Measurement of the Nuclear Dependence and
Momentum Transfer Dependence of Quasielastic
(eel P) Scattering at Large Momentum Transfer
by
Naomi C. R Makins
B.Sc., University of Alberta
1989)
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
September 1994
Massachusetts Institute of Technology 1994
Signature of Author
...
- C7 1 - -
Department of Physics
July 19, 1994
f
Certified by ............
....... j ..
Richard G. Milner
Associate Professor, Department of Physics
Thesis Supervisor
Accepted by ............
MASSACHUSETTS
INSTITUTE
OFTFCHNOinry
I(CT
141994
LIBRARIES
'
George F. Koster
Chairman, Departmental Graduate Committee
Acknowledgments
Summarizing the contributions of the many individuals who have made my graduate
years such a formative experience is infinitely more daunting than summarizing the contributions of deadtimes and radiative corrections to my graduate data.
First of all, I must thank my advisor, Prof. Richard Milner, for providing a learning
and working environment which has been truly ideal. I recall going for lunch some time
ago with several fellow graduate students; with us was a prospective new student who
was visiting MIT (and whose presence provided us willing volunteers with a free meal
in exchange for answering her questions). I remember rhapsodizing enthusiastically on
the exceptional research environment at MIT, and enumerating some of the details of
my own work group with which I was so delighted. I remember, too, the subsequent
stares of astonishment from my colleagues. Little episodes like listening to their equally
rhapsodic lists of grievances about their groups make one deeply grateful for having an
advisor who is so thoroughly concerned for his students' well-being. One must never take
for granted someone who conducts study sessions twice a week to prepare his students
for the oral exams, who comes up with scholarship opportunities out of thin air, who
directs his students from Day One towards a thesis topic, who is willing to drive in from
Arlington at ten o'clock at night to read yet another thesis draft. Richard's outstanding
research program has provided each of his students with exceptional projects of enviable
quality. I have greatly appreciated his ability to give one the trust and space to work
alone, while keeping a watchful eye on the direction and relevance of one's endeavors I
have countless times benefitted from his keen and astute perception. He seems able to
constantly keep the end of a project in sight, to see directly the essence in a morass of
detail and confusion, and to have an utterly accurate sixth sense of the way things will
turn out. So many times, he has pointed out the Forest to me, while I analysed and
plotted only the Trees. I fervently hope that his intuition and ability to see with a wider
eye is a contagious one. I have been touched by the generous hospitality of Richard and
Eileen Milner, who have many times invited me to their home for sumptuous dinners. It
2
has been a great pleasure knowing Eileen, who has always had an encouraging word for
me and is a fabulous story teller.
I have been fortunate all around in the people I had to work with on NE18. From the
outset of the experiment, Rolf Ent was my immediate superior in the chain of command
and managed to teach a starry-eyed youngster with her head full of the ineffable beauties
of field theories how to be an experimentalist. I have benefitted tremendously, by deed
and by example, from his efficiency, his vast knowledge and intuition in nuclear physics,
his friendship, his tireless assistance on everything from analysis to getting this thesis
written, and his ability as a swimmer. Tom O'Neill and Wolfgang ("The Phantom")
Lorenzon formed the West Coast half of the NE18 task force, and it has been a pleasure
to know and
ork with such talented, able, and congenial people. I have been working
closely with Tom since the beginning of the experiment and I couldn't ask for a more
skilled colleague, or one more willing to share and explain his work in deatil. I am also
tremendously thankful that his codes are so easy to read. I am grateful, too, to Brad
Fillippone for numerous, extremely helpful local and long-distance discussions, and for
his strange and wondrous ability to understand exactly what you mean even when you
don't. Henk Jan Bulten deserves special mention for his expert assistance on so many
stages of this experiment; for his truly unmatched ability with spectrometer optics he will
forever be dubbed Mr. Matrix in my glossary. Finally, let me include John Arrington and
Eric Belz, and we have the entire membership of the Hell Week Crew. It was a delight
to be part of a group of people who were willing to do what it takes, and to have a good
time at it too. (All we need is a faded group photo and squadron insignia to complete
the war buddies clich6).
Janice Nelson, now an Operator Extraordinaire, was an undergraduate at MIT when
she worked for our group, and I cannot express how much I missed her greased-lightning
efficiency when she graduated. She claims to have returned The Brain, but I have my
doubts as I can't find it anywhere. And Dave "C++" Wasson, responsible for our radiative corrections prescription, is the type of theorist that every experimentalist should
3
have at the end of a Red Phone direct line. He is a wizard, a joy to work with, brilliant
and patient with his explanations, and I believe a highly skilled golf player.
You couldn't ask for more perfect running conditions than what we experienced at
SLAC. It was a rare treat to spend a year at a facility entirely staffed by helpful and
efficient people who were so clearly happy with their work. "It must be the weather!", we
exclaimed, but for whatever reason the MCC operators delivered a beam so stable that a
Monte Carlo simulation could hardly have done better. The American University group
must get special mention for their tireless and skilled assistance to us newcomers at the
End Station. A heartfelt "thank you" and the hope that we can do it again sometime go
to Lisa Andivahis, Ray Arnold, Peter Bosted, Thia Keppel, Allison Lung, Steve Rock,
Linda Stuart, and Zen Szalata. Finally, no summary of the SLAC membership to whom
a debt is owed would be complete without a mention of Mr. 0. Katt, whose vigorous
efforts did so much to keep the experimenters on their toes.
I would like to thank my thesis committee members, Prof. Ernie Moniz and Prof. John
Tonry for their encouragement, valued comments, patience, and flexibility -
what ordi-
nary committee would consent to read a student's thesis during weekends, in unstapled
segments hastily squeezed under office doors, and ultimately consent to attend a defense
on a Sunday? I am particularly grateful to Prof. Moniz for allowing me to profit from
his vast experience and understanding via discussions and astute observations.
I am
also grateful for his appreciation of Canadian dialects and figures of speech. Despite the
weather, the MIT physics department is the most accessible, concerned, and thoroughly
human example of a bureaucracy that I have ever encountered. I have been deeply impressed with the series of faculty-student roundtables and seminars on the employment
issues facing young physicists which the department has sponsored. The willingness of
the graduate student office to accomodate students by bending rules and deadlines to
the breaking point is unprecedented in my experience. Here, I must express my deepest
gratitude to Peggy Berkovitz, who has done so much for all of us and makes the graduate office a delightful place to visit. Peggy knows just exactly how grateful I am for
4
the aformentioned flexibility, and by some means unknown to modern science or ancient
alchemy has managed to keep my affairs in order for five years despite all of my unwitting
efforts to thwart her. I will also miss Joanne Gregory, who keeps life in order here at
the Laboratory for Nuclear Science. We've had some great conversations and some good
laughs; I'll ne,,,er forget the words of encouragement that she has provided at exactly the
right moments.
My classmates are a wonderful group of people, consciencious, thoughtful, and so
many with widely varied interests and fascinating backgrounds. Studying for the general
exams produces yet another set of war buddies, and I remember fondly all our discussions,
study sessions, and the old Friday dinners. I know every one of them will go far. A special
thanks to Jordina, Ole, Mike T, and Mike Y for some truly great times. I'm thankful
that physics is a small world - I'm confident I'll see you all again soon, and often. Same
thing goes to Bryon and Eric up at Bates - keep the tradition alive, you two! Thanks to
Kevin, for patiently teaching me all about the polarized target when I first got here, and
for helping me pack on the very last crazy day! To Thia, the kind of friend that comes
along once in a very blue moon, you've always been there for me and I can only hope to
do the same. Finally, I'm giving a copy of this thesis to Deborah Scott, and another one
to Mark Davey, so that they know how much I enjoyed the late night rap sessions and
pep talks. Good luck to both of you!
To thank my family would be to thank a limb or a sense for its existence. What we
do, we do together; the best of what I am is what they have given me. This work, as
always, is theirs.
5
Measurement of the Nuclear Dependence and Momentum
Transfer Dependence of Quasielastic (ee'p) Scattering at
Large Momentum Transfer
by
Naomi. C. R. Makins
Submitted to the Department of Physics
on July 19, 1994, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Experiment NE18, performed at SLAC, has measured the coincidence quasielastic crosssection for (ee'p) scattering from Q of I to 68 (GeV/c)'. This extends the existing Q'
range of such measurements by over an order of magnitude. Five targets were used: 11,
2H, 12C, 56Fe, and 197Au. To test our understanding of quasielastic scattering, the data
were compared with a Monte Carlo calculation of the experiment based on a conventional
nuclear physics picture. This calculation included radiative effects, using a prescription
based on the work of Mo and Tsai and recalculated in a coincidence framework.
The elastic hydrogen data were found to be well explained by standard parametrizations of the proton form factor. Spectral functions were extracted from the nuclear data
and found to be in good agreement with the Plane Wave Impulse Approximation (PWIA),
the deForest offshell electron-proton cross section ,,,, and Independent Particle Shell
Model spectral functions based on measurements made at Q _ 0.2 (GeV/C)2.
The nuclear transparency was extracted from the data, and examined for evidence of
colour transparency. This phenomenon, motivated by perturbative QCD considerations,
is predicted to cause a rise of the transparency with Q2 . No evidence of such a rise was
observed in the data. Also, the A-dependence of the transparency was found to be well
parametrized by a classical model of transmission through the nucleus.
Thesis Supervisor: Richard G. Milner
Title: Associate Professor, Department of Physics
6
Contents
1
Introduction
15
1.1 The Plane Wave Impulse Approximation
16
1.2
The Off-Shell
1.3
Previous
(ee'p)
Cross-section
Data
1.3.1
Independent
1.3.2
Spectroscopic
. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
Particle
Shell Model
Sum Rule
. . . . . .
. . . . . . . . . . .
1.3.3 Evidence for Multi-body Processes . . . . .
1.3.4
Nuclear
Transmission
. . . . . . . . . . . . .
1.4 Exclusive Processes at High Momentum Transfer
1.5
of Perturbative
1.4.1
Application
1.4.2
Counting Rules and Other Results of PQCD
1.4.3
Comparison
1.4.4
Arguments in Favour of a Higher Perturbative, Threshold
with Data
Colour Transparency
1.5.1
Experimental
QCD
.
. . . . . .
............
............
............
............
............
............
............
............
............
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
22
22
25
27
32
34
34
37
40
. . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evidence
21
. . . .
2 Description of the Experiment
41
44
50
52
2.1
B eam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.2
Experimental
55
Layout
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Target ..
. . . . . . . . . . . . . . . . . . .
2.4
1.6 GeV/c
2.5
8 GeV/c
2.6
Trigger
Electronics
2.6.1
1.6 GeV/c
2.6.2
8 GeV/c and Coincidence Triggers
Spectrometer
. . . . . . . . . .
Spectrometer
2.7
Data
2.8
Kinematics
. . . . . . . . . . .
. . . . . . . . . . . . .
Trigger
Acquisition
. . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
.................
.................
.................
.................
.................
.................
.................
.................
3 Data Analysis
3.1
Overview
3.2
Tracking
56
58
66
70
70
71
73
74
79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
3.2.1
1.6 GeV/c
Tracking
3.2.2
8 GeV/c
Tracking
3.2.3
Multiple
Tracks
. .
. . . . . . . . .
. .
. . . . . . . . . .
. .
. . . . . . . . . . .
3.3
Reconstruction of Events to Target . . . .
. .
3.4
Determination
. . . . . . .
. .
3.5
Tim ing . . . . . . . . . . . . . . . . . . . .
. .
3.6
Corrections
of E,
and p
. . . . . . . . . . . . . . . . .
3.6.1
Electronic Deadtime Corrections
3.6.2
Computer Deadtime Corrections
3.6.3
Proton
Absorption
. .
. . . . . . . . .
..............
..............
..............
..............
..............
..............
..............
..............
..............
..............
. . . . . . . . .
O verview . . . . . . . . . . . . . . . . . . . . .
4.2
Off-Shell
Prescription
for
o,
I I I I I I I I I I
8
79
80
83
84
86
89
90
93
93
95
104
108
4 Description of the Experimental Simulation
4.1
79
...............
...............
109
110
4.3
. . .
ill
. . .
116
Models
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
Corrections
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
Model Spectral
4.3.1
Functions
Correlation
4.4
Spectrometer
4.5
Radiative
. . . . . . . . . . . . . . . . . . . . . .
Corrections
Internal
. . . . . . . . . . . . . . . . . . . .
4.5.1
First Order
Bremsstrahlung
4.5.2
Virtual
Photon
4.5.3
Higher
Order
4.5.4
Peaking
Approximations
. . . . . . . . . . . . . . . . . . .
. . .
141
4.5.5
External
Brernsstrahlung
. . . . . . . . . . . . . . . . . . .
. . .
150
4.5.6
Radiative Techniques Employed in the PWIA Calculation
. . .
152
Corrections
Bremsstrahlung
. . . . . . . . . . . . . . . .
125
. . . . . . . . . . . . . . . . . . . . .
131
. . . . . . . . . . . . . . . . . . . .
138
5 Results of the Experiment
5.1
Extraction
of Results
154
. . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1
Acceptance
Cuts
5.1.2
Extraction
of the Spectral
5.1.3
Extraction
of Transparency
5.1.4
Systematic
Uncertainties
154
. . . . . . .
155
. . . . . . . . . . . . .
. . . . . . .
157
. . . . . . . . . . . . . . .
. . . . . . .
157
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
159
Measurement
. . . . . . .
161
. . . . . . .
176
. . . . . . .
177
Hydrogen
5.3
Spectral
Function
5.4
Nuclear
Transparency
Function
. . . . . . .
. . . . . . . . .
5.2
Results
. . . . . . . . . . . . . . . . . . .
154
. . . . . . . . . . . . . . .
Measurement
with Glauber
. . . . . . . . . . . . .
5.4.1
Comparison
Calculations
. . . . . . .
5.4.2
Comparison with Colour Transparency Predictions
. . . . . . .
184
5.4.3
A Dependence
. . . . . . .
188
. . . . . . . . . . . . . . . . . . . . .
6 Discussion of Results
195
9
List of Figures
1-1
PWIA
1-2
E,
i-3
model of the (ee'p)
projection
of "C
projection
of
1-4
Energy
dependence
1-5
Spectroscopic
1-6
Effect of correlations
1-7
Bates measurement
reaction
. . . . . . . . . . . . . . . . . . . . .
19
spectral
function
. . . . . . . . . . . . . . . . . . .
23
C spectral
function
. . . . . . . . . . . . . . . . . . . .
24
of the nucleon-nucleon
. . . . . . . . . .
26
factor for various nuclei . . . . . . . . . . . . . . . . . . . .
27
. . . . . . . . . . . . . . . .
28
in the dip region . . . . . . . . . . . . . .
29
1-8 Saclay measurement of 2C(ee'p) in the quasielasticregion. . . . . . . . .
30
1-9 Bates measurement of
31
1-10 Nuclear
transmission
on spectroscopic
of
C(ee'p)
RT
at Q
and
RL
-_ 034
factor
cross-section
structure functions from 2C(ee'p)
.
(GeV/c) . . . . . . . . . . . . . . . . .
33
1-11 Schematic diagram of factorization in PQCD analysis of electromagnetic
form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
1-12 Born diagrams contributing to a PQCD evaluation of the nucleon form
factor .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-13 Example of Landshoff diagram in meson-baryon scattering
37
. . . . . . . .
39
scaling at high Q . . . . . . . . . . . . . . . . . . . . . . . . .
41
1-15 Proton-proton cross-section compared with PQCD prediction . . . . . . .
42
1-16 Elastic proton form factor G'
43
1-17 PQCD calculations of Fl'(Q'), using various proton wavefunctions . . . .
44
1-14 Form factor
10
Contribution of non-perturbative amplitudes to G' - - I . . .
45
1-19 Evidence for geometric scaling of hadron-hadron cross-sections
47
1-20 Colour transparency
calculations
et al . . . . . . . . .
49
1-21 (p,2p)
data of Carroll
et al . . . . . . . . . . . . .
51
1-18
transparency
diagram
of Farrar
2-1
Schernatic
of the A-beamline
. . . . . . . . .
54
2-2
Floor plan of End Station
A . . . . . . . . . . . . . .
57
2-3
Schematic drawing of the 16 GeV/c spectrometer
2-4
Schematic drawing of the 16 GeV/c detector stack
2-5
Arrangement of wires in the drift chambers of te
2-6
Schematic drawing of the
2-7
Schematic diagram of the
2-8
Formation
2-9
Formation of the
61
. . . . . . . . . . .
16 GeV/c spectrometer 65
GeV/c spectrometer . . .
67
eV/c detector stack
of the 16 GeV/c
trigger
69
. . . . . . . . . .
72
Q
2==
1(GeV/c)
GeV/c and coincidence triggers
73
2-10 Raw and corrected coincidence timing spectra for "'Au at Q of 68 (GeV /C)2
2-11
63
75
Experimental phase space in E, and p, for the carbon measurement at
.
3-1
Timing
3-2
Example of raw coincidence timing spectrum measured by TDC8-
3-3
Proton
4-1
E,, and p
4-2
Feynman diagrams contributing to first order Bremsstrahlung cross-section 127
4-3
Feynman diagrams contributing to the virtual radiative correction . . . . 133
4-4
Angular
5-1 E
windows
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78
used in selection
absorption
as a function
distributions
distribution
for
of coincidence
events
. . . . . . . . . .
92
- - - -
101
of Q2 . . . . . . . . . . . . . . . . . . . .
106
H(ee'p)
. . . . . . . . . . . . . . . . . . . .
of first order Bremsstrahlung
photons
. . . . . . . .
distribution of H(ee'p) events, compared with PWIA calculation
11
124
143
162
5-2 E, distribution of 'H(ee'p) events, compared with PWIA calculation
163
5-3 Extracted spectral function for
164
integrated over
2H,
d3p,
5-4
Extracted spectral function for 12C, integrated over
5-5
Extracted spectral function for
5-6
Extracted spectral function for "'Au, integrated over d3p,,,
167
5-7
Extracted spectral function for 2H, integrated over dE,
168
5-8
Extracted spectral function for 12C, integrated over dE,,,
169
5-9
Extracted spectral function for 56Fe , integrated over dE,
170
integrated over
56Fe ,
165
d3p,,,
166
d3p,
5-10 Extracted spectral function for "'Au, integrated over dE,
171
5-11 Extracted
p(p,)
for the 1p shell of 2C
. . . . . . . . . . .
174
5-12 Extracted
p(p,,)
for the Is shell of 2C
. . . . . . . . . . .
175
5-13 Measured
nuclear
. . . . . . . . . . . . . . .
178
transparencies
5-14 Measured transparency for '2C, compared with Glauber calculations . . . 180
5-15 Measured transparency for
16
Fe, compared with Glauber calculations
181
5-16 Measured transparency for "'Au, compared with Glauber calculations
182
5-17 Transparency
185
calculations
of Benhar
5-18 Correlation effects on transparency,
et al .
. . . . . . . . . . . . . . . . .
as calculated by Nikolaev et al. .
.
186
5-19 Measured transparency for 12C, compared with colour transparency calculations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
5-20 Measured transparency for "Fe, compared with colour transparency calculations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190
5-21 Measured transparency for "'Au, compared with colour transparency calculations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-22 Measured transparencies, vs. A, compared with best fit classical model
12
191
194
List of Tables
1.1 Transverse contribution to the free ep cross-section at NE18 kinematics.
32
2.1
Description
59
2.2
Description of materials in the path of incident and scattered particles
60
2.3
Summary
77
3.1
Correction
4.1
Minimum
4.2
Model spectral
function
4.3
Model spectral
4.4
Model spectral
4.5
Correlation corrections applied to IPSM model spectral functions
4.6
Single photon Brernsstrahlung in the soft photon approximation, at Q = I
(G eV /
4.7
of targets
of kinematic
factors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
settings
applied
to the data
. . . . . . . . . . . . . . . . . . . .
107
for several nuclei . . . . . . . . . . .
114
parameters
for 12C . . . . . . . . . . . . . . . . .
116
function
parameters
for
separation
energies
. . . . . . . . . . . . . . . .
117
function
parameters
for
. . . . . . . . . . . . . . .
118
56Fe
197
C)2
proton
Au
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
131
Single photon Bremsstrahlung cross-section in the soft photon approximation, for 100 MeV photon
4.8
. . . . . . . . . . . . . . . . . . . . . . .
energy
Radiative correction functions
vided into various
components
. . . . . . . . . . . . . . . . . . . . . .
131
for single photon Bremsstrahlung, subdi. . . . . . . . . . . . . . . . . . . . . . . .
136
4.9 Singlephoton Brernsstrahlungin the ultra-relativistic limit . . . . . . . . 137
13
5.1
E,
and p
cuts placed on results
5.2
Cuts placed on reconstructed spectrometers quantities
5.3
Summary
5.4
Summary of the measured nuclear transparencies
5.5
Input values used in classical model of transmission
. . . . . . . . . .
193
5.6
Fit values of
. . . . . . . . . .
193
of systematic
. . . . . . . . . . . . . . . . . . . . . .
uncertainties
. . . . . . . . . .
156
. . . . . . . . . . . . . . . . . . . .
160
. . . . . . . . . . . . . 177
ff using classical model of transmission
14
155
Chapter
Introduction
Quasielastic electron scattering from nuclei provides an excellent means of studying nuclear structure. One notable advantage to the use of electrons is that the interaction is
relatively weak, and so an electron probe is able to sample the entire nuclear volume.
Another is that the electron-photon interaction vertex is extremely well understood in
terms of QED A wealth of quasielastic (ee'p) data exists on a host of nuclei, at Q up
to about 03 iGeV/c)'.
(Here, Q denotes the square of the four-momentum transferred
to the struck proton.) These measurements have been interpreted with considerable success in terms of the Plane Wave Impulse Approximation (PWIA) and the Independent
Particle Shell Model (IPSM). In reference to this latter model, (ee'p) measurements
have provided conclusive evidence for the existence of a shell structure in the nucleus,
and have been able to measure precisely the properties of individual shells. Several phenomena requiring further explanation have also been identified, such as violations of the
spectroscopic sum rule and evidence of contributions from multiparticle currents.
If we take (ee'p) measurements to higher Q, it is likely that, as the wavelength of
the scattering probe becomes smaller and individual nucleons are more precisely singled
out, the PWIA description will improve. On the other hand, it is possible that such an
15
approach will cease to be valid because, at energies of the same magnitude as the proton
mass, one might expect quark degrees of freedom to begin to play a role and eventually
dominate the reaction dynamics. At sufficiently high energies also, the asymptotic freedom of the strong interaction should allow perturbative methods to come into play and
permit the exact calculation of QCD processes. The threshold energy at which PQCD becomes a valid approach is at present under debate. A recent prediction of PQCD known
as "colour transparency" provides us with a signature that can be effectively sought using
coincidence electron scattering.
Experiment NE18 was performed to investigate all of these questions by taking (ee'p)
data at Q values up to 68 (CeV/c)
-
an order of magnitude higher than previously
achieved.
1.1 The Plane Wave Impulse Approximation
Elastic electron scattering from a free nucleon in lowest order QED (i.e. one-photon
exchange) is described by the Rosenbluth cross-section:
where
Iq
12 _
Q2
and
do,
do,
dQ
dQ Mott I
=
1
1+2(1+-r)tan2
I
[GE' +,rE-'G M
2]
This expression involves the Mott cross-section
2
(for a spinless relativistic electron scattering from a fixed point charge), and two form
factors GE and GM which describe the electric and magnetic structure of the nucleon.
For reference, these form factors can also be expressed in the Dirac-Pauli form:
KQ2
GE
F - 4M2F2
Gm
F + F2,
16
(1.2)
where
is the anomalous magnetic moment of the nucleon 1-79 for the proton and 1.91
for the neutron). A further reprentation is in terms of the transverse and longitudinal
response functions:
RT
2
= 27-GM
RL
=(I
)G2E'
(1-3)
These form factors have been measured up to momentum transfers of
(GeV/c)' for
the proton and 4 (GeV /C)2 for the neutron [1]. (Neglecting the small contribution of
GP
E at very high
Q2
, GP
M has in principle been measured up
to
Q2
= 31
(GeV/C)2
[21).
However, when one places the nucleon in a nucleus and attempts to describe quasielastic
scattering in a similar way, one is immediately faced with several fundamental difficulties.
First, the nucleon is bound and so is off-shell; the relevance of free form factors to this
situation is not clear. Also, the nuclear electromagnetic current which couples to the
virtual photon now depends on the dynamical structure of the nucleus: it is not simply
that of an isolated charge/current distribution, but depends on the complex interactions
of the nucleon with its surroundings. Determining a complete, coherent solution to this
problem requires a thorough understanding of the nuclear wavefunction and should be
calculated using a relativistic theory. This task is sufficiently formidable that it has not
yet been accomplished for even the simplest nucleus,
2H.
In practice, one adopts an approximate approach known as the impulse approximation
(IA), wherein the nucleons are treated as independent entities moving in a mean potential.
This approach, though straightforward, leads at once to an inconsistency. The current
conservation relation
d
dt
s7 J = -P
17
(1.4)
required of any isolated system can only be satisfied if one includes exchange currents
between the moving charged particles[3]. Such currents are by definition neglected by
the impulse approximation.
Nevertheless, the IA has historically proven remarkably
successful in describing (ee'p) data.
A second approximation helps to simplify the description. This is the Plane Wave
Approximation, which assumes that the scattering matrix elements can be evaluated using free unperturbed particle wave functions (i.e. plane waves). Also assumed is that
one-photon exchange is sufficient to describe the ep scattering vertex (the Born approximation). The validity of each of these assumptions should improve as the reaction energy
increases. First, the Feynman amplitude for virtual photon emission is proportional to
, where q. is the -momentum of the emitted photon. The relative strength of higher
order to first order diagrams thus decreases with increasing momentum transfer and the
one-photon exchange approximation improves. Second, high energy particles are less affected by either the electromagnetic or strong fields of the residual nucleus than particles
of lower energy. The plane wave assumption of using unperturbed particle wavefunctions
should therefore improve as well.
The assumptions described above constitute the Plane Wave Impulse Approximation,
or PWIA A schematic diagram of this model of the (ee'p) reaction is presented in Figure
1-1. The variables used throughout the text to describe the reaction kinematics are as
follows:
le,k]
= 4-vector of incident electron
IC/,k/]
= 4-vector of scattered electron
[E, p]
= 4-vector of initial state proton (bound)
[E PI]
[ErecPreel
4-vector of scattered proton
= 4-vector of recoiling A-1 system
18
--a.
[E", Pl
[ C, k]
--M.
E rec Prec I
Figure 1-1: Schematic diagram of the PWIA model of the (ee'p) reaction.
1W,q]
q., 4-vector of exchanged virtual photon
Q 2
-qjjqt = q q - w2
IVIP
proton mass
MA
target mass
Note that it is only by virtue of the plane wave assumption that we can define a momentum for the iitial state proton at all. Using all the information available in a coincidence
measurement and assuming this model of the reaction, we can deduce the initial state
proton 4-vector from the "missing" energy and momentum (i.e. the 4-vector missing
from the energy-momentum conservation relation):
EM = E C - (E - Mp - E, - Mr),
(1-5)
pm =
(1-6)
p'- q.
19
Since the PWIA is assumed throughout our analysis, these variables will often be used
interchangeably with E and p. Note that the recoiling A-1 system is not observed. Its
kinetic energy is evaluated by observing that momentum conservation in this picture
requires Pree :-- -pn,, and by assuming that M,
= MA-,. This last relation is not
strictly true, as the recoiling system may be in an excited state and have a mass larger
than its ground state mass. However this variation makes only a small contribution to E,,
one that cannot be detected without very high energy resolution. The components of the
vector pm are generally defined relative to the momentum transfer q and the scattering
plane. They are referred to as "parallel" (pm 4), "perpendicular"
"out-of-plane"
(pm Wx k).
(pm-Wx Wx 0)), and
As will be explained in Section 28, NE18 took measurements
in so-called "perpendicular kinematics". This refers to the fact that the large values of p,
detected by the experiment had large perpendicular components. A sign was therefore
attached to the variable p,
corresponding to the sign of pm, , as defined above. Note
that positive p, indicates missing momentum vectors on the larger-scattering-angle side
of the q vector.
A direct consequence of the PWIA model is the factorization of the differential doublearm cross-section into nuclear structure information and bound eN cross-section 4:
d6o,
dE'dQdE'dQp
K is a kinematical factor, equal to Pp'.
= KO',NS(Ep).
The spectral function S(Ep)
(1.7)
is interpreted
as the probability density of finding a nucleon with momentum p and energy E in the
nucleus. It is defined to be
S(p, E)
< 1la+b(E'
+H
P
20
-
- Ei)ap It >
Here, H is the nuclear Hamiltonian, E is the energy of the initial nucleus in eigenstate
Iand
a
and a are the usual particle creation and annihilation operators.
0,N
is the
fundamental off-shell electron-nucleon cross-section, and is discussed in the next section.
1.2 The Off-Shell Cross-section
The problems involved in determining a consistent model of quasielastic scattering have
been addressed by many workers. One of the prescriptions most commonly used in that
of deForest 3], who imposes the current conservation requirement by a suitable choice of
gauge field. He expresses the off-shell cross-section
N
in terms of four nucleon structure
functions:
O'eN
O'Mott
A2w C
+
A
-
2
tan 2(-O)
WT
2
A A tan 2(_10)
2
,Q--7
= arccos(k k,
IqlI
WI Cos
2
+
where A
1/2
and
ACOS2+ tan 2(i 0) Ws
2
arccos[(4 x k) (4 x p)].
expressions for the structure functions are given in ref
(1.9)
Two different
3 based on two different off-
shell extrapolations of the nucleon current. These are named
and
O',,2",
where
the subscript, "cc" refers to the current conserving nature of the forms; both versions
were used in our calculations. The structure functions are expressed in terms of the
nucleon form factors, and another assumption involved in this prescription is that free
form factors can be used.
21
1.3 Previous (ee'p) Data
1.3.1 Independent Particle Shell Model
Coincidence (ee'p) experiments to date have been performed at Q up to about 03
(GeV/c)', and have produced a wealth of information about the nucleus. In particular, dramatic evidence of the shell structure of the nucleus has been provided. Figure
1-2 depicts the results of two measurements of the missing energy distribution for 12C.
In panel (b), one can clearly see the two shells expected in the ground state, with the
1S112
peaking at zero momentum, and the IP3/2 dominating in the higher momentum
slice. Panel (a) presents results from a higher precision experiment performed at Saclay,
with missing energy resolution of I MeV. This precision reveals additional structure in
the spectrum, corresponding to the various possible excitation states of the residual
nucleus. These states affect E, via the term T, -
B
the kinetic energy of the recoiling
A-1 system which depends on the mass MA-,. Figure 13 shows the p, spectrum for
carbon as measured by three different experiments. One can isolate p(p"') for individual
shells by placing missing energy cuts around the appropriate peak. The solid line in the
figure does an excellent job of describing the distribution, and is the result of solving for
the nucleon wavefunction in a Woods-Saxon mean field potential. Such an approach is
referred to as the Independent Particle Shell Model (IPSM). To produce such agreement,
however, it is necessary to evaluate the distortions of the scattered proton wavefunctions
caused by interactions with the spectator nucleons. Corrections for these effects constitute the Distorted Wave Impulse Approximation model, or DWIA. An Optical Model
(OM) potential is used to compute the proton distortions, containing both a real term
which shifts the measured particle vectors relative to their true vertex values, and an
imaginary term which represents the possibility of inelastic scattering in the final state
22
I IQC. CP)
(4
h
1
1
C4
E
U
en
a-
I
"i
I
U
1
O
VI
W
2
A
U
-10
W"M
0V
W
M
W
W
EWRGY(MM
a)
b)
Figure 12: E distribution for carbon, Figure 76 from ref 4.
and causes the "absorption" (or loss) of scattered protons. A further enhancement of
DWIA involves the evaluation of electron distortions as well, calculated in the Coulomb
field of the nucleus. These calculations are referred to as CDWIA (Complete DWIA),
and are especially important for heavy nuclei.
