detector simulation: The fast detector simulation used in this study is most suspect
in its treatment of low energy particles. To examine this cuts on the minimum energy
deposition per counter were imposed on the data and MC. Fig 4.7a shows the measured
and predicted variation of EV from these cuts, and Fig 4.7b shows the resulting variation
in the a 8 determination, which is minimal. Similarly, using only the information from
the inner calorimetry causes at most slight shifts in the results (marked "ABC only"
in Fig 4.7), as does use of the GHEISHA detector simulation (marked "GHEISHA" in
Fig 4.7).
The largest source of systematic uncertainty comes from the difference between the results
for the two fragmentation models. Averaging the two results in terms of a8
24
(fin)
and taking
the difference into the systematic error, the strong coupling is determined using the fraction
of large OB events to be ct,(V'o;
4.2
O)
0.004 (stat.)
= 0.144
0.015 (syst.).
a. from the Jet Multiplicity
In this section the number of partons and hence the rate of gluon emission is deduced from
the number of observed jets. An "observed" jet is a high density of energy flow surrounded by
a region of lower energy flow, such that the width of the jet is less than the angle between jets.
Experimentally this is quantified with the use of a cluster algorithm. Many such algorithms
exist [4-4], here one designed for the MARK-J calorimeters is used.
In this algorithm the largest energy hit is taken as the seed for the first cluster.
within an angle
jet/2 of the seed vector are added into the cluster:
Hits
the cluster direction
equalling the vector sum of the Ei and the cluster energy being taken as the algebraic sum of
Ej I. The seed for the second cluster is taken as the largest energy hit not added into the first
cluster, and hits within Sj.t/2 of it are added to it, and so on, until all hits have been added
into clusters. Depending on bjet this yields an average of 10-30 clusters/event. Clusters with
an energy less the ejet - EV are then "dissolved", their constituent hits added hit by hit to
the nearest (in angle) cluster. Finally, for cluster pairs separated by an angle less than 2 3j,,t,
the lower energy cluster is dissolved. The jet multiplicity is taken as the number of surviving
clusters. The (fejt,jet) used in the algorithm correspond to the Sterman-Weinberg (Ccut ,Sut)
parton resolution cuts used in the QCD calculations, Eqn (3-9).
Counting the number of observed 3-jet events is a straight forward attempt to measure
the rate of hard gluon emission, which has advantages and disadvantages. On the positive
side, the intuitive interpretation of observing an event with 3-jets is much cleaner than, e.g.,
observing an event with 0 B =0 .35. On the negative side, the resolution criteria are tighter.
For a q-qg event where 0. can tag a gluon jet partially merged into a quark jet (the 'broader
"quark + gluon" jet' of 4.0) a cluster algorithm has trouble.
Using a sufficiently loose
(Ejet,Sjet) to identify 3 jets in such an event leads to jet multiplicities of 5 or 6 in other events,
which are difficult to interpret. Related to this is the difficulty in estimating the effects of the
parton level resolution cuts. With these provisos in mind the method is applied.
Fig 4.8a-c show the measured jet multiplicities and the MC predictions as a function of
center of mass energy for a typical value of (fjetet). Again using the Ali model at \/s = 35
GeV as an example, Table 4.3a gives the fractional jet multiplicities for each event type.
Comparing the rate of 3-jet events between data and MC determines the strong coupling to
be a, = 0.12. Table 4.3b gives the resulting predicted fractional contribution of each parton
type to each jet multiplicity.
A similar number of MC events as in 4.1 were generated and the strong coupling strength
was determined by comparing the predicted and measured rate of 3-jet events. The results
are given in Fig 4.9 and Table 4.4.
For the simultaneous fit to the entire
25
fV
range the
values a(N/s; JM) = 0.123
cs(Fo; JM)
=
0.003 (stat.)
0.143
0.035 (syst.) when using the Ali Model and
0.003 (stat.)
0.035 (syst.) when using the Lund Model gave the
best description of the data. The systematic uncertainties, summarized in Table 4.4, were
estimated as for the rate of large 0B determination, but the interplay of the (E, 6) cuts on the
parton and experimental levels is much stronger than the corresponding relationship between
Ycut and the cut on
0
.. Fig 4.10 shows the relationship between the a, value determined
and the fraction of 3-jet events predicted to have come from qzg initial states for a range of
(EjetSjet) and (Ecut,bcut). The points which are plotted as crosses yield a poor description of
the rate of 2- and 4-jet events. The errant points have either a very tight (fiet ,jet) compared
to (Ecut, cut) ® (fragmentation smearing) or a too loose an (Ejettjet). The errors quoted in
Table 4.4 do not include these points. Fig 4.11 shows the stability of the result with respect
to different criteria used to define the event sample. Fig 4.12 shows the variation of the result
with
Uq.
Fig 4.13 shows the disappointing sensitivity of the method to a minimum energy
per counter cut. The observed jet multiplicity decreases rapidly with increasing min(Ei) and
indicates this method is dependent on the details of the detector response. This is despite
Eqet
being taken as a fraction of the energy used. Combining this with the systematic error from
the difference between the two fragmentation models the QCD scale parameter is determined
to be a,(A/so; JM) = 0.133
4.3
0.003 (stat.)
0.037 (syst.) using the observed rate of 3-jet events.
a. from the Energy-Energy Correlation Asymnetry
In this section the modification of event shape due to gluon emission is extracted from
the correlations between the energy flow in different parts of the event. This is achieved with
the Energy-Energy Correlation (EEC) function, Z1, first introduced by Basham, Brown,
Ellis, and Love [4-5]. Experimentally, the pairwise products of fractional energy deposition,
Ej Ej/E' , are plotted versus the cosine of the angle between them, cos Xij, for all events:
1 d~c (cos X)
01
dcos
xdcs
a X
1for
XN
Acos
V
Nevents i~j.A Ey2
Acos~
Icos X - cos Xij < -Acos X)
.
(4-3)
The intuition is best served by considering the parton level: A 2-parton event has entries
only at cos X =
1. A 3-parton event has a reduced entry at cos X = + 1 (the self correlation
term), no entry at cos X = -1, and three entries spread across -1
< cos x <
+1. A 4-parton
event has a further reduced entry at cos X = +1 and six entries spread over -1
< cos X < +1.
Both 3- and 4-parton events have an entry at cos X = cos 26, where cos 26 was used in the
definition of the Sterman-Weinberg parton resolution cuts, Eqn (3-9).
For this reason the
Sterman-Weinberg cuts are used with the EEC and the EECA, defined below.
The utility of the measure comes from the fact that qqg events contribute more at negative
cos X than at positive cos X, while qq events contribute only at cos X =
26
1 and 4-parton events,
being more isotropic than qqg events, contribute more symmetrically about cos X = 0. This
is enhanced by forming the EEC Asymmetry (EECA), A, defined as:
A(cos X) = 1 [d~c(cos(,r - X))
01
d COS (r - X)
cs7-X cs~ --1<5cos X50. (4-4)
d.c(cos(X)) 1 ,
d cos( X)]
_
Still at the parton level, qq events are zero for the entire distribution, while 3- and 4- parton
(the self correlation term) and cos X = - cos 26.
events have negative entries at cos X = -1
Hadronization distorts the parton level considerations. The back-to-back jets from a q-q
event bleed inward from cosX =
1 in ZC, but this smearing is symmetric and cancels in
A. The entries from a qqg event are likewise smeared, but this smearing is again symmetric.
Averaging over events the EECA still bears the imprint of the underlying QCD matrix elements. Specifically, the region away from cos X = -1 is populated by 3-jet events. Therefore,
to determine the strong coupling strength, the distribution is examined for cos Xo < cos X 5 0,
where the functional dependence is given by the QCD matrix element and the area is ~ proportional to the rate of hard gluon bremsstrahlung, i.e., to a.. If - cos 26cut < cos X0 , this
has the added advantage of excluding the region of the parton level resolution cut, where the
perturbative calculations are least sure (this is further examined in the next chapter). Indeed,
parton level calculations indicate that the EECA distribution away from cos X = -1 is stable
with respect to the parton level resolution cuts
(Cc
ut), even in the limit of very loose cuts
[3-12c], [4-6]. Another advantage of the EECA compared to the methods of 4.1 and
4.2, is
that it requires no complicated algorithm to pick out event or jet axes (these algorithms can
become confused by, for example, holes in the detector response).
