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Probing the Proton’s Quark Dynamics in
Semi-inclusive Pion Electroproduction
Wesley P. Gohn, Ph.D.
University of Connecticut, 2012
Measurements of pion electro-production in semi-inclusive deep inelastic scatter-
ing (SIDIS) have been performed. Data were taken with the CEBAF Large Ac-
ceptance Spectrometer (CLAS) at Jefferson Lab using a 5.498 GeV longitudinally
polarized electron beam and an unpolarized liquid hydrogen target during the
E1-f run period in 2003. All three pion channels (π+, π0 and π−) were measured
simultaneously over a large range of kinematics (Q2 ≈ 1-4 GeV 2 and x ≈ 0.1-0.5).
Single-spin azimuthal asymmetries from all three pion channels were measured as
functions of x, z, PT , and Q2, from which AsinφLU were extracted. This new high
statistical data could provide access to transverse-momentum dependent parton
distribution functions (TMD’s), which are thought to be important in understand-
ing of the physics underlying the spin structure of the nucleon.
Probing the Proton’s Quark Dynamics in
Semi-inclusive Pion Electroproduction
Wesley P. Gohn
B.S., Indiana University, Bloomington, IN, 2004
M.S., University of Connecticut, 2007
A Dissertation
Submitted in Partial Fullfilment of the
Requirements for the Degree of
Doctor of Philosophy
at the
University of Connecticut
2012
Copyright by
Wesley P. Gohn
2012
APPROVAL PAGE
Doctor of Philosophy Dissertation
Probing the Proton’s Quark Dynamics in
Semi-inclusive Pion Electroproduction
Presented by
Wesley P. Gohn,
Major Advisor
Kyungseon Joo
Associate Advisor
Harut Avakian
Associate Advisor
Peter Schweitzer
University of Connecticut
2012
ii
This dissertation is dedicated to my wife, Lindsey,
for your infinite patience.
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ACKNOWLEDGEMENTS
I would not have achieved the accomplishment that this document represents
without the support of many other people in my life, whom I would like to thank
here.
To begin I would like to thank those who directly contributed to this work,
first and foremost my thesis adviser, Prof. Kyungseon Joo. Kyungseon is the
hardest working person I have ever met, and I hope that his work ethic will stay
with me through the various phases of my career. He told me once that the most
important thing a student learns during their PhD is how to solve problems, and
he has helped me to significantly improve my abilities in that area. Kyungseon
makes it a priority to support his students at the highest level possible. Without
his support over the past six years, I would not be at this point in my life, and I
owe him a great debt of gratitude.
I would also like to thank my two associate advisers who have also con-
tributed strongly to my work. Dr. Harut Avakian has helped me a great deal,
and much of what I have learned about data analysis and simulation came from
him. He would always make time to discuss my analysis, and he provided many
valuable insights. I have met few physicists as competent in both experiment and
theory, and I hope to one day achieve the same. I also owe a great deal of thanks
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to Professor Peter Schweitzer, who contributed strongly to my understanding of
the theoretical background for my experiment. His door was always open, which
led to many impromptu questions with meaningful answers about TMDs and the
interpretation of my data. He helped me with the theoretical sections of every
conference talk I gave or paper I wrote (including this one), and I would not know
half of what I do about TMDs without his explanations.
I also owe a great deal to Dr. Maurizio Ungaro. He was a postdoc in our
group when I began my research, and he was the one I went to with everyday
questions about my analysis, and he always pointed me on the right track. I
learned a great deal from him. Although he did not officially hold the title, he
was very much like another adviser to me. He has always been a great source of
advice and encouragement, and I do not know if I would have made it through
this process without his help. I hope that one day he has students of his own,
because he would make a fantastic adviser.
I would also like to thank my fellow students in our research group and in
CLAS, who all along have proved to be a valuable source of knowledge, and a
much needed outlet to vent. Thanks to Nikolay Markov, Taisiya Mineeva, Erin
Seder, Nathan Harrison, Puneet Khetarpal, and Sucheta Jawalkar, who have all
done their parts to make this process much easier. Nick in particular, as a senior
student when I was just beginning to learn, helped me a lot by providing me
with a lot of tips on how to effectively use ROOT to analyze our data, and also
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teaching me how to perform momentum corrections. I would really like to thank
Nick, Taya, Sucheta, and especially Puneet for their hospitality during my many
trips to JLab. I was always able to count on a ride from the airport or to the
grocery store. They would help me navigate the JLab bureaucracy while I was
working, and invite me to socialize on the weekends, which always made extended
stays away from home so much easier.