It is important to note that the need for these complex corrections should decrease as
one proceeds to higher energies. Clearly, a higher energy electron will be less affected by
a fixed change in electric potential. As for the proton, Figure 14 shows the energy dependence of the NN interaction and reveals that the inelastic portion of the cross-section
dominates at nucleon momenta greater that about
23
GeV/c. Consequently, at such en-
IVW
-1
w
ap
4L
P.
8
-
(Mew/0
Pe
Figure 13: P,,, distribution for carbon, Figure 91 from ref. [4].
24
MMICI
ergies the proton final state interactions will be largely absorptive -
the proton will be
lost rather than deflected. This simplifies things greatly, making it a good approximation
to correct for final state interactions (FSI) with a single factor.
1.3.2 Spectroscopic Sum Rule
The theoretical distributions in Figure 13 have been normalized to agree with the magnitude of the data, and so say nothing about the total amount of strength actually found
in the extracted spectral function. From the IPSM, one expects that the integral of the
spectral function over a shell with total angular momentum
eracy (2j+l) -
will equal the shell degen-
i.e. the number of protons in the shell. This relation is often referred
to as the spectroscopic sum rule. In fact, one finds empirically that the integral has a
lower value. The left panel of Figure 1-5 shows the strength found in the valence orbitals
of several nuclei relative to the sum rule expectation. Only about 70% of the expected
strength is found.
Attempts to explain this depletion focus on the nucleon-nucleon correlations that are
neglected by a mean field theory. These many-body effects can be grouped into two
broad groups. Weaker (sometimes called "long range") correlations shift nucleons to
deeper binding energies, causing for example filling of the "empty" orbitals above the
Fermi energy (see the right-hand panel of Figure 1-5). "Short range" correlations on the
other hand are caused by the strongly repulsive nature of the nucleon-nucleon force at
short separations. An interaction with such a strongly correlated pair will greatly increase
the measured missing momentum, shifting strength to well beyond the Fermi momentum
kF. These correlations will therefore cause an overall decrease in the spectral function
below
kF-
alculations of correlation effects are frequently done using nuclear matter
models (an approximation to the nuclear medium using infinite boundary conditions).
25
400
II
I. I. II II 11II I
I
.I .I .I
I.
II I. II I, ..,,1 1
.I
I
I . II I
I.
.I .I .I .I -I I I I
.I
. II-
I I I I I 11
102
O'total
'Z
E
?Iio
if
b
-
_
41, -4 44-
.
j
t
.#ff4
-- -
a
ti I444t
4
I
f #
-
.
.
0
.
I
I
f , ,
t
-, 4
I
10
I-_
r-
J
I I I I I, I
b
10'I
- I
I
I , I I I 11
100
1.9
(7eaafic
, f; +
I?,
I
f f I
II
I
102
I
101
I
3
2
Pbeam (GeV/0
I ! I I I I
I I . I . 10.
6 6 7
I
4
I I
I I , , , ,I
I I I I I
103
I
20
104
I
I I
.
40 50
I
30
.
.
I
I
. . . I1001
ECM (GeV)
2000
...
I .I -. I I . . ,
I
.
I
I
. I . I''I
I I I I . 11
I
I
I
I I IIII
I
.1
V
t-
'3 -2
,= lu
(7total
7
b
-41
f_ztr + f I
f
I
10
atotal
I 1
f
I
I
I
I
I , I , I
I
10'
fII--"t*- - --
,
I
, 1.
6elwtic
I I . I , .,I
100
I
2
I
2.9
3
I
I
It '++t
I . I , , ,,I
102
10I
(GeV/c)
Pbeam
1.9
+ .1 f
j
4
3
1
4
5
I
I
5
EC.
T_
6 7
I I
__
7
I __1
910
20t
910
I I I
20
30
30
40I
I
1
40 50 60
(GeV)
Figure 14: Energy dependence of the nucleon-nucleon cross-section (ref. [5]).
26
U.0
t
'F=MPTV'r)PA1T.q
(.8
1
-
0.6
0.6
-
0.4
0.4
z
(n
30si
A
0.2
(.2
12c 160
(D
nn
ID
lo,
102
TARGET MASS
I . ,,.I
40,48Ca9oZr
(D D
(D I
lo,
. . CD
. . _(DI
o,
b.
Figure 1-5: Evaluation of the spectroscopic factor for the valence orbitals and "emtpy"
orbitals (above the Fermi energy) of various nuclei (ref. 61).
Such models have been quite successful in explaining the depletion of valence orbitals (see
for example Figure 16). Note that correlation effects do not appear to alter significantly
the form of the p distribution below the Fermi momentum, a evidenced by results such
as those of Figure 13 showing excellent agreement with mean field distributions.
1.3.3
Evidence for Multi-body Processes
Inclusive (ee') experiments such as that of ref. 7 have indicated evidence of processes as
yet unaccounted for in quasielastic scattering. Measurements in the so-called "dip" region
(between the quasielastic peak and first nucleon resonance) have produced cross-sections
larger than those predicted by the combined effects of quasielastic nucleon knockout,
resonant A production, meson-exchage currents, and pion production [8]. Also, measurements of the transverse and longitudinal structure functions in the quasielastic region have revealed
R
RL
ratios which are -_ 60% larger than the predictions of the impulse
approximation with free nucleons [9]. One calculation 7 of scattering in the dip region
27
E
0)
C
P
V;
a)
Y
I-
9
fA
M
0
0
-EM -
F
[MeV]
P.
Figure 16: Spectroscopic factor for valence orbitals of various nuclei, accompanied by
predictions of correlated nuclear matter spectral functions (dotted line). The solid line
includes surface effect corrections for 'O'Pb (ref. 61).
manages to explain some of the missing strength by including the probability of scattering from multi-nucleon currents. These currents are caused largely by short-range
correlations, a direct consequence of the strongly repulsive nature of the nucleon-nucleon
interaction at short distances. This strong interaction makes it possible for an incoming
electron to scatter off a coupled system composed of two or more quarks.
Physically, such a process might be expected to knock out both nucleons rather than
just one. The net effect would be the addition of the second nucleon's momentum and
kinetic energy to the measured missing momentum and energy. These quantities can
be measured directly in an exclusive reaction, and such studies have been performed.
Ref. [101details a measurement of the (ee'p) reaction from `C in the dip region; the
experiment was performed at Bates at Q = 012 (GeV/c)'. The resulting missing energy
distribution is shown in Figure 17, and clearly shows an almost constant distribution
28
500
400
Ir
i
210
300
L
-.5
0
z
,% 200
0.0
P
C.)
100
n
20
20
60
100
140
180
MissingEnergy (MeV)
Figure
17: Missing energy distribution for
Bates at Q
=
of strength at E
12
C(ee'p) in the dip region, measured at
(GeV/C)2.
up to 150 MeV (the maximum value detected by the experiment). A
similar measurement performed at Saclay 111on "C in the quasielastic region, however,
measured up to 80 MeV in E, at
Q2
= 0 16
(GeV/C)2
and detected no evidence of
strength unexplained by single-particle knockout (see Figure 1-8). It is possible, of course,
that the relatively small contribution from quasielastic scattering in the dip region enables
one to detect the presence of additional effects.
The measurement of the transverse to longitudinal structure function ratio has also
been investigated with coincidence experiments.
"C(ee'p)
Ref.
performed at Bates which separated the
RT
121 details a measurement of
and
RL
structure functions at
quasielastic kinematics. Figure 19 depicts the extracted structure functions, measured
out to > 60 MeV in missing energy at
Q2
= 0 14 (GeV/C)2. The differenceST
-
SL
is
shown in panel (c); here, STandSLrefer to the spectral function of equation 17, but with
29
M15SING ENERGY MeV)
Figure 1-8: Missing energy distribution for "C(ee'p)
sured at Saclay at Q = 06 (GeV/c)'.
in the quasielastic region, mea-
only the transverse or longitudinal term of o,p included respectively. The IPSM, PWIA
formalism makes no provision for any difference in these two versions of the spectral
function. The data indicate that
ST
and SL are about equal at the valence 1p shell,
but that the transverse portion dominates increasingly as one moves to higher missing
momentum and eventually the difference appears to level off. As pointed out in ref. 12],
since the density varies from shell to shell one might suppose that excess strength seen
in the ls but not the 1p shell is indicative of a density-dependent modification of the
bound nucleon current. However, the fact that this extra strength appears only in the
transverse reaction channel suggests the presence of an additional process.
If indeed these measurements are indications of coupling to correlated multi-nucleon
currents, it is of great interest to repeat them at high momentum transfer. First, the
cross-section is predominantly transverse at high Q.
Table 1.1 shows the percentage
transverse contribution to the free ep cross-section at the momentum transfers measured
by NE18: above 80% at the lowest Q and above 98% at the higher settings. This
should enhance our sensitivity to multinucleon processes. However, any cross-section for
30
1.00
(a)
x-
3
0.75
0.50
0.25
0.00
2.0
IT
.7.7 -
(b)
3
- - - - - - - - - -
1.5
1.0
0.5
0.0
V
1
>
-- - - - -
30 7 (C)
44
2-Body
Thresh.
7
20 7 X
lo
- - - - - - - - - - - - 10
Figure 19: Measurement
RL) from "J(ee'p)
at Q2
transverse and longitudinal
depicts RL, and (c) depicts
20
30
40
50
60
Missing Energy (MeV)
of transverse and longitudinal structure functions (RT and
= 04 (GeV/C)2, performed at Bates. ST and SL are the
spectral functions, defined in the text. (a) depicts RT, (b)
the difference ST - S.
31
Table 1.1: Percentage of the free ep cross-section which is transverse, at NE18 kinematics.
I Q
I transverse
1.04
3.06
5.00
6.77
contribution
(%) ]
82.6
98.6
99.5
99.7
scattering from a composite object necessarily drops with increasing momentum transfer:
the wavelength of the scattering probe decreases until it is able to resolve the substructure
of the ob.ect, and at this point the large scale structure of the object ceases to play a role.
It is thus reasonable to suppose that at Q values of the same order as the nucleon mass,
the coupling of the virtual photon to correlated nucleon pairs is substantially suppressed.
1.3.4 Nuclear 9ransmission
Final state interactions were mentioned above, and it was indicated that knockout protons
are not only deflected but also absorbed by the spectator nucleons. Processes of this type
are caused by the cross-section for inelastic final state interactions, and can be expected
to increase with energy until the inelastic NN cross-section reaches its asymptotic value of
about 28 mb. The quantity typically studied in this context is the "nuclear transmission",
the probability of escape of a quasielastically scattered proton.
Figure 1-10 shows transmission data for a variety of nuclei taken at Bates at Q'
0.34 (GeV/c)'.
The transmission in this experiment was determined using the ratio of
the coincidence (ee'p) rate to the rate of electron singles. A series of calculations are
also depicted in the figure. The basic theoretical technique involved is that of Glauber
[13], consisting of a semi-classical calculation of proton multiple scattering in the nuclear
medium. The theory curves show the cumulative effect of incorporating various subtleties
32
z0
05
Ln
2
z
Cn
Cr.
I-
NUCLEON NUMBER
Figure 1-10: Measurement of nuclear transmission at Q2 = 034 (GeV/c)2 14]. The
lines refer to a calculation using the free Np cross-section (dotted), the adding Paull
blocking (dashed), density-dependent effects (dot-dashed), and a correlation hole (solid).
into the calculation. These are the Pauli blocking of protons scattered into filled states
below the Fermi momentum, the density-dependence of the nucleon-nucleon cross-section
in the nuclear medium, and the inclusion of two-body short range correlations. The full
calculation appears to account successfully for the measured final state losses. These and
other elements of FSI calculations, along with their importance at higher energies, will
be discussed in detail in Section 54. 1.
33
1.4 Exclusive Processes at High Momentum Transfer
Dramatic evidence of the existence of sub-nucleonic degrees of freedom was provided by
the celebrated parton experiment of 1967. The observation of Bjorken x-scaling above
Q2 of
I
(GeV/c)' (and for missing mass W > 2 GeV) in deep inelastic electron-proton
scattering indicated the presence of point scatterers inside the nucleon.With the fact thus
established that inclusive electron scattering at relatively low Q2 can resolve this parton
substructure, one is led to wonder to what extent quark kinematics contribute to nuclear
structure and to the dynamics of quasielastic scattering.
Exclusive processes at large momentum transfer test both the internal dynamics of
hadrons and the detailed structure of hadronic wavefunctions at short distances. Electron
scattering at
Q2
>
(GeV/C)2, for example, corresponds to a spatial resolution of less
than about 0.1 frn and so is clearly able to distinguish between the component partons
of a nucleon. Calculating the properties of the nucleus directly from
CD, however, is
a formidable task. The strong coupling constant a, diminishes with increasing energy,
and is larger than I until one reaches the region of so-called asymptotic freedom. At
sufficiently high energies, a, becomes small enough that one can apply perturbative
methods, of the type that have proved so successful in QED. This technique is referred to
as PQCD. The energy threshold at which PQCD becomes a viable description, however, is
not clear 15][24]. In the attempt to identify this threshold, the measurement of exclusive
processes plays a dominant role.
1.4.1
Application of Perturbati've
CD
An excellent presentation of the application of PQCD to exclusive processes is found in
ref. 16]. As an example of the manner of analysis employed, we consider the calculation
of the nucleon magnetic form factor GM. Treating the problem in the familiar infinite
34
X1
0-4
O
Figure 111: Schematic diagram describing the factorization of PQCD scattering amplitudes into initial and final quark distribution amplitudes and O*,and a hard scattering
amplitude TH. This diagram specifically describes the calculation of an electrornagentic
form factor.
momentum frame, the momenta of the quarks are taken to be parallel to the nucleon's
direction of motion and are parametrized in terms of the Bjorken scaling variable x.
This variable denotes the fraction of the nucleon momentum carried by each quark
that F_ x
(so
1). This assumption is equivalent to requiring that the average transverse
momentum of the quarks, < k' > 2 is much less than Q, the only momentum scale in
the problem. The nucleon is described in terms of a Fock state expansion over its valence
and excited states, where the valence state consists of the minimal number of quark fields
(qqq) and the excited states contain additional elements (qqqg, qqqqq, ...). The reaction
itself is then factorized into three pieces, as is depicted in Figure 1-11: the incoming and
outgoing nucleon wavefunctions O(xi, Q) and O*(yi, Q), and the hard scattering amplitude
TH(xi, yi, Q). The wavefunctions (or "quark distribution amplitudes") are assumed to be
in their 3-quark valence states, an approximation which is supported below.
TH
is the
amplitude for the initial nucleon to absorb the momentum q of the virtual photon, and
redistribute it among the component quarks to produce a final state where the quark
momenta are also approximately parallel. This redistribution is required in an exclusive
reaction: both the initial and final hadrons are measured, and so the quarks are bound in
known configurations. The "spectator quarks" in the reaction must thus "catch up" to
the struck
ark so that the nucleon retains its identity without multiparticle emission.
35
To leading order, TH is calculated by summing the connected Born diagrams shown in
Figure 1-12, which depict the incoming virtual photon and the gluon exchange described
above. We see now why in the limit of high momentum transfer only the minimal Fock
state of the nucleon contributes: the number of gluon lines increases with the number of
elementary fields present, and according to the Feynman rules each of these propagators
contributes a factor of Q-' to the amplitude. One explicitly takes the nucleon wavefunction, then, to be an integral of the 3-quark wavefunction Ov(xj, k i)over transverse
quark momenta less than Q:
O(xi, Q = I
d'k-Lid2kl2d 2k.L30V(xi,
k -)0(k2
-Ls
I
< Q2).
(1.10)
This restriction is equivalent to ensuring that the transverse spatial separation (or "quark
impact separation")
b-L
is at least of order Q-1 - so that the virtual photon will resolve
individual quarks. The final result for the form factor is
Q2)
GM(
(Q2
Q2
2
E anm
Q 2
-Yn -Ym
log A2
1
+ 0(a,(Q)) + 0
nm
The logarithmic terms can be considered corrections to the leading order
behaviour; they are parametrized in terms of the "anomalous dimensions"
depend on the nucleon wavefunction used. a,
Q2
(Q2)
Q 2
Am, An
scaling
which
is the running coupling constant of
and also depends logarithmically on Q2.
a, (Q2)
In Q2
A2
36
(1.12)
(a)
(c)
W
I
V
I
I
I
C9)
d)
(F)I
I
I
I
I
I
a
i
I
I
(q)
I
X
Figure 112: Leading order diagrams contributing to a PCD
form factor.
1.4.2
evaluation of the nucleon
Counting Rules and Other Results of PCD
In the mid-1970's, Brodsky and Farrar 171demonstrated that in the asymptotic limit
of high momentum transfer, application of dimensional analysis to a renormalizable field
theory such as QCD leads to a set of so-called "counting rules" for the energy-dependence
of fixed-angle electromagnetic and hadronic scattering. The derivation of the counting
rules is most straightforward for exclusive reactions, and only these will be discussed
below. The analysis is based on the following assumptions: First, one postulates that a
composite hadron can be replaced by point-like constituents carrying finite fractions of
the hadronic momentum. This, of course, is the essence of the quark model and of PQCD,
and an implicit assumption here is that the number of constituent fields in mesons and
baryons is accurately given by the quark model to be 2 and 3 respectively. Next, it is
assumed that the only scales (dimensional quantities) in the system are particle masses
and momenta. Further, the asymptotic limit of high center-of-mass energy
s is taken
(thus the masses can be neglected), and all invariants are fixed relative to s (in a two-
37
particle reaction, this corresponds to fixing the center-of-mass angle). The result of this
is that only one scale, s, remains in the problem. The crux of this assumption is that
binding effects do not contribute any additional scales - i.e. that the scaling behaviour
of bound quarks is well-described by that of a collection of free quarks. With these two
assumptions, the leading-order (Born) diagrams involved in a perturbative evaluation of
the scattering amplitude can be written down.
One now performs dimensional analysis on the scattering amplitude M. From the
Feynman rules, one knows that dimensionally, each external fermion line contributes
[length] (via the normalization of the Dirac spinors), each photon or gluon propagator
(internal line) contributes [length]-', and each fermion propagator contributes [length]-'.
With some reflection one sees that the net dimension of the amplitude is entirely described
by the minimum number n of elementary fields (i.e. leptons, photons, and quarks) in
both the inital and final states: [M] = [length]n-4 . Given that Vs is the only length
scale, one obtains the counting rule for two-particle scattering:
da
dt
AB-CD
:= dom.tt
Im
dt
2
,,82-n f (S),
-t
t
I
(S --+00,S fixed).
(1.13)
This can be generalized to the case of multiparticle production (see ref. 171). A corollary
is the counting rule for electromagentic form factors of hadrons:
FH(t) , tl-nff,
where
nH
is the minimum number of elementary fields in the hadron. Note that this
agrees with the dominant
Q2
dependence of Equation 1.11 for the nucleon form factor.
A third assumption is being made somewhat implicitly here, which is that the class
of so-called Landshoff diagrams [18] can be neglected. An example of such a diagram for
38
11
0V
Figure 113: The left panel depicts a Landshoff diagram for meson-baryon scattering,
while the right panel shows two characteristic connected Born diagrams for the same
process. The off-shell quark propagators are indicated by dots.
meson-baryon scattering is shown in the left panel of Figure 1-13(a), while the right panel
depicts typical Born terms for this reaction. The Landshoff diagram is characterized
by a relative absence of off-shell quarks (i.e. of internal quark lines), and physically
corresponds to the independent, elastic, on-shell scattering of pairs of constituents from
different hadrons such that the final momenta are properly aligned. These diagrams have
a slower fall-off with Q than those described before, but are presumed to be strongly
suppressed -- for example by the sheer number (tens of thousands) of connected Born
diagrams involved in a purely hadronic PQCD calculation. Fortunately this complication
is not present in lepton-hadron scattering.
Other PCD
predictions of the dynamics of exclusive reactions exist, notably those
concerning helicity conservation. An example of this is the prediction that non-helicity
conserving amplitudes are suppressed relative to helicity conserving ones. A case in pc
is the proton form factor
2 which is not helicity conserving, unlike the form factor Fl'
which describes a quark spin flip transition. The PQCD expectation is that
F,(Q2)
F2(Q2)
(1.15)
Q2
at high
Q2
. These ideas can be further explored through the use of polarized beam and
target experiments, and the measurement of spin-dependent asymmetries 19].
39
1.4.3 Comparisonwith Data
How well do these predictions fare when compared with experiment? The success of the
counting rules in predicting the momentum-transfer dependence of many elastic form factors is demonstrated in Figure 1-14. These form factors appear to achieve the expected
scaling behaviour at Q of around 5 (GeV/c)'.
The free proton-proton cross-section is
shown in Figure 1-15, and is seen to exhibit the s-" falloff predicted by the counting
rules. The fluctuations of the cross-section around the leading-order PQCD form appear
to be explained by the model shown as a solid line in the figure. This is the heavy-quark
threshold model of ref. 20], where the contribution of scattering from hadronic resonances at certain kinematics is included. Unlike a hard-scattering reaction, a reonance
couples to the large-scale structure of the proton, hence the deviation from the simple
PQCD scaling picture. The predicted falloff of 2(Q') relative to FI(Q2) was verified by
the recent experiment NE11 [1], which separated the form factors for Q2 up to 8 (GeV/c)'
for the proton and 4 GeV/c)' for the neutron. The signature looked for is that the ratio
G&
GM
approaches a constant; this can be deduced from equations 12 and 1.15. Another
notable success is the calculation of the proton form factor G'
it is compared in Figure
1-16 with the data of ref 2 The gradual drop in the calculation above Q2 -_ 7 GeV/c)'
is caused by the appearance of the running coupling constant a,(Q2) in Equation 111.
One must point out, however, that these predictions are arbitrarily normalized. The
normalization (and sign) of the calculation is strongly dependent on the choice of quark
distribution amplitude for the proton and so provides a measure of this distribution (assuming that the theory can be applied validly at all). This sensitivity as well as the
fact that certain choices of wavefunction are able to obtain the correct normalization are
demonstrated in Figure 117. The conclusion of these successful comparisons would seem
to be that PCD
is a viable theory of exclusive processes at momentum transfers of
40
100
lo-,
ib"
C
U_
lo-,
TC
CY
,a
10-2
10-2
10-3
10-4
0
2
4
C)2
6
(GeV2)
Figure 114:: The measured elastic form factors of several quark systems; the form
factors are mltiplied by the power of Q predicted by dimensional scaling, and appear
to approach this scaling behaviour as Q increases.
(GeV/c)' and. higher.
1.4.4
Arguments in Favour of a Higher Perturbative Threshold
From one point of view, it is somewhat surprising that the counting rules are so successful in these kinematic regimes. An essential assumption in their derivation is that all
invariants in the reaction are sufficiently large that the particle masses can be neglected,
and a momentum transfer squared of
(GeV/c)' corresponds to only
times the mass
of the proton. A more serious argument in support of a far higher threshold for the
41
10-2
N
a)
O
1-1
.0
10-3
61E
10-4
0
a
I'D
10-5
6
8
P lab
10
12
14
(GeV/c)
Figure 1-15: Measurement of the proton-proton cross-section at 90' in the center-ofmass 21]. The dotted curve is the - s-10 PQCD prediction; the solid line is the prediction
of ref. 22] including the effects of strange- and charmed-particle production thresholds.
validity of perturbative methods is presented by Isgur and Llewellyn-Smith 241. These
authors demonstrate that calculations involving only "soft" non-perturbative amplitudes
can achieve a magnitude of the same order as the proton from factor data (Figure 118).
Their argument centers on the selection of proton wavefunctions, which was shown previously to be crucial in the normalization of PQCD calculations. Equ. 1.10 indicated
that in the calculation of the scattering amplitude M, transverse quark momenta k are
restricted
to values less than Q. In other words, the amplitude
is determined
by soft,
low k_Lportions of the wavefunction. Their calculations suggest that a quark distribution
with < k 2 >I=2
300 MeV produces a value for GM which is two orders of magnitude
below the data. Much larger values of this mean transverse quark momentum are possible, but using them then violates the
Q2
> < k 2 > requirement on which the validity of
the perturbative techniques rests. Physically, one can picture this as follows: Essential
to the perturbative approach is the hard gluon exchanges required to redistribute the
momentum transfer Q among the quarks. If, however, the nucleon wavefunction has sig42
I
r
-
\.,.%.j
r---"
V
I
I
r,%--- '_
I
I
I
M-
I
0.5
W
(D
0.4
i
0.
0.2
0
a
IT
03
-0
0
0.2
8
- --IIII
0
--I --- I
10
(2
20
30
PGOV/62]
Figure 116: Measurement of the elastic proton form factor GI at high momentum
transfer (ref. 21). Note that the (small) contribution of GE was neglected in the extraction of the result.
nificant strength at k_Lof the same order of magnitude as Q, this redistribution may not
be required since the struck quark may be accompanied by other quarks already moving
along its redirected path. Furthermore, Isgur and Llewellyn-Smith find that wave functions with < k' > 2 = 300 MeV can generate soft non-leading which are as large as the
data. This fact alone brings the application of perturbative techniques into question. It
is clear that the energy threshold at which PCD
yet to be answered definitively.
43
can be applied is a question that has
1-5
P-V
0
11-1
V> 1.0
1-1
0
CZ
CF.
a
0.5
tv
-0
0
a
0
0
M
a
TnsiAs Tnionval
-S. -
'-0.3(GeV/c")2
I
10
Q"
I
I
20
30
((GeV/c)2]
Figure 117: PQCD calculations of Fl'(Q'), using various proton wavefunctions[23].
The predictions use distribution amplitudes of Chernyak and Zhitnitsky (CZ), King and
Sachrajda (KS), and Gari and Stefanis (GS).
1.5 Colour 'Transparency
About 10 years ago, Mueller and Brodsky suggested that at sufficiently high momentum
transfer, the final (and initial) state interactions of hadrons with the nuclear medium
in exclusive processes should be reduced, leading to the phenomenon termed "colour
transparency". The argument is based on three assumptions 251. First, a hard (high
momentum transfer) elastic scattering vertex involving a hadron should "select" a particle
configuration of reduced size. In a PQCD picture of the reaction, the hard virtual
photons/gluons which carry the transferred momentum scatter directly from the quarks.
Also, since exclusive processes are being considered, one can demand that the scattering
is elastic. Thus if a hadron is involved, it must remain in one piece despite the fact
that momentum is delivered to its component parts independently. The momentum
imparted to the hadron must be then be distributed among its component partons, and
44
a4 Gm Q
G' ((
M
(GeV4)
0.
W
(Gew)
Figure 1-18, Demonstration that in certain calculations [241,contributions from soft,
non-perturbative amplitudes to GP
(Q2) (dotted lines) are found to be of the same order
M
as the data.
45
this is accomplished through the exchange of gluons. Each of these gluons will carry
on average an equal fraction of the total momentum; since this is high, the uncertainty
principle demands that the distance the gluons have to travel be correspondingly small
-
of order
Q
in fact. In other words, the hadron state selected at the hard scattering
vertex is compressed in size. The object of reduced size is often referred to as a point-like
configuration (PLC). Another way of looking at this is that an accelerating colour current
causes gluon radiation and so multiparticle final states. To exclude this possibility, one
must postulate that the struck particle is point-like and therefore colour neutral. The
second assumption, of "colour screening", is directly related to this last argument. One
postulates simply that small particles have small cross-sections. This is the QCD analogue
of the QED Chudakov effect, which predicts that ee-
pairs have a small separation at the
production vertex and consequently a reduced interaction with the surrounding medium.
This effect has been observed experimentally 261; in QCD, evidence for such an effect
is provided by the geometric scaling of hadron-hadron cross-sections: the total crosssection Oh,,h, is proportional to < r2>< r2
hi
h2 > (see Figure 119). The third assumption
is that the distance over which a PLC expands to its dressed (free) size is at least as
large as the nuclear radius. Suppose the expansion time in the nucleon rest frame is to.
The expansion time in the nuclear frame is then -ft = M
Eto, and so becomes larger with
increasing energy. At sufficiently high energy, and making the previous two assumptions,
the hadron's interactions with the nuclear medium are certain to be reduced over a
substantial fraction of its exit path, and will ultimately disappear leading to complete
C4colour transparency"
An example of one of the early calculations
of the size of this effect is depicted in
Figure 120 from ref. 28]. The final state interactions are modelled using an effective
46
1-9
.0
E
I-j
0
40
30
20
10
0
Figure 119: Evidence for geometric scaling of hadron-hadron cross-sections 271.
hadron-nucleon cross-section:
eff
O'hN W
01hN
tot
Here,
1h
z
-
Ih
T
+
< n 2 k2
>
t
t
z
I
T
-
((1h - Z) + O(Z - h)-
Ih
(1.16)
denotes the full expansion distance of the hadron, z the hadron's distance from
the interaction point, () the usual step function, and
verse area of the PLC. Both
1h
which can assume several values;
a measure of the trans-
OhN < 2 k2>
t
and z are measured in the lab frame.
gives no colour transparency,
is a parameter
produces the "quantum
diffusion" model, and 2 the "naive parton" model. The equation assumes a geometric
interaction -- i.e. that the cross-section is proportional to the transverse area of the
hadron. That area, in turn, is assumed to expand at the rate xt2
,,z
T.
The "naive par-
ton" model assumes that the quarks separate at the velocity of light, giving xt - to
Similarly, the total expansion distance in this model is proportional to the radius of the
dressed hadron times the time dilation factor:
h
Yv'-rL
-f M.
The "quantum dif-
fusion" model provides a second model of the expansion. As in any diffusive process, the
47
rate of expansion is given instead by x2 , z (consider, for example, the rms displacement
of a particle moving at fixed velocity through a medium and executing a random walk
in the transverse direction). Accompanying this picture is an estimate to
for the expansion time. Here,
Eh
< E, -I Eh >
refers to the energy of the free hadron, and E, to the
energy of the intermediate (excited) hadron state produced at the vertex. This value of
to corresponds to the typical quantum oscillation period of a wavefunction consisting of a
superposition of two energy eigenstates. As is seen in Figure 120, these models produce
widely different estimates of the magnitude of the effect: from a 20 to 40% increase in
transparency over the
to 68 (GeV/c)' range of NE18.
It should be noted that although colour transparency was originally postulated from
PQCD arguments, recent work has demonstrated that PCD
is not necessary for the
existence of the effect. For example, one can show 291 that the emission of mesons
from a baryon in a PLC is suppressed by -
( rPc
Y.
I'meson
The interaction of the PLC with
the medium is thus reduced in the same geometric way as described earlier, but in a
conventional meson-exchange picture of the nucleus.
Colour transparency thus provides us with a clear signature to look for in quasielastic
(eelp) scattering: As the momentum transfer of the reaction increases, evidence for CT
effects will appear as a drop in FSI and so a rise in transparency relative to the Glauber
calculations described earlier. Around proton momenta of I GeV/c, the free NN crosssection (cf. Figure 14) approaches an asymptotic value of
40 mb. At these energies,
then, Glauber calculations may be expected to be Q independent, leaving a clear CT
signal.
48
1.0
0.8
A
0.6
Z
a
0.4
0.2
0.0
10
20
30
Plab (Geft)
Figure 120: Colour transparency predictions from ref. 281, using Equation 116. The
dashed line is for =
("quantum diffusion" model), and the dotted line for = 2
("naive parton" model). The = results for no CT is constant with energy and equal
to the value of the
curves in the Pab --+ 0 limit.
49
1.5.1 Experimental Evidence
One experiment already performed expressly to look for colour transparency effects was
that of ref. 30]. This was a p,2p) experiment performed at the Brookhaven National
Laboratory. Note that the transparency found in (p,2p) scattering cannot be directly
compared to (ee'p) results because in the former case, strong final state interactions are
present in three channels rather than one. The BNL data was taken at incident proton
momenta from 6 to 12 GeV/c and at a center-of-mass scattering angle of 90', providing
a Q range of 48 to 10.4 (GeV/c)'.