Looking at the definition of the EEC, Eqn (4-3), and remembering that hadrons enter the
detector packed together in jets, a positive bin-to-bin correlation is expected. Considering
an event, a large entry in one bin is accompanied by entries in the neighboring bins. This
correlation carries through to the EECA. These correlations are accounted for as described
below.
The measured and predicted EEC and EECA distributions at the five center of mass
energies are shown in Fig 4.14 and 4.15. Note that the MC histograms were generated with
a single value for a,(v%), adjusted according to Eqn (3-7), and this is not the "best fit"
value for the EEC(V ), Fig 4.14. The EECA falls quickly with cos X (note the logarithmic
scales). Fitting the integrated EECA,
A,(cosXo) =
o
A dcosX would reflect primarily
the contents of the bins near cos Xo. Instead the strong coupling strength is determined using
the EECA by minimizing the following X 2 :
X 2 (a)
=
(Alat
-
AM 0
)
(ata
_ AMC(&,))
k,L
where the prediction for A in cos X bin k is, from Eqn (4-1),
AMC(c,) = Z
A'AF(a.)/acceptance
% a,('s) see Eqn (5-1),
and where the covariance matrix V is defined from the EEC as:
27
(4-5)
+[same
~L~)1M
k
+
i
MC
at a
2()D
2(E)
,(pc
o 0_Lc)_
Vk
k
cos Xj =cos(7r -Xk)
Z
or
kevent
events
Acos X . .
ev1eNt1
Zevent\
ZkN
IN
(z
N
WE)
(event
events
2
E2
I
N
events
for ICOSXk - COsXij < IAcos X
V accounts fully for the bin to bin correlations. Fig 4.16 and Table 4.5 give the results of the
fits at each energy. Fig 4.17, the contribution to the X 2 versus (k, 1) for one of the fits, shows
the importance of the off-diagonal terms in V. To determine
a. (,/59) the X 2 was summed
over the five energy points. The best fits (one parameter, 88 degrees of freedom) were obtained
at: a8 (/Fs
0
; A) = 0.108
and ci(vF/o; A) = 0.121
0.005 (stat.)
0.006 (stat.)
0.010 (syst.) with a X 2 /d.f. = 1.54 using the Ali Model
0.010 (syst.) with a X 2 /d.f. = 1.50 using the Lund
Model. These X 2 /d.f. are discussed below.
The systematic errors were estimated as in 4.1 and 4.2. Particular attention was paid to
the variation of the measurement and the chi-square with cos Xo. Fig 4.18a shows the X 2 /d.f.
for different (Ecut, cos Xo) and Fig 4.18b shows the resulting variation in the determination of
cs (VFo). For cuts away from the 2-jet region, cos X Z -0.9, both models show no variation
with COS Xo. The results of the Lund model is also stable with respect to variation of the parton
level resolution cut, Ecut. The results of the Ali model display a ~ linear dependence on ccut.
As mentioned in
3.3, this discontinuous response of the model when varying the parton level
resolution cut results from the anzatz of independent fragmentation. This deficiency appears
despite the use of the EECA, which is relatively free from fragmentation effects. Because,
within reasonable limits as explained in 3.2, no particular Ecut is preferable, this dependence
is taken into the systematic error for the model.
Other estimations of systematic uncertainty were made as detailed in 4.1, the attendant
plots are presented in Fig 4.19 (Acceptance cuts), Fig 4.20
(O-q)
and Fig 4.21
(min E1/counter).
The determination is quite stable against variation of all three. The main single source of
systematic error comes from the discrepancy between the two fragmentation models. Taking
the simple average of the results from the two models, and increasing the systematic error
to include the discrepancy, the strong coupling constant evaluated at \/% = 34.6 GeV is
0.018 (syst.).
determined using the EECA to be: c,(\5 0 ; A) = 0.114 0.005 (stat .)
28
4.4
Comparison
In the preceeding sections determinations of the strong coupling constant, a,, as a func-
tion of center of mass energy have been presented using three very different methods. The
variation of the coupling strength with energy, as predicted by perturbative QCD to complete
second order, has been used to simultaneously fit the data at all energies. Before directly
addressing the running of the coupling constant and whether the dependence is that given
by Eqn (3-7), a set of a, values must be selected. Therefore, the relative merits of the three
methods will be discussed.
Comparing the results given in Tables 4.1, 4.4 and 4.5, the three methods show a systematic offset between the results when using the Ali and Lund models, the
with Lund model are ~ 14% higher in
a values obtained
a, than those determined with the Ali model. Because
the two models represent extreme viewpoints on how fragmentation proceeds, the results are
averaged and the offset taken into the error.
More difficult to estimate, especially as a function of energy, are method related systematics.
Because the three methods used are well defined at the level of observation and
at the level of partons, it is reasonable to look for systematic variations at the parton level.
These include the effects of omitting higher order terms and imposing resolution criteria in
the perturbative calculation.
The method of jet multiplicity has a large systematic uncertainty associated with the
resolution criteria (e.g. see Fig 4.10). The number of jets is a discreet measure so that parton
configurations on the resolution boundary make a discontinuous contribution as the criteria
are varied. This could be avoided by setting the algorithm parameters well above (parton level
resolution)((fragmentation)((detection). From Eqn (3-3) hard 3-jet events are rare, so the
result of such a hard criteria is that the number of three jets tends to statistical insignificance.
In addition the method is dependent on low energy hits as shown by comparison with the MC
calculation (Fig 4.13). The utility of the method was that it is a straight forward attempt
to tag gluon bremsstrahlung. In conclusion, it qualitatively demonstrates QCD, but is not
sufficiently sensitive.
Use of the Energy Flow variables, specifically the rate of events with large
0
B,
yields
results which are stable with respect to a variation of the parton level resolution criteria over
the experimentally relevant region. In particular the results were found to be insensitive to
finite size Yeat, as given in Table 4.1. However, an estimate for the limit Ycut --+ 0 showed a
change in the relative rate of large 0 . events leading to an order of magnitude 20% reduction
in a,. This could indicate a sensitivity to higher order contributions when the cut is set to
small values, and these have not been calculated.
The third method does not rely on the specific event topology.
The results from fit-
ting the EEC-Asymmetry function have been shown to be relatively free of detector related
29
systematic uncertainties and from fragmentation effects. Additionally, the parton level prediction agrees well with the observed distribution and the effect of imposing the parton level
resolution criteria is less than 2% on the parton level (again, see [3-12c],[4-6]), although the
implementation of the method using the Ali model showed a larger variation. So this method
is free of the problems found in the other two methods.
Returning to Table 4.5, the chi-square per degree of freedom is
~ 1.5 for the fits using
both the Ali model and the Lund model and at both of the high statistics points, (0)
=
35, 44 GeV. Comparing the data taken at the same - but at different times ( 1 year) yields
a X 2 /d.f at or below unity. The fit residuals versus cos X and versus -,F do not indicate any
systematic trend. To account for the high X 2 the quoted statistical errors in Table 4.5 have
been increased by 25%(
=
[(X2/d.f.)]I
- 1). Because the use of either model yields fits of
equal quality, the a, values are averaged.
An independent analysis of the data and MC, internal to MARK-J but using separate
calibration, selection and reconstruction procedures to those presented in
2.2.2, obtained a,
values that were consistent with those presented in Table 4.5. The two sets of results are
compared for the different energies in Table 4.6. For each model at each energy the statistical
error on the averaged result is taken to be the larger of the two errors.
The difference is
accounted for in the systematic error. The combined result from the two models,the bottom
row in Table 4.6, is taken as the most accurate and precise determination of the strong coupling
constant at each of the five center of mass energies. The error has been increased to include
the uncertainty due to fragmentation as reflected by the two models.
A similar comparison can be made for the results obtained from fitting the MC predictions
to the data over the entire energy range, with the running of the coupling constant given
by Eqn (3-7). The results from the independent analysis are consistent (for the Ali model
ci,(W 0) = 0.108
0.005 from Table 4.5 compared with 0.114
analysis; for the Lund model a,(Vs-) = 0.121
0.004 from the independent
0.006 compared with 0.131
0.006).