Five nuclear targets were used, (along with a CH2
target for calibration), and T was defined to be the cross-section ratio
T
(daldt)(p - p quasielastic in nucleus)
A(duldt)(p - p elastic in hydrogen)
(1.17)
The data are shown in Figure 121 and exhibit two striking features: a clear rise in
transparency up
to Q2
= 8.5
(GeV/C)2
and a subsequent dramatic fall-off. Though it
is tempting to interpret the initial rise as evidence for CT effects, the fall-off is clearly
not explained by this picture and must be understood first. Various attempts to explain
this behaviour have been made. Ref. 311observes that the fluctuations in transparency
are correlated with the oscillations of the free proton-proton cross-secton around the
S_1' dependence predicted by PQCD (see Figure 1-15). Dividing the quasielastic crosssection by the free pp cross-section as in Equation 117 is liable to produce oscillations
in transparency, especially when one considers the fact that only one of the scattered
proton vectors was completely measured in the BNL experiment. The second proton
was detected in a non-magnetic spectrometer, and so the initial proton momentum (and
consequently the effective center of mass energy s at the vertex) had to be deduced using
the assurntion E
= . The interpretation of the (p,2p) data is thus somewhat unclear,
and the data have yet to be satisfactorily explained 25].
50
1.00
F]
T T I
r I
I I
I
I I
f _TT_ t
IIII-
(a) Various Nuclei
>_1
C)
C:
0.50
+ -Li
0 -C
I- -
a)
(t
0.
X
0.20
V)
C:
It
-Al
C
0.10
E__
A
0.05
III
I 1
4
.11,
)r
CU
-Pb
I
I
I
I I
I I I I I I I I I I I I I I I I I
IIII
0.5
U
C
0)
co
0.
W
r.
(t
0.2
E_
0.1
0
2.5
5
7.5
10
12.5
15
Incident Momentum Gev/c
Figure 121: (p,2p) transparency data from ref. [301.
51
Chapter 2
Description of the Experiment
2.1
Beam
The electron beam was generated by the Nuclear Physics Injector (NPI) at the Stanford
Linear Accelerator Center (SLAC). The NPI became operational in 1985, and uses the
last six klystron sectors of the SLAC linear accelerator to deliver high intensity electron
beams of up to about 5.8 GeV to End Station A. The beam is pulsed, with a pulse length
of 16 Ms and rate of 120 pulses/sec, yielding a duty factor of 2xlO-'. The number of
electrons delivered in each pulse (or spill) varied from 2x1O'Oto NW',
providing average
beam currents from 03 to 9.5,vA. The SLAC energy doubler (SLED) was not used during
the experiment.
The experiment was performed in End Station A ESA), located just beyond the end
of the linear accelerator and between the arms of the SLAC Linear Collider (SLC). The
beam is directed to the End Station from the switchyard via the A-beamline (see Figure
2-1). The first section of the A-line is known as the A-bend and consists primarily of
dipoles (B 10 17) which defined the beam energy. Focusing was provided by 4 quadrupoles
(Q10-13), and the energy spread was determined by the slits S10 and SUL At a setting
52
of ±0.5%, over 90% of the beam passes through the slits. At the lowest energies, a setting
Of 0.1
w
used to limit the spread and
aximize the energy resolution; at higher
energies where the reaction rate was lower, the slits were opened to ±0.2% in order to
increase the beam current and maintain an acceptable count rate.
The beam spread
consequently dominated the missing energy resolution at the higher Q settings.
Beyond the A-bend, two pairs of horizontal and vertical steering coils (A 10 13) provide
fine-tuning of the beam position at the target and can be adjusted by the experimenters.
Roller screens RSI and RS2 consist of ZnS-coated plastic plates and can be moved in
and out of the beam line by remote control from the counting house. Cameras directed
at the screens allow the experimenters to view the beam position between runs since
these screens fluoresce visibly when struck by the incoming electrons. Monitoring of the
beam position and width during data taking was accomplished using another series of
detectors, read out online by a DEC Microvax 11(referred to as the BCS, or Beam Control
System). The first of these devices was a wire array positioned 20 m upstream of the
target. The array consists of two planes of thin Al wires strung 06 mm apart; in one
plane the wires are horizontal, in the other, vertical. Electrons striking the wires caused
secondary eission
that was measured by a series of ADCs (one per wire). The signals
from the array thus provided a picture of the beam profile. Second, a pair of resonant
microwave cavities located further upstream monitored the offset of the beam from the
cavity center. Because of the distance between the wire array and the cavities, a measure
of the beam angle was also obtained. The BCS provided continuous online monitoring
of these detectors, and through a feedback loop was able to fine tune the steering coils.
The position of the beam at the target was kept to within 2 mm of center and the beam
angle to less than .05 mr. The beam shape at the target was an ellipse roughly 1 mm
wide and 2 mm tall.
53
A-Bond
TOMWO
RS2
Tawts
, A13
Al I \
012
Cl
0
:
co PMI
PUS
a.s.
slo GI I
914 -
1
13
lo
I
MM will$
C12
1
12-
4
C
C0111mawr
PM PUnd ap
a
Oim*UPOIO
I 0
a
1 ffe"
SL Sk
A S"
FIB o
ingot
mm
I
Figure 21: Schematic diagram of the A-beamline
54
The quality of the beam was also monitored using two scintillators placed in the
vicinity of the -target. The "good spill" scintillator was located approximately
0 m
from the target, where it could monitor beam-induced scattering from the target. Its
signal was displayed on an oscilloscope as a function of time, and so its shape provided a
measure of the distribution of charge in each pulse. The beam tune was adjusted to keep
this distribution as uniform as possible. The "bad spill" scintillator was placed near the
beam pipe and slightly upstream of the target. It was shielded against scattering from
the target, and so the signal in this scintillator indicated the amount of halo around the
beam. The beam tune was adjusted to keep the bad spill level to a minimum.
The charge in each beam pulse was measured by two toroidal coils surrounding the
beam pipe, forming a transformer circuit with the beam itself as the primary winding.
The measured charge was read and integrated over each run by the BCS. Also, a toroid
calibration was performed before each run by sending a pulse of known charge down a
wire running through the coils and adjusting the gain parameter in the BCS software to
provide the correct reading. The uncertainty in the charge measurement was determined
to be 0.5
b experiment NES performed previously at SLAC 32].
2.2 Experimental Layout
Figure 22 shows a floorplan of End Station A. The beam enters the hall from the west,
intersects the target at the indicated pivot point, and continues on to a beam dump
outside the End Station. Three spectrometers are shown in the figure. The 16 GeV/c
was used to detect scattered electrons, and the
GeV/c measured scattered protons in
coincidence; the 20 GeV/c was not used in NE18. The spectrometers are positioned
on circular rails and mounted on motorized support structures, and their angles with
respect to te
beam line can be adjusted by remote control from the counting house.
55
The spectrometer angle position is measured by an encoder which is read out in the
counting house. The encoder determines the angle by keeping track of the cumulative
rotation of the carriage drive gears. Surveys of the spectromters were done both before
and after the experiment, and the difference between surveyed and encoder values for
their angular position was found to be less than 0.01' for the 16 GeV/c and 002' for
the
GeV/c. The point at which the magnetic axes of the spectrometers intersects the
beam line was also measured, for a range spectrometer angles. The intersection points
varied by about 0.1 cm with angle, and the 16 GeV/c centerline was found to point 1.09
cm upstream of the pivot on average. No such offset was found for the
GeV/c.
2.3 Target
The target consisted of two parallel vertical ladders, holding the cryogenic liquid targets
and solid targets respectively. The selection of targets wa's accomplished by moving the
ladders remotely from the counting house. The ladders were enclosed in a vacuum sealed
aluminum chamber, which imposed thin aluminum windows between the targets and the
incoming and outgoing particles.
The liquid target ladder held two 'H and
two
2
H target cells, of 4 cm and 1 cm
length respectively. (The long hydrogen target was not used in NE18). The cells themselves consisted of aluminum cylinders 32 cm in radius with rounded endcaps, and were
constructed from Coors beer can blanks to provide a minimal and constant thickness
of material between the enclosed liquid and the detectors. The liquid 'H and
2H
was
circulated between the targets and a cryogenic cooler at a flow rate of 2m/s. Platinum
resistors and vapor pressure bulbs were located near the inlets and outlets of the target
cells to monitor the target temperature and pressure. These values were very stable and
were used to determine constant target densities for the experiment. The uncertainty
56
--I
I
i
I
I
I
\
Figure 22: Floor plan of End Station A
57
in the target thickness due to the combined uncertainties in density and target length
(the latter due to shrinkage of the target upon cooling) was estimated to be < 0.7%.
Also on the liquid target ladder were two aluminum "dummy" targets, consisting of solid
aluminum pieces positioned to simulate the endcaps of the long and short liquid targets.
The thickness of the dummy targets however was 95 times that of the empty liquid
cells. Measurements of the scattering rate from the dummy targets were reduced by this
factor in the analysis and subtracted from the liquid target results. The purity of the
liquid 'H and 'H was measured after the experiment and found to be 99.94% and 99-68%
respectively.
The solid target ladder provided two targets of different thickness for each nucleus
studied. The purity of these targets was found to be effectively 100%. The uncertainty
in the thickness of the targets was estimated at 02%.
Details of all targets used are presented in Table 21. The total amount of material
encountered by the incoming electron and scattered particles is summarized in Table 2.2.
The materials in this table cause the particles to lose or gain energy between the point
where they are measured and the scattering vertex. Corrections for these energy shifts
are made in the analysis.
2.4 16 GeV/c Spectrometer
The 16 GeV/c spectrometer, shown in Figure 23, is a 90' vertical bend spectrometer
consisting of a single dipole magnet surrounding a vacuum-sealed chamber and topped by
a shielded detector hut. The vacuum chamber is sealed at either end by mylar windows
(12 mil and 14 mil thickness for the entrance and exit foils respectively). The opening
angle is limited by a fixed slit made of 12 inch thick lead and positioned 1959 m from the
target pivot. The slit is flared in the dispersive (vertical) direction, with a front opening
58
Table 21: Description of targets. The radiation lengths for the solid targets are from
the formula of Tsai (ref. 33]). For the liquid targets, the values come from the Particle
Data Booklet, where a correction for molecular binding is made.
Name
Material Density Rad. Length
Thickness
(g/Cm')
4cm-LH2
4cm-LD2
15cm-LD2
C-2%
C_6%
Fe-6%
Fe-12%
Au-6%
Au-12%
1H
2H
2H
12C
12C
56Fe
56Fe
197
197
Au
Au
(g/CM2)
(CM)
(g/CM2)
(% rad. len.)
61.28
122.6
122.6
42-66
42.66
13.88
13.88
6.46
6.46
4.03
4.03
15.75
0.4097
1.1730
0.1064
0.2098
0.0206
0.04013
0.2840
0.6853
2.6782
0.8985
2.5724
0.8330
1.6425
0.3946
0.7688
0.46
0.56
2.19
2.11
6.03
6.00
11.83
6.11
11.90
0.0705
0.1701
0.1701
2.193
2.193
7.829
7.829
19.157
19.157
of ±9.07 cm vertically and ±9.30 cm horizontally. Thinner movable slits just before the
fixed slit were also available but were kept open during all but a few checkout runs. The
acceptance of the spectrometer was determined to be 340 msr (for a point target). The
1.6 GeV/c is described in detail in ref. 341.
The field of the dipole magnet was measured before each data taking run by remotely
inserting an NMR probe into the center of the magnet. The central momentum was then
determined using the following formula:
k'(central = 0075 * (B
where
326)
(2.1)
is te magnetic field in Gauss and k' is in MeV/c. The maximum momentum
that could b detected with the 16 GeV/c spectrometer was 147 GeV/c.
The optical properties of the 16
eV/c are defined by the single dipole magnet.
Slanted and urved pole faces introduce quadrupole-type fringing fields, producing pointto-point focusing in the dispersive (vertical) direction and line-to-point in the non-
59
Table 22: Description of materials interposed in the paths of particles (between the
scattering vertex and the points at which they are measured). The superscript "L"
indicates materials only relevant to liquid targets.
Item
Material Density
Thickness
(g/CMI)
(CM)
(g/CM2)
(% rad. len.)
seen by incoming electron
wire arrays
target entrance foil
liquid
target
upstream
endcap L
Al
Al
Al
2.70
2.70
2.70
0.0041
0.0025
0.0076
0.0111
0.0068
0.0206
0.046
0.028
0.086
Al
2.70
0.0122
0.0329
0.137
Al
mylar
2.70
1.39
0.0127
0.0064
0.0343
0.0088
0.143
0.022
Al
mylar
2.70
1.39
0.0127
0.0356
0.0343
0.0494
0.143
0.124
Al
Al
2.70
2.70
0.0305
0.0254
0.0824
0.0686
0.343
0.286
I
seen by both scattered particles
liquid
target
downstream
endcap L
liquid target side wallL
liquid
target
side wall
insulation L
seen by scattered electron
1.6 GeV/c target exit foil
1.6 GeV/c entrance foil
seen by scattered proton
8 GeV/c target exit foil
8 GeV/c entrance foil
60
FOCAL
PLANE
6
I
- -M*-i-,-KAM LINE
-
I
I
I r-----j
I
Figure 23: Side view of the 16 GeV/c spectrometer
61
I
dispersive (horizontal) direction. "Point-to-point" indicates that particles with the same
momentum are focused to the same position at the focal plane; "line-to-point" refers to
focusing based instead on a particle's angle in a given plane. The magnet is designed
so that the focal planes in both dispersive and non-dispersive directions are coincident.
The spectrometer had a nominal momentum bite of ±5%. Larger values of
could be
detected, but TRANSPORT calculations performed before the experiment determined
that the acceptance ceased to be flat (i.e. independent of particle angle) outside this
range. Consequently, ±5% cuts in
were consistently applied to the data. Details of the
optical models used in the analysis are given in Sections 33 and 44.
The detector stack used in NE18 is shown in Figure 24. It consisted of four detection
systems: a Cerenkov counter and shower counter calorimeter for particle identification, a
series of drift chambers for precise tracking, and scintillator planes for fast timing. Both
the scintillators and calorimeter were segmented and were used in track selection. In
the figures and text below, Cartesian coordinates x, y, z are used to describe positions
in the stack. These variables are defined using the TRANSPORT convention: the x
axis runs in the dispersive direction toward higher momentum values (thus down at
the spectrometer
entrance),
z is the magnetic axis, and y is defined so as to provide a
right-handed coordinate system. Also,
refers to the percentage deviation of a particle's
momentum from the central momentum of the spectrometer. The positions of all detector
elements were precisely measured during a post-experimental survey.
The rate of pions entering the 16 GeV/c spectrometer can reach 00 times the electron
rate at the highest Q value measured by NE18, and so it is critical that pions be rejected
efficiently at the trigger level. To make this particle identification, one can use the fact
that highly relativistic particles travelling faster than the speed of light in a refractive
medium emit Cerenkov radiation. Since the speed of light in a medium of refractive index
62
SHIELOING
j
I
.
4
SHOWERCOUNTERS
P8
PA
XUYU
CXCY1
UPPER SCINTILLATORS
V
1 zI zI zI zI zI /I /I zI 41 41
...........................
WIRE
-
CHAMBERS
BXBy I -.........................
AxAy I ...........................
XDYD
WWER SCINTILLATORS
16 1 1 1 1 5
5
Z 1
rn
C02
Cherenkov
Counter
0
Figure 24: Diagram of the 16 GeV/c detector stack. The detector positions are to
scale, but their thicknesses are in general exaggerated.
63
n is /n, one obtains a threshold momentum below which a particle of mass m will not
radiate Cerenkov light:
p(threshold)
M
-
n1-
(2.2)
I
As this momentum is higher for a pion than an electron, pions and electrons of the same
momentum can be distinguished by selecting a material with a value of n that will cause
only the electrons to emit Cerenkov radiation. The radiation is emitted in a characteristic
cone of half-angle
0 = os-'(1/nO).
(2.3)
The Cerenkov detector used in the 16 detection stack consisted of an aluminum cylinder
1.3 m in height, with aluminum entrance and exit windows 04 mm thick, and filled with
C02
gas of refractive index 100045 (at room temperature).
A concave mirror of very
thin aluminized mylar (0.08 mm) was positioned about 12 cm below the exit window,
directly in the path of the scattered particles. This mirror directed emitted radiation
onto a smaller spherical mirror at the side of the detector, and a photomultiplier tube
(PMT) was placed on the opposite side at this mirror's focus. A wavelength-shifting
coating was applied to the tube face to increase it's sensitivity to ultraviolet rays.
Three drift chambers provided precision tracking. These chambers were used in the
previous experiment NEII, but were found to be inefficient. Consequently, they were
carefully cleaned and repaired before this experiment. Each drift chamber consisted of a
14 cm high box containing four wire planes separated by thin foils. Two wire planes were
strung in the x direction and two in the y direction; the planes in a given direction were
separated by only 09 cm and so were treated as a single plane. The arrangement of wires
is depicted in Figure 25. The wire spacing was 1.0 cm, with cathode and anode wires
alternating and charged to 500 V and 1850 V respectively. The cathode wires provided
uniform field shaping, while the anode (sense) wires collected the signals induced by the
64
Anode (sense) wires
Kapton foils
,. 'A
I
'
*
0
0
0
? i
ii*%
*
010,
0
0
0
0
0!
'11
---
*
0
00
0
a
0
0
I
0
0
0
Cathode (field shaping) wires
Figure 25: 'The arrangement of wires in the 16 GeV/c drift chambers; each chamber
consisted of two such pairs of wire planes, strung alternately in the x and y directions.
ionization trails of fast charged particles. The chambers were equipped with gas input
and output valves, and were connected in series to a flow system that circulated isobutane
gas at a rate of 20 cc/minute. Each sense wire was attached to a channel of a common
start TDC module; the 16 trigger (which does not include the wire chambers) provided
the start, while each wire signal provided the stop for its own channel.
Precise eent
timing was provided by four planes of plastic scintillator bars, two
segmented in the each of the x and y directions. The scintillators were 1.1 cm wide
and their segmentation was used to assist with track identification. For this reason, their
transverse edges were precisely mapped out relative to the wire chambers using tracks
reconstructed. from the data. Each scintillator bar had phototubes attached at either end,
and each of these was connected to both a TDC and an ADC. These measures permitted
careful determination of the time a charged particle passed through each scintillator (see
Section 35).
At the top of the detector stack were two layers of 14 lead glass blocks each. They
measured 10.6 cm along each horizontal side by 25 cm vertically, and were each viewed
by a PMT. A.pion with momentum greater than I
eV/c has energy loss similar to that
of a minimum ionizing particle (
this is much less than that of a I
2 MeV/g/cm');
65
GeV/c electron which loses energy predominantly by Bremsstrahlung radiation. In fact
the measured electrons typically deposited all their energy in the calorimeter. The total
signal deposited in the shower blocks during an event consequently provided a second
means of particle identification. The segmentation of the shower blocks was also used in
track identification, but the precision of the device in this context is sharply reduced by
the transverse size of the electromagnetic shower that is precipitated by a high energy
electron. This transverse size was found to reach a radius of about 4 cm at the top of the
shower counter, and so energy was generally not localized to a single block in any row or
column.
2.5
The
8 GeV/c Spectrometer
GeV/c spectrometer is depicted in Figure 26. It is a QQDDQ spectrometer, which
refers to the sequence of quadrupole and dipole magnets seen by a particle emanating from
the target; it's optical axis rotates a total of 30' over the total magnetic path, culminating
in a shielded detector hut inclined at this angle to the floor. The spectrometer is described
in detail in ref. 35]. In the NE18 experiment, a new "large acceptance" magnet tune was
adopted. This involved reversing the polarities of the first two quadrupoles from their
normal acceptance" configuration, and resulted in an increase in solid angle of more
44
than a factor of 4 Like the 1.6 GeV/c, the
GeV/c provided point-to-point focusing
in the bend plane and line-to-point focusing in the non-dispersive (horizontal) direction.
It should be noted that a different coordinate system is used to describe positions in
the
GeV/c than in the 16 GeV/c: z is directed along the magnetic axis, x points
horizontally toward higher scattering angle, and y is defined to produce a left-handed
coordinate system (and so points vertically upward at the spectrometer entrance).
The
GeV/c detector package is depicted in Figure 27 and consists of a Cerenkov
66
QC
0 0za
Q
WW
ln
0a
kA
0 z0
C
D
11
&A
J
Ai
O
W
-i
0.9
(n
Figure 26: Side view of the
67
GeV/c spectrometer
detector,
planes of segmented scintillators, and 10 multi-wire proportional chambers
(MWPC's). The 31 m long Cerenkov detector was filled with freon 114 (n=1.00140 I
atm of pressure) only for the highest
two
Q2
settings. It was intended that the Cerenkov
provide pion rejection at these energies, while at the lower energies the long drift distance of 386 m between the front and back scintillator planes would permit particle
identification by measured particle velocity. In practice, the pion rate in the
GeV was
found to be sufficiently low relative to the accidental coincidence rate that pions could
be efficiently analyzed as accidentals, and particle identification was not needed. Timing
information was entirely provided by the five scintillator planes. The SF layer was segmented horizontally, into eight bars of 6 cm width framed by two of 15 cm width. The
NBS hodoscope consisted of three closely spaced planes: the outer planes consisted of
ten scintillator bars 48 cm wide by
cm tall, stacked in two adjacent columns of five
each, while the middle layer was formed from 22
cm wide bars placed side by side. The
SM and SR planes were vertically segmented, into three 42 cm tall bars. Finally, the
10 MWPC, positioned between the SF and NBS scintillators, contained wires of I mm
separation oriented in alternate planes at O' and 60' to the horizontal plane. The wire
angle of alternate 60' planes also alternated to the left and right. The 00 planes were
referred to as "P" planes and those at 60', as "T" planes; these labels refer to the planes'
sensitivity to particle momentum and in-plane angle ("theta") respectively. A "magic
gas" composed of 65.75% argon, 30.0% isobutane 025% Freon 13BI, and 30% methylal
was circulated through the chambers.
68
r'Mf-^K
I
I
I
A
/-
-I
n
I
A
N
e Chambers
loscope
entrance window
of Cherenkov
SM
SR
384 cm
I
0
I
1 M
Figure 27: Diagram of the GeV/c detector stack. The detector positions are to scale,
but their thicknesses are exaggerated.
69
2.6 Trigger Electronics
2.6.1 16 GeV/c Migger
The formation of the 16 trigger is depicted in Figure 28. All 16 GeV/c detector elements
except for the wire chambers participated in the trigger, after being grouped together in
the following way. The scintillator tube signals were discriminated and
Red by plane,
producing signals XU, YU, XD, and YD. (The labels correspond to the plane names
indicated in Figure 24.) These in turn produced SU (XU and YU in coincidence), SD
(XD and YD in coincidence), and SC (a 34
coincidence between XU, YU, XD, YD).
The shower counter singals were amplified by a factor of 10, and summed by FAN IN
modules in stages to form PASUM and PBSUM (referring to the lower and upper shower
planes respectively). These were summed in turn to produce SHSUM- Discriminators
then converted PASUM to two signals: PA Lo and PA Hi, with threshold settings of
155 mV and 500 mV respectively. SHSUM was discriminated only once (at 600 mV to
produce SH. The Cerenkov output was immediately discriminated to produce signal CK.
All discriminated signals had widths of 20 ±
ns.
Two electron triggers were defined, both intended to fire at every electron while
providing some degree of pion rejection. EL Hi was formed from a 33 coincidence of
PA Hi, SC, and SH. This trigger provided the stronger level of pion discrimination,
using both the summed information from all shower blocks (SH) and a relatively high
requirement on the energy deposited in the first layer (PA Hi). The latter condition was
motivated by the fact that the rate of energy loss of hadrons in matter peaks toward
the end of the hadron's path (described by a "Bragg curve" 36]). Electrons radiating
away their energy, in contrast, do so with -dEldx
- E and so should deposit relatively
more energy in the first shower layer. These shower counter tests, however, are not 100%
70
efficient for electrons, because of cracks between the shower blocks, the low energy tail of
electron energy loss, and a sporadic malfunction in one of the shower block ADCs (PA4).
Thus a second electron trigger, EL Lo, was also used. EL Lo required a 23 coincidence
between SU, SD, CK, and PA Lo. Thus, the shower requirement is weakened, and was
not required for the formation of the trigger. The high efficiency of the Cerenkov and
scintillator planes in detecting electrons makes this test 100% efficient to well within
experimental statistics. The final electron trigger EL20 was formed from an OR of these
two signals and given a pulse width of 20 ns.
The 16 GeV/c pretrigger was formed by an OR of three signals - EL20, RANDOM,
and PION PRESCALE -, arriving in coincidence with the beam gate. RANDOM was
generated by a low rate pulser, also vetoed by the inverted beam gate. Its purpose was
to provide "empty" events where hardware information such as ADC pedestals could be
periodically monitored. PION, designed to provide a sample of pion events, consisted
simply of SC vetoed by the Cerenkov CK. The high PION event rate required it to be
prescaled before being included in the pretrigger. This was accomplished by forming
PION PRESCALE from PION in coincidence with the prescaled beam gate (by up to
a factor of 2').
Finally, PRETRIGGER 16 was converted to TRIGGER 16 by forcing
the final trigger rate to at most one event per spill. This requirement is imposed by the
processing speed of the data acquisition. It was accomplished by vetoing the pretrigger
with a long (1.0Ms)gate started by a delayed copy of the pretrigger. This gate effectively
blanked out any further pretrigger signals arriving before the next spill.
2.6.2
8 GeV/c and Coincidence rh-iggers
The formation of the
GeV/c and coincidence triggers is shown in Figure 29. Of the
8 GeV/c detectors, only the scintillators participated. The PROTON signal was formed
71
Figure 28: Formation of the 16 GeV/c trigger.
from a coincidence of the 3 planes SF, SM, and SR. Another trigger, called 2/3
or
PION, required only 23 of these planes; its purpose was to provide an event sample
with which to test the efficiency of individual scintillator planes. Since virtually any
particle entering the
GeV/c hut would cause any scintillators it encountered to fire,
the 2/3" event rate was extremely high and had to be prescaled to prevent prohibitive
dead times. Factors of 2
2
to 2'
were used, for all but the 11 targets.
The PROTON signal did not contribute directly to the final trigger. Instead, it was
used to form the coincidence triggers. The 16 pretrigger started 2 gates, one 100 ns and
the other 31 ps long. A PROTON trigger arriving within the short or long gate cause a
COIN or LONGCOIN signal respectively. COIN provided the true coincidence trigger;
LONGCOIN was defined to provide a selection of PROTON singles. The PROTON
signal was delayed by an amount that was adjusted at each kinematic setting to position
the coincidence peak roughly 20 ns after the gate start. Note that the leading edge time
72
TYAWER
memoC.&
Figure 29: Formation of the
GeV/c and coincidence triggers.
of COIN and LONGCOIN was that of the PROTON signal. The
GeV/c pretrigger was
then formed from and OR of COIN, LONGCOIN, 2/3", and RANDOM (shared with
the 16). As in PRETRIGGER 16, PRETRIGGER 8 was vetoed by the inverted beam
gate, and TRIGGER
(7
was formed by vetoing PRETRIGGER
on a delayed widened
s) copy of itself. Two coincidence TDCs were setup to measure the relative time
between the electron and proton triggers. TDC8 was started by the
GeV/c COIN
trigger and stopped by the 16 GeV/c EL20 trigger, while TDC16 was started by EL20
and stopped by COIN.
2.7 Data Acquisition
Signals from all detector elements were delivered to the counting house on heliac cables
and processed by a DEC PDP-11 computer. All 16 and
GeV/c triggers were recorded
to tape. At the beginning and end of each run comprehensive begin and end headers
73
were recorded, and every
- 3 minutes, a checkpoint was stored containing a summary
of charge and scaler information. The online analysis package available at SLAC ran
continuously on a VAX cluster during data taking and provided real time information
on not only the detector systems but also the results from single-arm tracking. The
data acquisition package was also modified to write coincidence events (COIN triggers)
to compressed disk files. The relatively small size of these files permitted coincidence
information from the entire experiment to be stored and made available for analysis by
the oine
coincidence analysis program. This arrangement permitted us to monitor the
coincidence rate even at the highest Q kinematic setting, when the true to accidental
rate was so low that no timing peak could be discerned from the raw coincidence TDC
distributions. (see Figure 210).
2.8 Kinematics
The kinematics of the experiment were chosen based on several considerations. A broad
range in
Q2
over which to study quasielastic (ee'p) scattering was desired, and in par-
ticular, we wished to take data at the maximum possible value of Q in order to search
for colour transparency. The limiting factor in this case was the
GeV/c momentum,
which could only be set to 45 GeV/c. Beyond this, the solid angle acceptance in the
large acceptance tune dropped away sharply, as
of Q2
Consequently the maximal value
that the experiment could achievewas 68 (GeV/c)'.
We also wished to sample as much of the nuclear spectral function as possible, and
in particular to detect p
up to the Fermi momentum for each nucleus and at each Q.
This was achieved by varying the proton angle at each
to see the peak of the spectral function at p
Q2
around the value required
= . In so doing, we employed so-called
perpendicular kinematics: changing the angle of the proton changes the component of
74
r
I"II
1
.... I....................................
.
.
.
.
.
.
.
.
.
I
Raw TDC, spectrum
"I
V)
-+l
C
I
0
0
-1--
I'l
I
I HI I
0
. .....
. ..... . ....
-I5
0
-
b-
IT
. . . . . . -F-
. . . . . . . I. . . . . . . . .
I0
t8 -
t 1.8 (ns)
to -
,, (ns)
5
I
10
20
15
V)
-.1
C
I
0
0
ic
C
Figure 210: Raw coincidence TDC and corrected coincidence timing spectra for 197 Au
at
Q2
of 68
(GeV/C)2.
75
the pm vector. perpendicular to q in the scattering plane, while the detection range of the
other components remains unchanged. This choice of kinematics is the motivation behind
the sign we apply to the magnitude of the pm vector (see Section 1. 1). As Q
(and so
P') increases, a kinematic focusing of the "Fermi cone" of proton momenta occurs. This
refers to the fact that a given change in p, requires an ever smaller change in P' as p'
increases. Thus a given setting of the proton spectrometer momentum provides a larger
detection range in p, at higher Q, and the number of proton angle settings required to
cover the Fermi cone decreases. The relatively high energies of the experiment and large
momentum bites of the spectrometers ensured that missing energies up to the pion mass
at 140 MeV would be detected. Figure 211 depicts our experimental phase space in E,
and p, for carbon at Q = I GeV/c)', and graphically illustrates the effect of Op,on the
missing momentum coverage.
Our kinematic settings are summarized in Table 23.
76
Table 23: Kinematic settings used in NE18.
Q2
Targ
ts
(GeV/c)'
1.04
lH
c
f
Et
pI
0,
OP
(GeV)
(GeV)
(GeV/c)
(deg)
(deg)
2.015
1.41
1.39
1.24
1.20
37.3
35.5
1.36
1.28
38.8
3.188
1.47
2.49
2.45
49.0
47.7
4.212
1.47
3.56
3.54
54.2
53.4
5.120
1.47
4.49
57.0
56.6
43.3
35.0, 37.8, 40.6, 43.4,
46.2) 49.0, 51.8, 54.6
43.4, 46.27 49.0, 51.8,
54.6
35.9, 39.1, 41.3, 43.5,
46.7, 50.2
26.5
27.7
27.7, 30.5, 33.3
27.7, 30.5
19.6
19.5
18.5, 20.9, 22.6
20.9, 22.6
20.9
15.9
15.9
15.9, 17.3
15.9, 16.7, 17.3
16.7
Fe, Au
1.21
2H
3.06
lH
2H
5.00
C, Fe
Au
lH
2H
6.77
c
Fe
Au
lH
2H
c
Fe
Au
77
400 -
' ' :....... . .
,=55'
. . . . .3...
300
. -
-
-=52'
0,,=49'
0,=46'
0,=43
2
a
200 100
U
'-I
0
I
.......
....
Q)
' ::.
. . . . . . :: . ,
................
.. ::: .. . i
E
a_
I I I: : .
...
. : : . -I I I I I I
-100
:'
:.........
gig
. :: :
-200 -
: :
-300
-1(
I
. . . . . . . .. :
...
: I ' '
1
)O
-50
1
50
0
I
100
- I--
150
I
200
E M (M eV)
Figure 211: The NE18 experimental phase space in E and p for the carbon Q = I
(GeV /C)2 measurement. The eight diagonal bands that can be distinguished in the figure
correspond to the coverage of each of the proton spectrometer settings.
78
Chapter 3
Data Analysis
3.1
Overview
The reduction of the raw experimental data to useful physics information took place in
two stages. First, detector information was read in event by event from the coincidence
disk files described in Section 27, and the raw measurements from each spectrometer
hut were reduced to tracks. Using optical models of the spectrometers, these were reconstructed to particle vectors at the target. The tracking algorithms were modified
versions of programs developed at SLAC over the lifetime of the detectors. Secondly, the
nuclear structure quantities E, and p, were determined from the single particle vectors,
and true coincidence events were identified. This step included corrections to the single
particle vectors for their energy loss in the target, fitting of the coincidence timing peak,
and random subtraction.
3.2
Tracking
The tracking algorithms in both spectrometers basically proceeded by constructing tracks
from all combinations of wire chamber hits, then eliminating candidates using a sequence
79
of cuts and purges. At the end of the sequence of track reduction tests, it is possible that
several equivalent candidates remain; these are referred to as multiples. The tracking was
operated in the so-called "one track" mode, where a single track was always identified
from a collection of multiples by selecting the track which arrived first in time. The
possibility of the wrong track being chosen from a set of multiples and the motivation
for this selection criterion are dealt with in Section 32.3.
A "no track" correction is applied to the results to account for events where an
actual coincidence event occurred but no candidate track was found. This correction is
determined by examining the raw coincidence TDC spectrum for events where a given
spectrometer was unable to find a track, and calculating the coincidence content of this
TDC peak. The corrections used are < 3
3.2.1
and are described below.
1.6 GeV/c ']Tracking
At the outset, a cut on the Cerenkov ADC value was imposed on all events considered
bythe 16 tracking algorithm. This pion rejection measure was made possible by the excellent efficiency of the Cerenkov (>99.9%). Tracks were first constructed by considering
all possible combinations of wire hits, in the x and y directions separately. Particles
passing through a wire plane generally caused a pair of wire hits - one on either side
of the track's position. Occasionally, however, only a single wire hit would be recorded.
Consequently two different criteria were required for the construction of a candidate track
from the wire hits: a "good" track was constructed from at least two pairs and a total
of
or more wires, but any track consisting of at least 4 wires was also considered. If a
"good" track was found for a given event, only "good" tracks were considered for that
event. Candidates were also required to point inside the vacuum chamber exit of the
spectrometer and within the boundaries of the shower counter. Furthermore, at each
80
scintillator plane in which both phototubes of any scintillator bar fired, tracks were required to pass through a scintillator bar in which edher tube fired. At the end of this
procedure, the number of possible x and y tracks were counted, and if either was greater
than one, a prging algorithm was called to decide between them.
A series
purges was applied to the tracks in the sequence described below, until
only a single candidate remained. At each stage, the x and y tracks were combined to
form candidate pairs, and it was these pairs (rather than individual xly tracks) that
were discarded. Any purge that discarded all tracks was dismissed, so that no events
with candidate tracks were discarded at this stage. (This is with the exception of the
test on reconstructed quantities, which was applied as a hard cut).
shower counter The software gains applied to the shower counter ADCs were tuned
so that a signal of "I" in any block corresponded to the electron energy loss peak
for that block. Since most of the electron loss occurs in the lower ("A") layer, the
gains o
the A blocks were lower than those in the
block. This test required
that the sum of energies deposited in shower blocks along a track be greater than
0.6. This threshold value was determined by applying the test as a hard cut and
progressively lowering the value until no significant gains in the final number of
coincidences was obtained. The determination of which blocks could be considered
to lie on a track involved the use of a slop factor of 4 cm, to account for shower
spreading; this value was determined in a similar way. It should be noted that
a few of the shower counter ADCs malfunctioned sporadically during the earlier
part of the experiment.
For these events, the shower purge was not used. This
choice was supported by examining runs where the malfunction affected a limited
percentage of events, and verifying that the number of extracted coincidences was
the same if one neglected this test or applied it as a hard cut and later corrected
81
for the fraction of bad ADC events.
target cuts Candidate tracks were reconstructed to target at this stage (see Sec 33)
and loose cuts were placed on the reconstructed vectors to eliminate very improbable tracks. These cuts were about a factor of two larger than the spectrometer
acceptance in each coordinate, and so did not bias the event sample in favour of
tracks away from the acceptance edges.
purging of "nearby" tracks Due to the high event and background rate in the chambers, a handful of wires which were either noisy or dead (and therefore forced on
in software), and the relatively spare requirements for the construction of a track
(e.g. only 2 chambers required), a common source of multiple tracks was the construction of two or more "nearby" tracks which actually corresponded to the same
event. To eliminate spurious choices, tracks within 2 cm of each other at the focal
plane were defined as "nearby" and those with less the highest number of wires
and pairs in the group were discarded. Finally, a purge designed to identify the
most common patterns of spurious "nearby" tracks was employed. These patterns
included tracks separated by no more than a wire at one plane, tracks constructed
using hits in only 2 out of 3 available chambers, and tracks involving a dead or
hot wire. The graphics capabilities of the oine
software, which enabled single
events and associated tracks to be displayed on screen, were extremely helpful in
the design of these tests.
The "no track" correction factor for the 16 GeV/c was found to vary from I to 2
and is listed in Table 3.1.
82
3.2.2
8 GeV/c Tracking
The tracking algorithm used for the
eV/c was very similar to that used for the 1.6.
However, it was called repeatedly, with three increasingly lenient track criteria, until at
least one track was found. First, a track was required to include at least 3 P chambers, 3
T chambers, and 7 chambers in total. Second, 2 P 2 T, and 6 total chambers, and finally
only 2 P 2 T, and
total chambers were demanded. Candidates were determined first
in the y (vertical) direction, using only the P chambers, then companion x tracks were
identified using the y tracks already found to resolve the horizontal positions indicated
by the slanted. T chambers.
The purging tests in the
GeV/c were applied in the following sequence. We should
note that a hard cut on the extracted
of the track was applied at the outset, as a pion
rejection measure.
X2 cut A maximal X2value
of 30 was required.
fiducial cut Cuts were placed on track position at the SF, NBS, and SR scintillator
planes which defined a fiducial (contributing) region in the hut.
scintillators
on track Tracks not matching the information from planes which fired at
least one tube were purged. Only the SF, SM, and SR planes were used in this
test.
NBS hodoscope
Tracks not matching the NBS hodoscope information in at least one
direction were purged. If no tubes fired in a given direction, the track was passed.
The relative looseness of this test was due to the inefficiency of the NBS scintillators.
scintillators
in combination
Tracks that failed to satisfy either a strict scintillator or
a strict NBS cut were purged. These strict tests were each about 1-5% inefficient,
but the efficiency of the OR is about 0.1%.
83
x-dth correlation
Due to the line-to-point focusing of the
GeV/c, the horizontal po-
sition and angle at the focal plane of a track emanating from the target should be
closely correlated. Tracks falling outside this correlation band were rejected.
target quantities
As in the 16 GeV/c purging, loose cuts on target quantities (well
outside the spectrometer acceptance) were applied. The reconstructed position at
target along the beamline is more accurate in the
GeV/c than in the 1.6 and so
a loose cut on this position based on the length of the target was also applied.
For the
GeV/c, the vast majority of "no track" events registered as coincidences by
the coincidence TDC were caused by the fact that the wire chambers do not quite cover
the acceptance of the
GeV/c hut, while the trigger-forming scintillator planes did. This
effect is taken into account by the PWIA Monte Carlo which is used to normalize all of
our results. We apply a % systematic uncertainty to account for any tracking losses.
3.2.3
Multiple h-acks
After the sequence of purges described above is applied to a set of tracks, there may still
remain several equivalent choices; in this case the event is referred to as a "multiple".
In the "one-track" mode, a single track is always selected from such a sample. This is
accomplished by determining each track's time relative to the 1.6 or
GeV/c trigger,
after projecting back to the focal plane. The track with the earliest time is then selected.
The reason for this choice is that tracks in a multiple sample very often share one or
more scintillators; since the scintillators are used to determine the event time and since
only the first track to pass through a scintillator will have the correct time assigned to
it, selecting the track that is earliest in time avoids the possibility of losing coincidences
because their coincidence time has been computed inaccurately.
The number of multiples in the
GeV/c is in general low 84
less than 2
for all
kinematics except the deuterium runs at Q = I GeV/c)' and at forward proton angles.
At these kinematic settings, the number of background events in the
GeV/c increases
dramatically, and up to 10% of the analyzed events are multiples (at the most forward
angle,
08=
36"). This high rate of multiples can be explained by the rate recorded by the
PROTON scalers; in other words, the multiple tracks correspond to real trigger-causing
particles. The! loss of coincidence protons due to earlier background protons arriving in
the same beam burst can then be calculated as a component of the computer deadtime.
This is examined in detail in Section 36.2.
The fraction of multiple events in the 16 GeV/c, however, is much higher -
ap-
proximately 5% at low Q settings, and reaching 10% during some Q = 7 GeV/c)'
runs. The 16 GeV/c is always subject to a high pion background rate, and at high Q,
this rate becomes several hundred times larger than the electron rate; the source of the
multiples is clearly pions accompanying trigger-causing particles into the spectrometer.
The question is, do the multiple events contain the same fraction of coincidences as the
single track events, or is there something innately "bad" about them? The answer lies
in the purge on the amount of shower energy found along each track. The large majority
of the multiple events are caused when all the candidate tracks failed the shower purge;
in this case, the purge is discarded and the tracking must use only its remaining tests
to refine the sample down to a single choice. Although a Cerenkov cut is placed on
all events analyzed by the 16, there is a substantial possibility that a pion produced a
knock-on electron which then caused the Cerenkov to fire. To determine a correction for
the 16 multiples, the fraction of coincidences found in single-track samples which passed
and failed the shower purge was computed. This fraction is indeed much larger for the
sample of good shower events. The multiple tracks were then discarded, and those which
passed and failed the shower track test were counted separately. Different corrections
85
were applied for each of these sets of multiples, using the fraction of coincidences found
in the single-track samples. In practice, the "one-track" method of selecting the first
event turns out to be very successful at choosing true electrons. The net corrections
applied to the "one-track" results to compensate for multiple track losses is at most 2.
Upon reflection, this is not terribly surprising. The rate of multiples in the 16 is much
larger than the number one would expect from the rate of 16 EL20 triggers, and so it is
clear that the extra tracks are not pions which leaked through the electron trigger. They
are therefore not the particles which caused the trigger. Since the 1.6 trigger starts all the
TDC's, the pions will in general have later times at focal plane than the true electron.
3.3 Reconstruction of Events to Target
Tracks determined in each spectrometer were reconstructed to particle vectors at the
interaction point by means of matrix models of the spectrometer optics. During data
taking, the online analysis used matrices determined by TRANSPORT 37] a program
developed at SLAC which computes optical matrices describing the passage of charged
particles through a magnetic system. Input to the program is provided by "decks", files
describing the position, shape, and properties of all magnetic elements in the system.
During the post-experimental analysis, however, a powerful tool for understanding the
spectrometer optics was available in the form of elastic data. The matrix elements were
fit to this data using an iterative procedure, as described below 38]. The technique
involves correlating target quantities which are known (by virtue of fixed target positions,
collimators, or kinematic correlations) with focal plane quantities (or combinations of
them). Any correlation indicates the need for a matrix element; the element's value is
determined by eye from the slope of the correlation. The necessity for doing this by eye
rather than with a fitting procedure comes largely from the presence of radiative tails,
86
which distort the shape of an elastic stripe, for example. By hand, the true stripe can
be identified.
Two coordinate systems are used in this discussion. The first refers to particle coordinates at target, with Z pointing downstream along the beam line, Y directed vertically
upwards, and X defined to form a right-handed system. Z
of Z onto the spectrometer entrance plane. The angle
refers to the projection
is the difference between the
in-plane angle and the central spectrometer angle, with positive
angles.
at larger scattering
is the out-of-plane angle, with positive (Dcorresponding to angles above the
plane of the lab. Positions at the focal planes (in the hut) are described instead using the
TRANSPORT convention: x along the dispersive direction toward larger momentum, z
along the magnetic axis in the direction of particle travel, and y forming a right-handed
system. Track angles in the x and y directions (relative to z) are denoted dx and dy
respectively.
1. The Z matrix elements are fit first, using the 15 cm dummy target cell. This target
consists of two aluminum plates at precisely known positions and with effectively
zero width. The Z ly, Z Idy, and
ZLJO
(offset) matrix elements are fit so that
the reconstructed positions are correct and the resolution is optimized. In practice,
resolution in this coordinate was poor (FWHM >
cm), and only very loose cuts
on target position were used in the analysis.
2. (D16 matrix elements are determined using the 4D16distribution for 'H. The center
must be at
the edges must correspond to the vertical edges of the fixed slit at
the front of the 16 GeV/c, and the distribution should be filled uniformly because
of the idependence
of the cross-section on this coordinate. Also used here were
checkout runs where the movable slit at the spectrometer entrance was positioned
at ± 21 and 27 mr.
87
I % and DN matrix elements are tuned using H and requiring that the out-of-plane
component of pm is independent of these target angles.
4. For 'H data, E
= -
- E - M, - T.. is fixed at the binding energy of 22
MeV. Selecting a sample of events with p
less than about 10 MeV, the effect of
TI,, on E, can be kept to less than about
MeV so that, to this accuracy, E,
depends only on the difference of spectrometer energies. Such an event sample
can be used to tune the secondary
primary dispersive
68
and
16
matrix elements (i.e. other than the
x elements), because any correlation between E, and dx, for
example, must be due to a 1dx matrix element.
5. The 'rimary dispersive
blx) and
matrix elements were fit using a series of H
runs, performed during checkout, where the central momenta of the 16
and 8 GeV/c were varied independently in steps of 3.
eV/c
Handles on these matrix
elements are obtained using the exact correlations between all scattering angles
and momenta determined by elastic kinematics.
A momentum scan moves the
elastic peak across the spectrometer'sfocal plane in known steps and thus provides
information on the spectrometer dispersion.
6. Matrix elements were checked using an angle scan of the
GeV/c, also done on H
during checkout.
7. Offsets in all target quantities can be determined using H data, available at all
Q2.
These offsets were fit at all Q2 , and were found to vary somewhat. Varying
offsets in the beam energy as its value changes, and offsets in central spectrometer
momentum as the magnets are retuned are believed to be responsible.
Ultimately, only first and second-order matrix elements were required in the 16
GeV/c, but three third-order elements had to be introduced in the
88
GeV/c.
These
were Olyx', Oxy', and a very small term in bxy'. If one uses the width of the extracted
E, and p. peaks in hydrogen as a measure of the net resolution of the reconstruction,
one finds that the spread in beam energy (which contributes to each of these measured
quantities) dominates the resolution at all Q.
This is determined using the fact that
elastic kinematics are fixed by the beam energy and only one other scattering angle or
momentum. Using this overdetermination, one can compute the beam energy for each
event using, for example, the proton scattering angle and then examine the widths of the
E,.,, and p, spectra. The resulting resolution is substantially less than that incurred by
'dEbe.,.1Ebe.,..
3.4
The reconstruction optics can thus be considered well-known.
Determination of E and m
Before E, and p
were determined for each event, two corrections were applied to the
measured particle energies to obtain a good estimate of their values at the scattering
vertex.
The ionization losses of the particles as they passed through the materials listed in
Table 22 were calculated using formulae based on the Bethe-Bloch equation. For particles
heavier than an electron, this equation gives the most probable rate of energy loss as
dE = 03071z2
dt
In
1
A
2m'02
2
I(
-
-
_ 02
(3.1)
2)
where dEldt is in MeV g CM-2 , z is the charge of the particle, m, is the mass of the
electron, and I is an ionization constant. I depends on the material being traversed:
for Z >
I = 16Z'-' eV, while for liquid
correction
for the charge screening provided by atomic electrons was neglected because
H2
and D2 I = 21.8 eV. The density
it contributed less than 5% at our highest proton energy. For electrons, the Bethe-Bloch
89
formula must be modified to take into account the substantial deflections the electrons
undergo because of their lighter mass, and the fact that their interactions are now between
identical particles (incident and atomic electrons). This calculation leads to the following
expression in the limit
1: 391
dE = 03071 Z 1 19.26
dt
A2
I N ]
kpjj
(3.2)
Here, t is the material thickness in g cm- 2 and p is its density in g cm- 3
The incoming and outgoing electrons are affected not only by the struck proton, but
are also accelerated and decelerated respectively by the Coulomb field of the spectator
protons in the target nucleus. For an electron in either arm experiencing a hard scattering
at a radial position r inside the nucleus, the total change in potential energy it sees
between this position and a location infinitely far away is given by
AEc
(Z _
)e2
41rEORO
3
2
2
(3.3)
2
where R is the charge radius of the nucleus, given approximately by Ro = 1.18A1/3 fM.
Averaging this over a sphere of radius Ro, one finds a mean energy change of
6 (Z -
)e2
2K-E = 5 47reoRo-
3.5
(3.4)
Timing
Thin plastic scintillator bars produce signals characterized by quick rise times that make
them ideal timing devices. All of the scintillator times found along a given track were
used to determine the time of event production at the target, after applying careful
corrections. These corrected single-arm times were then combined with the information
90
from the coincidence TDCs to provide a distribution of relative event times.
In each spectrometer, the time at which each particle track crossed the focal plane as
well as the particle's speed were determined by fitting these quantities to the particle's
trajectory and. recorded times at the scintillator planes. The timing at each plane was
corrected for wo effects. First, the scintillation light produced when a particle passes
through the scintillator takes time to travel to the phototubes. The difference between the
times measured at either end of a scintillator can be used to correct for this, using the light
travel speed and the length of the scintillator. The light travel speed is about
4 cm/ns,
and was determined for each scintillator bar by a fit to the "sum time" distributions.
(The sum time refers to the sum of the times recorded at either end of a scintillator).
Second, one notes that a phototube signal with a larger pulse height will cross its TDC
discriminator threshold sooner than one with a smaller pulse. Accordingly each TDC
time was corrected by a function of the ADC signal, with parameters fitted to optimize
the timing resolution.
Once the time at focal plane is determined, orbit corrections for the exact particle
path through the spectrometer must be performed to determine the event time at target.
This path length was calculated from a track's coordinates at the target using a series of
matrices base n TRANSPORT models of the spectrometers. The matrix elements were
fit to the data to optimize the coincidence time resolution.
The coincidence (relative) time is given by the simple formula
ctime = tproton
and
-TDC8
- tle
where
tproton
and
6 GeV/'c triggers, after projection back to the target. The delay time
tlex
(3-5)
+ tdelay,
refer to the proton and electron event times relative to the
tdlay
was
fit at each kinematics to center the coincidence timing peak. We should note that two
91
k
I k
10
R Window
B Window
ctime
C Window
31: Diagram representing coincidence timing peak and windows used in selection
of "true" events.
Figure
coincidence TDC's were available: "TDC8" started by the proton trigger, and "TDC16
started by the electron. TDC16 malfunctioned during the latter half of the experiment,
and so TDCs is used throughout the analysis. However, a non-linearity in TDC8 at small
values (which was corrected for in software) caused the coincidence peak to be set too
near the TDC underflow. Using the results from TDC16 at the first two Q settings,
a correction factor of 1.5
for the portion of the coincidence peak lost by TDC8 was
determined. This correction is referred to as deff
in Table 31. The resolution of the
experiment is demonstrated in Figure 210: the coincidence peak had FWHM widths of
around 06 ns.
Figure 31 illustrates the procedure used for random subtraction. For each kinematic
setting, a range over which the ctime background was flat (the "R window", typically ±
8 ns) was determined, and the ctime peak was fit to a Gaussian with a flat background.
92
The extracted FWHM produced two further timing windows, the "C window" at ± 2
FWHM, and the "B window" at ± 1.5 FWHM. Histograms of the event distribution in
E, and p, were taken from the
window, to increase the coincidence to accidental rate,
but were corrected for the randoms and for the total number of events in the wider C
window:
Ntrue(EmIpm = [NB(Em7pm - R,.t,,ATBNR(E,.7pm)]C7
C
Nc(Em7pm
- RrateATC
(3-6)
NB (E., pm - RrateATB'
where R(E,7pm) denotes the limits of the arrays, AT refers to the size of the timing
windows, and the random rate is given by
Rrate
ER(E,,,,p
NR(Em, pm)
(3.7)
ATR
3.6
Corrections
The correction factors applied to the data are summarized in Table 31. The NOTRK,
multiple track, and coincidence timing efficiency (deff)
corrections have already been
discussed. The computer deadtime and proton absorption corrections are described below. The electronic deadtime corrections discussed in Section 36.1 are not applied directly to the data but instead are needed to correct scaler rates used elsewhere in the
computer deadtime calculations.
3.6.1
Electronic Deadtime Corrections
A scaler counting logic pulses of duration -r and mean rate r will record counts at a
smaller rate T, The reason for this is that two counts arriving within a time interval less
93
than
will overlap and only one will fire the scaler. A scaler is a so-called paralyzable
device in that it does not "reset" itself after each count, and successive overlapping pulses
consequently cause deadtime to accumulate. (This is unlike a non-paralyzable device,
such as a simple data acquisition system, which is dead to incoming events while it is
processing, but is ready to accept new information as soon as it is finished with the
current event.)
The behaviour of a scaler can be analyzed in terms of Poisson statistics. The probability of finding n counts of mean rate r within a time interval t is
P (n = (rt )n,-rt
(3.8)
n
The probability of finding no events in an interval t is then
from which we can infer
the (normalized) probability distribution for the time intervals between events:
P(t = re -rI
(3-9)
A paralyzable scaler will only record events that occur more than 'r apart. The fraction
of counts recorded by the scaler is then
00
r
re tdt
e
-rT
(3-10)
What one actually measures however are not rates but numbers of counts integrated
over a run. To convert between a rate r and a scaler measurement n, one needs the
effective total beam time T delivered during the run (n =_rT). Basically, T = TpjnNrpjjj,
the product of the number and length of the beam bursts, but one cannot simply use
Tpjl = 1.6ps since the spills certainly did not possess a perfectly uniform charge distribution. One can effectively avoid this problem by sending a signal of interest to two gate
94
---
generators of ifferent widths and then to separate scalers. Specifically, if two rates ,
and F2, are measured, the true rate can be extracted as follows:
ii =
_nr
ne
ji2
n- - T .
T
(3-11)
n2-r
In other words, the electronic deadtime correction for fi is
elec
KIr
n2,r
(3.12)
Several scaler rates needed in the analysis were indeed measured with more than
one pulse width, but many were not. To deadtime correct these, one requires the total
effective beam time T so that Equation 310 can be applied. Using any pair of scaler
rates and equations 39 and 311 one can determine this value:
ii2
n2,
In
(3.13)
Equation 311. then yields the true number of events, but cannot be solved analytically.
Due to the high rates of some of the event types (e.g. up to
in the
23" triggers per spill
GeV/0, the equation cannot be approximated by a first order expansion inrr.
However a simple iterative procedure can be used to determine n.
3.6.2
Computer Deadtime Corrections
Computer deadtime refers to the loss of events incurred by the fact that the data acquisition system can process at most one event per beam burst. The computer deadtime
correction is defined to be the ratio of the actual number of events of interest to the
number analyzed. Generally, this is given by the ratio of pretriggers to triggers, but
because of the design of our coincidence trigger this is not an appropriate approach here.
95
The experimental trigger was formed from the OR of the 16 and
GeV/c triggers; the
1.6 trigger consisted of EL20, PION PRESCALE, and RANDOM, while the
trigger
was composed of COIN, LONGCOIN 23 (PION) PRESCALE, and RANDOM. The
random trigger is not included in the deadtime calculations because its very low rate <
.0.002/spill) gives it a negligible chance of blocking a coincidence event. LONGCOIN can
also be neglected because it cannot occur without COIN and so cannot block coincidences
either. Three sources of lost coincidences remain: the 16 trigger rate, the prescaled 23
rate in the
GeV/c, and the chance that a background proton blocked a coincidence
proton in the formation of the trigger.
1.6
Yigger Rate
Every coincidence requires an electron trigger. Thus if we neglect any complications from
the
GeV/c side, our deadtime correction is simply the total number of EL20 events
over the total number which formed the trigger.
Note that the possibility of PION
events blocking electrons is taken into account in this ratio, by the denominator. The
number of electrons forming triggers is directly measured by a scaler, E40-V,
which
counts EL20 events (widened to 40 ns pulse width) vetoed by an inverted 16 trigger (so
that the only events counted are those which coincided with a 16 trigger). The total
number of electrons can be determined via Equation 311 from the scalers EL40 and EL80,
measuring EL20 events widened to pulse widths of 40 and 80 ns respectively. From now
on, a deadtime corrected scaler measurement will be indicated with an asterisk, as in
EL40*. A better estimate of the number of real electrons (without pion contamination)
is provided by the CAB20-80 series of scalers, which measured coincidences between CK
(Cerenkov), PA Low, and PB. The Cerenkov requirement, in particular, is the same as
that present in the 16 analysis. However, the ratio
96
CAB40*
CAB40-V
ought to be equal to
EL40*
EL40-V
or
indeed to a similar ratio for any component of the total 16 rate. This is indeed the case,
to within 1%. To summarize, the 16 trigger contribution to the coincidence computer
deadtime is
Kc
Prescaled 23 Miggers in
omp,1.6
EL40*
EL40-V
CAB40*
CAB40-V'
(3-14)
GeV/c
The only remaining component of the experimental trigger able to block COIN's is the
2/3 trigger of the
GeV/c (also referred to as PION). Designed to provide a check on
scintillator efficiency, this event type requires only a coincidence between two scintillator
planes and can count at a tremendous rate (up to 8/spill). It was consequently prescaled,
by blocking te 23 from the
GeV/c pretrigger during a fraction
of the beam bursts.
During the reaining 1-,E)bursts, coincidence triggers might be blocked by earlier 2/3's.
Let
be the computer deadtime correction for these bursts. The total livetime unaffected
by the 23 rate is then
+
yielding a net, contribution to the total deadtime of
Kcomp,2/3=
(I _E +
To evaluate tc, we consider only the fraction
(3.15)
of bursts where 23 might contribute to
the pretrigger. Further, let rOIN and r2/3 be the true COIN and 23 rates respectively.
In the absence of 23 interference, we would expect the number of
involving COIN to be
,ENpjjj(l -
-rC01NT6i11).
97
eV/c triggers
The actual number of COIN triggers, however, is
PIPRTRIG
rCOIN
r2/3
where PIPRTRIG is a scaler measuring the number of triggers involving 23.
This is
understood by observing that every COIN requires a 23; the number of COIN signals
participating in the trigger is then just the number of 23 triggers times the fraction of
the 23 rate provided by rcOIN- PIPRTRIG itself is given in Poisson statistics by
PIPRTRIG
and so our correction
(3-16)
N.Pill( - e-12/3Tspill)
can be written
r2/3
rcolN
The rates r2/3 and rOIN
-rCOINTspill)
(1
(3.17)
e-12/3T-pill
can be determined from the scalers PION and COIN respec-
tively, after deadfirne correction and division by the total effective beam time T derived
earlier. The result is then
COIN'
PION*
Tr2/3
COIN*
TrcOIN
PION* (I
K
COIN* I
e N.pill
-
PION*
(3-18)
Npill
As a consistency check on the Poisson relations employed and on the scaler measurements themselves, the COIN sand PION values can be replaced by equivalent expressions
computed from other scalers. Note that the latter concern - the accuracy of the scaler
readings themselves - is largely due to the possibility of double-pulsing in scalers counting pulses of 20 ns width. The raw signals produced by photornultiplier tubes and the
like generally consist of an initial voltage spike followed by a long tail. Secondary voltage
98
spikes may occur along this tail, and superimposed on the partially decayed tail 20 ns
or so after the nital spike, may be of sufficient height to cause a discriminator module
to fire twice. Most logic pulses were in fact about 20 ns long. Some scalers however,
such as the CAB40-CAB80 series, were immune to this problem because the decay of
a voltage tail after > 40 ns is sufficient to prevent superimposed spikes from reaching
discriminator thresholds. Our approach is to use the redundancy of the
GeV scalers to
provide alternate measurements for necessary rates; the difference between the various
results is then included in the systematic uncertainty of the experiment.
From Equation 316 and 318,
PION* = -Npili In
I
PIPRTRIG
(3-19)
,ENrpin
Note that the use of PIPRTRIG, which counts at most once per beam burst, avoids
the need for an electronic deadtime correction. The coincidence rate, similarly, can be
estimated fror CIN-V,
which measures the number of COIN causing
GeV triggers
and also counts at most once per spill. For bursts where the 23 signals are blocked, the
probability of a COIN trigger is
I
-rCOINTpill
_
while for the rearnining bursts, the total number of COIN triggers is
PIPRTRIG
rcolN
PIPRTRIG
PION*
r2/3
rcOINT.
Consequently,
COINV = Npju(l _
_
-rcOINT.Pill)
+PIPRTRIG rcOINT.
PION*
99
(3.20)
To extract
rcOIN
readily from this expression, one can take a first order approximation
of the exponential, assuming rOINTpi < 1. In practice, this product is at most 0.1,
making the approximation valid to better than 1%. The result is an alternate expression
for COIN*:
COINV
COIN*
(3.21)
PIPRTRIG'
PION-
Using these various expressions as input to Equation 315 and 318, one seeks to
compute an uncertainty on K ,,p,2/'.
The correction itself is only significant A.5%)
for hydrogen and deuterium runs at Q2 <
generally set between I and 2'.
making the contribution of
(GeV /C)2, where the prescale fraction. was
For the remaining runs,
was set between 2 8 and 2`0,
to K"'P,2/' extremely small. For hydrogen and deuterium,
the uncertainties were at most 1%, and this only at those few kinematic settings (forward
proton angles) where K"'P,2/3 reached 1.10. (High 23 rates are due entirely to background processes, which vary inversely with scattering angle and momentum transfer).
Blocking of Coincidence Protons
A COIN trigger was formed when a PROTON trigger fell within a 00 ns gate opened
by the EL20 trigger. The proton was delayed so that the coincidence peak fell - 20
ns after the opening of the gate. Note that a COIN trigger was also produced if the
delayed proton signal was already present when the electron arrived. Consequently the
20 ns width of PROTON must be included to determine the total time window before
the peak in which a proton could fall and produce a coincidence. This arrangement is
illustrated in Figure 32, which shows a sample spectrum recorded by TDC8.
KcompPROTON
is the final computer deadtime correction we must evaluate. It takes
into account the possibility that a background (or "singles") proton may arrive in the - 40
ns window between a coincidence electron and proton, thereby blocking the coincidence
100
- - 2000
- IIIIIIIII
.
I
I
I
I
I
I
I - 1
I
I
L 1- 1
I
I
I
I
,
I
L .
.
.
.
I
I
I
I
I
I
I
I
I
I
1500
Cn
D" 1000
0
U
500
0
.
0
I
I
I
I
,
I
I
I
I
I
I
7
I
20
10
30
40
TDC,, (ns)
Figure 32: Example of raw coincidence timing spectrum measured by TDC8. This
TDC is started by PROTON and stopped by EL20 (delayed) - inverse to the sequence
which formed the COIN trigger. Consequently, larger time differences between electron
and proton produce smaller values in TDC8- The peak at 35 ns is caused by EL20 signals
overlapping with PROTON signals at the openng of the coincidence gate. The counts at
0 ns are due to TDC underflows and were excluded from the analysis.