Averaging these values yields the strong coupling strength to complete second order in
QCD:
ci(,(F= 34.6 GeV) = 0.119
0.005 (stat.) t 0.020 (syst.)
where the uncertainty from the fragmentation models has been placed in the systematic error
along with 0.008 to account for the difference of the two analyses. Evaluating this result for
the QCD scale parameter yields:
A = 78 i2 (stat.) i12 (syst.) MeV
from Eqn (3-7), that is, in the MS renormalization scheme.
This value of the scale parameter in agreement with the earlier analysis of the MARK-J
data [4-7], which also used the second order calculation outlined in y3.2.
30
In the three years that second order calculations have been available, many other groups
have presented determinations of the strong coupling constant using multihadron data from
e+e- annihilation and several topological measures, including the EECA. Unfortunately for
the field, the handiest second order calculation was that of [3-12b]. As mentioned, this calculation was too approximate and the a, values reported have been shown to be 20-30% too
large. This has been understood and the increased values are reproducible using their method.
However, since this was incorrect, it precludes showing these determinations for comparison
[4-8].
Recently, determinations of the scale parameter using an accurate second order calculation have been published by the PLUTO collaboration [4-9] and reported for a combination of
the PLUTO and TASSO data [4-10]. Both of these determinations used the EECA, and both
find a A value in agreement with that given above, the PLUTO analysis finding A between
100 and 300 MeV.
The QCD scale parameter can also be determined from deep inelastic scattering, where
the running of the coupling constant appears as a logarithmic dependence of the structure
functions on q 2 , the momentum transferred by the lepton probe squared. Several difficulties
with the determination are discussed in [4-11]. A recent summary of the results is A between
100 and 500 MeV at a mean q 2 of 100 GeV2 [4-12]. The measurement of the strong coupling
constant from onia decay is discussed in the following section.
4.5
Measurement of the Running of the Strong Coupling Strength
There is no existing model of a non-running strong coupling constant. A first attempt to
test the validity of Eqn (3-7) was made with the ad hoc assumptions that both the fractional
partial cross sections, F', and the coupling, call it aNR, are independent of center of mass
energy. Following this assumption the F used for the non-running analysis were calculated
without including the effects of photon radiation in the initial state. In all other respects the
analysis was identical to that of 4.3. The resulting X 2 /d.f. using either fragmentation model
was 10% larger than that obtained assuming Eqn (3-7), which does not favor or disfavor a
running coupling constant. Over the energy range considered the expected variation in a, is
simply not discernible, given the size of the statisticalesystematic errors.
A detectable variation is provided by including the lower energy measurements from
the decays of the T and T'. The QCD understanding is that the bbs bound states decay
at rest into three gluons, as shown in Fig 4.22a.
This is an as process and so allows a
fairly sensitive measure of the coupling. As in the continuum analysis presented above, the
difficulty in separating the non-perturbative and perturbative is overcome by factorization.
The prediction for the hadronic decay width is a non-perturbative term including the wave
function of the bound state times a perturbative term from the decay [4-13]. The leptonic
decay width (Fig 4.22b) can be similarly factored. Forming the ratio of these two widths the
31
non-perturbative term cancels and the perturbative terms are calculable, indeed calculated
P(T -+ hadrons)
P(T-+
p)
10(7r 2
-
9) a(q)
817rQ2
a
f1+
2
2q
7r
'
[4-14]:
IM
where Qb = 1/3 is the charge of the b-quark and bo is from Eqn (3-5) with Nf = 4.
The choice of renormalization scale, q, is a bit more flexible than in the continuum
analysis, where the scale was set by the energy of the virtual photon, q = \,
in the calculation
of the closed loop diagrams of Fig 3.4. This is despite the application of the MS scheme in both
calculations. Usually the scale suggested in [4-14] is chosen, so that the term inside the square
bracket equals zero, i.e. the dependence on the higher order, a4, term is minimized. Using
this and the data from the CLEO collaboration [4-151 yields: a,(q
from the T decay, and a.(q = 0.48MT,) = 0.151 +0.026
=
0.48 Mr) = 0.165 0:08
from the T' decay.
Another determination is possible using the radiative decay width of the upsilon, Fig
4.22c. Again this is calculated and the scale determined in [4-14]:
-+
+ hadrons)
r(T -hadrons)
36Qb
5
a
a,(q)
f
a(q) [ 2
1
2
0 6
]
'
r(T
where the scale is taken at q = 0.157 Mr. The data from the CUSB collaboration yield [4-16]:
a,(q = 0.157Mr) = 0.226 0067 from the T decay, and a,(q = 0.157MT,) = 0.197+0123
from the T' decay. Again, other choices of scale are possible, but, practically, they yield the
same result for A evaluated in the MS scheme.
Two systematic uncertainties in these determinations must be mentioned. First, the
a,
values would increase if the T decayed into glueballs [4-11]. This has not been observed [4-17].
Second, the radiative decay width measured has a dependence on the model used to fit the
photon spectrum [4-18]. Below neither of these effects are included in the error; the published
numbers are taken at face value.
One point remains to be clarified before testing the running of the coupling constant:
how to compare measurements involving different numbers of active flavors, Nf. The strong
coupling constant,
a., is the value directly accessible by experiment and it should be contin-
uous and monotonic. The QCD scale parameter, A, can only be derived from the data using
formulas like Eqn (3-7), but it is the one free parameter of QCD and should be independent of Nf. Remembering that Eqn (3-7) is from a perturbation expansion which starts with
massless quarks and is only valid for V = q
> 2Mf, the mass of the heaviest active flavor,
it should not be applied at and just above flavor thresholds. The measurements from the T
decay have Nf = 4. The measurements using the EECA, with F
the b-threshold and have Nf = 5 (threshold effects falling as
;> 22 GeV are well above
3f3). In either region the same
value of A is used. Between the two regions the coupling is assumed to decrease smoothly,
but, as no measurements from this region are considered, the functional form is unimportant.
32
To test the running of the coupling constant and determine the validity of Eqn (3-7), the
combined a, values obtained with the EECA (Table 4.6) were used with the points from the
T decays. The best fit to the 9 points using Eqn (3-7) was obtained at:
A = 98i1 MeV, with a X2 = 5.2 for seven degrees of freedom.
This fit and the
a, points are plotted in Fig 4.23. Evaluating the x 2 , the fit corresponds to
the 59% confidence level (CL). Fitting for a constant aNR, treating the EECA measurements
as described above, yielded:
0.004 with a X2 = 44.5 for seven d.f.,
which has a CL below 10-
4
.
a R = 0.151
Clearly the hypothesis of c, running according to Eqn (3-7) is strongly favored over the
hypothesis of a constant a,.
To quantify this statement, use is made of the F-test, which depends on the ratio of
the two x 2 .
This has the advantage of separating the statistical spread of the measured
points from their deviations to the given hypothesis [4-19]. For seven degrees of freedom the
evaluation is: the running hypothesis is favored over the non-running hypotheses at more
than the 99% CL. This conclusion and the value of A determined are effected only slightly
when dropping the "good" points at q = 0.48 MT and
fi
and running is favored over non-running at the 92% CL.
33
= q = 34.6 GeV, then A = 92 MeV
Chapter 5
A Stringent Test of QCD Versus Energy
The major difficulty in testing perturbative QCD though the analysis of hadronic final
states is the connection of the calculable parton level predictions with the observed hadrons.
The systematic offset in a, values obtained when using the Ali model and the Lund model
reflects this difficulty (see Tables 4.1, 4.4 and 4.5 or 4.6). To arrive at a quantitative test
perturbative QCD versus energy, the Vfs dependence of the event topology is examined with
the use of the Energy-Energy Correlation Asymmetry.
To first order in a, the parton level prediction for the EECA has been derived analytically
from Eqn (3-3) [4-5]:
-
X
f (C) =
The
a(C),
6Crl(1 -
a8
a(C) = f (1 - C) - f (C),
[2(3
-
1
= -(1 - cos X),
2
(5-1)
6C+ 2C2) ln(1 - C) + 3C(2 - 3C)]
(
A(cos x) =
F dependence enters only through a,. Extension to second order in a, and inclusion
of the effects of the parton level resolution cuts (EcUt
&ut)
and quark masses has of necessity
been done numerically [5-1] using the techniques and calculations of [3-12a] mentioned in 3.2.