101
proton. Note that tracks are certainly recorded for events within several tens of ns of
the trigger which starts the TDC's and ADC's attached to the detectors - in other
words, the tracking can certainly "see" several particles arriving within say 20 ns of each
other. However, in the "one-track" mode, a group of equivalent tracks is reduced to a
single choice by taking the track with the earliest time. Coincidence protons are then
truly blocked by earlier singles. This choice is motivated in Section 32.1, but is also
convenient for the evaluation of the proton blocking correction which follows.
The central observation to make is that only singles protons can cause deadtime over
and above what has already been corrected for: no matter how high the coincidence rate,
K`P,"
has already corrected for unanalyzed coincidences. Let us use r,, r, , and r
to
refer to the coincidence, electron singles, and proton singles rates respectively; "singles"
here refers to non-coincidence events only. Further, let t, refer to the effective - 40 ns
window in which a PROTON may arrive before the coincidence timing peak and still
form a COIN event. There are 4 possibilities for what causes a coincidence trigger, and
each is restricted to a certain part of the timing spectrum: a coincidence electron and
proton (ep) will fall in the peak, a coincidence electron and single proton (e p.) will fall
before the peak (p. must come before p) a single electron and coincidence proton
e,)
falls after the peak, while a pair of singles (e. p,,) may fall anywhere in time. Thus only
events involving a p., fall before the timing peak. The total number of events in this
region, labelled NA, is simply the number of spills with an electron event (N,) times the
probability of a p. arriving withing the t, coincidence window:
NA
=
1 _ e-rptc).
(3.22)
The number of events in the coincidence peak (N,) is the number of electron triggers IV,
times the probability that e arrived before e, times the probability that no p. occurred
102
in the t, window:
N, = N
r,
(3.23)
e-rp,, tc
r,, + r,
The number we are interested in is the true number of coincidences, n, =- rT. From the
previous equations one obtains
nc == N n, + n,
C
Here, Nc is te
N,
(3.24)
1
N
A
N,-
number of coincidences extracted from the corrected coincidence timing
spectrum by random subtraction, and so the remainder of the expression is the net
computer deadtime correction. The first term , n,,,N,+nc is just the total number of electrons
over those giving triggers -
i.e. the factor Kc"Pl
.6
from Section 36.2. The final term
is our proton blocking correction:
K coinPROTON
(3.25)
N
N,
NA is the number of events in front of the timing peak and can be accurately determined
without recourse to the value of tc by simply summing that portion of the ctime spectrum.
N, is the number of electron events, but as
NA
is determined from the final coincidence
timing spectrum, any restrictions the analysis places on electrons must be included in this
number. In fact, the 16 tracking places a Cerenkov cut on every candidate electron and
so N, is better estimated by the CAB40-V scaler than by the larger number E40-V.
The
CAB rates also involve shower counter cuts PA Low and PB, and though they are not very
restrictive, there are no hard shower counter cuts imposed in the analysis. CAB40-V will
then tend to somewhat underestimate N. Fortunately, however, CAB40-V and EL40-V
record very similar values at exactly those kinematics where K ci,,PRITOI
is large -
at
low Q and forward angles. At the higher Q, the increased pion rate in the 16 GeV/c
103
causes a greater discrepancy between the two scaler rates, but the background rate of
proton singles in the
eV/c is much lower. The net result is that the difference in
correction factors computed using the two extreme values for N, is at most 1%.
3.6.3
Proton Absorption
The detected proton rate is reduced not only by final state interactions with the residual
nucleus, but also by similar interactions with materials encountered on the proton's
path to the detectors. Information on the cross-section for nucleon scattering in various
materials is available in ref. [5]. The total cross-section is composed of three components:
(3.26)
O'total ::--::O'elastic + Oquasielastic + Uinelastic-
Of these, only the inelastic portion (which will cause at least m = 140 MeV of energy
loss) is certain to remove a proton from the experimental acceptance. The elastic and
quasielastic cross-sections are strongly peaked at small scattering angles at these energies, and may deflect a proton only minimally. Consequently, extreme upper and lower
estimates of the percentage of protons lost may be obtained using O't,,t,,land
Uinelastic
re-
spectively. The mean free path A is a convenient quantity to use for such calculations, as
the classical probability of transmission through a material of thickness t is given simply
by
t
Ptrammission = e
In unitsof
(3.27)
g CM-2,
A = 1.661A
(3.28)
01
where A is the nuclear mass of the material in amu and
is in mb. From Equation
3.27, we see that the total transmission probability through a sequence of materials is
104
simply e-
X. The resulting correction factors are listed in Table 3 , after averaging
between the upper and lower estimates given by AT and Al. The difference between the
two estimates provides a systematic uncertainty of ± 2.
consistently arger at Q =
Note that the absorption is
and 68 (GeV/c)'; this is due to the presence of gas in the
8 GeV/c Cerenkov counter, which was filled only at these settings.
These results were verified for hydrogen at all Q by comparing the electron singles
rate in the 16 GeV/c with the rate of coincidence protons detected by
GeV/c. Ap-
propriate corrections were made to the proton rate (e.g. for coincidence deadtime and
identification efficiency). By placing cuts on the electron kinematics, one can ensure that
the elastic proton cone corresponding to these electrons lies entirely inside the
GeV/c,
and so the probability of proton absorption is directly measured. The comparison is
shown in Figure 33, and demonstrates that the absorption is understood to within the
systematic error of the data.
105
............ I.................
. . .
1.00
I
I
I
I . . . . . . . . . I . . . . . . . . .
I I
r.
0 0.95
.0.4
rn
U)
. P.4
- - - - - - -.-
- - - -
41
5V)
0
- - - - - - -
II
------------
0 0.90
$.4
E-
- - - - - - -0
0
0
0 0.8 1.4
.4.)
04
0.80
I I
0
2
4
Q2
(GeV/c
6
8
)2
Figure 3-3: Proton absorption as a function of Q1 40]. The dotted lines represent the
results of a calculation using the total (lower) and inelastic (upper) cross-sections of the
proton in the various materials encountered. The data point represent the ratio of (ee'p)
coincidences to (ee') singles measured using a hydrogen target.
106
Table 31:
Correction factors applied to the data.
K comp,1.6
K comp,2/3 , and
K compPROTON
are the electronic and computer deadtimes discussed in Section 36.2. Note
that these corrections were computed for each kinematic setting using the formulae given
in that section; ranges of these values are given here for reference.
Target
1H
2H
12C
56Fe
197Au
Q2
I
3
5
6.8
I
3
5
6.8
I
3
5
6.8
I
3
5
6.8
1
3
5
6.8
I
NOTRK1.6 p abs.
1.00
1.00
1.00
1.00
1.01
1.01
1.01
1.02
1.01
1.02
1.03
1.03
1.01
1.02
1.03
1.03
1.01
1.02
1.03
1.03
1.09
1.09
1.11
1.11
1.09
1.09
1.14
1.14
1.09
1.09
1.13
1.13
1.08
1.08
1.11
1.11
1.08
1.08
1.11
1.11
def f
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
1.015
KcOmP,1
.6
1.040
1.003
1.003
1.003
1.204 - 1436
1.018
1.014
1.004
1.061 - 1153
1.016 - 1023
1.014 - 1016
1.008
1.075 - 1108
1.016 -1.022
1.006 - 1.012
1.003 - 1.006
1.037 - 1.051
1.010 -1.012
1.013
1.009
107
Kcomp,2/3 KCOMPIMIT117
1.039
1.041
1.039
1.021
1.01 - 1095
1.000
1.099
1.002
1.001
1.001
1.002
1.002
1.001
1.001
1.001
1.001
1.001
1.001
1.002
1.002
1.002
1.001
1.001
1.000
1.007 - 1167
1.004
1.015
1.008
1.009 - 1027
1.003 - 1.008
1.01 - 1.068
1.010 1.019
1.013 - 1.026
1.006 - 1.014
1.00 - 1.007
1.005
1.009 - 1023
1.004 - 1012
1.010
1.005
Chapter 4
Description of the Experimental
Simulation
Each of the two forms of results derived from NE18 (spectral functions and transparencies) requires a Monte Carlo model of the experiment. As will be seen in the next chapter,
the extraction of spectral functions requires a model of the coincidence phase space of
the experiment - the probablility of obtaining a given E, and p,.
As explained in the
introduction, the transparency is defined to be the ratio of measured to PWIA yield, and
so in this case a full model of the PWIA cross-section described by equation 17 is required. Since radiative corrections can be modeled accurately using such a Monte Carlo
calculation, one is also able to compare the extracted spectral functions with the Monte
Carlo results, thereby compensating for the distortions to S(Enpn)
caused by radiation
and obtaining a good estimate of the success of the PWIA model in this context. Each
component of the PWIA calculation is discussed in detail below.
108
4.1
Overview
The Monte Carlo program, named SIMULATE, randomly generated the momenta and
angles of the scattered electron and proton vectors (i.e. the 6 quantities in terms of which
the differential cross-section is defined) with a flat distribution over limits calculated to
exceed the experimental acceptance. The energy and position of the incident electron
at the target were also generated randomly, to match the energy and spatial spread of
the beam, and the beam energy was corrected for ionization losses in the target. With a
basic event at-,the scattering vertex now determined, the possibility that any or all of the
particles emitted real or virtual photons was modelled and the particle vectors were adjusted accordingly (see Section 45). The scattered electron and proton vectors were then
transported through the target, applying ionization losses and a multiple scattering distribution, and subsequently transported through the spectrometers. "Single arm" Monte
Carlo models of the optics, apertures, and interfering materials of the spectrometers were
employed and are described in Section 44. Both forward and backward sets of matrix
elements were used, to simulate the optical resolution of the magnetic systems. Once the
particle vectors were reconstructed back to target, they were corrected to the scattering
vertex using the mean energy loss calculations employed in the analysis, and E
and
p, were determined. The successful events were stored in histograms, with each event
being assigned a weight of Ko,,pS (1-6h.rd)
1 Wg,.,,. The factor
(
-
h,,d)-'
is a correction
for radiative diagrams involving hard virtual photons, and is explained in Section 45.
The "generation weight" (Wg,,,) comes from the following source. To increase computer
speed, the limits in which event quantities are generated can be refined once partial information about an event is known. These refinements are based on the acceptance limits
of the spectrometers, the cuts imposed on reconstructed E,
p,
and particle vectors,
and the range in E, (at the vertex) over which the spectral function is defined. These
109
refined limits are especially important in the generation of radiation. For example, to
take into account the possibility that a scattered electron "radiated into" the 16 GeV/c
momentum acceptance from a higher momentum, one must use generation limits in
1.6
which are much wider than the acceptance. However, once the electron's momentum
has been generated, one can determine the range of photon energy required to produce
a successful event. The generation weight reduces the event weight to compensate for
the restricted limits employed. Finally, the results histograms were normalized so that
the number of events in each bin would correspond to the number of counts expected
from the experiment. The results were thus multiplied by
L(AF_'AQ"'AE'A0P')g_Ngen
where C
is the experimental luminosity, and the other terms refer to the phase space volume and
total number of events used by the generation.
Each histogram bin was assigned an
inverse fractional error equal to the square root of the number of Monte Carlo events
contributing to the bin.
4.2
Off-Shell Prescription
for
p
As discussed in Section 12, the deForest prescription 3] for the fundamental cross-section
for quasielastic electron scattering from a bound nucleon is used. His 0',,, prescription
for the nucleon structure functions is used in our final results. However, the effect of the
uncertainty in o,P was estimated by performing additional calculations, using
0',,2
as well
as a simple on-shell prescription obtained by replacing the off-shell kinematic variables
in o,,, by their on-shell values. The net PWIA cross-section was found to change by less
than 2.
The structure functions are computed in terms of the form factors
the free proton.
We used the dipole form for the proton electric form factor GP
E and the parametrization
of
ari and Kriimpelmann 411 for the proton magnetic form factor GP
110
These forms
were determined by experiment NEII [1] to provide the best description of inclusive
electron scattering data up to Q of
4.3
(GeV/ C)2.
Model Spectral ]Functions
The model spectral functions used in the PWIA calculation are based on the Independent
Particle Shell Model (IPSM) - i.e. the nucleus is described by a sum over nucleons occupying distinct shells. Furthermore, we assume the customary first-order approximation
that the spectral function can be factored into separate functions p(p,,) and CE,).
S(E.,p.
= 2' + 1) 1: Pn,1j(Pm),Cn,1j(Em)
(4.1)
For hydrogen, of course, the spectral function can be described trivially by delta functions
around Em =
and p, = . In practice, the Monte Carlo procedure is altered to generate
only the electron angles; all other kinematic quantities can then be computed.
The nucleon momentum distributions p(p,) were computed by the program DWEEPY
[42], obtained from NIKHEF-K, which solves the Schr6dinger equation in an optical
model potential.
(It should be noted that DWEEPY provides many additional op-
tions and output modes, including extensive calculations of distorted wave effects CDWIA) and full five-fold cross-section computation.) In the case of deuterium, the wellestablished Bonn potential was used 431. For the other nuclei a sum of Woods-Saxon
and Coulomb potentials was employed 4:
Yom =
Vof(r, Ro,ao) + V
h
MC
22
df
-- I s + V(r).
r dr
(4.2)
The Woods-Saxon potential describes the average nucleon-nucleon interaction and is
composed of a volume term depending only on the nucleon's radial position r and a
III
spin-orbit term which takes into account the coupling between the nucleon's spin and
angular momentum. The radial shape of each term is described by the Woods-Saxon (or
two-parameter Fermi) form:
f (r, R, a)
I+e
(4.3)
aR
The radius parameter R is in practice replaced by another parameter ro (or r,,), defined
so that the bulk of its dependence on the number of nucleons in the residual system is
removed:
R = ro(A -
(4.4)
1)1/3.
The Coulomb term is simply an expression for the potential produced by a uniform
spherical charge distribution of radius RC:
(Z - I)e
Vc (r =
where RC =_rA11'.
4rcoRc
2
3
r2
2
2RC
(4-5)
2
The optical model provides a strictly local potential, while the true
potential is in general non-local, depending on the value of the nuclear wave function at
other locations. This complication is typically approximated by modifying the nuclear
wavefunction computed from the local potential. One such prescription is that of Perey
[44], involving a range parameter
V)NL(r)
V)LM
(4.6)
[1 + 2M02VOM(r)]1/2
The set of parameters
I (O I ro, ao, Vso, rso, a.9o, I provides a complete description of the
nucleon wavefunction in a particular shell; the binding energy
EB
of the shell can thus
be computed from these values. One can alternatively choose to provide DWEEPY with
a value of
EB
as input, and have it compute the well depth V. This option was only
used when no information on
could be found. If both parameters were available, o
112
was used in p(p,) while EB was used inC(E,,,).
The binding energy distributionsC(E,,,) were described using the Lorentzian (or BreitWigner) shape associated with an isolated resonance:
L(E)
where
EB
r/2
7r(E - EB
)2
(4.7)
(IF/2)21
refers to the central binding energy of the shell. The minimum proton sepa-
ration energy Emin
of a nucleus is given by the mass difference MA-,
3
Mp - MA. We
incorporate the lack of spectral function strength at binding energies lower than this by
imposing a cutoff onC(E) at E.,min and renormalizing the Lorentzian to 1. Values of E.min
for nuclei relevant to the discussion below are listed in Table 4.
The width parameter
r was either taken from fits to JOWQ2data (as explained below), or from the formula of
Brown and Ro
45]
f'(E)
(24MeV (E - EF)2
(4-8)
(50OMeV2)
+ (E - EF)2
with E taken to be the central binding energy EB of each shell, and E, the Fermi energy.
This formula is a parametrization of the widths of single particle states, used to fit data
on A <
8 nuclei. In Ref. 46], however, (ee'p) data on 201Pb demonstrated that the
expression also works well for heavy nuclei. An alternative
where the fixed width
version of L(E) was also used,
in Equation 47 was replaced with a running width I'(E) from
Equation 48. The difference in results between the two forms of L(E) was found to be
less than 2
for all targets and kinematics.
The Woods-Saxon parameters
for 2C
weretaken from Saclay12C(ee'p)
data
47 at
Q 2 of 016 (GeV/C)2, and are listed in Table 42. Ref. 4 also contains a convenient
summary of the Saclay experiment. No non-locality correction was applied to the wavefunctions used to fit the parameters, and so we also do not apply it. The binding energy
for the Is shell was taken to be the centroid energy of 38.1 MeV reported in 4
113
The
Table 41: Minimum proton separation energies for nuclei discussed in the text.
I Nucleus.1 E 3mi.n(MeV
12C
15.96
56Fe
10.18
197
Ni
Au
8.17
5.78
208
Pb
8.01
58
I
Ip shell energy was set at 16.2 MeV, from a combination of various experimental results
detailed in Table X of 4 and an inspection of the E
spectrum found at Saclay. The
shell widths ' were also determined from an examination of this spectrum.
The
16 Fe
model parameters were taken from (ee'p) data on the nearby nucleu 5Ni
(also from ref. 47]), and are listed in Table 43. The Woods-Saxon parameters were
obtained from Table VII of 4
and again the non-locality correction was not applied.
However, p(p,,,) extracted from NE18 data at Q =
(GeV/C)2
showed a narrower
distribution than this model. A better fit was found by simultaneously widening the
radius parameters ro and r,,, from 126 fm to 13 fm, and applying the non-locality
correction with range parameter
=
5. The results presented in the next chapter
are calculated using this best-fit model, but the low-energy model was also used, to
obtain a model dependent uncertainty. For L(E), the shell energies were taken from the
same source (Table X of 4.
However, the experimental data on the deepest shells (15
and p) are quite uncertain, and a second set of values, obtained from the Hartree-Fock
calculation of 48], we also used. These theoretical values were found to provide better
agreement with our data. All of these binding energies were increased by 20 MeV to
correct for the difference in separation energy between 18 Ni and
16 Fe
(see Table 4 1). The
shell widths were calculated from Equation 48, using a Fermi energy of
MeV.
The model spectral function for "'Au (see Table 44) was obtained using the 'O'Pb(ee'p)
114
data of ref. 46]. Fits of ro were only performed for shells near the Fermi surface and are
found in Table 45 of 46]; discussions with a collaborator on the experiment 49] revealed
that the version of the DWIA corrections presented in column (b) of the table is the best
one to use. For the deeper lying shells, the typical value for heavy nuclei of ro = 118 fm
was used. Or Q =
(eV/c)'
data on (p,) were again found to be narrower than
the model. The radius parameters were consequently increased by a factor of 1.1 in
our best fit model. Potential well depths Vowere not fit by the
108 Pb(ee'p)
experiment,
and so the shell energies were provided instead as input to the Woods-Saxon calculation.
These are found in Table 49 of 46] for all but the deepest shells, which were beyond the
experiment's detection range. Energies for these shells were taken from theory 48] A
second set ws determined from the collection of separation energy measurements presented in ref. [50] (and also in Figure 117 of ref 4
These measurements extend only
to A < 60; values at A = 197 were obtained by extrapolating the trends in the energies
for each shell. These extrapolations are certainly accurate to no more than 20%, but
as they affect-,less than 10% of the protons in "'Au, their effect on the results is small
compared with other uncertainties. The 22 MeV difference between E"'
211
for
97
Au and
Pb was subtracted from all energies. The remaining Woods-Saxon parameters are
shell-independent, and taken from the discussion of the optical potential on p.15 of [46].
The widths
for the four shells most near the Fermi surface were taken from Table
4.9 of 46]. The rest were not obtained in that experiment, and so we computed them
from Equation 48, using a Fermi energy of 44 MeV. This value of EFwas that used by
Quint corrected for the 22 MeV mentioned above) in his calculations using the Brown
and Rho formula- as mentioned earlier, he found that the formula agreed well with the
measured widths.
115
-Table
42: Model spectral function parameters for 12C
Woods-Saxon parameters for p(p,), from data of 47]. No non-locality correction was
applied.
11
V0
ro
mev
fm
6
55
1.36
1.36
ao
she
1sj/2
IP3/2
qo
Lorentzian parameters for CE,),
rso
aso
MeV fm
0.55
0.55
9
1.36
rc
fm
0.55
1'3
1.3
from the same source. The Fermi energy used was EF
15.96 MeV.
EB
r,
MeV
MeV
IS1/2
38
20
IP3/2
16.2
5
shell
4.3.1 Correlation Corrections
The IPSM provides a good approximation to the nuclear wave function, but by construction fails to account for any correlation between nucleons. In fact these correlations have
a measurable effect on the spectral function. They are primarily due to the strongly
repulsive nature of the nucleon-nucleon interaction at short distances, and this repulsion tends to accelerate the affected nucleons to higher momenta. Our approach to this
problem is as follows: We assume that strength shifted to higher p, by these shortrange correlations only becomes comparable to the IPSM strength well past the Fermi
momentum, and so beyond the detection range of our experiment. In other words, we
assume that the IPSM correctly describes the shape of the spectral function we observe
but should be corrected for the integrated correlation strength outside our phase space.
This assumption is supported by both nuclear matter calculations [51] 521 and spectral function measurements made at low Q [6]. Ultimately, of course, the test of our
assumption will be the agreement between the model and the data.
116
Table 43: Model spectral function parameters for 56Fe
Woods-Saxon parameters for p(p,), from data of ref. 471 on "Ni. To improve the
fit to NE18 data, ro was widened to 13 fm and the Perey non-locality correction was
applied with range parameter
=
5. Calculations using ro = 126 and no non-locality
correction (the fit determined in ref. 47]) were also performed, and used to determine a
model-dependent uncertainty.
shell
VO
ro
ao
MeV fm
1S1/2
IP3/2,1/2
ld5/2,3/2
2s,/2
1f7/2
80.7
69.0
58.2
52.7
52.7
1.30
1.30
1.30
1.30
1.30
0.60
0.60
0.60
0.60
0.60
Vso
r,,,
MeV
fm
a.,
40
23.5
1.30
1.30
0.60
0.60
13.8
1.30
0.60
rc
fm
1.3
1.3
1.3
1.3
1.3
Lorentzian parameters for CE,,). The 1d, 2s, and f binding energies are from the
Hartree-Fock calculation of ref. [481-those for s and Ip are from ref. 471. As both of
these sources are descriptions of "Ni, the binding energies were corrected for 20 MeV
difference in proton separation energy between "Ni and "Fe. The energy widths are
given by Equation 48, with E = MeV. Calculations were also done using the Is and
Ip binding energies from ref. 47]; these are indicated in parentheses.
117
Table 4
Spectral function parameters for 197 Au
Parameters for p(p,) and (E), from data of ref. 461 on 211 Pb. Note that EB rather
than
is provided to the Woods-Saxon calculation, and that these values are also used
to determine LE). Also, as all EB values are for 208 Pb, they are corrected form the
2.2 MeV difference in proton separation energy between 208 Pb and 197 Au. The Perey
non-locality correction was applied, with range parameter
=
5. To improve the fit
to NE18 data, ro was widened by a factor of 1.11, but calculations using the original
ro values were also performed and used to determine a model-dependent uncertainty.
Two versions of EB for the deep shells are also provided. The first values are from the
Hartree-Fock calculation of ref. 48], while those in parentheses are derived from the data
of ref. [50).
shell
181/2
IP3/2,1/2
1d5/2,3/2
2sl/2
1f7/2,5/2
2P3/2,1/2
199/2,7/2
2d5/2
IhjI/2
2d3/2
EB
F
MeV
'ro
fm
ao
MeV
46 65)
41 (50)
32 40)
28
18 21)
17 19)
13 17)
11
1.31
1.31
1.31
1.31
22.9
17.2
12.5
8.3
7.7
6.2
9.4
7.9
6.0
3.7
4.0
3
1.31
1.31
1.27
1.32
1.29
1.36
118
Vso
MeV
r,,
fm
ajo
rc
fm
0.65
0.65
0.65
0.65
6
6
1.15
1.15
0.65
0.65
1.2
1.2
1.2
i.2
0.65
0.65
0.65
0.65
0.65
0.65
6
6
6
6
6
6
1.15
1.15
1.15
1.15
1.15
1.15
0.65
0.65
0.65
0.65
0.65
0.65
1.2
1.2
1.2
1.2
1.2
1.2
Table 45: Corrections applied to the IPSM spectral functions to correct for strength
shifted outside our detection range by short-range correlations.
I Nucleus
12C
Fe
"Au
I
F,,orrel
I
1.1 ± 003
1.26 ± 008
1.32 ± 008
We divide the IPSM spectral function by a correlation factor
SjpsmdpdE
F,orrel
Forrel:
(4.9)
Scorreld'pdE
The overall normalization f SdpdE
this normalization.
= Z is maintained since Sorrel is also defined with
Two spectral functions Scorrel were used for carbon and are from
ref. 53] and [51]. The latter is a calculation of the correlated spectral function for the
nearby nucleus `0
using the Brueckner theory of finite nuclei. For the heavier nuclei
iron and gold, we use correlated nuclear matter spectral functions, corrected for finite
nucleus effects; these are from ref. 52] and 541. The correlation corrections applied to
the IPSM spectral function are listed in Table 45. The indicated errors account for the
different values produced by the various models. Note that, as mentioned earlier, we use
several IPSM models to obtain a model dependent uncertainty in our results, and the
value Of correlmust be recomputed for each Sjpsm. The values listed in the table apply
to the best-fit IPSM models used to generate our final results.
4.4
Spectrometer Models
The single arm spectrometer models employed sets of forward and reverse matrices to
transport particles between the pivot point and the focal plane. The reconstruction
119
matrices derived from the data and used in the analysis (see Section 33) could not be used
in this context for two reasons. First, inverting a set of > second order matrices cannot be
done analytically, and if done numerically is not guaranteed to produce reliable results.
Second, the magnetic field needs to be modelled in stages so that magnet apertures can be
checked at intermediate locations in the spectrometers. Consequently, the spectrometer
models used in the Monte Carlo calculation were derived from TRANSPORT decks. The
following criteria were used to evaluate the models:
* The forward matrix elements should reproduce the data distributions at the hut
(focal plane).
* The reverse matrix elements should reproduce the data distributions at the target.
* The elastic cross-section determined by dividing (normalized) experimental counts
by the spectrometer acceptance, as calculated by the Monte Carlo, should have the
correct magnitude and be independent of the cuts applied.
* The forward and reverse matrix elements should invert each other as nearly as
possible.
As described below, the
GeV/c TRANSPORT model was modified to meet these cri-
teria.
The spectrometer Monte Carlos proceeded by transporting particles through the spectrometer in steps, stopping at each aperture to check the acceptance and at each intervening material to compute energy losses and multiple scattering. Energy straggling was
calculated using equations 31 and 32. Multiple scattering was simulated via Gaussian
distributions in each of the planar angles dx and dy, each with a standard deviation (in
radians) of [5]
O'dxdy '---
13.6 z vft7d(I + 0.088 1091o(tad))
Po
120
(4.10)
Here, z is particle charge, p is particle momentum in MeV/c, and tad is the thickness of
the material in -units of radiation length. Once transported to focal plane, particle tracks
were required. to intersect the fiducial detector volume. The tracks were also adjusted
for multiple scattering in the detector materials before the wire chambers, and for the
resolution of track reconstruction due to wire spacing. Finally, the adjusted tracks at
focal plane were reconstructed to target.
The 16 GeV/c optical model was taken from the work of ref. [55]. This model was
developed for, a previous experiment at SLAC which used the same configuration of the
1.6 GeV/c as NE18. The 16 dipole is modeled in
segments, producing
sets of forward
matrix elements.
The
GeV/c model was based on a second-order TRANSPORT model developed
at SLAC 561 for the large acceptance tune used by NE18. Careful assessment of the
uncertainties in measured magnet currents was made, and their values were adjusted
within these limits to provide optimal forward-backward agreement. The forward model
was found to reproduce the data at the focal plane but cut dependences were noticed in
the E) and D coordinates. In D, the Monte Carlo distribution was found to be narrower
than the data.
A solution to the problem, which reduced the cut dependence of the
elastic (ee'p') cross-section to < 2,
was found by multiplying the
matrix elements by
a factor of 1025. Some sacrifice in forward-backward resolution was necessary with this
change, but this sacrifice is not critical; The primary problem induced by such a mismatch
is the incorrect assignment of vertex quantities to reconstructed target quantities in the
PWIA calculation. As the out-of-plane cross-section varies slowly and multiple scattering
smears the resolution, the indicated compromise is entirely reasonable. In 0, it was
concluded that the trouble was related to the third-order matrix elements needed in the
data reconstruction (see Section 33). Replacing the
121
matrix entirely with the elements
from the data. set produced the desired result of reducing cut dependences again to <
2%.
The final transparency results in general show a higher cut dependence of 5%. The
uncertainties in the
GeV/c spectrometer model are largely responsible for this, and
we invoke a 5% systematic uncertainty at all targets and kinematic settings to take into
account this lack of understanding.
4.5 Radiative Corrections
The PWIA model described thus far involves only one Feynman amplitude, for the exchange of a single virtual photon between the electron and struck proton. However, one
must also consider the possibility of emitting additional photons, both real and virtual.
The emission of real photons is referred to by the
miliar term "Bremsstrahlung"
or
"braking radiation". These photons are emitted when the charged particles involved in
the reaction are accelerated by the fields of either the nucleus involved in the primary
hard scattering ("internal radiation"), or by the other nuclei encountered by the incoming/outgoing particles as they travel through intervening material ("external radiation").
The emission of real photons causes a discrepancy between the detected particles' momenta and their actual momenta at the scattering vertex, and so cause distortions in
the extracted experimental spectra. Conversely, amplitudes involving the emission of
additional virtual photons affect only the magnitude of the measured cross-section. As
will be demonstrated below, the Bremsstrahlung cross-section is strongly dependent on
W. In the search for a Q-dependent
signature like that of colour transparency, then,
radiative effects must be computed precisely.
In the following discussion, we will use
to represent the four-vector of an emitted real
photon. This appears in the energy-momentum conservation relation as an additional
122
four-momentum in the final state:
PA
=
P'
w+ *
(4.11)
All of these variables are four-momenta, representing respectively the initial electron, the
inital target nucleus, the scattered electron, the knockout proton, the emitted photon,
and the recoiling (A- 1) system (possibly in an excited state, as indicated by the asterisk)
If one now denotes the values one measures for the missing momentum and energy by
PM and
, and their actual vertex values (in the absence of radiation) by p. and Em,
one obtains
pm = pw-q=Pm+w
Em + Tre
=
E
E-
E - M - wo
= km + Tec WO,
(4.12)
and so
PM = PM-W
E.
= Em + T
Em
- Trec+ W0
w0
Note that the measured value of the recoil kinetic energy,
(4.13)
cj depends on the measured
missing momentum and so is also distorted by Bremsstrahlung photons. However, the
contribution of Trec to the missing energy is, in general, small (and non-existent in the
case of elastic ep scattering). This approximation is not used in the radiative description,
but merely serves to illustrate the overall effect of radiation on a measured (Em,pm)
distribution: the real photons produce long "tails" which, at very high photon energy
(W > Em, pm), are described by the relation Em '-- ,,,
123
wo. Elastic ep scattering
I
$
0
a
I
_IgSbN_
-------------------------- - - - - - - - - - -___
------Figure 41: Distribution of counts in E, and p, for (ee'p) from hydrogen at Q =
(GeV/c)', demonstrating the existence of "tails" due to Bremsstrahlung radiation. The
E, axis runs in the bottom-right direction, from 25 to 125 MeV in bins of 25 MeV; the
p, axis runs towards bottom-left, from 160 to 160 MeV/c in bins of MeV/c.
provides a clear demonstration of these tails, since in the absence of radiation, all strength
is localized at E = ,, =
(see Figure 41).
The coincidence variables E, and p
thus provide a natural coordinate system in
which to evaluate radiative effects. By contrast, radiative corrections have generally been
calculated in the framework of inclusive (ee') experiments - in terms of their effect on
the measured electron energy c = P.