The result is expressed as:
AQCD
a(
a R
,)+
where Ra is the second order correction to a and
(5-2)
f is the quark velocity. Over the PETRA
energy range the dependence on the quark velocities is linear and small
(
8% for f = b, and
so much less on average).
Again, the primary
i/
dependence of AQCD enters logarithmically through a,.
To facilitate the comparison of prediction and observation the measured EECA distributions, Fig 4.15, were corrected for initial state radiation, selection criteria and detector
acceptance:
A(Perfect Detector)
A(Data) = A(Measured) A(Full MCSmlto)
A(Full MC Simulation)
(5-3)
The correction ratio on the right hand side of Eqn (5-3) was found to be ~ 0.7 for cos X >
-0.80 and
~ 0.85 for -0.96 < cosX < -0.88.
More important than the value, the ratio was
found not to depend significantly on the fragmentation model, on Vs- for Ns > 30 GeV, or on
the parton type (i.e. it does not depend on a,). To obtain equal statistical significance the
bins in cos X are coarser than in 4.3.
Avoiding the use of any fragmentation model, the perturbative prediction of Eqn (5-2)
was fit directly to the corrected data for -0.72
< cosX < 0.0 and 22 < ,/F < 47 GeV. The
best fit was obtained at A = 85+1 MeV with a X 2 /d.f.
towards -1
=
1.1. If the cosX range is extended
the quality of the fit decreases, reaching a X 2 /d.f. = 1.7 for -0.88 < cosX
34
0.0.
The perturbative prediction consistently yields too large an asymmetry for cos X ! -0.60,
that is, outside the range of hard three jet events.
To probe whether the theory can match the observation over a wider range of events, the
expected non-perturbative effects need to be considered, preferably avoiding the complicated
fragmentation models of 3.3. For the EECA, using only the assumption of limited transverse
momentum fragmentation yields a simple parameterization [4-6], [5-2]:
A(QCD ® Fragmentation) = AQCD (1 + C(
)
(5-4)
Roughly C can be taken as a measure of the net fragmentation effect, so the exact functional
dependence of C on cos X is, as yet, unknown. Eqn (5-4), if valid, displays the utility of testing
perturbative QCD versus energy, the fragmentation effects having a different evolution with
Vs than the existing perturbative calculation, namely 1/fl compared to 1/log F. C was
estimated to be ~ -8 GeV for -0.9
< cos X by applying the Ali independent jet model to
the second order calculation [4-6] or by extrapolating the application of the Lund color string
model to the first order calculation [5-2].
The parameterization (5-4) does not include the non-perturbative contribution of q-q
fragmentation to the EECA. This contribution, Agq, is largest for cos X
significant for cos X > - cos
AData(X,
/)=
26
cut.
--
-1
but is still
Including this the full prediction is:
AQCD(XlA)(1 +
2-)
+ F 2 (V, A)A q(X,
where F 2 is the partial cross section for qq events as explained in
3.2.
)
(5-5)
In practice Agq
estimated with either the Lund or Ali model agrees. The average is used.
The data, corrected by (5-3) versus ,F for different cosX bins are plotted in Fig 5.1a-g.
Also shown are the best fit values from Eqn (5-5) to the region -0.88
22 <
< cos X < 0.0 and
fV < 47 GeV. The values determined for C(cos X) are in Table 5.1, the QCD scale
parameter was determined with this method to be:
A = 114+" MeV, with a X 2 /d.f. = 0.81
Allowing a different C in each cos X bin while requiring one overall value for A to fit the entire
distribution A(X, Vi) has two advantages. First, the data is well described up to the parton
level resolution cut jcosXJ L I cos28eua| = 0.9; fitting for a C independent of cosX yielded
twice the X 2 /d.f. Second, the error on A covers the uncertainty from fragmentation without
resorting to complicated models. Fragmentation is accounted for in a general but sufficient
fashion.
The C values determined above are in agreement with expectation, they are large and
negative for cos X near -1
and go to zero with increasing cos X. The ,
data in the bin closest to cos X = -1,
dependence of the
(Fig 5.1a), could not be described with Eqn (5-5). This
was expected as this bin has a large qq contribution and straddles cos X = - cos 2Scut.
35
To summarize the results of this chapter, the
fi
dependence of event topologies is well
described by perturbative QCD for 3-jet configurations and without fragmentation corrections. A similar conclusion was reached in [4-9,10]. This description is improved and can be
extended nearly to the resolution boundary with the inclusion of a simple power correction.
36
Chapter 6
Conclusions
In this study the strong coupling constant, ao, has been determined to complete second order over the center of mass energy range 22
< -,F < 47 GeV using the MARK-J
data collected at PETRA and several measures of gluon bremsstrahlung. Present theoretical
understanding prefers the determination using the Energy-Energy Correlation Asymmetry:
C,(5 = 34.6 GeV) = 0.119
0.005 (stat.)
0.020 (syst.),
where a 14% systematic uncertainty has been included to account for the discrepancy between
the results from the two fragmentation models. Within the MS renormalization scheme this
is equivalent to a value of the QCD scale parameter
A
=
78
t-
(stat.)
ii22
(syst.) MeV.
A value of A has been extracted from the data over a restricted region but without any
fragmentation model which supports the above result, A(frag. = 0) = 85
i"
(statistical) MeV.
Therefore, the relative effect of fragmentation models to pure QCD is small in this region.
These data do not conclusively favor or disfavor the expected variation of the coupling. By
including the
a, measurements from the T and T' decay widths the strong coupling constant
has been shown to run with the center of mass energy. The scale parameter determined in
this way is A(V/8 dependence) = 98 t+
The .
MeV, which confirms the value quota above.
dependence of the topology of hadronic final states has been studied with the
use of the EECA. Perturbative QCD to second order has been found to provide an adequate
description of this dependence for 3-jet configurations, specifically for cos X > -0.60, without
the inclusion of any hadronization effects. Other configurations, -0.90
! cos X < -0.60, are
well described by perturbative QCD plus a simple parameterization of the non-perturbative
effects. These effects have been determined. They range, at Fs = 35 GeV, from about 30%
for -0.88
< cos X < -0.80 to less than 10% for -0.60
< cos X.
The A value which best
describes the data in this fashion is A = 114 jr MeV, consistent with the values quoted. The
error brackets possible non-perturbative corrections.
Looking ahead, two ways to proceed in the study of the strong interaction at high energies
suggest themselves. Extremely difficult but necessary is the extension of the calculations to
higher orders of a,. More promising is the availability of data at higher energies, fs = 60 GeV
at TRISTAN and .
= 90 to 160 GeV at LEP. This study has attempted, in part, to show
that, even at PETRA energies, the uncalculable effects can be simply described, and that the
detailed predictions of the theory can be tested, provided proper variables are used. With
the increased center of mass energy these effects should cease to inter pose themselves between
observation and calculation. This should allow more rigorous tests, which will hopefully lead
to a more quantitative understanding of the strong interactions.
37
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38
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R. Brandelik et al., Phys. Lett. 86B (1979) 243;
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B. Adeva et al., Phys. Rep. 109 (1984) 131.
[2-3] The specific phototubes used are: AMPERX XP2230 for the A and B counters, RCA
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[2-4] M.-C. Ho, Ph.D. Thesis, NIKHEF, Amsterdam (1983).
[2-5] H.-G. Wu, Master Thesis, Hefei University (1984);
H.-S. Chen, Ph.D. Thesis, MIT (1984).
[3-1] My favorite: G. Kramer, Theory of Jets In Electron-Positron Annihilation
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S.
A.
S.
L. Glashow, Nucl. Phys. 22 (1961) 579;
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[3-4] This formula was first derived by J. Ellis et al., [1-17].
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commonly used: G. t'Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189.
[3-6] This theorem, originally stated for QED processes, has two parts: (1) The cross section
for e+e- -+ X, such that X is a sum over all indistinguishable configurations, is finite.
(2) The fully inclusive (i.e. total) cross section is finite in the limit of vanishing quark
masses.
T. Kinoshita, J. Math. Phys. 3 (1960)650;
T. D. Lee and M. Nauenberg, Phys. Rev. 133 (1966) 1594.
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P. M. Stevenson, Phys. Lett. 78B (1978)451;
B. G. Weeks, Phys. Lett. 81B (1979) 377.
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M. J. Marciano, Phys. Rev. D29 (1984) 580.
39
[3-9] M. Dine and J. Sapirstein, Phys. Rev. Lett. 43 (1979) 668;
K. G. Chetyrkin, A. L. Kataev and F. V. Tkachov, Phys. Lett. 85B (1979) 277;
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Phys. Rev. D21 (1979) 3112.
[3-10] A partial calculation is given in: B. Lampe and G. Kramer, DESY 85-030 (1985).
[3-11] A. Ali et al., Phys. Lett. 82B (1979) 285; Nucl. Phys. B167 (1980) 454;
J. G. K6rner, G. Schierholz, J.Willrodt, Nucl. Phys. B165 (1981) 365;
K. J. F. Gaemers and J. A. M. Vermaseren, Z. Physik C7 (1980) 235;
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R. K. Ellis and D. A. Ross, Phys. Lett. 106B (1981) 88;
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Nucl. Phys. B168 (1980) 490
A. Ali et al., DESY 79-86 (1979); Phys. Lett. 93B (1980) 155.
Important groundwork for this model was lain in:
G. Altarelli and G. Parisi, Nucl. Phys. B126 (1979) 298;
D. J. Gross, Phys. Rev. Lett. 32 (1979) 298.
[3-15] R. D. Field and R. P. Feynman, Nucl. Phys. B136 (1978)1.
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[3-17] T. D. Gottschalk, CERN TH 3810 (1984).
[3-18] Though starting from different ideas, both models specify fragmentation via the same
four quantities: a longitudinal fragmentation function, f(z), where z is the fraction of
the momentum carried by the sea quark parallel to the fragrienting quark (in the Ali
model) or fragmenting string (in the Lund model); the mean t ransverse momentum of
the sea quark, Uq; the fraction of pseudoscaler mesons produced, r; and the sea quark
flavor. The parameters which best describe the data used in thm study are:
For the Ali Model:
For u, d, s quarks:
For c, b quarks:
For all
quarks:
f(z) = 1 - a + 3a(1 - z) 2 , with a = 0.7,
f(z) = 1,
-q ~ 0.30 GeV.
For the Lund Model:
40
all
u, d
s
c
b
all
quarks:
quarks:
quarks:
quarks:
quarks:
quarks:
f(z)
a
a
a
a
Uq
=
=
=
=
=
~
(1 + a)(1 - z)a,
0.50,
0.35,
0.15,
0.05,
0.42 GeV
.
For
For
For
For
For
For
For both models r is taken as I and sea quarks are given flavors in the ratio u:d:s = 2:2:1,
based on inclusive particle spectra. The uncertainty on oq for either model is ; 12%, as
determined using the methods of:
R. B. Clare, Ph.D Thesis, MIT (1982).
[3-19] W. Bartel et al., Z. Physik C21 (1980) 37; C25 (1984) 231.
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G.
R.
T.
B.
C.
D.
D.
R.
Fox and S. Wolfram, Nucl. Phys. B168 (1980) 285;
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[4-2] see A. DeRdjula et al., [1-17] and D. P. Barber et al., [1-16].
[4-3] Oblateness was suggested by H. Gorgi.
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[4-8] However, a recent summary of PEP and PETRA results on a. is given in:
R.-Y. Zhu, Proc. of the DPF Conf. (Portland,Oregon, 1985), to be published.
[4-9] Ch. Berger et al., Z. Physik C28 (1985) 365.
[4-10] F. Barreiro, DESY 85-086 (1985).
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to be published.
[4-13] This factorization holds to at least the first non-leading order in a,. The subject is
reviewed in: E. Remiddi, E. Fermi Int. School of Phys. (Vareniii, 1980).
41
[4-14] P. B. Mackenzie and G. P. Lepage, Phys. Rev. Lett. 47 (1981) 1244;
S. J. Brodsky and G. P. Lepage, Phys. Rev. D28 (1983) 228.
[4-15] P. Avery et al., Phys. Rev. Lett. 50 (1983) 807; and references therein.
[4-16] R. D. Schamberger et al., Phys. Lett. 138B (1984) 225;
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and see [3-12c] and [4-6].
[5-2] L. S. Brown and S. D. Ellis, Phys. Rev. D24 (1981) 2383;
S. D. Ellis, Phys. Lett. 117B (1982) 333.
42
Data Summary
(F)
f dt
14*
22
14.0
1.6
22.0
3.2
33-36
39-43
43-45
45-47
GeV
34.6
87.5
41.3
6.7
44.1
18.2
46.1
6.0
GeV
pb-1
#
thousand
1.1
3.5
1.4
22.7
2.0
2.3
Events
Acceptancet
64.
73.
76.
<
77.
* uncertainties about b-quark production and decay limit the utility of the 14 GeV Data
t model uncertainties lead to a systematic error of ~ 2% on the acceptance
-
-%
Table 2.1
Summary of the event sample upon which this study is based.
43
Strong Coupling Strength
From: Rate of Large
vFs (GeV)
22
rate(OB ;> 0.3)
.145
B
Events
33-36
39-43
43-45
45-47
.130
.116
.120
.128
Model
from:
0
as
22-47
as We)
Ali
.138
.140
.122
.122
.128
.134
stat. error
.017
.005
.012
.007
.012
.004
syst. error
.015
.010
.017
.017
.019
.012
YcUt
.008
.006
.013
.0132
.016
OB cut
acceptance
.008
.007
.003
.003
.002
.007
-
.008
.002
.003
Og
-
min(Ehit)
-
.006
.004
-
.002
.006
.007
.007
.003
.007
-
.006
.006
-
as
as(V%)
Lund
.162
.157
.145
.141
.141
.154
stat. error
.02
.005
.013
.008
.013
.004
syst. error
from:
Y,,ut
.019
.014
.011
.004
.013
.005
.015
.010
.016
.010
.011
.005
.003
.001
.002
.002
.000
.001
accept. cut
.009
.007
-
.006
.008
.009
Uq
.003
.006
.003
.007
OB
cut
Combined
-
a5 (v/so) = .144
.004 (stat.)
-
.015 (syst.)
Table 4.1
The strong coupling constant determined from the rate of large 0 B
events at the five center of mass energies and determined over the
entire energy range 22 <; %f < 47 GeV and evaluated at V/f = 34.6
GeV, and the estimates of systematic uncertainty.
44
Different Acceptance Cuts
used in Fig 4.5, 4.11 and 4.19
Cut#
E
>
Ev/i <
EI/Ey <
Ez/Ev <
cos 0 <
1
.50
.60
.60
-
1
2
1
3
I
.70
1.30
.30
.30
-
.70
I
-
.50
.50
I
-
Acceptance at \is=
using Ali
using Lund
.84
.86
1
1
.76
.78
.69
.71
1
1
6
7
8
.70
.50
.50
.85
.70
1.30
.30
.30
.80
1.20
.20
.20
.80
1.20
.20
.20
.85
-
.85
.62
.63
.58
.59
.50
.51
.43
.43
4
5
.50
.60
.60
.85
35 GeV
.66
.67
Standard Cut
Table 4.2
Different acceptance criteria used to define the event sample and so
to estimate the systematic uncertainty on a. and a ( so) from a
variety of sources, and the resulting acceptance at Vs = 35 GeV.
45
Distribution of Jet Multiplicities From Each Parton Type
2 :
3 :
3':
4 :
qq
qqg
qqg
qq(gg or q'4')
#
Jets
00(a0)
0 (C,)
0 (a2)
2
3
4
.81
.36
.47
.12
.17
.50
.43
.52
.02
.12
.09
.31
>5
-
Parton Type =>
.01
.01
.05
Table 4.3a
Predicted percent contribution of a given parton type to each jet
multiplicity (using the Ali model at fi
= 35 GeV, with cluster
algorithm parameters of (Ejet = 0.13, 6 jet = 19.40) and parton
resolution cuts of (Ecut = 0.10, 6 cut = 12.90). For example, 50% of
the first order qqg events are observed as 3-jets.
Composition of Each Jet Multiplicity by Parton Type
Parton Type 42 :
3 :
3':
4:
q-j
q~qg
qqg
q-q(gg or q'q')
#
Jets
0 (c,)
0 (a2)
0 (a2)
2
3
4
>5
.77
.18
.04
.01
.33
.50
.08
.09
.16
.54
.07
.24
.07
.54
.32
.07
Table 4.3b
The estimated fractional compostition of a measured jet multiplicity by parton type (parameters as Table 4.3a). For example, 58%
of the observed 3-jet events come from first and second order qqg
events (50% and 8% respectively).