The effect of radiation on this quantity depends
on the direction of the emitted photon: Consider elastic scattering, with differential
cross-section dQi
d
The reaction amplitude is thus fixed by the direction of the scattered
electron (and, of course, the incoming electron energy). If we treat this direction
as fixed, the radiation of a photon parallel to k simply decreases the energy E' by the
124
photon energywo. if, however,the photon direction is parallel to the incoming electron,
E'
is affected by an amount that depends on the electron scattering angle. (Note that
the scattered proton vector is also affected). Thus, when one comes to evaluate the
total probability of emitting radiation that affects
by less than some cutoff energy
AE, one has to perform integrals over photon energy and direction with interdependent
integration limits. In the case of coincidence scattering, independent integrals can be
performed as the measurement of both scattered particles enables one to select a more
"natural" choice of variables.
The formalism described in this section 571 is based on the work of Mo and Tsai
[58] 59], which has provided the standard radiative corrections prescription for three
decades of inclusive electron scattering experiments. The basic formulae of Mo and Tsai
have been reevaluated in a coincidence framework: one can no longer integrate over all
final states of the scattered proton as in (ee') measurements, but must calculate the
radiative effect on both the scattered electron and proton. The resulting distributions
are then included in the event generation of our PWIA Monte Carlo and folded with the
experimental detection range in k' and p as described earlier.
4.5.1 First Order Internal Bremsstrahlung
The probability for radiating a single Bremsstrahlung photon is represented by the four
Feynman diagrams of Figure 42. Since each of these diagrams involves the same final
state, the amplitudes must be summed coherently:
da
d3kld3W
"IIM',
+ M.f + MP,
Mpf 12.
(4.14)
These four matrix elements refer to the emission of a photon by the incident electron,
scattered electron, incident proton, and scattered proton respectively. To evaluate them,
125
one requires a knowledge of the coupling of the electron and proton to the photon. The
electron coupling is given exactly by QED and is specified by the electron current
J,'(q = eie(k + q)-ylu,(k).
Here, e =
(4.15)
r4ra is the electron charge and u, is the electron spinor, normalized to
U,(k)u,(k
= 2m (m is the electron mass). The proton-photon coupling is complicated
by the fact that the proton is in general bound and off-shell, and the description of such
a proton is only approximately known (see Section 12). Consequently, we neglect these
effects and assume elastic scattering from an on-shell proton:
JI(q = -eiip(p + q)P(q)up(p).
(4.16)
The deviation of the proton from a point particle is described by
I" (q = F (q'), I + -
I
2M
F2(q')Zo,`q,,
(4.17)
using the free proton form factors. Again, the proton spinor is normalized to the proton
mass:
p(k)up(k = 2M. In support of this neglect of off-shell effects, one can point
out that the overall contribution to the radiation from the proton arms is expected to
be small. Using these couplings, one obtains the following expressions for the first-order
Bremsstrahlung matrix elements:
i,"(k
Me
=
zue(lk'),A (k -
Mf
=
1ii,(k')e-y',E,
A4p
= 'iip(pp(q)
- w,) +
2
e-"E'U'(k) q2 - P2
_U
W)2 - MI
(kl+
+ w,) + m
L;)2
-t u, (k)
M2
e2
(p')F,.(q)up(p)
UP(p')rjq)up(p)
e2
(p - W)2
_)F'(W)'E'UP(P)
M2
q
126
2-
A
2U
,(k')7.u.
(k)
M'f
M'ei
I
_F
+
Mpf
Figure 42: Feynman diagrams contributing to first order Bremsstrahlung radiation
cross-section.
M
Pf
up(p)
LI
(W),E iy"(P'
w)- +
(p + +
O)2
M2M
e
q - Y2
Here, c, is the polarization of the Bremsstrahlung photon, q = p - p is the momentum
transfered to the proton if the electron emits the photon, and q = k- k is the momentum
transfered to the proton if the proton emits the photon.
the photon
is a parameter representing
ass, which will ultimately be taken to .
The single hoton emission cross-section can be calculated from these expressions,
with no further approximations.
However, the formulae simplify greatly in the limit
that the photon energy wo is much less than the momenta of the initial and final state
fermions. In this case, the basic one-photon exchange amplitude MM
ep factorizes from
the Bremsstrahlung amplitudes, giving
M
= eM ep
127
w
k
k
= eM (1)
ep
M'f
,E k'
k'
Mpi = -Ov I ep1)
Mpf =
-(,
-em(l)
ep
P
(4.19)
W-P,
This limit is referred to as the soft photon approximation (SPA); it can be seen to be
reasonable from the distinctive
energy dependence of the emission amplitudes. Part of
this approximation is the use of the elastic (unradiated) values of the fermion momenta k,
k p, and p in the above expressions. These elastic values are also used in the evaluation
of the one-photon exchange amplitude,
M(1)
ep = i,(k')-y"u,(k)_
e2
q2-
9 2 ii
p') r, (q) up p).
(4.20)
The resulting total cross-section for single-photon Bremsstrahlung is thus given by
do,
dQd3 w
d
dQ,
a
ep
47r 2WO
k'
w k'
P
w p
k
w k
p
w p
2
(4.21)
For later convenience, we write this as a product of photon energy and angle distributions,
do,
dQdQYdwO
du
dQ,
A(Q,)
P
(4.22)
W0
A
(Q_
=- -- -+
where
aw 02
k'
4r2
w k
PI
w p
k
w k
p
-p
2
(4.23
depends only on the photon direction L. Integrating Equation 421 over photon angle
and energy, one obtains the cross-section for emitting a photon of energy less than AE:
do,
dQ, (WO< A E
AE
= Jo
do,
d3WdQd 3W
128
do,
dQe
2a)
0 (pi) E)(pj) B (pi, pj, A E),
(4.24)
ep
where
AE
B (pi, pJ, A E)
3W
Pi
d
87r2w,
P
Pj
pi) P
pi),
(4.25)
Here, two pieces of convenient notation have been introduced. pi for i = 1,...
to represent the four fermion momenta k, k, p, p in turn; the constants
4 is used
(pi) denote the
signs accompanyingeach term, 0(k = 0(p' = -1 and O(P = 0(p = 1. This integral
can be evaluated using the expression
w k = w(ko - Jklcos 0),
as well as introducing
(4.26)
a new variable x as indicated in Equation
P., = Xp + (I
111.19) of Tsai 59]:
XPj.
(4.27)
One then obtains
B (pi, p, A E)
P'2rpj
Pi
01 dx IAE
0
'p I
2r
dx
o
2
PX
0
Wo P2W2+
X
In
AE
0
PX
0
+
P2 (P 0)2
X
2
2p2X
In P
P
2
+PX - 1PX1In PX- 1P.1 + In _2pX
POX
+ 1P.1
21pxl
We note that the sum Ej 0pi)0(pj)B(pj,
0
(4.28)
POX+1P.1
pj, AE) is negative, making the total cross-
section (and the angular distribution A(L)) positive.
One observes that this expression contains two non-physical divergences: when the
C4
photon mass" p -- 0 and when the energy cutoff AE --+ 0. Both of these are due to
approximations made so far, and will be addressed in later sections. Before continuing,
129
however, it is worthwhile to try to evaluate the validity of the soft photon approximation.
As mentioned above, the one photon Bremsstrahlung calculation can be computed without this approximation. Accordingly, the ratio of the full to the soft photon calculation is
presented in table 46 for Q = I GeV/c)' and a variety of photon energies, and in table
4.7 for a photon energy of 100 MeV and a range of Q from I to 15 (GeV/c)'.
Qual-
itatively, one sees that the SPA improves at low photon energies and high momentum
transfers, as expected. At Q = I GeV/c)', the discrepancy between the two calculations is less than 1% for photon energies less than 10 MeV, while for a photon energy
of 100 MeV the discrepancy drops to 5% at Q = 9 GeV/c)'.
The discrepancies are
considerably higher at the other settings listed, however. Two effects are involved: the
shape of the Bremsstrahlung energy spectrum, and the evaluation of the matrix elements
using elastic (wo = 0) particle vectors (i.e. neglecting the difference between q and q in
Equation 418). In an attempt to separate these effects, table 46 also contains the SPA
to full ratio using a point-like proton, i.e. a proton without form factors but with the
correct magnetic moment. At Q = I GeV/c)', one sees that most of the discrepancy
is removed along with the q-dependent form factors. To resolve this one must evaluate
the cross-section using a value of q which is corrected for the effect of radiation.
In other
words, one must distinguish between photons emitted before and after the hard scattering, a task which is complicated by the interference terms between the Bremsstrahlung
amplitudes
j, Mf,
Mpi, and Mpf. However, such a correction can be built into
the calculation, as is explained later on. The maximal E
range considered by NE18
is about 140 MeV, and so the wo = 100 MeV results in table 46 can be considered a
worst-case scenario. Assuming that the correction to q at the hard scattering vertex
can be accomplished, one is faced with a SPA inaccuracy of at most 2
the direction k and 7
for radiation in the direction P
130
for radiation in
We point out in passing that
Table 46: Ratio of single photon Bremsstrahlung cross-section calculated in the soft
photon approximation to the full calculation, at Q = I GeV/c)'. Various photon
energies wo are considered; the photon angle is taken to be in the direction of either the
initial (i) or final (f) electron. The values in parentheses are the SPA/full ratios using a
point-like proton in the calculations.
I Wo
(MeV
I
1
10
100
200
i
1.0023
1.023
1.26
1.59
f
1.0002)
1.002)
1.02)
1.04)
09993
0.993
0.93
0.87
0-9993)
0-993)
0.93)
0.87)
Table 47: Ratio of single photon Bremsstrahlung cross-section calculated in the soft
photon approximation to the full calculation, for photon energy
=
MeV. Various
momentum transfers Q are considered; the photon angle is taken to be in the direction
of either the initial (i) or final (f) electron.
IQ' (GeV/c)
I
1
5
9
15
f
1.26
1.14
1.05
1.03
093
093
099
099
these percentage discrepancies are given in terms of the radiative corrections, which are
themselves small; the effect of these discrepancies on the final cross-section is thus much
less than the quoted percentages.
4.5.2 Virtual Photon Corrections
One of the non-physical divergences observed in Equation 428 was found in the limit
y ---+ 0. This is known as an "infrared divergence", and is a direct consequence of the fact
that the one photon Bremsstrahlung cross-section is of order a 3 and that other diagrams
of the same order have not been included yet. These are amplitudes for the emission of
131
two virtual photons, collectively referred to as M(').
ep These must be summed coherently
with MM which represents the same final state:
M2 = IM(1)12+ M(2)tM(l) + M(l)tM(2) + O(C,4).
ep
ep
ep
ep
ep
ep
(4.29)
Figure 43 contains a summary of the second-order amplitudes. Unfortunately, several of
these depend implicitly on the strong interaction via the poorly known proton current.
The point of view advocated by Mo and Tsai and espoused here, is to include only those
terms which do not unambiguously depend on the strong interaction. Certain amplitudes
such as M(2.1)
in the figure are calculated, but only infrared divergent terms necessary
ep
to cancel those from the Bremsstrahlung cross-section are kept; the rest are left buried
in the electron-proton cross-section. It should be noted that other workers 60] have
derived alternative expressions for the virtual radiative correction, by including some of
the components left out by Mo and Tsai. However, the point to be made here is that the
evaluation of MM includes the use of proton form factors extracted from previous data.
The radiative corrections applied should thus be onsisted with whatever corrections
were used in extracting these form factors. The standard prescription of Mo and Tsai is
thus the appropriate choice [I].
The second order diagrams depicted in Figure 43 are grouped into three categories
depending on their sensitivity to the strong interaction. We use the same evaluation of
these amplitudes as Mo and Tsai, and restate them here. Also used is the notation
K(pi, p
= pi p
1 dx In
3
10
132
2
Pr
p x2
2 1
IL
(4-30)
M (2-1)
M (2.2)
ep
ep
M 2.3)
ep
Figure 43: Feynman diagrams representing virtual photon corrections to one-photon
exchange ep cross-section included here and in ref. 591.
describing the form of the infrared divergent terms. Note that
K(pi, pi) =In
(4.31)
Mil
Y2
and that the IR divergent term of Equation 428 has this form.
./"
the
ep2. )
limit
is the electron-photon vertex correction and is known exactly from QED. In
Q2
>
(which is well satisfied by momentum transfers in the GeV/c range),
M2
one obtains
a
M(2-1) - ep
M(2.2)
ep
2ir
-
K(k, P) In(-Tn2) + -3In(-- q2
) - 2 M(1)
IL2
2
rn2
(4-32)
ep
represents the vacuum polarization correction, and contains contributions from
both lepton and hadronic loops. Only the former are known unambiguously from QED,
and contribute
M ep2.2 = a E 6VP1
M ep
(1)
i
i
133
(4-33)
whereEi sumsover the different flavors of leptons with mass mi. Again applying the
limit Q2 > m2 ,
one obtains
bVP
5
-
37r
3
+In
q
(4-34)
M?I
As there are no IR divergent terms in the vacuum polarization amplitude, all contributions from the strong interaction are neglected. Finally,
M(2.3)
ep
includes two-photon
exchange and nucleon self-energy graphs, both of which depend intrinsically on the strong
interaction. Only the IR divergent terms are used:
M(2.3) - _a
ep
2ir
K (k, p) + K (V, p - K (V, p - K (k, p')
- K(p, p') - In (M2)
(4.35)
92
The total cross-section for emitting a photon with energy less than AE is now obtained
by adding all of these terms to Equation 424. The dependence on the photon mass cancels
as required, leaving
do, 0
P < AE)
dQ,
doO)
lep (I - bsoft(AE - 6haxd) i
dQ,
(4.36)
where
bsoft(,AE = 2a
0 (p 0 (p 13(pi, pA E)
(4-37)
and
shard
2
-
3
4r
ln(-q 2/M2) +
1
ir
2)
bvp(q
(4.38)
Here, d7
Jp represents the one-photon exchange ep cross-section, bhwd i the contribudQ,
tion from the second order virtual photon diagrams, and bsoft(AE) is due to one photon
Bremsstrahlung. f(pi, pj, AE) is simply B(pi, pj, AE) of Equation 428 without the IR
134
divergent term.
In order to seperate out the contribution of the proton we divide &,ft(AE) into three
parts,
bsot(AE =
ee
soft
(AE)
+ bep
soft
(AE) + 6PP
soft (AE).
(4-39)
6ee
soft is the electron Bremsstrahlung contribution, involving f3(k, k, AE), P(P, V, AE),
and B(k, V, AE). bP
soft includesthe electron-protoninterferenceterms B(k, p, AE), B(k, p', AE),
B(k p, AE), and B(k , p, AE); while bsoftis entirely due to proton radiation and includes
the remaining terms B(p, p, AE), P(p, p', AE), and B (p, p', AE). Table 48 contains values of these terms as well as bh,,,dat various kinematics. Note first that
bh"rdis
negative,
and so causes a net increase in the total ep cross-section. Its magnitude is also small: less
than 10% up to Q of 15 (GeV/c)'. The direct proton contribution
(lowest
Q2)
to 8% (highest
Q2)
of the electron contribution
soft .
bee
varies from 2%
However, the electron-
proton interference is about twice the size of the direct proton term, leading to a net
6-20% contribution of proton Bremsstrahlung. It is clear that proton radiation, though
afflicted by strong interaction uncertainties, cannot be neglected at these energies.
The evaluation of the functions f(pi, pj, AE) must be done numerically. Numerous
Spence functions
must be computed, where
(X =
x - In(Il - yj) dy.
10
(4.40)
Y
As an aside, the contributions of these functions turn out to be important only when their
arguments are large (jxj > 1), and in this case an excellent approximation is provided
by
.D(x)
2
In 2(IXI).
(4.41)
The formulae simplify, however, in the "ultra-relativistic (UR) limit" where the momen-
135
Table 48: Values for the radiative correction functions 6, evaluated at various momentum transfers and for cutoff photon energies of 10 and 140 MeV. Note that the virtual
correction 6hard i independent of this cutoff parameter. Note also that the results depend
on the choice of electron scattering angle as well a.,3 on Q, and that NE18 kinematics
were used for these calculations. The final colum n is the percentage contribution of
the proton-proton and electron-proton interference terms to the total Bremsstrahlung
correction ( 6,Pft+6,PPfI
).
650ft
Q2
AE
bh.d
bee
soft
bep
soft
6PP
soft
bsoft
(GeV/C)2
MeV
I
10
140
-0.07
0.332
0.158
0.015
0.007
0.007
0.003
0.354
0.169
6.2
5.9
3
10
140
-0.08
0.377
0.190
0.038
0.020
0.019
0.009
0.434
0.219
13.1
13.2
5
10
140
-0.08
0.398
0.205
0.056
0.030
0.028
0.014
0.482
0.249
17.4
17.7
7
10
140
-0.09
0.424
0.226
0.070
0.038
0.035
0.019
0.529
0.283
19.8
20.1
136
%
ep+pp
Psoft
Table 49: Single photon Bremsstrahlung spectrum, evaluated at two NE18 kinematic
settings and integrated up to photon energies of 10 and 140 MeV. bs.ft is calculated
using the full SPA expressions of Equation 428; ' is from the closed form expressions of Equation 442 found in the ultra-relativistic limit. The final column present the
percentage discrepancy of the UR calculation relative to the full SPA.
Q2
AE
oft 6'
discre
6S
soft
(GeV/C)2
MeV
I
10
140
0.354
0.169
0.347
0.166
-2.0
-1.8
10
0.413
0.474
-1.7
140
0.249
0.246
-1.2
5
%
P.
turn transfer and intial final fermion momenta are large compared to both the nucleon
and electron mass. In this limit, one obtains the following closed forms:
bur
ee
Ur
7
Z2
PP
bur
ep
kk I
(AE)2
a
-In
a
7r
z
_'
7r
In
P0P 01
In
(AE)2
0 Of
In P P
AE2
-q 2
M2
In
k
In -
k'
q2
I
M2
+ In
kki
AE2
-In
2
2
P Of
M
k
In -
k'
I
kk'
2
M2
+-In
k
In -
k'
.(4.42)
These forms reveal the essential features of one-photon emission: all of the dependence of
bsoft(AE) on AE takes the form In AE but additional terms independent of the photon
energy cutoff are also present. These expressions will prove very useful later on, and
so it is worthwhile to see how good the UR approximation is at NE18 kinematics
A
comparison of 6,.ft(AE) computed using Equation 428 and Equation 442 is presented
in table 49. One sees that at NE18 kinematics, the approximation is accurate to at least
2%.
137
4.5.3
Higher Order Bremsstrahlung
In the previous section, we removed the infrared divergence from the first order Brernsstrahlung
cross-section. The other divergence that needs to be understood occurs in the limit
AE --+ 0. This indicates that there is an infinite possibility for emitting a photon of infinitesimal energy. In other words, the first order perturbation expansion breaks down as
AE becomes very small, and one must include the possibility to emit many soft photons.
In actuality, the probability of scattering without losing any energy to Bremsstrahlung
is zero so the actual cross-section should approach zero as AE -+ 0.
It was originally determined by Yennie, Frautschi, and Suura (ref
6
that the
emission of soft photons can be summed to all orders via exponentiation:
do, (WO< AE = duM
dQ,
'
dQe
e- 6,.ft (A E) (
-
(4.43)
bh.d)
ep
The notation (2) indicates that this expression represents the cross-section for emitting
any number of soft photons, each with energy less than AE. In practice, however, one
is interested in the total photon energy emitted. This case is discussed below, and found
to agree with the preceeding formula to within a correction of order a 2
Recall that the probability for emitting a single Brernsstrahlung photon has a
energy dependence that factors from the angular distribution A(L) (Equation 422)
order to maintain a handle on the AE -
In
divergence for the moment, we write the cross-
section to emit one photon with energy wo > AE, along with any number of photons
each with energy less than AE:
do,
(n = , AE)
dQdL,),OdQ,
Here,
du
dQ, ,P
e-
&.ft(AE)
(I -
h.d)
0
0(w - AE)
(4.44)
W 1
represents the usual step function, and dQ1 indicates the emission angle of the
138
photon wl. Similarly, the cross-section to emit two photons with energy Wo
1 > AE and
W2 > AE, along with any number of photons with individual energy less than AE is
do,
_(n
= 21,AE = do,
dQdw0dQjdLo0dQ2
1
2
dQe
(4.45)
I - 6h.d)
P
I A Lj )0(w
2
Lo0
1
- AE)
LIJ0
2
0(w - AE).
2
Generalizing this to the case of n photons of "large" energy, one obtains:
do,
ffiedwoffij
(n, AE = do,
... dwOdQ,
n
dQe
(4.46)
e- 6..ft(AE)( I - 6h.d)
ep
A j
n!
O(Lv - AE) ...
u 0
I
Wo
n
O(wo
- AE).
n
At this point, we can impose a requirement Ejw9t < E on the total rather than individual emitted energies by replacing the step functions above with a delta function and
integrating over individual photon energies and angles. Also, we sum over all numbers n
of emitted photons:
do, JAE)
Ef
dQdE
n=O
E
AE
0
dw dQj-
E
AE
dwodQn
n
da :Td-jj (n, A E)
dQdw'dQj
1
n
n
6(w1 + - + wo
n - E).
(4.47)
One observes that the angular integration can be done at once for each photon, and for
convenience we write
A
dQ, A ).
(4.48)
We then combine equations 446 and 447 to obtain
do, (AE = do,
dQdE
dQ,
n
e_6..ft(AE)
( - h.d) c'o
ep
n=O
n
i=1
E &
A 6(w + - + w - E).
E
UP
n
(4.49)
139
This is a form that we will encounter again later on. It can be evaluated by substituting
an integral form for the delta function:
6(E wo
i - E =
I
00
2z
eiX(F
WO-E
)dx,
(4.50)
i
and observing that Equation 449 has the familiar form
n=O
ez. Carrying through
the computation, one finds that the AE -- 0 divergence in e&-ft(AE) is cancelled by
the similar terms due to the AE lower integration limit. Taking the limit AE --+ 0, the
following relatively simple form is obtained:
du (AE = do,
dQdE
-b'.ft(E)) e6.ft(E)F(A).
(I - 6h.d)
dQ,
(4.51)
S
The function F(A) is expressed in terms of the gamma function and Eulers constant
C '_ 0577; if we recall that A is of order a, we can expand this function in powers of A:
F(A)
r(
e-c'\
A)
r2A 2
12
+
(4.52)
Integrating Equation 451 over dE, one thus gets
du (1:
dQ
LO0<
E = do, lep( - bh.d)e- &.ft(E)[+ O(Cf2)],
(4.53)
dQ,
which agrees with the previous exponentiated formula, Equation 443, to within a correction of order a 2
Exponentiating &ft thus provides a good approximation to the Bremsstrahlung crosssection for emitting a total photon energy up to a certain cutoff value. The exponentiated
cross-section also has the correct limiting behaviour, liME-0 dQ,
d (E
140
Lo'
< E = 0, since
bsoft(E - B(pipjE)
- ln(E).
Note, however, that
6hwd
i
not exponentiated.
Mo
and Tsai [58] take the point of view that whether or not to exponentiate this term is an
open question. As with the choice of which second order diagrams to include in
bhd,
the crux of the matter is that experiments comparing results with one another must use
the same prescription. In the case Of
the correction. itself is small:
(
and
- 6had)
4.5.4
c6h.,d
6hwd,
however, this is generally a moot point since
varies from 007 to 009, and so the difference between
bh,,,d
is at most 04%.
Peaking Approximations
We have now calculated the energy distribution for multi-photon Bremsstrahlung to
all orders, given the soft photon approximation and to within an order a 2 correction.
However, to alculate radiative effects in a coincidence framework, one must know the
effect of the emission cross-section on all measured particle vectors. The integrated
probability up to an energy cutoff is not enough, and one needs to know the angular
distribution
f photons as well.
The angular distribution of single photon Bremsstrahlung is given by Equation 423,
and is plotted in Figure 44 for Q =
7 and 15 (GeV/c)'.
One salient feature of the
distribution is immediately apparent: the radiation is strongly peaked along the directions
of the incoming and outgoing electron. Only a very broad peak is seen in the direction
of the scattered proton at
Q
=
I
(GeV/C)2, but it becomes more sharply defined as
Q2
increases. These features suggest a simple approach to the angular distribution, known
as the "peaking approximation": the single photon Bremsstrahlung spectrum may be
divided into three discrete photon directions, along each of the vectors k, k , and p' In
other words, we replace A(c) in Equation 422 with the simple form
Apeaking(L
= A,6(C
- k + A,,6(c,- k + Apb(L- '/),
141
(4.54)
where f dQ,b(c)
The terms of the one photon angular distribution A(c) may be divided into three
groups, due to the electrons, the electron-proton interference, and the protons respectively:
kf
2
A )
awo,
w
47r2
ki
kf
-2
2
W ki
kf
ki
w kf
w k-
+
Pf
W.
Pi
f
LO Pi
2
Pf
PI
W - Pf
W Pi
(4.55)
In order to better understand the structure of the peaks, consider the expansion of the
first term in an angle
describing the direction of photon emission relative to the
direction. In the region
<
k
Jkl,
2
kf
k,
2
LO
4-
'."
w kf
w
ki
Iki 14o2,
(4.56)
M4
indicating that extremely close to the k peak), the emission probability actually drops
to zero. This feature is too small to be seen in the electron peaks of Figure 44, but is
apparent in the much broader proton peak at Q2 = 7 (GeV/ C)2 (since
Further away from the peak, in the region -
IWI
<
VI
is of order 1).
the angular distribution falls off
quadratically with :
kf
LO2
We will refer to this
ki
kf
L, - ki
2
4
02'
(4.57)
shape later on.
We next need to determine the values of A,, A,,, and A, by integrating the various
terms of A(L) and distributing the results among the three peaks. The first (electron)
142
loI
10
10i
lo
10
i01
101
10
10I
10
lo
I
10I
10
10
10
10
10
10
10
10
10
10
10 11
io9
108
107
1(6
105
104
103
102
10I
T0
10 -1
lo -2
10-3
10. -14
photon angle
Figure 44: Angular distribution of first order Bremsstrahlung photons from Equation
4.23, calculated at Q =
7 and 15 (GeV/c)' and showing the improvement in the
peaking approximation with increasing momentum transfer. The photon angle is measured with respect to the direction of the incoming electron and given in degrees. The
directions of the scattered electron and proton are indicated by dotted lines and the
notation
, O
143
term of Equation 455 produces two terms of the form
k2
Cew02
j( dQy
(W
a
k)2
- - 7r
(4-58)
(one for each of k and k). Since the integrand is highly peaked in the direction k (or W),
it is assumed that all this strength contributes in the k (or W) direction. Next consider
the integral of the cross term,
aw 02
2
k V
47r2
IdQ1f
(4.59)
(w k)(u., k)'
In this case the integrand peaks in the k and k' directions. We evaluate it using
I &21
(w
k k'
k)(w
k)
1`1
k - k'
= I dQ1fP
k)(k k wO
= 2r In
LA)2
k - kl
Jkl
+ 2r
+ I d,
In
LIJ2
k - k'
(k'
k1 +
WI
klo -
k/I
k)(w
IWI
k) wO
(4.60)
This expression approximately integrates over the two peaks separately; the first and
second integrals are assumed to contribute to the k and k' peaks respectively. Cornbining these equations, one obtains the "typical" peaking approximation for electron
Bremsstrahlung:
A
=
a
4k2
- [In
_ 11
r
M2
Ael
=
-
a
r
[In
40
M2
_
11.
(4-61)
We can further assume that the third term of Equation 455, although only broadly
peaked at intermediate energies, contributes entirely to the final proton peak, yielding
Apt =
a [In
7
ePO+
0
- 10
144
- 21.
(4-62)
Some Bremsstrahlung strength still remains, due to the electron-proton interference
term of Equation 455 and to the non-peaked contributions missed by the approximation
of Equation 460. This is true even in the ultra-relativistic limit, where one expects the
peaking approximation to be the most valid (cf. 44).
If one uses the closed form UR
limit expressions of Equation 442 to determine the difference
t (El) - &ft(E2) between
two energies, and compares this with the result using only the peaked strength described
by Equation 4.61 and 462, one finds two missing terms. These are
In
7r
E2
4 In
El
due to the electron-proton inteference term
a In
E2
7
El
due to the non-peaked strength in
jkj
(4.63)
jk1j
and
2 In
1 - Cos(0,)
(4.64)
2
.Our approach is to preserve the total strength (as
bee
evaluated in the UR limit) by distributing the contributions of these non-peaked terms
among the three photon peaks. We choose to split the two terms evenly between the
electron peaks:
A,
a
k
- 2 In(-)
7
P
=
A,
Ae =
Ae
Apt
AP/.
=
+
a
7r
k
2 In(-)
ki
In
In
1 - cos(O,)
2
I
Cos (O,)
2
(4.65)
This set of formulae can be termed the "extended peaking approximation" for single
photon Brenisstrahlung. In the text below, the notation A will be assumed to mean
.
From Section 45.3, we know that including higher order Bremsstrahlung is critical in
145
evaluating the energy spectrum for low photon energies. One is then led to consider its
effect on the angular distribution. Calculating such higher order contributions directly
from Equation 423 is a formidable task. Instead, we observe that the single photon
peaking approximation,
do,
dQdw
k + Ae'6(C - k) + Ap'6(L
do,
doe epwO
(4-66)
effectively provides us with three independent single photon energy distributions, each
for radiation in a fixed direction. We can then proceed in the manner of Section 45.3
and determine a multiphoton spectrum, this time in terms of three energies: the total
photon energies E,, Ee,, and Ep, emitted in each of the three peaked directions. The
total radiated three-vector is then simply
Wtotal--:":Ej
+ Eek + EP.
(4-67)
Furthermore, radiation along the direction of a given particle can be interpreted as radiation due to that particle. In this way we correct the q vector used to evaluate
d,
dQ,
ep
at the scattering vertex for energy radiated before the scattering (i.e. radiated by the
incoming electron). This was seen in Section 45.1 to be the source of the largest discrepancy between the soft photon approximation and full calculation for single photon
radiation.
By analogy with Equation 449, we obtain the cross-section to all orders for radiating
a total energy Ee along k, E
along W, and Ep, along p', as well as any number of soft
photons with energy less than AE:
do,
do,
dQdEdEedEp, (AE = dQ,
ep
e-
6soft
EEL
':'o
(I -
h.rd)
'o
00
1=0 m=O n=O
146
A
E
b(We
C dw"O
E
i Oi
M
E i
n
n!
1
A
Leto
aE
+ L',e _ Ee)
+
bpe, 0I +...
+ Le 10
Eet
i Welo
EP,
A
1
dL,)P
i=l
0
10
Io 6 (L"P I+
-- +
)Pn
L')P
Using the same technique as in determining Equation 451, one obtains:
da
do,
dQdEdE,,dEp,
e- 6..ft (afl (I -
dQ
hwd)
e AC1n(EC/AE)
p
•
eAe11n(E,,/aE)
•
eAP1
1n(EP,/aE) '\PF(Ap,).
A,'F(Ae,)
Et
Ep,
Again, the As are of order a, and so F(Aj) (Equation 452) is
of order a 2
ACF(Ae)
E,
(4-69)
to within a correction
1
We see that the In(AE)
dependence of b,.ft(AE) will be cancelled by the
other terms of the expression, taking care of the AE -* 0 divergence of the single photon
spectrum.
By construction, the
subdivision of the terms of
s of the extended peaking approximation provide a
ft(AE) which depend on AE:
b.oft(El - boft(E2) = In E2
(
+
+
(4.70)
El
(Note that this is true only in the UR limit). However, 6,.ft contains additional terms.
Using Equation 442, we find that these terms can also be subdivided in terms of the As:
6,(AE)
A,In(. AE )
bel(AE)
A,,ln(
bp (A E)
AptIn(
VkO k'O
147
)
AE
AE
(4-71)
Employing these definitions, we can take the limit AE
0 to produce our final result
for the multi-photon peaking approximation:
do,
dQdEdE,,dEPl
do,
dQ, ep
Xe _1,!,E,,
do,
dQ,
(I
6P, EP,)
be (Ee)
P
6h.d)
ep
(4.72)
Ae A,, AP,
X
Vkk
I) A.!(/kk,)A_,
(,,Me)
AP'Ele +A"El+
el
Al-/EplI+AP'
The cross-section thus factorizes neatly into three independent functions, for the total
energy emitted in each of the three radiative tails.