46
Strong Coupling Strength
From: 3 Jet Multiplicity
(Ejet = .13,
5
cut =
VfI (GeV)
22
rate of 3 Jets
.382
12.90), (Ejet = .13, bjet = 19.40)
33-36
39-43
43-45
45-47
.301
.296
.291
.304
Model
as
22-47
as(WS-0)
Ali
.136
.116
.127
.125
.130
.123
stat. error
.027
.004
.007
.005
.008
.003
syst. error
.046
.041
.035
.021
.025
.035
5
.018
.008
.011
.008
.015
.014
acceptance
.009
.004
.007
.002
.003
.003
Uq
-
.012
.039
(e, ')cut, (E,
)jet}
min(Eit)
-
.005
.019
-
.010
.030
as
Lund
.134
.144
.146
.149
+.042
.015
.005
05
.010
.10
.006
06
.010
.10.003
syst. error
.053
.033
.033
.034
.035
.035
{(E, 6 )cut, (E, O)jet}
acceptance
.020
.010
.008
.014
.016
.012
.038
.004
.007
.001
.003
.002
aq
-
.013
.008
-
.008
stat. error
_____________
Combined
.090
as (VF')
-
as(Ne) = .133
.003 (stat.)
.037 (syst.)
Table 4.4
The strong coupling constant determined from counting the number of 3-jet events for the different center of mass energies and simultaneously from a fit to all the data 22 < F < 47 GeV (evaluated at -Fs=34.6 GeV), and the estimates of systematic uncertainty.
47
.143
Strong Coupling Strength
From: Energy-Energy Correlation Asymmetry
N/s (GeV)
22
33-36
Model
Ali
39-43
43-45
45-47
as
.068
22-47
as We)
.119
.071
.094
.122
.108
.005
.017
.010
.015
.005
.018
.014
.018
.013
.013
.010
.002
.010
.006
.002
.003
.015
.003
.009
.006
.009
.005
.003
acceptance
.014
.010
.008
.007
.005
.003
aq
-
.008
-
.004
-
.006
-
.002
-
.002
-
.002
0.7
1.5
stat. error
syst. error
from: Ecut
cosXO
min(Ehit)
x 2 /d.f.
+.02
0.8
1.3
1.5
1.5
cis
Lund
.080
as(Vo)
.133
.097
.106
.123
.121
stat. error
+:050
.007
.020
.012
.019
.006
syst. error
.036
.010
.023
.014
.018
.010
.022
.019
.020
.003
.004
.004
.002
.017
.014
.003
.012
.004
.002
.011
.013
.001
.004
.007
from: Ecut
cosXO
acceptance
Og
x 2 /d.f.
Combined
-
.006
0.9
1.3
-
.006
1.6
1.8
cs(F = 34.6 GeV) = .115
-
.006
1.0
1.5
.006 (stat.) t .012 (syst.)
Table 4.5
The strong coupling constant determined by fitting the EEC Asymmetry distribution using the X 2 defined in Eqn (4-5). The values
are for the five center of mass energies and the fit to all the data
22 < F < 47 GeV, evaluated at \F = 34.6 GeV. The prediction is made with the full MC simulation, including the Ali or
Lund fragmentation models. The statistical errors given have been
scaled to reflect the mean X 2 /d.f. Nominally cos X > -0.72 and
Ecut = 0.10 were used. The estimates of systematic u ncertainty
the sources considered are also given. The systematic error on the
combined value has been increased to reflect the uncertainty due to
fragmentation.
48
Comparison and Summary of a. Results
Determined with the EEC Asymmetry
39-43
33-36
22
,Fs (G eV)
43-45
45-47
using the Ali model
.119
.005
.071
.017
.094
.010
.122
.015
.119
.004
.090
.010
.105
.007
.115
.012
+.28
.119
.005
.081
.020
.100
.010
.119
.015
From Table 4.5
.080 +.050
.133
.007
.097
.020
.106
.012
.123
.019
Indep. Analysis
.110
.040
.144
.006
.102
.012
.116
.008
.118
.015
Average
095
8
.139
.007
.100
.020
.111
.012
.120
.019
-
.129
.012
.090
.023
.105
.013
.119
.019
From Table 4.5
.068
Indep. Analysis
.085
Average
.076
+.025
.030
using the Lund model
4
Combined
(Ali.086
Table 4.6
Comparison of the results obtained in 4.3 ("Table 4.5") and an
independent analysis of the data and MC, for a. at the five center
of mass energies, as determined from fitting the EEC Asymmetry.
The larger relative error was taken as the relative error on the simple average of the two sets of results. The combined value is the
simple average of the results for the different models, its error has
been increased to include the resulting uncertainty. The estimates
of systematic uncertainty from Table 4.5 need to be increased to
reflect the difference.
49
MeV
+5'
s114
Determination of AMg
and Fragmentation Effect from Vs Dependence
A(Dat) at Ns
Bin
<
C(cos x) (GeV)
-. 8 8
.304
.006
-. 88 < cos X < -. 80
.153
.003
-9.3
72
.0883
.0021
-6.5
+2.4
-. 72 < cos X
-. 60
.0527
.0013
-5.1
+2.6
-. 60 < cos X
-. 48
.0307
.0010
-0.5
-. 48 < cos X < -. 24
.0155
.0006
0.8
+3.4
+3.
2.9
.0006
-2.6
-. 80 < cos X <; -.
-. 24 < cos X
-. 00
A__
.0044
=_114
_MeV-4.4
no fit
+
-. 96 < cos X
=
Table 5.1
Results from fitting the EEC Asymmetry as a function of center
of mass energy directly to the QCD prediction with the possible
fragmentation effects in each bin accounted for by C(cos X)/ i,
as given in Eqn (5-5). The error given on Afs includes the uncertainty from the C parameters. The EECA measured at Ve= 34.6
GeV and corrected as in Eqn (5-3) is also given.
50
Figure Captions
2.1 Layout of the PETRA storage ring and experiments.
2.2 The integrated luminosity collected with the MARK J detector over the last six years
versus the center of mass energy, Vs.
2.3 The MARK-J Detector (a) end view, (b) side view, (c) cross section at constant q.
2.4 Energy deposited by hadronic events in the inner calorimeter (A, B and C in Fig 2.3)
divided by .F at (a) 35 GeV and (b) 44 GeV. The remaining energy is deposited in the
outer calorimeter (K). In this figure and in Fig 2.5-2.7, 2.10 and 2.11 the solid symbols
are the measured values and the connected open symbols or histograms refer to the two
different detector simulations discussed in
3.4 as noted. For this figure the GHEISHA
values are smoothed owing to poor statistics.
2.5 Energy weighted distribution of the inner calorimeter hits in cos 0 at (a) 35 GeV and (b)
44 GeV. The rectangles at the bottom of the figure indicate the radial and longitudinal
extent of the different counter layers.
2.6 As 2.5, except in the outer calorimeter. The rectangles indicate the extent of the different
layers and the amount of iron in between them, showing, despite adjustment with the
counter track 0, the decreased angular acceptance compared with the inner calorimeter.
2.7 Energy weighted distribution of the counter tracks in 4 at (a) 35 GeV and (b) 44 GeV,
showing that the corners at 45*,135*,..., are correctly modeled.
2.8 Schematic production diagrams of the signal (a) and the backgrounds (b-f)
-+ hadrons
(a) One photon multihadron production, e+e- -+
(b) Tau pair production, e+e- -+ (r -* hadrons + v)(r -+ hadrons or e + v)
(c) Two photon hadron production, e+e- -' Y -4 e+e- + hadrons
(d) Bhabha scattering, e+e- -- e+e- or e+e-(e) Electron-Beam gas scattering, eA -+ junk
(f) Cosmic Rays, pA -
junk.
2.9 Computer reconstruction of a "typical" one photon multihadron event (compare with
Fig 2.3):
(a) The entire detector: views from the end, side, top and an expanded view of the A
and B layers. The heavy black lines indicate the counter tracks constructed from energy
depositions.
(b) top and side view of hits and tracks fit in the vertex detector. Typically, when events
need to be scanned, this display and (a) suffice for classification. (c) Expanded end view
giving the energy deposited (in GeV) in each of the calorimeter elements and the hits in
the S and T drift chambers.
51
2.10 The fractional visible energy for hadronic events at (a) 35 GeV and (b) 44 GeV. The
smooth dot-dash curve is a gaussian fit to the data Ev/s
> 0.70 with a width of
- ~ 18%.
2.11 Mean values of detector related quantities versus Vs-: (a) the fractional visible energy;
(b) the fractional energy imbalance perpendicular to the e+e- beams; (c) the fractional
energy imbalance parallel to the beams; (d) the fractional energy measured in each of the
calorimeter layers.
2.12 The distribution of the 35 GeV data in AE., AE 1 , showing the final acceptance cuts
(the area of each box is proportional to the number of events).
2.13 The number of accepted hadronic events in
3.1 Conceptual view of the process e+e- -+
F which constitute the basis of this study.
hadrons.
3.2 Feynman diagrams for the order a, 1 calculation: (a) zeroth order pair production; (b) single gluon bremsstrahlung; (c) second order, virtual correction, 2-parton, which interferes
with (a).
3.3 Feynman diagrams for the order
a, 2 calculation: (a) tree level 4-parton; (b) third order,
virtual correction 3-parton, which interferes with 3.2b; (c) uncalculated, fourth order,
virtual correction, 2-parton.
3.4 Diagrams involved in calculation of the total cross section to order a,2. The scale, q,
is taken as the energy of the incoming = outgoing photon in the MS renormalization
scheme, q = V.
3.5 Schematic showing the application of the fragmentation models to a of a qqg event. In the
Ali independent jet model, (a), each parton (q or g) is fragmented into a jet of hadrons
with out considering the energy and direction of the other partons. In the Lund color
string model, (b), color strings which run between the partons are fragmented, as the
strings are assumed to have a ~ constant energy density per unit length, this correlates
the results of fragmentation i.e. the hadron energies and directions, with the overall
configuration of the initial partons.
4.1 Schematic showing the coordinate system of the Energy Flow Variables for a 3 jet event.
4.2 Energy Flow variables versus
c-,Fs)
F for the data (dark circles), the MC predictions with
= 0.144 using the Ali model (open squares connected by dashed lines), and the
Lund model (open diamonds connected by solid lines). The dot-dash curve is the qq
contribution estimated using either model. (a) the mean value of 1 - Thrust, (b) the
mean value of Major, (c) the mean value of minor, (d) the rate of accepted events with
Thrust < 0.8, (e) the rate of accepted events with
OB > 0.3.
4.3 The values of the strong coupling constant determined using the rate of large 0
events
versus Vi using the Ali model (open squares) and the Lund model (open diamonds). Also
plotted are the results of the simultaneous fits for ca(V/s?0 ) or the QCD scale parameter,
52
I
< 47 GeV using the Ali model (dashed curve) and
4.4 (a) Effect of varying the 0. cut on the
a (VO) determination from the data 22 < 5F < 47
A, to the entire energy region 22 <
the Lund model (solid curve).
GeV for the Ali model (open squares) and the Lund model (open diamonds). (b) The
measured variation of the rate of accepted events passing the
4.5 Effect on the
as(F0;
0
B
cut at 35 GeV.
OB) of eight different sets of cuts used to define the event sample
(see Table 4.2) versus the estimated acceptance at 35 GeV, using the Ali model (squares)
and Lund model (diamonds). The statistical error is ~ the symbol size, not including the
point-to-point correlations.
4.6 The
a,(O.) results obtained with different
Uq
values for the Ali and Lund models at
F = 35 GeV (dark symbols) and \/s = 44 GeV (open symbols). From [3-18]
a,
is reliably
determined to be one of the 3 central points for each model, excluding the leftmost point
which was determined using the Ali model with
Uq =
0.
4.7 Effects of imposing a cut on the minimum energy deposition per counter: (a) on the mean
value of Ey at 5F = 35 GeV (the dark circles are the data and the solid bar graph is the
MC prediction) and at ,
= 44 GeV (open circles, dashed bar graph). (b) on the a (0,)
determination. Also plotted are the results using the GHEISHA detector simulation and
using only the inner calorimeter (ABC).
4.8 The measured (dark symbols) and predicted (Ali model: open squares connected by
dashed lines, Lund model: open diamonds connected by solid lines) jet multiplicities
versus
F for (jet,6 jet) = (0.13, 19.40) (circles), (a) 2-jets, (b) 3-jets, (c) 4-jets. The
MC prediction is for (Ecut =
0 10 6
.
,
cut
= 12.90) with a,(,F/o) = 0.133.
4.9 The strong coupling strength and QCD scale parameter determined from the rate of 3-jet
events, plotted as in Fig 4.3.
4.10 The a, results using the rate of 3-jet events for different sets of (Cjet,,jet) and (cut,6cut).
The horizontal scale is the fraction of matched 3-jets, i.e. the fraction of 3-jet events
inferred to result from 3-parton events. For the nominal cuts, (et ,,6 et) = (0.13, 19.40)
and (ecut = 0 . 1 0 , 6 cut = 12.90), from Table 4.3b, this was found to be 58%. The points
plotted as crosses have a large discrepancy between the predicted and measured 2- and
4-jet multiplicity.
4.11 The effect of varying the acceptance cuts on
a,(\F/o) determined using the rate of 3-jet
events, as Fig 4.5.
4.12 The effect of varying
Uq
on a,(%,5 0 ; JM) values determined at %Is= 35 and 44 GeV, as
Fig 4.6.
4.13 The effect of imposing a cut on the minimum energy deposition per counter, as Fig 4.7b.
4.14 The measured (dark circles) and predicted Energy-Energy Correlation using the Ali
Model (dashed histogram) and the Lund model (solid histogram). The plots at different
53
energies are offset by a factor of 10'/2 starting at the bottom, e.g. the (Vs) = 41 GeV
plots are multiplied by a factor of 10.
The MC histograms were all produced with
ia,(V0) = 0.114, not the best fit to these distributions.
4.15 The measured and predicted Energy-Energy Correlation Asymmetry as in Fig 4.14, except the MC prediction is the average of the best fit to the two models a,(v')
= 0.114,
and, as the vertical scale indicates, the plots for each energy are offset by 1} decades.
4.16 The strong coupling constant determined by fitting the EECA with the full covariance
matrix defined in Eqn (4-5) using the Ali model (open squares) and the Lund model
(open diamonds). Also plotted are the results of the simultaneous fits for
a,(/Fs) using
the Ali model (dashed curve) and the Lund model (solid curve).
4.17 The x 2 contribution from each bin, as defined in Eqn (4-5), for the best fit at 35 GeV
to the Ali model with cosXO = -0.72,
(a) the positive contributions, (b) the negative
contributions. The area of each box is proportional to the magnitude of the contribution.
4.18 The effect of different (Ecut, cos Xo) on (a) the X 2 /d.f. and (b) c,50;
A) using the Ali
(x's) and Lund (diamonds) models.
4.19 The variation of a,Fs;
4.20 ci,(A) at V
A) when varying the acceptance criteria, plotted, as in Fig 4.5.
= 35 and 44 GeV for different Oq values used in the fragmentation models,
as Fig 4.6.
4.21 The effect of imposing a cut on the minimum energy deposition per counter on
a,(A) at
N/i= 35 and 44 GeV, as Fig 4.7b.
4.22 Lowest order diagrams important in this study for the decay of the T, (a)-+ ggg, (b)-. pp,
(c)--+
Ygg.
4.23 Result of the fitting for "running" coupling constant, 4.4. The two lowest energy points
hadrons)/(T, T' -+
-y
+
(squares) are from the ratio of the decay widths for (T, T' --
hadrons), [4-10]. The next two highest points (diamonds) are from the ratio of the decay
widths of (T, T' -+
hadrons)/(T, T' -+ pp), [4-9]. For all four of these points the scale
V, or more precisely, q, used is that suggested in [4-8]. The five highest energy points
(circles) are the a. points determined using the EECA from Table 4.6.