The angular distribution implied by the above equation can be evaluated easily by a
Monte Carlo program such as our PWIA calculation, by randomly generating the energies
emitted in each direction and adjusting the fermion vectors accordingly. However, it
is worth studying the multiphoton angular distribution analytically, to determine the
approximate shape of the multiphoton peaks. For our calculation to be valid (or useful),
we must confirm that these peaks are substantially broader than the single photon peaks,
which were approximated as delta functions in Equation 454. To accomplish this, we
employ a change of variables: from E,, E,,, and Ep, to E, u, and v Here, E is the total
radiated energy E, + E, + Ep,, while the emission direction is fixed by u and v:
U
V
Eg
Ei
Eg
EP,
Note that u and v vary from
to oo with u, v
(4.73)
oc corresponding to emission in the e
direction, u --+ 0 corresponding to emission in the e' direction, and v - 0 corresponding
to emission in the p direction. The Jacobian between these two sets of variables is
148
straightforward:
dEdE,,dEp, - dEdudv
EE,,Ep,
(4.74)
Euv
Consequently, the multiphoton emission cross-section 472 can be rewritten easily in terms
of the new variables. The dependence on the total energy E factorizes completely from
the angular distribution, and the integration over emission angles can be accomplished,
yielding
do,
-
dQedE
d
API
- - '
(I -
d,,p
h.d)
(VMP
v/kk I)A Vkk 1)),
x
x (A,
A,,
Ap,)r(I + A,)r(I + A,,)r(
r(
+ A,
A,,
+ Ap,) 4.75)
Apt
9
Recalling that the As are of order a, one finds that the ratio of gamma functions on
the last line is I to within the usual O(a') correction. To within this accuarcy, this
distribution agrees with the previous multiphoton formula, Equation 451.
The analysis of the photon distribution simplifies greatly if one neglects proton radiation. Taking
to be the angle between the photon and k, and 0,, to be the usual
scattering angle between k and W, one finds that for
do,
< 1,
sin(O,,)-'--'
dQdEdO
(4.76)
01-A'/
and for 0 -- 0, < 1,
do,
dQdEdO
sin(O,,)-",
(
_
The photon spectrum thus drops away from the peaks at the rate
more gradual than the
(4.77)
")1-A,,
As this is
falloff of the single photon peaks, our calculation of the
multiphoton distribution from perfectly peaked single photons is reasonable.
149
In the case of proton radiation, of course, the peaking approximation is suspect from
the very beginning. Its use hinges on the relatively small Brernsstrahlung contribution
of the proton, and on the resolution of the experiment.
Also, as pointed out at the
beginning of this section, at sufficiently high photon energies all radiative tails converge
on the same k
= , kinematic path. The sensitivity to the precise angular distribution
is thus most apparent at low photon energies. The effect of the peaking approximation
on the NE18 PWIA calculation will be examined in Section 45.6.
4.5.5
External Bremsstrahlung
One element of the radiative calculation remains to be described. This is the spectrum
for the emission of Brernsstrahlung photons in the field of nuclei other than the one
participating in the hard scattering.
These losses occur as a charged particle moves
through the target material and traverses vacuum chamber windows and air gaps. The
probability distribution a particle of momentum k to radiate at total energy of E"'
when traversing t radiation lengths of material is found to be 33]
1
r(i+
bt
E"t
bt) Eext
Jkl
Eext
bt
text
Jkl
(4-78)
where the parameter b depends on the atomic charge Z of the target material:
b =
LI =
L2
1
9
12
+
ZLI + L2
I
n(184.15 - - n(Z)
3
2
= ln(1194. - - n(Z).
3
150
(4.79)
The function V` is a correction for large photon energies, expanded to second order in
E ext
IkI
ext
(X)
X + 3 X2
(4.80)
4
External radiation is far simpler to treat than internal.
First of all, the particles
radiate independently and so incoherently, and this eliminates the non-peaked strength
cause by the interference terms of internal Bremsstrahlung. Furthermore, proton radiation is suppressed relative to electron radiation by the factor
(_)2
M
and so can
10-6 ,
be neglected ntirely. Equation 472 can thus be expanded in a straightforward way to
include the contributions from external radiation along the k and k' directions:
do,
dQdE"1tdE'PxtdEintdE"t =
i
f
f
do,
doe ep(I -
h.,-d)
Eext
X
r(I
+ bI
X
I
X
btf
F(I + btf) EXt
f
bti
A-
nt
E,
X Sent
Iki
Efext
IkI
k
btf
Af
I
E int
X -
f
Eint
f
Nkki
Af
(4.81)
Here, the internal proton contribution has been omitted for convenience, and the subscripts i and f have been introduced to indicate the initial and final electron arms. Since
both
it
and Ext are emitted in the same direction, we would like to rewrite te distri-
bution in terins of the total energies Ei and Ef radiated along k and W. This problem is
exactly analogous to the transformation made between Equations 472 and 475, where
a change of variables was made from three energies E,, E,,, Ep, to a total energy E and
angle variables u and v. The result is
do,
dQdE,nYExtdEintdEext
I
f
f
-
do,
(I
doe eP
151
6hard)
F(I + bti) (I + btf)
(bt + Ai)
X
kbt
(vl-kkl),\i
(btf + Af)
dEj
k1btf (Vlrkk1)Af Eil-Ai-btj
dEf
r'I-Af
Ef
.(4-82)
-btf
We thus see that the As of internal radiation play much the same role as the material
thickness bt of external Brernsstrahlung.
One can also express the external radiation
contribution in terms of the usual Bremsstrahlung functions 6,,ft. One obtains forms
which are very similar to those of Equation 471:
6ext
(AE
e
=
bex
el t(AE
bti In( AE)
k
k'
= btf In( AE).
(4-83)
These functions can simply be added to the corresponding b(AE)s for internal radiation
in Equation 472, yielding the same result as Equation 482.
Thus far, the correction function
the ratio
E-1
jkj
Oext
e-t) has been neglected. At NE18 kinematics,
Jkl
in which the function is expanded is small
only the first order term of
Oex, (x)
0.1). Consequently we take
and include it in Equation 481. Carrying through
the angular integration, one obtains multiplicative factors Vt
and Vxt
f to include in
Equation 482:
Dext (Ei)
-
i
(The same form applies for
4.5.6
(Dext with
f
bti
bt
i
Ej
A
t
(4.84)
ki
f everywhere).
Radiative Techniques Employed 'in the PWIA Calcula-
tion
Two models of the radiative corrections are included in the PWIA Monte Carlo SIMULATE. The first uses the multiphoton energy distribution of Equation 453, evaluated
using the full SPA expressions of Equations 437 and 438. The angular distribution is
152
taken to be the pure peaking approximation of Equation 454. The strength is distributed
among the three tails Z=
subscripts e,
23 using the fractions
A' (I is shorthand for the usual tail
I '). The second method tries instead to obtain the correct multiphoton
angular distribution by generating the total photon energies E, E,
Ep, emitted along
each direction., and summing the resulting photon vectors according to Equation 4.67.
The distributions are generated according to the independent forms found for each tail in
Equation 472. These energy distributions were calculated using the approximate closed
form expressions of Equation 442, found in the ultra-relativistic limit of high momentum
transfer and particle momenta. These two choices represent a trade-off between the bt:st
available forms for the photon energy (first technique) and angular (second technique)
distributions. The first method can thus be referred to as the "peaking" technique, and
the second as the "multiphoton" technique. Note that these names are somewhat misleading: the "peaking" formalism certainly involves contributions from Bremsstrahlung
radiation to all orders, and the "multiphoton" prescription involves the peaking approximation at the one photon level. One hopes, of course, that the two prescriptions give
very similar results and this indeed turns out to be the case. The distributions of counts
calculated by- SIMULATE using the two techniques are sufficiently similar that one is
hard pressed to see any differences on a plot of the projections in E, and p,.
The
integrated yields are less than 1% different at all Q. This excellent agreement indicates
the lack of sensitivity of our results to the fine details of the photon angular distribution,
and the validity of the UR limit at our energies.
Comparisons of the radiative calculation with the data will be presented in the following chapter.
153
Chapter
Results of the Experiment
5.1
Extraction of Results
5.1.1
Acceptance Cuts
The symbol
is used throughout the following discussion to denote the experimental ac-
ceptance. This acceptance was defined by cuts placed on both the reconstructed particle
vectors and on the extracted values of E
The p
and p,.
coverage is, of course, different for each setting of Q and ',P and to a lesser
extent for each target. Cuts in p, were chosen at each of these settings, to exclude low
statistics bins at the edge of the phase space. Overall kinematics-independent E, and
p,,, cuts were also imposed on the results and are listed in Table 5.1. For the carbon,
iron, and gold targets, p, values on only the positive side of the q vector are selected
because the negative side was not fully sampled at all Q' settings. The 250 MeV/c choice
of upper limit was based on the maximum coverage available at all Q, without including
regions where the phase space did not extend to the Fermi energy in E,.
This cutoff
is reduced to 210 MeV/c for the gold data, because of the slightly smaller phasespace
available for the
Q2
of 7 (GeV/c)' data. Note that alternative choices of overall cuts
154
Table 5.1: E, and p, cuts placed on results at all kinematics.
Target
111
2H
12C
56Fe
197
Au
E, cuts
MeV
p,,, Cuts
Mev/c
-25
-25
-25
-25
100
100
100
100
-160
-160
0
0
160
160
250
250
-25
100
0
210
were tested as well and used to provide uncertainties on the results. These are discussed
in a later section.
The cuts on reconstructed spectrometer quantites are designed to keep these vectors
inside the well-understood regions of the spectrometer acceptances. Such cuts are particularly important in the momentum variable , as the acceptance is not flat outside ± %
in either spectrometer. Perhaps more importantly, the optical matrix elements were only
tuned in this region and are known to produce unreliable results at larger momentum
deviations. Also, evaluating our results using several cut settings supplies an estimate of
the systematic uncertainties caused by our spectrometer models. Four cut settings were
used in the analysis, and are listed in Table 52; CUT 3 was used for the extraction of
our final results.
5.1.2 Extraction of the Spectral Function
The spectral function was extracted from both the data and the results of the SIMULATE
Monte Carlo. Both sources produced histograms of experimental counts, binned in E,
and p,.
Such a distribution is related to the differential reaction cross-section by the
relation
N(Em7pm)
)CKeff
I6(E,,,,,-)
155
d6o, d" V,
d6V
(5.1)
Table 52: Cuts placed on reconstructed spectrometer quantities. Note that the same
cuts were imposed on both the 16 GeV/c and GeV/c.
refers to the in-plane angle
(relative to the magnetic axis), and D to the out-of-plane angle. CUT 3 was used for all
final results.
CUT
(D
%
where N(E,
mrad
mrad
I
2
±
±
± 25 ± 60
± 25 ± 40
3
±
± 15
± 40
4
± 4
± 15
± 40
PM) is the array of true coincidences corrected for experimental efficien-
cies, and dV denotes the six-fold differential phase space of a coicidence experiment
dc'dQ,,dE'dQP1. V(E,,,, p,) indicates the total phase space volume contributing to a
given (E,,,,p,) bin. Using the PWIA factorization assumption of Equation 17, one can
remove the spectral function S(E,,p,)
from the integral and obtain
N (E,,,, PM = L S (E,, PM)K oep I Empm) P (EM I PM)
Here, Ko,,p
(Empm)
(5.2)
is the average off-shell ep cross-section, averaged over the events
collected in an (Em, PM) bin. Taking this average and removing KO',p from the integral is
reasonable because this function is slowly varying over the quasielastic peak. 'P(EM, p"')
is the so-called coincidence phase space, given by
P (Em, PM =
JV6
d 6V,
(E,.,,,Pm)
(5-3)
and represents the probability density for obtaining a given Em and PMover the six-fold
phase space of the experiment. It is evaluated by running the Monte Carlo program SMULATE without radiative effects, and weighting each event with Wg,, only (see Section
156
4. 1). The physics is thus removed from the calculation, leaving only an integral over the
phase space. The P(E,,,, p,) array is finally multiplied by the 6-fold phase space volume in which the generation is performed, and divided by the total number of generated
events. The extracted spectral function is thus defined as
S (EM PM) =
N (Em, M)
(5.4)
IcKo,,p (E,,p,,,) P(EmpM)'
5.1.3
Extraction of Transparency
The kinematics of the experiment involve data taking at several setting of the proton
spectrometer angle for each Q. The transparency at each of these settings (denoted )
is defined to
e
T =
E R (E,,,,p,,,)Ndata(Em, M)
(5.5)
E Ri(E,,,,p,,,) NPW IA Em PM)'
The values at individual angle settings were then combined to obtain the most probable
result for the net transparency at a given Q:
I
T-
T = Ei
I:i dY.7
-1 .
(5.6)
dT.7
The errors dTj are the sum in quadrature of the statistical and systematic errors. The
fractional statistical error is taken to be the familiar VN standard deviation of a Poisson
distribution. The systematic uncertainties are discussed below.
5.1.4 Systematic Uncertainties
A summary of the systematic uncertainties of the experiment and their effect on the
extracted transparency is given in Table 53. Most of the uncertainties have been presented and explained in previous chapters, but some appear here for the first time and
157
require additional comment. The error in random subtraction was evaluated by looking
for systematic trends in the number of coincidences found between E, of 25 and -5 MeV.
There should be no spectral function strength in this region. Events found in this region
should be taken into account by statistical errors; the uncertainty listed here is an upper
value on any systematic deviation from zero. The acceptance error was determined by examining the fluctuation in results for different spectrometer cuts. The error on radiative
corrections was computed from two sources. First is the variation of the transparency
with the upper E, limit used in the sums of Equation 55. Upper limits of 50, 80, 100,
and 130 MeV were applied to the hydrogen and deuterium data, and produced variations
of < 2 %. This is referred to in the table as the error on "internal radiation", as radiative
corrections for these thin liquid targets is dominated by the internal portion. A similar
test done with carbon and E, cutoffs of 80, 100, and 130 MeV confirmed this result,
although the Lorentzian tail of the carbon missing energy distribution also contributes
at high E,
preventing the isolation of purely radiative effects. The "external radiation"
error was determined using test runs performed at
iron targets of different radiation length 2% and 6
Q
=
I
(GeV/c)' with carbon and
r.l. for carbon, and 6
and 12%
r.l. for iron). Assuming that the target thicknesses are well known and the experimental luminosity properly computed, external radiation is the only element of the analysis
affected. An error of 2
was determined from these comparisons. The uncertainties
due to the model spectral function and o,p were simply determined by performing the
PWIA calculation with various formulations of these. The choices of prescription for
the ep cross-section and model spectral function are explained in Sections 42 and 43
respectively.
The individual systematic uncertainties are summed in quadrature at the bottom of
the table. The net uncertainties for 'H, 'H, and `C are 78% in magnitude, and are
158
dominated by the experimental acceptance errors. For the heavy targets
56
Fe and
197
Au,
however, the systematics are considerably larger (I 1 13%). The relatively poor knowledge
of the model spectral functions for these targets is responsible, as well as the sensitivity
of the transparency to modifications of these input functions. The transparency depends
on the number of experimental counts determined by the PWIA Monte Carlo, and this
in turn is the result of folding the spectral function with the coincidence phase space.
The IPSM spectral function is normalized to
Z
S(EM Pm)dEd
S(Em, pm)dEmp2
M dpmdQpm,
3P,
(5.7)
and so produces a constant result when folded with p2M . The coincidence phase space,
however, has a different p
dependence in general. Consequently, adjusting shell model
parameters which affect for example the width of the spectral function may cause large
changes in the number of counts computed by the Monte Carlo of the experiment.
5.2 Hydrogen Results
The spectral function is not a meaningful quantity for elastic scattering from free protons,
as the differential cross-section is inherently two-dimensional. However, the distribution
of hydrogen ata counts in Em and pm compared with the Monte Carlo calculation provides a precise test of many aspects of the calculation. The true distribution of elastic
events is precisely localized at E
=
and p
=
; any deviation from this must be due
to the improper subtraction of randoms, experimental resolution, and particle radiation.
The last two effects should be correctly modelled by the Monte Carlo. In particular,
a comparison of the data and Monte Carlo on the hydrogen radiative tail provides a
precise test
f the radiative procedure, unclouded by other physics. Finally the hydro-
159
Table 53:
ment.
Summary of systematic uncertainties affecting the transparency measureItem
dead times
% Uncertainty in T
1.4
deff
1
proton absorption
target thickness
liquid
2
solid
charge
tracking, 16 GeV/c
tracking, 8 GeV/c
multiples, 16 GeV/c
multiples, 8 GeV/c
random subtraction
acceptance
internal radiation
external radiation
O"P
0.7
0.2
0.5
1
1
1
1
1
5
2
2
2
model spectral function
12C
2
6
9
12Fe
12
Au
correlation corr.
12C
3
6
6
12Fe
12 Au
sum in quadrature
- 1H
2H
- 12C
- 12Fe
-
7.0
7.0
7.9
11.0
12.9
12 Au
160
gen transparency must be one and this provides the ultimate test of the experimental
normalization,.
The distribution of hydrogen data counts as a function of E, is presented for all Q2 in
Figure 5-1. Superimposed on these figures is the corresponding Monte Carlo calculation.
Also included here are the E, distributions for deuterium (Figure 52), as the single
deuterium bound state is very sharply peaked at the binding energy of 22 MeV and so
'H data in this coordinate provide the same precise test of the radiative procedure as 'H.
The figures clearly demonstrate that the radiative prescription perfectly describes the
data to within its statistical uncertainty (note that the statistics available for 'H and2 H
are higher than those for any other nucleus).
The hydrogen transparency is listed in Table 54 depicted in Figure 513 and is consistent with unity at all momentum transfers. As a quantitative measure of the E
and p,
dependent agreement discussed in the previous paragraph, one can evaluate the hydrogen
transparency with a variety of E
cuts. One finds that the transparency varies by an
amount well within the statistical error of the data for upper E
cutoffs from 50 to 130
MeV. The statistics provide a precision from 1% at Q = I GeV/C)2 to 4
at
Q2
= 7
(GeV/ C)2.
5.3
Spectral ]Function Measurement
The spectral functions extracted from our data are represented here in their projected
forms, p(p,) - f SdE, andC(E,,) =_f Sdp,,
in figures 53 through 5-10. The PWIA
calculation is normalized to the measured transparency.
In all cases, only statistical
errors are shown. These can be inflated by the amounts indicated in Section 51.4.
First, we see that there is reasonable agreement between the measured E
distribu-
tion and the PWIA calculation. In particular, the carbon spectrum shows no evidence of
161
103
104
V)
1
C 10
:3
0
0
-2
C
2
:3
0 10
0
100
10-
I
E,,, (MeV)
E,,
(MeV)
10'
102
101
U) 102
0
S
0
0
S
0
U
100
10-1
0
50
100
0
E .. (MeV)
50
100
Ern (MeV)
Figure 5-1: Distribution in E, of coincidence events recorded for the hydrogen target,
compared with the prediction of the Monte Carlo program SIMULATE
162
I
lo"
10 3
102
10 2
U)
-!2
C
73
0
U
1
:3C
10
0
0
10 0
100
lo-
I
E .. (MeV)
E
M
(MeV)
103
10 2
102
rn
-3C:
U)
-.1
-3C10
I
0
0
0
U101
100
100
0
50
100
0
Er,, (MeV)
50
E
rn
100
(MeV)
Figure 52: Distribution in E, of coincidenceevents recorded for the deuterium target,
compared with the prediction of the Monte Carlo program SIMULATE.
163
.08
.08
.06
.06
I
I
ID
:2
a)
M
604
04
a11
In0-
V)
(A
'--.02
"-.02
.00
.00
a
a
Em (MeV)
EM (MeV)
.08
.08 I
.
I
.
.06
I I .
Q2
-1
I
I
= 68
I
I
(GeV/C)2
.06
I
I
Q)
:2
20
604 -
eO4
a-
CL
0
M
V)
(n
'--.02
'--.02
.00
.- -
a
nn . . . . . .. . .- .- . .- .0
E.. (MeV)
50
.- -I .---- --
100
E M (MeV)
Figure 53: The extracted spectral function for 2H, integrated over dp,,,. The solid line
represents the result of the PWIA calculation, normalized to the measured transparency;
the data points are shown with statistical errors only.
164
.06
.06
.05
.05
04
-,.04
0
M 03
9
Q)
M 03
C
C1.
-11
0
(n 02
v).02
CL
n
'a
.0 1
.01
.00
.00
E.
(MeV)
E,,, (MeV)
InC
Ij:
.06
-
.05
.05 -':
.
. .
I I I .
.
Q2
I I I I I I .
= 6
.
.
(GeV/C)2
- 04
04
:Z
0
M
4)
:2
503
03
E
011
na-
(n.02
v).02
11-1
a
a
I-N
.01
-:
.01:
I
.00
.00
E.. (MeV)
. - .-II
0 I- -50 I
E.
-
II
100
(MeV)
Figure 54: The extracted spectral function for "C, integrated over d3p,,,. The solid line
represents the result of the PWIA calculation, normalized to the measured transparency;
the data points are shown with statistical errors only.
165
.10
.10
.08
.08
I
I
Q).06
Q).06
E
I0.04
a-E
aI0.04
V)
V)
.02
.02
.00
.00
E,,, (MeV)
.10
.10
.08
.08
I
I
0.06
0).06
E.
(MeV)
E
(MeV)
aE
E
0-
I0.04
I0.04
V)
V)
1-1
.02
.02
.00
.00
50
0
E
-,
100
50
0
(MeV)
I - I
M
100
Figure 5-5: The extracted spectral function for "Fe, integrated over dp,. The solid line
represents the result of the PWIA calculation, normalized to the measured transparency;
the data points are shown with statistical errors only.
166
.12
.12
.10
.10
I .08
1 .08
Q)
M
4)
:2
- E06
506
0011
CL
110
v).04
v).04
.02
.02
.00
.00
0
50
100
E,,,, (MeV)
E M (MeV)
.12
.12
.10
.10
I .08
1 .08
Q)
:2
0
:2
- E06
506
aa
am
in.04
(n.04
I--,
.02
,02
.00
.00
0
50
100
0
E.. (MeV)
50
E
M
100
(MeV)
Figure 56: The extracted spectral function for "'Au, integrated over d'p,,,. The
solid line represents the result of the PWIA calculation, normalized to the measured
transparency;
the data points are shown with statistical errors only.
167
11-1
lu
I... I.-I .... I''..
:
Q2
10 -6
10
:
1-1 -7
>01
0
:2
-
E
E
LU
W
0
'10
0 -a
Inio
:
r
10 -9 .
................-
. I
-2 DO
p,,, (MeV/c)
-0-I
...... I.........
Q2
10.
= 3 GeV/c)
-6
I
2
10
-
0
p,,, (MeV/c)
100
2( 10
-0I ..................
= 68 (GeV/C)2
10. -6 _
r-
I
U
'-I
110
E
,,lo
. I I I . I I I . . . I . . . I
-100
Q2
I1
0
1-1 .-7
110
W
'D
(GeV/ C)2
I
U
U
I--,
110
'!e
10
=
.-7
E
Li
0
-8
'-8
1110
1-1
k--
-9
-9
10
10
-200
F--
-100
.............
0
p"' (MeV/c)
100
I
200
-200
-100
p,
1
.6 . . . .ido' ' ' 200
(MeV/c)
Figure 57: The extracted spectral function for 2H, integrated over dE,. The solid line
represents the result of the PWIA calculation, normalized to the measured transparency;
the data points are shown with statistical errors only.
168
p .. (MeV/c)
p"' (MeV/c)
10-7
II
"I
I
U
'I-,
111
Q)
-8
4)
-a
7 0
-:M0
E
L'i
E
W
a
0
V)
(n
I--,
I--,
10-9
10-9
r) .. (MeV/c)
p.. (MeV/c)
Figure 5-8: The extracted spectral function for "C, integrated over dE,,,. The solid line
represents the result of the PWIA calculation, normalized to the measured transparency;
the data points are shown with statistical errors only.
169
10
n
40
U
'_1
Q)
E
LLJ
a
V)
I___%
10
P,,, (MeV/c)
p .. (MeV/c)
lo-'
11
-Io
'_1
-7
4)
t
E
LLJ
a
cn
I--,
10-9
-200
100
0
100
200
300
p,,, (MeV/c)
p,,, (MeV/c)
Figure 5-9: The extracted spectral function for 'Fe , integrated over dE,,,. The solid line
represents the result of the PWIA calculation, normalized to the measured transparency;
the data points are shown with statistical errors only.
170
10
I1
lu
1-1
:t
:,Do
E
W
a
cn
10
pi", (MeV/c)
p,,, (MeV/c)
10
10
0
Q
11-1
4)
-
ZO
4)-
ZO
E
E
W
Uj
a
a
(n
(n
11
10
11-N
10
P,,, (MeV/c)
p.
(MeV/c)
Figure 5-10: The extracted spectral function for "'Au, integrated over dE,. The
solid line represents the result of the PWIA calculation, normalized to the measured
transparency; the data points are shown with statistical errors only.
171
strength beyond that predicted by the PWIA calculation. This is in contrast to some previous measurements 10][12][14]made at Q of about 02 (GeV/c)' which found evidence
for extra strength at high E,; this was interpreted as evidence of scattering from multinucleon currents which would tend to shift portions of the IPSM spectral function to higher
missing energy (see Sections 13.2 13.3, and 54.1 for further discussion). As described
in Section 13.3, it is possible that scattering from multinucleon currents is suppressed
at
Q2
above 1 (GeV/C)2 - a high energy probe is less likely to scatter from a compos-
ite system that one of larger wavelength. However, we have seen how Bremsstrahlung
radiation also shifts strength toward higher missing energies, and it is also possible that
part of the high E, strength is due to radiation not completely accounted for. It would
be interesting to use the Monte Carlo calculation and attendant radiative prescriptions
developed for NE18 to reanalyze the older data.
One also notes that there are some substantial discrepancies between the PWIA
calculation and the data at low missing energy. This is the region dominated by the
nuclear shells. The Is shell in carbon, for example, seems to be a good deal broader than
that used in the model spectral function. The data distributions for deuterium at
I
Q
=
(GeV/ C)2 and for iron also seem broader than the Monte Carlo result. It is possible
that the energy resolution of the experiment was underestimated by the Monte Carlo, a
hypothesis supported by the discrepancy seen in some of the spectra at E
below the
Fermi energy (i.e. in the region where there is no spectral function strength and all
contributions must be due to experimental resolution). Fortunately, the transparency
measurement is largely insensitive to the precise shape of the E
distribution, as the
coincidence phase space of the experiment varies only slowly with missing energy. (This
insensitivity was quantified in Section 51.4 above).
The missing momentum spectra show good agreement with the PWIA calculation for
172
all nuclei and momentum transfers. Again, there is no evidence of contributions from
processes other than those described by the PWIA. This observation also validates our
use of a single absorption factor (the measured transparency) to describe final state interactions: Discrepancies between the calculated and extracted p(p"') would suggest the
presence of significant elastic rescattering of the knockout proton, specifically rescattering
which is sufficiently forward peaked to shift the proton momentum within our spectrometer acceptance. Such effects would have to be unfolded from the distributions using
DWIA corrections (as is done in precision coincidence measurements at lower energies).
Figures 5-11 and 512 show the spectral functions integrated over the approximate
E, ranges of the 1p and ls shells. One sees that the good agreement with the Monte
Carlo is maintained even over these restricted integration regions (i.e. there is no E,-p,
correlated source of disagreement which might be hidden by the integrals over the full
acceptance). The familiar shell structure is clearly in evidence up to Q of 7 (GeV/C)2
pip(p,) drops to zero at p = , while the s-shell strength peaks at p = .
We emphasize at this point that the extracted spectral functions are not radiatively
corrected. Rather, radiative effects are taken into account by the PWIA calculation,
and the extracted distributions are always compared with the Monte Carlo results. The
contributions to the spectral function from the various settings of the proton angle have
been statistically averaged on a bin-by-bin basis, but one should note that the radiative
corrections for a given E, p, bin are different for different kinematic settings. The spectral functions cannot be analyzed in a model independent way unless they are radiatively
unfolded. Tis is the next step we plan to take in our analysis. The ability to radiatively
unfold the data is a direct consequence of the radiative Monte Carlo, for SIMULATE
can accurately determine the radiative tail shape in the measured quantities
cor-
responding to spectral function strength in any actual E, p,,, bin. This Bremsstrahlung
173
7
IL)
'I-,
:Z
:)O
W
E
V)
10
p.
(MeV/c)
p,,, (MeV/c)
p,,, (MeV/c)
p.. (MeV/c)
10
7
0
"I.
0
ZO
-
E
W
0
(n
10
Figure 5-11: Extracted p(p,,,) for the 1p shell of "C. The spectral function has been
integrated over -1 < E,, < 25 MeV. The solid line represents the result of the PWIA
calculation, normalized to the measured transparency; the data points are shown with
statistical errors only.
174
10
10
M
I
0
11-1
U
4)
Q)
:Jo
:10
E
W
E
LU
V)
V)
a
'a
,--I
1C
PI,,,
(MeV/c)
p,,, (MeV/c)
10-7
10
I
0
"I
I
"I
1-1
0
4)
-
:N
-8
-:: 0
E
E
Ui
Li
I--,
1-1
'a
(n
0
(n
10-9
10
P"' (MeV/c)
p,,, (MeV/c)
Figure 512: Extracted p(p,) for the Is shell of "C. The spectral function has been
integrated over 30 < E < 0 MeV. The solid line represents the result of the PWIA
calculation, ormalized to the measured transparency; the data points are shown with
statistical errors only.
175
shape has all th e experimental acceptances and resolutions correctly folded in. The only
model dependence in such a deradiating procedure would come from the radiation of
spectral function strength into the acceptance from regions outside. A model spectral
function is required to determine these contributions, but at the
Q2
settings where miss-
ing momenta on both sides of the q vector were detected, most of the spectral function
is detected and these contributions should be small.
5.4 Nuclear ransparency Measurement
The measured transparencies for all Q and all targets are presented in table 54 and
plotted in Figure 513.
uncertainties.
The figure depicts both statistical (inner) and total (outer)
Small differences between these numbers and those appearing in NE18
publications 62][63]are due to the averaging that was performed between these results
and those of a parallel analysis performed at Caltech 40].
A detailed comparison of the transparency values with theoretical predictions follows,
but some important conclusions can be drawn immediately. First, one sees that the
expected signature of colour transparency - a rise of the results with increasing Q - is
not observed within experimental errors. A gradual rise with
Q2
is certainly permitted,
but the results are consistent with a Q independent model of final state interactions.
A notable exception to this, however, are the values at Q = I GeV/c)'.
A dramatic
rise in the transparency is exhibited at this momentum transfer for all A>2 nuclei. An
immediate explanation of this effect is suggested by Figure 1-4, depicting the energydependence of the free pN cross-section. If one surmises that the pN interaction in the
nuclear medium has a similar energy-dependence, one sees that final state absorption
will be less for our
Q
=
I
measurement. At
Q
=
1, p is about 12 GeV/c and pp is
significantly lower than the asymptotic 40 mb approximately achieved at our higher p'
176
Table 54: Summary of the NE18 transparency measurements. Statistical errors are
given in parentheses after the total uncertainties.
Target
Q' (GeV/c)'
1
3
1H
0-99±.07
2H
0.91±.06 (.01)
0-65±.05
02)
0-51±.06 (.02)
0.42±.06 (.02)
12C
-16Fe
197 Au
(.01)
0-99±.07
5
(.01)
0.89±.07
03)
0.64±.06
03)
0.38±-05 (.02)
0-28±.05 03)
6.8
104±-08
03)
107±.09
(05)
0.91±.07
0-62±.06
0.41±.06
0.23±.04
03)
03)
04)
03)
0.93±-08
0-68±-07
0.44±-06
0.31±.07
(.04)
(.04)
(.04)
(.06)
settings. If one looks at 513 with an eye to identifying a rise with
Q2,
one is tempted to
neglect the error bars and remark on the rise of the actual measurements. However, such
an observation would conclude that a rise also exists in elastic scattering from hydrogen.