5.1 The EECA data, corrected with Eqn (5-3), versus
F for different cos X bins (open
circles) compared to the best fit from Eqn (5-5) to the region -0.88 < cos x 5 0 (dotted
line), and its component terms: the pure perturbative QCD prerliction, Eqn (5-2), (solid
line); the qq contribution (dashed line); and the perturbative predIction times the simple
fragmentation correction, AQCD(1 + C(X)//s), Eqn (5-3), (doi dash line).
54
r
Exp.JADE
10 8 m
HF-Hallen N
PLU>
Halle NO
Halle NW
DORIS
cJ1
(Ji
[INACH
E
Halle W
Halle 0
DESY
H NACTI
Figure 2.1
Halle SW
Exp. MAF KIJ
-Halle SO
HF-Hallen
4-108m-
S
syExp.TASSO
Luminosity
102
I
I
I
I
I
E(fLdt) =
I
I
I
I
I
I
I
I
I
132.4 pb -1
7
-4
Q.
4
.
.
101
CA1
4~)
r.O
100
'-4
t
* -..-...*
101'
I
10
..-.-.-
I I I
I
- -..
I
20
30
Vs
Figure 2.2
-
40
(GeV)
50
MARK J DETECTOR
(Cross Section)
r 7
a
(D
(1
( D
no
s
m
is
t
PARTICIPANTS:
RWTH - Aachen
SHOWER COUNTERS
DESY -Hamburg
MIT - Cambridge
TRIGGER COUNTERS
DRIFT TUBES
BEAM PIPE
Amsterdam
NIKNEFMAGNETIRON
CHAMBERS, MEDIAN
ORIF(
CHABER,
RIF
MEIANHEM
(I)
- RING
At
OUTER
DRIFT CHAMBERS,
MULTIPLIERS
DRIFT CHAMBERS, INNER
-Peking
JEN-Madrid
CALTECH - Pasadena
Figure 2.3a
57
WEIGHT Itotal : -. O0t
MAGNETIC FIELD: II T
"EN
-Madrid
MARK J- DETECTOR
..-.......I
N,
.1Xx5
-
A.1 SHOWER COUNTERS
TRIGGER COUNTERS
0k E
M MINI-BETA QIADRUPOLES
K CALORIMETER COUNTERS
OT DRIFT TUBES
S.TU.V DRIFT CHAMBERS.INNER
4 DRIFT CHAMBERSMEDIAN
P.R DRIFT CHAMBERS,OUTER
AL-RING
2 MAGNETIROC
3 BEAM PIPE
4 ROTATIONAL SUPPORT
COILS---
--
6
VACUUM PUMP
00
---mbridge
-EY-Hmbr
i
Figure 2.3b
*
WEIGHT:~4Oit
MAGNETICFIELO:.8
PARTICIPANTS.
RW TH -Aachen
OESY- Hamburg
MIT -Cambridge
NIKHEF -Amsterdam
EPI -Peking
J
?I. DESV 6181
10 PLanes
Driftchambei
ii 17 KGx M
ve-.v~v~v ...... ~ .......
-
45cm Fe
,ounter
Ir
tchamber
15 cm Fe
-
10 cm Fe
.
y.....v
. ............ .. .. ..
...
2.5cm Fe
12 Planes
Icm thick
calorimeter
counters, K
S,)T
Driftchamber
C..-12 Xo
mm Pb
B: 3 Xo
+ 5mm
A:= 3 X
scintiIlator
drift tubes
e+
em
Figure 2.30
59
Inner Cal. Energy
I
I
I
-
2.5
11111111
'
I
I
I
I
I
F1
ld N
Nd(C)
I
0 Data
--- Newman--- Gheisha-
33 -36 GeV
2.0
r
1.5
1.0
(a)
1<
- TCYT t
0.5
0.0
2.5
SData
- - - Newman-
43 -45 GeV
Nd(C)
1.5
Gheisha
1.0
(b)
-L
-r
0.5
-
2.0
-
ld N
L
II I
7-
~~
I
0.0
0
0.25
0.5
EABC
Figure 2.4j
60
I
i
1
0.75
S
1.25
1.5
Inner Cal.
I
I
I
I I
1
1
1-
-33 -36 GeV
:
--
1
1
Data
Newmar
-
I
0.8
1
d
EAB
N d cosO
0.6
0.4
0.2
*-
-
(a)
--
I
-
0.0
Data
-Newman
143 -45 GeV
0.8
E
EABC
N d cosO
0.6
0.4
OC-
(b)
0.0
--
-
0.2
-1
0
-0.5
Cos 0
Figure 2.5
61
0.5
1
Outer Cal.
- I
I
I
I
I
I
I
I
I
II
33 -36 GeV
I
-N
I
I
I
ama
0.8
1 dE1 /EK
NdcosO
0.6
0.4
m -Ne
-
0.2
-
(a)
Fe
0.0
0.8
--
1 dE,/EK
e
Fe
NdcosO
0.6
0.4
(b)
0.2
Fe
0.0
-1
-0.5
0
cos 0
Figure 2.6
62
0.5
1
Counter Track <
-33 -36 GeV
0.004
1
dE/Ev
Nd
5 0.003
0.002
(a)
0.001
Data
Newman
I II II I II I II II
I
I
0.000
I
II
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I
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1 1
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0.002
(b)
0.001
0.000
Data
Newman
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IllIllIll
I
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III
180
90
Figure 2.7
63
I
I
III
270
360
Signal
(a) One Photon
Backgrounds
hadrons
e
L
e+
e+
(btau pair
T
~"
or
r
e-
e
e
e+
(c) Two Photon
Multi hadrons
hadrons
e
e-
e +e
(d) Bhbh
e+
Scat ering
e
e-
e+
e+
e
e
e-
+-
hadronic junk
-
(e) Electron
Beam gas
(undisturbed)
YZ
hadronic junk
(f)Cosmic Rays
(undisturbed)
e-
Figure 2.8
64
-
ha drons
s
hUd rn"
-
Multi hadrons
I I
7
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..
7_________
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22.05 5
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qq only
0.6
0.5
I
00
I
I
I
I
I
0.4
A
0.3
0.2
0.1
0.0
10
Figure 5.1a
40
30
20
Vs
(GeV)
50
Corrected
Data
-. 80
5
-
QCD(1+C/j )+qq
- -- -- -
QCD(1+C/v-)
)
EECA( -. 88 < cos X
QCD
- - - --- -
qq only
0.25
0.20
A
...........
0.15
.........
0.10
r -
0.00
10
Figure 5.1b
20
40
30
Vs
(GeV)
-
0.05
50
EECA( -.
Corrected
Data
-
--
-
-- - -
<
80
cosy
X<
-. 72)
QCD
QCD(1+C/ -)+qq
QCD(1+C /
-)-
--- -- -
- -
qq only
0.150
0.125
0.100
A
01
0.075
I
0.050
I
I
I
I
I
I
I
LF~---[--~--kl-
0.025
0.000
10
Figure 5.l c
30
20
Vs
40
(GeV)
50
)
EECA( -. 72< cos Xi5 -. 60
QCD
------------ NQCD(1+C/V-)+qq
Corrected
Data
-
--
I
I
-
QCD(1+C/--)
- - -- - - -
qq only
0.08
0.06
A
0.04
0.02
I
I I-L -----
0.00
10
Figure 5.1d
30
20
NS
40
(GeV)
50
EECA( -. 60 < cos X:!; -. 48)
Corrected
QCD
QCD(1+C/-)+qq
-
Data
-
--
- --
QCD(1+-C
--qq only
0.05
0.04
A
-
I
I
0.03
0.02
0.01
0.00
10
Figure 5.1e
40
30
20
Vs
(GeV)
50
EECA( -. 48 < cos X < -. 24)
QCD
QCD(1+C/)+qq
Corrected
-- - - -
Data
QCD(1+C/---
- --
- - -
qq only
0.030
0.025
00
A
0.020
0.015
0.010
I
mC
Li
r[ -vJ
R
I~~~
0.005
0.000
10
Figure 5.lf
40
30
20
s
(GeV)
50
-
Data
-. 00
QCD
QCD(1+C/V )+qq
-
Corrected
5
)
EECA( -. 24 < cos y
-- - - -
QCD(1+C/ )-
- - - - - -
qq only
0.010
0.008
I
A
I~-
-a
.-.
-
-
I
0.006
0.004
0.002
0.000
10
Figure 5.1g
20
30
Vs
40
(GeV)
50