It seems likely that a
Q2
dependent systematic error in the normalization is involved,
but is properly accounted for by the experimental uncertainties of Section 5.4.
5.4.1
Comparison with Glauber Calculations
A search for an effect such as colour transparency must be made not only by comparing
the measurements with CT predictions, but also with calculations made without including
the effect. It was pointed out above that no evidence for a rise in T( Q2) is observed;
one is then led to ask whether "conventional" calculations can explain the results. Such
calculations are performed using the "Glauber" technique described in Section 1.3.4. Also
demonstrated in that section was the variety of effects which can alter the calculated
results and which must be taken into account.
A large nmber of Glauber calculations have been carried out for a variety of nuclei,
notably at A = 12 and - 200. A comprehensive selection of these is presented in Figures
5-14, 515 ad 516, along with the NE18 transparencies. For the sake of convenience,
the following discussion will focus on the results and calculations
177
for "C.
Before we
1.2
i
1.0
0. I--,
N
0
I
0.6 -
I--
0.4 -
6OFe
197Au
0.2 0.0
0
I
1
I
2
I
I
3
I
4
Q2
(GeV/c
5
Q2
for ease of illustration.
178
I
7
8
)2
Figure 513: The measured transparencies for all targets and
points are offset in
I
6
Q2
values. Some of the
continue , however, we should mention the deuterium transparency measurement, which
is consistent with a Q independent value of about 090. The applicability of a
lauber-
type multiple scattering calculation of final state absorption to the 'H nucleus, with
its single spectator nucleon, is clearly questionable, and few estimates of the deuterium
transparency exist. Two such estimates suggest values of 092 64] and 095 65], both
of which are in good agreement with the data. The significance of our
2H
transparency
measurement remains an open question.
Two immediate observations can be made from an examination of Figures 514 - 516.
First is the large variation in the calculations. Much of this is due to the omission of
different effects from the various calculations - in fact, it will be seen in the discussion
below that no calculation includes all of the aspects of the problem that have been
identified to ate. Second, one sees that the calculations are in general lower than the
results. This suggests that the few theories producing high enough values of T( Q2 ) are
identifying a important aspect of the FSI description.
Only two of the calculations shown in Figure 514 are
Q2
dependent. These are from
Frankfurt, Strikman, and Zhalov (FSZ93) 701and Jennings and Miller 71] account for
the energy dependence of the (free) pN cross-section over the NE18 kinematic range, while
the rest use asymptotic values for this cross-section. The
Q2
dependence of this effect
must be understood if one is looking for a colour transparency signature. The asymptotic
values of upArused also vary among the remaining calculations. The largest difference
here is between the use of the total or the reaction (i.e. inelastic) cross-section. Most
calculations use Ototal (with values varying from 40 - 43 mb). However, most calculations
effectively work with a semi-inclusive transparency, integrating over all proton final states,
and as is highlighted in the work of Kohama et al. (KYS93)
66], elastically rescattered
protons will be recovered by such an integration. The reaction cross-section is less than
179
1.0
0.8
0.6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
N
0
I_'
I---
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.4 -
-
-
-
-
-
-
-
-
KYS93 a,
-
-
-
-
-
-
-
-
-
Nik93
KYS93 at.,
Ben92
CGA
Far88
0.2
JM92
FSZ93
Ben9
GA
0.0
0
2
4
Q2
(GeV/C)2
6
8
Figure 514: The measured transparency for C, compared with several Glauber calculations. The references denoted by the legend are as follows: (i) KYS93 = Kohama et
al. of ref. 66]; calculations using the reaction (o,,) and total (ut.t) free NN cross-sections
are shown (ii) Nik9
= Nikolaev et al. of ref. 671, which gives the same result as KYS93
(iii) Ben9 = Benhar et al. of ref. 68]; GA = glauber approximation, CGA includes
the correlation hole effect (iv) Far8 = Farrar et al. of ref. 28] (v) JM92 = Jennings
and Miller of ref. 69] (vi) FSZ93 = Frankfurt, Strikman, and Zhalov of ref. 70]
Ur
180
I
I
I
0. -
--- -----
I
I
I
KYS93a / Nk93
Ben92
CGA
FSZ93
Far88
0.6
- - - - - -
KYS93 att
Ben92
GA
I-'
N
'11
0
_ 04 -
LL
--- ---------- :Tt__ ___
-- ----- --- T------- - - - - --------
--------
---------------- ----------------------
- - - - - - - -
0.2 0.0
I
0
I
I
2
I
I
I
4
Q2
I
(GeV/c )2
I
I
I
6
I
I
8
Figure
15: The measured transparency for "Fe, compared with several Glauber calculations. Te notation in the legend is the same as that of Figure 514.
181
I
.8
I
JM92
KYS93 a,
Ber92 CGA
.6
I
A-
Nik93
FSZ93
Far88
Ben92
GA
KYS93 atot
I--,
CN
4-
I--
--
- - - - - - --- - - - - - - - - - - - --- - - - - - - - - - -- -- - - - -- -- -- -- -- - - -- -- - - - - - - - -- - - - - - - -
----- -------------------------------------------------
.2
.0
0
2
4
Q2 (
eV/c )2
6
8
Figure 516: The measured transparency for "'Au, compared with several Glauber
calculations. The notation in the legend is the same as that of Figure 514.
182
70% that of the total cross-section in the high energy limit, and KYS93 and Nikolaev et
al. of ref. 64] indicate that the resulting difference in transparency for carbon is about
0.1. This is a large effect, and should be evaluated carefully by computing the fraction
of the elastically rescattered protons which ends up inside the experimental acceptance.
Precise information about the acceptance is thus required. KYS93 points out that the
strong forward peaking of the pN reaction will tend to make the reaction cross-section
more appropriate. Ref. 641has used approximate values for the NE18 momentum cutoff
to evaluate tis effect, and finds the resulting transparency to be midway between the
two extreme cases. KYS93 further points out that the dependence of the cross-section
on the relative momentum should be taken into account, and evaluate a Fermi smearing
correction.
his is found to be small (5%) in the case of 'He, but can be expected to
increase for eavier nuclei and for the restrictive p. acceptance of NE18.
The effect of short range correlations is also variously included in the theories. Benhar
et al. (Ben92) 68] suggest that the short-range NN repulsion causes a "correlation hole"
around the sruck nucleon. This is evaluated using an expression of the form
T
z
Id 2bdz'p(r') e,
f "' dz11aG(z11
-zl)p(rl)
(5.8)
Here the initial position of the struck proton is described by an impact parameter b and
position z' aong its momentum vector; the integral over z" describes its path through
the nucleus. The correlation function G(z - z) is negative and effectively clears a space
around the travelling nucleon, reducing its interaction with the medium. The magnitude
of this effect is found to be quite large: in Figure 517 the carbon transparency is shown
to increase y 0.1. However, very different effects of correlations have been found by
other workers. Nikolaev et al. (Nik93) 671point out that the so-called "spectator effect"
largely cancels the "hole effect". Simply, the spectator effect accounts for the fact that
183
a "hole" in the nucleon density at one location must be balanced by a bunching of
nucleons elsewhere. The effect is described by replacing the negative function G(z" - z)
in Equation 5.8 with the expression (I + 1,,,,op(r")) in terms of the (positive) correlation
length
Figure
A net increase of 2
18 shows the magnitude of the two effects found by this calculation.
is determined, a result which is consistent with the independent
calculation of KYS93. However, the magnitude of the individual effects is less than 5%,
and so inconsistent with the - 20% hole effect of Ben92. The expressions found for the
hole effect in Ben92 and Nik93 are identical, and so the discrepancy must come from
the input to the calculation. Possible explanations are the nuclear density distributions
employed, and the use of the total pN cross-section (Ben92) rather than the reaction
cross-section (Nik93 and KYS93).
The inclusion of Pauli blocking and effective mass corrections was shown (Figure 1-10)
to contribute substantially to the transmissionat Q < 04
eV/c)'. These corrections
are expected to be less important at NE18 energies 72]. One evaluation of Pauli blocking
in the (GeV/c)' range is found in Ben92 for iron. As shown in Figure 517 the iron
transparency is found to increase by 005 (- 12%) a non-negligible effect.
5.4.2 Comparisonwith ColourlYansparency Predictions
The previous section demonstrated that there is no firm theoretical baseline for the nuclear transparency expected in the absence of PCD
effects. A large range in magnitude
is covered by the Glauber calculations, leaving us with only the Q-dependence of the
data to provide a clear signature of new effects. However, the few Glauber estimates
which incorporate the Q dependence of the free pN cross-section demonstrate that a
substantial rise of T with momentum transfer can be expected in the absence of colour
transparency as well (- 10% for 12C over the
184
Q2
= 3 to 7 range). One can thus expect
F_
:1%
Q
C
0I..
aCL
In
C
0
.t:
'a
F0
1%
2.
*+
V
o2((GeV/c)?]
a
IV
Figure 517: Transparency calculations of Benhar et al. (Ben92) 681. GA and CA
denote the Glauber and correlated Glauber approximations respectively, CT indicates
the inclusion of colour transparency effects. For iron, two additional versions of the CGA
+ CT prediction are included: the dash-double-dotted line approximates the correlation
function G(r') from Equation 5.8 with the pair correlation function; the dot-double-dashed
line includes Pauli blocking.
185
I
CZ
_
I
10 &A
%_--I
c0
5-
0
0L_
0-
0
U
..........
-5-
-
0-
-
1.c -
hole effect
--------------------------
0
net en ect
.....................................................
.......................
.................
spectato'i "
I
50
____
100 150 200
atomic number
I
I
I
250
Figure 518: The correlation "hole" and "spectator" effects, as evaluated by Nikolaev
et al. (Nik93) 671.
that the interpretation of the data in terms of colour transparency will be ambiguous at
best.
The NE18 transparency measurements are compared in Figures 519 through 521
with a representative selection of calculations including colour transparency. The estimated size of these effects covers a very large range, and it is at once clear that only the
most extreme prescriptions can be excluded by the data. Again we focus on the carbon
data, for simplicity. The most drastic rise with Q' is found in the naive parton model of
Farrar et al.
ar88) 281 which was discussed in Section 1.5; the indicated > 40% rise
over the NE18 kinematic range is clearly not observed. The quantum diffusion model
from the same paper, however, predicts a more gradual rise of - 20%. The Benhar et al.
(Ben92) 681CGA+CT calculation uses the same quantum diffusion prescription for the
evolution of the struck hadron as it leaves the nucleus, but incorporates the correlation
186
hole effect described in the previous section. A 25% rise is indicated.
Note that the
Ben92 GA+CT calculation shown in Figure 517 indicates a larger rise of 30%. This
can be understood by recalling the large size of the correlation hole effect found by these
authors, and onsidering its effect on the colour transparency picture. The hadron of
reduced size and interactions postulated by CT is now surrounded by a correlation hole,
and so the early stages of its expansion go "unnoticed" by the nucleus. The difference
between FSI with and without CT is thus reduced by the hole effect. Noteworthy also
is the discrepancy between the 20% rise of the quantum diffusion model of Far88 and
the 30% rise of the GA+CT curve, although these two calculations use the same CT
formalism. The results are clearly dependent on the details of the calculation, particularly the input parameters such as the nuclear density distribution and in-medium pN
cross-section. The other curves depicted in Figure 519 are those of Jennings and Miller
(JM92) 69] and Frankfurt, Strikman, and Zhalov (FSZ93) 70]. Both are based on an
expansion of the struck nucleon wavefunction in terms of excited hadronic basis states.
JN192 is an extension of an earlier calculation 711,evaluated using only one such excited
state. In JM92 an expansion over a spectrum of excited nucleons N* is considered; the
density of these immediate states is parametrized by a function g(M2) where M is the
mass of the N*. Various choices for the form of this function are considered, based on
measurements of the N* spectrum from inelastic scattering and diffractive dissociation
experiments. Two extreme CT predictions are provided by a power law form g(M2)
(indicated by - M.,,O in Figure 519), and a distribution with a sharp mass cutoff. The
former version determines a relatively weak CT effect
10% rise of NE18
Q2
range).
The latter suggests a larger rise, and gives similar results to the calculation of FSZ93
- The FSZ93 calculation is distinguished by its use of an energy-dependent
above Q2 of 3.
free pN cross-section, and so is the only one which exhibits a rise at low
187
Q2.
Deciding which prescriptions can be excluded by the data is an exercise of limited use,
given the uncertainty in the "standard" FSI prescriptions on which the CT calculations
are based. By shifting the calculations up or down by the 10% indicated by Figure 514,
one can bring almost any CT curve into reasonable agreement with the data. It is clear
that higher precision experiments in the Q >
(GeV/c)' range are required to settle
these questions and provide a sharper baseline for quasielastic scattering in this range.
Only with more a exact understanding of "traditional" nuclear physics effects can one
hope to look effectively for the signature of a new process.
5.4.3
A Dependence
As pointed out in Section 1.5, one of the assumptions involved in the colour transparency
effect is that the PLC produced at the scattering vertex expands to the size of a dressed
proton over a distance at least as large as the nuclear radius. CT effects may therefore
show up in the A dependence of the transparency
as well as the Q dependence.
Once
again, the critical question is what to compare to: are CT effects required to explain the
A dependence of the data?
To answer this question, we invoke a simple model of the transmission probability.
Classically, if a particle travels a distance x through a medium of containing p scatterers
per volume and interacts with these scatterers via a cross-section O',the probability that
the particle escapes without any interaction is
P,.c., = "'.
The combination
Pa
(5.9)
is the mean free path, A. If one then considers the knockout of a
proton from a spherical nucleus of radius R and uniform nucleon density po, assuming
that the scattering probe samples the entire nuclear volume, one obtains the following
188
1.0
0.8
0.6
0
I--I
0.4
0.2
0.0
Q2
(GeV/ C)2
Figure 519: The measured transparency for 12C, compared with several calculations
including colour transparency. The references denoted by the legend are as follows: (i)
Far8 = Farrar et al. of ref. 28]; calculations in the naive parton and quantum diffusion
models are sown; (ii) Ben9 = Benhar et al. of ref. 68]; CGA+CT indicates inclusion
of the correlation hole effect (iii) JM92 = Jennings and Miller of ref. 69]; two forms
of the density-of-states function gM,,,) (described in the text) are used (iv) FSZ93 =
Frankfurt, Strikman, and Zhalov of ref. 70].
189
I
I
I
I
I
I
I
I
I
0. -
I
I
--------
I
I
I
I
I
I
I
I
T
I
I
Ben92 CGA+CT
FSZ
0.6 0
C-4
0
1-1
-
0.4 -
L
--
0.2 0.0
I
0
IIIIII1
2
I
I
Q2
I
4
(GeV/c
I
I
I
1
6
I
8
)2
Figure 520: The measured transparency for "Fe, compared with several calculations
including colour transparency. The notation in the legend is the same as that of Figure
5-19.
190
I
Q
.U
I
I
I
I
I
t ---- .
1-1
I
- - - - -
JM92
--------
Ben92 CGA+CT
JM92 g(M.2 _ M.-,$
-
I
FSZ93
.6 -
CN
g(M
CUt
I
4-
I
IIIIIIII
.0
0
IIIIIIII
4
2
Q2
(GeV/c
III
6
8
2
Figure 521: The measured transparency for 197 Au, compared with several calculations
including colour transparency. The notation in the legend is the same as that of Figure
5-19.
191
expression for, the transmission probability:
T= 3
8'3
with F
R.
A
(2k - 1) + e-"4(2f + 1)
Given the rms charge radius R,,,(A)
(5-10)
from previous data 73], the effective
radius for a sphere of the same charge but uniform density is R
number density of nucleons is simply po =
'A
4xR3
[5 5R,,,.
3
The uniform
The values used for the nuclei studied
by NE18 are given in table 5.5. The remaining unknown is the effective pN cross-section
in the nucleus o,f f, and can be fit to the data at each
Q2
using T(A). Only A > 2
are used in the fit as this model is clearly inaccurate for the single spectator nucleon
of
2H. The fits to the data as a function of A are shown in Figure 522; the resulting
values for
ff are presented in table 56, along with the X2 per degree of freedom for
the fit. The X values are all less than 1, indicating an excellent fit. One finds, then,
that the classical attenuation model provides a reasonable parametrization of the data.
Note that in the limit of complete colour transparency, o,ff --+ 0. Thus, the parameter
aq f provides another measure of the nuclear transparency as a function of Q2, one that
conveniently takes into account the data taken at all A. The fit values of 0,ff are also
compared to the free pN cross-section in table 56. J
obtained by averaging the
free pp and pn cross-sections of ref. [5], weighted respectively with the number of protons
(Z-1) and neutrons (N) in the spectator system. One sees that the 0',ff is about 60%
lower thandfr,,
at all
Q2.
Such a reduction could be expected from quantum effects
not accounted for in the classical calculation, as well as nuclear effects such as Pauli
blocking, short-range correlations, etc. We note that ref. 74] uses a similar classical
model to parametrize nuclear transmission data at
Q2
= 034 (GeV/c)', and also finds
that the effective cross-section is substantially lower than
192
They parametrize o,,ff
Table 5.5: Input values used in classical model of transmission.
Target
12C
Fe
197 Au
56
fm
R (equiv. uniform radius)
fm
2.50
3.76
5.33
3.23
4.85
6.88
Rrm's
Table 56: Fit values of aq f using classical model of Equation 5.10. The isospin-average
of the free pN cross-section is included for comparison.
Oleff
mb
x 2 /deg. freedom
1
3
5
18 ± 2
27 ± 3
28 ± 3
0.7
0.4
0.6
34.9
43.0
41.8
0.56 ± 06
0.62 ± 07
0.67 ± 07
7
23 ± 3
0.1
41.4
0.5
Q2
(GeV/c)2
df ree
a-ff
dfee
mb
±
10
in terms of an assumed density dependence:
17eff (p, r =
I
O'free _
Kp(r)'
(5.11)
Here, K rather than aq f itself) is the free parameter. This form is based on the work
of ref. 75], which evaluates some of the nuclear effects mentioned above and arrives at a
substantial dnsity dependent reduction of in-medium cross secion.
193
-
L___]
.
I I I
I
I
I
I I I I'll
I
.
.
.
I -
I
I id
02 = 1 (GeV/C)2
_ I I I 1 11.1
I
I I . . .I. I
= 5 (GeV/c)'
Q
1
I
I_'I
0.1
n
.
I
I .
I I
I
I I
. .
r-. 0. 1
10,-
lo-
loj
I,,
10"
,
. I .-
II
I
I I I
10j
A
.
I . . I II
.
= 3
(2
I
104
A
I
.
.
.
.
I .
. . . .
GeV/C)2
1
, ,
-
02
-
I . I . 1.111
11
, .
= 6.8 (GeV/C)2
-
1-1
F__
nV.
I
11
I II
I . I . . 1.112
10
lou
A
I
-
-
-1
0.
lo"
1
I
lo"
I . . I .11-
lo'
.
I II -
lo"
A
Figure 522: NE18 measured transparencies for A> 2 targets. The solid lines indicate a
one-parameter fit (at each Q2) to the classical transmission model described in the text.
194
Chapter 6
Discussion of Results
Experiment NE18 has measured the coincidence quasielastic cross-section for (ee'p) scattering from te
nuclei 11, '11,
12 c,
6
Fe, and
197
Au in a Q range of
to 68
(GeV/C)2.
This extends the previous momentum transfer range of such measurements by over an
order of magnitude.
The elastic hydrogen data were found to be well explained by the dipole form for
the proton electric form factor, the parametrization of Gari and Kriimpelmann 41] for
the proton magnetic form factor, and a radiative corrections prescription based on the
work of Mo nd Tsai 58][591. This prescription included radiation by the relativistic
knockout proton, and consisted of a reevaluation of the Mo and Tsai formulae in a
coincidence framework. The measured hydrogen E
distribution provided a precise test
of the radiative cross section, and was found to be in excellent agreement with the
calculation.
Data analysis was performed entirely in a traditional nuclear physics framework,
consisting of the Plane Wave Impulse Approximation (PWIA), the Independent Particle
Shell Model (IPSM) of the nuclear structure, and the deForest off-shell ep cross-section
[3]. These elements were programmed into a Monte Carlo simulation, which folded the
195
PWIA/IPSM cross-section with the experimental acceptances and the radiative crosssection; the data were then compared to this calculation. A momentum distribution
determined using the Bonn potential was provided as input to the calculation for 2 H,
and found to produce excellent agreement with the data. The IPSM spectral functions
used for the heavier nuclei were based on fits to measurements made at lower momentum
transfers. Certain parameters were also varied to provide a better fit to the NE18 Q2
= I
data, and the effect of this variation on the results was included in the experimental
systematics as a model dependent uncertainty.
The data were found to be in good
agreement with the calculation. The momentum distributions for the individual shells
of the
12C
Up to
Q2
nucleus were separated, providing direct evidence of the nuclear shell model
of 7
(GeV/C)2
. Furthemore, no evidence was found for additional or missing
strength in the nuclear energy and momentum distributions. This is in contrast with
measurements made at
Q2
- 02 (GeV/c)': DWIA/CDWIA calculations of initial and
final state distortions must be made to explain the data 6 and evidence of scattering
from multinucleon currents has been observed at high E[10][12].
The primary quantitative result of the experiment was the measurement of the nuclear
transparency T over a sizable range in
Q2
and A. A classical model of the transmission
probability was found to provide a good parametrization of the A-dependence of the
results. This parametrization yielded effective values for the in-medium pN cross-section
about 60% as large as the free cross-section. The
Q2
dependence of T showed little evi-
dence of colour transparency (CT) effects, expected to show up as a rise in transmission
relative to the "standard" expectation. The data are consistent with a
transparency above
Q2
Q2
independent
= 3 GeV/C)2, and extreme models of CT predicting rises of 30 -
40% over the NE18 kinematic range can be ruled out. However, the value of T expected
in a conventional nuclear physics picture is not well established; values differing by -
196
20% have been suggested for the carbon nucleus, for example. Furthermore, the Q dependence of T in the absence of CT is not flat over the NE18 Q range due to the energy
dependence of the pN cross-section. This
Glauber calculations at
Q
>
I
Q2
dependence is not well known, for most
have been carried out using a constant value for
pN,
equal to the asymptotic value reached in the high energy limit. These uncertainties cause
the colour transparency signature to be obscured, and make further interpretation of the
data in terms of CT effects ambiguous. The NE18 results have instead provided a baseline with which to constrain models of quasielastic scattering above
Q
=
I
GeV/C)2.
Whether or not CT effects are involved, the NE18 results cleary demonstrate that PQCD
effects are not, large in quasielastic scattering up to momentum transfers of 7 (GeV/ C)2.
The surprise is then that the constituent counting rules of Section 14.2, also a result
of PQCD, have proved so successful at describing form factor data and other exclusive
measurements at similar kinematics. The arguments of ref. 241, demonstrating alternative non-perturbative means of accounting for the magnitude of the data, present a
possible explanation and suggest that the momentum transfer threshold at which PQCD
can validly b applied is much higher.
The most striking result of this experiment has been the demonstration that our understanding of quasielastic scattering in a conventional nuclear physics picture extends
over two orders of magnitude in momentum transfer. Further analysis of the data in this
picture is possible. For example, the radiative Monte Carlo calculation provides us with
the means to deradiate the extracted spectral functions in a largely model-independent
way. The transparency for individual shells in
12C
could then be determined, for ex-
ample, providing a measure of the importance of density dependent effects and Fermi
smearing in FSI calculations. Also, analysis of the asymmetry of the deuterium momentum distributions is underway 381. This asymmetry is sensitive to the structure
197
function WI of Equation 19. Several calculations of this structure function exist, taking various approaches to the difficult question of relativistic corrections. The future of
these measurements clearly lies in higher precision and higher energy experiments. The
quasielastic analysis performed for NE18 provides a good basis for the analysis of coincidence experiments at CEBAF, for example, where high intensity beams will be enable
high statistics data taking in a similar Q range. Our results also demonstrate that to
observe sizable colour transparency effects one must proceed to higher momentum transfers; for the foreseeable future, an investigation of this nature can be made only at SLAC.
NE18 establishes the necessary baseline to which such higher energy measurements can
be compared.
198
Bibliography
[1] P. E. Bosted et al., Phys. Rev. Lett. 68, 3841 1992); A. Lung et al., Phys. Rev.
Lett. 70, 718 1993).
[21 R. G. Arnold et al., Phys. Rev. Lett., 57, 174 1986).
[3] T. de Forest Jr., Nucl. Phys. A132, 305 1969).
[4] S. Frullani and J. Mougey, Advances in Nucl. Phys. 14, Plenum Press, New York
1984.
[5] "Particle Properties Data Booklet", ed. by M. Aguilar-Benitez et al., North-Holland
(April 1992); Particle Data Group, Phys. Lett.
[6] L. Lapiks,
Nucl. Phys. A553, 297c 1993).
[7] P. Barreau
et al., Nucl. Phys
239 1990).
A 402, 515 1983).
[8] J. W. Van Orden and T. W. Donnelly, Ann. Phys. 131, 451 1980); T. W. Donnelly,
J. W. Van Orden, T. deForest, Jr., and W. C. Herman, Phys. Lett. 76B, 393 1978).
[91 J. M. Finn, R. W Lourie, and B. H. Cottman, Phys Rev. C 29, 2230 (1984).
[10] R. W. Lourie et al., Phys. Rev. Lett. 56, 2364 1986).
[11] J. Mougey et al., Nucl. Phys. A262, 461 1976).
199
[12] P. E. Ulmer et al., Phys. Rev. Lett. 59, 2259 1987).
[13] R. J Glauber, "Lectures in theoretical physics", vol. 1, ed. W. E. Brittin and D. G.
Dunhan, Interscience 1959).
[14] G. Garino et al., Phys. Rev. C 45, 780 1992).
[15] G. R Farrar, Phys. Rev. Lett, 53, 28 1984).
[16] G. P Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 1980).
[17] S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 1973); S. J. Brodsky
and G. R. Farrar, Phys. Rev. D 11, 1309 1975).
[18] P. V. Landshoff, Phys. Rev. D 10, 1024 1974).
[191 S. J. Brodsky and G. P. Lepage, Phys. Rev. D 24 2848
1981).
[20] S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett. 60, 1924 1988).
[21] C. W. Akerlof et al., Phys. Rev. 159, 1138 1967).
[22] S. J Brodsky and G. F. de Teramond, Phys. Rev. Lett. 60, 1924 (1988).
[23] V. L. Chernyak, A. A. Ogloblin, and 1. R. Zhitnitsky,
Novosibirsk preprints INP
87-135,136; 1. D. King and C. T. Sachrajda, Nucl. Phys. B 279, 785
and N. Stefanis, Phys. Lett.
987) M Gari
187, 401 (1987).
[24] N. Isgur and C. H. Llewellyn Smith, Phys. Rev. Lett. 52, 1080 1984); N. Isgur and
C. H. Llewellyn Smith, Nuc. Phys. B317, 526 (1989).
[25] L. Frankfurt, G. A. Miller, M. Strikman, Comments Nucl. Part. Phys., 21, 1 (1992).
[26] D. Perkins, Phil. Mag. 46, 1146 1955).
200
[27] B. Povh, J. Hfifner, Phys. Rev. Lett. 58, 1612 1987).
[28] G. Farrar, H. Liu, L. L. Frankfurt,
and M. 1. Strikman,
Phys. Rev. Lett. 61, 686
(1988).
[29] L. L. Frankfurt
and M. I. Strikman, Phys. Rep. 160, 235 1988).
[301 A. S. Carroll et al., Phys. Rev. Lett. 61, 1698 1988).
[31] J. P. Ralston and B. Pire, Phys. Rev. Lett. 61, 1823 1988).
[32] D. H. Potterveld,
Ph. D. Thesis, Caltech.
[33] Y. S. Tsai, Rev. Mod. Phys. 46, 815 1974).
[34] R. Anderson et al., Nucl. Instr. Meth. 66, 328 1968).
[351 P. N. Kirk et al., Phys. Rev. D8, 63 1973).
[361 W. R. Leo, "Techniques for Nuclear and Particle Physics Experiments", SpringerVerlag
1992).
[37] K. L. Brown et al., SLAC Report No. 91, Rev. 2 1977.
[38] H. J. Bulten, private communication.
[39] J. T. O'Brien, H. Crannel, et al., Phys. Rev. C 9 1418 1974).
[40] T. C. O'Neill, Ph. D. thesis, Caltech 1994).
[41] M. Gari and W. Kriimpelmann, Z. Phys. A322, 689 1985).
[42] C. Giusti and F. D. Pacati, Nucl. Phys
Pacati, Nucl. Phys A 485, 461 1988).
201
A 473, 717 1987); C. Giusti and F D.
[43] K. Holinde and R. Machleidt, Nucl. Phys A 256, 479 1976).
[44] F. C. Perey and B. Buck, Nucl. Phys. 32, 353 1962); H. Fiedelday, Nucl. Phys. 77,
149 1966); H. P. Blok and J. H. Heisenberg, from "Computational Nuclear Physics,
vol. 1", ed. by W. E. Brittin et al., Interscience
1959).
[45] C. E. Brown and M. Rho, Nucl. Phys A 372, 397 1981).
[46] E. N. M. Quint, Ph. D. Thesis, U. Amsterdam
1988).
[47] J. Mougey et al.. Nucl. Phys A 262, 461 1976).
[48] J. W. Negele, Phys. Rev. C1, 1260 1970); J. W. Negele and D. Vautherin, Phys.
Rev. C5, 1472 1972).
[49] L. Lapikas, private communication.
[50] G. Jacob and Th. A. J. Maris, Rev. Mod. Phys. 45 6 1973).
[51] J. W. Van Orden, W. Truex, and M. K. Banerjee, Phys. Rev. C21, 2628 (1980).
[52] X. Ji, private communication.
[53] 1. Sick, private communication.
[54] S. Liuti, private communication.
[55] W. Atwood, Ph. D. thesis, SLAC-Report
[56] G. G. Petratos,
no. 185 1975).
private communication.
[571 D. Wasson et al., to be published.
[58] L. M. Mo and Y. S. Tsai, Rev. Mod. Phys. 41, 205 1969).
202
[59] Y. S. Tsai, Phys. Rev. 122, 1898 1961).
[601 C. de Calan, H. Navelet, and J. Picard, Nucl. Phys. B348, 47 1991).
[61] D. R Yennie, S. Frautschi, and H. Suura, Ann. Phys. (N.Y.) 13, 379 (1961).
[62] N. C. R. Makins et al., Phys. Rev. Lett. 72, 1986 1994).
[63] T. C. O'Neill et al., submitted to Phys. Rev. Lett.
[64] N. N. Nikolaev, private communication.
[65] D. Wasson, private communication.
[66] A. Kohama, K. Yazaki, and R. Seki, Nucl. Phys A 551, 687 1993).
[67] N. N. Nikolaev et al., Phys. Lett.
317, 281 1993).
[68] 0. Benhar et al., Phys. Rev. Lett. 69, 881 1992).
[69] B. K. Jennings and G. A. Miller, Phys. Rev. Lett. 69, 3619
1992).
[70] L. Frankfurt, M. Strikman, and M. Zhalov, to be published.
[71] B. K. Jennings and G. A. Miller, Phys. Rev. D 44, 692 1991).
[72] S. Pieper, private communication.
[73] "Structure of the Nucleus", M A Preston and R. K. Bhaduri, Addison-Wesley
(1975).
[74] D. F. Geesaman
et al., Phys. Rev. Lett. 63, 734 1989).
[751 V. R. Pandharipande and Steven C. Pieper, Phys. Rev. C 45, 791 1992).
203