I also owe much gratitude to the entire staff of Hall-B and the CLAS col-
laboration. First and foremost I thank them for the availability of the data, and
for running the experiment. I also have received help and support along the way
from very many people. I am sure I cannot name all of them here, but several
who stand out as providing me with significant help are Volker Burkert, Latifa
Elouadrhiri, Valery Kubarovsky, Paul Stoler, Stepan Stepanyan, Ken Hicks, Dan
Carman, Brian Raue, Peter Bosted, Mher Aghasyan, F.X. Girod, and Keith Grif-
fioen. I have always been particularly impressed with how Volker, as the leader of
Hall-B, is always attentive to the needs of students. I believe that he replied with
comments to every set of slides that I sent out to our working group, and he always
gave meaningful suggestions for improvements. Also, thank you to Chris Keith
and the Jefferson Lab target group for allowing me to work with their group dur-
ing the eg1-dvcs experiment in CLAS. I had an enjoyable experience and learned
a lot in the process.
For further theoretical assistance and the help with the interpretation of my
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results, I thank Barbara Pasquini, Alexei Prokudin, and Marc Schlegel.
This concludes the section of acknowledgements for those who contributed
directly to my thesis work in CLAS, but I also owe much thanks to those in my
life who have helped me get here.
To begin I will thank the faculty and staff in the UConn physics depart-
ment, in particular Cynthia Peterson and Richard Jones. I have spent many
years working as a TA for Dr. Peterson’s astronomy course. Not only did she
help me to learn a deeper appreciation for astronomy, but she has always been
a consistent source of advice about grad school in general, and she has done a
lot to help me along the way. Dr. Jones helped me a great deal when I first
came to UConn, especially in regards to preparing for the preliminary exams. We
had several marathon problem sessions, which mostly involved me trying to solve
problems on his white board and him pointing out everything I was doing wrong.
I am very grateful for all of the time and patience that he put into that process.
The criticism definitely helped, and I know that passing the preliminary exams
would have been a much bigger struggle without his coaching. I would also like
to thank the professors who taught my courses (mostly Juha Javaneinen and Ron
Mallett, who taught six of my first eight courses at UConn). Also Quentin Kessel
for allowing me to spend a semester working in his lab while I was waiting for my
research assistantship to become available, and Winthrop Smith for helping me
resolve a last minute issue to make sure my dissertation proposal was turned in
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on time. Also, for our office staff, who have made every part of this process easier,
thank you to Dawn Rawlinson, Loraine Smurna, Kim Giard, Barbara Styrczula,
and all of the other office workers. Also thanks to Michael Rozman for all of the
computer support.
Thanks also to my classmates and friends I made in the UConn physics
department. First, thanks to Sam Emery and Don Telesca for many great mem-
ories. With them, the many hours studying for prelims in P401 was much more
bearable, and once we separated into different labs, I always enjoyed our weekly
lunches and afternoon coffee breaks. Also, thank you very much to all of the Psi
Stars and Lollygaggers softball players for all the great times (and the legendary
Manchester Men’s C league softball championship of 2008). Thanks to Nolan,
Ting, Don, J.C., Brad, Fu-Chang, and everyone else who played on those teams.
I would also like to thank all of my teachers who helped me reach the
point in my life when I could consider a goal of achieving a Ph.D. in physics.
In particular, thanks to all of my physics professors at Indiana University, most
notably Scott Wissink, from whom I learned a great deal by working in his lab
on the STAR experiment, and who first stimulated my interest in the topic that I
eventually chose to study for my thesis research. I also would like to acknowledge
Renee Fatemi, who was a postdoc in our STAR group. She really taught me a
lot and was a great source of advice. Also, thank you to Alex Dzierba, Adam
Szczepaniak, Ben Brabson, Steve Vigdor, Rick van Kooten, and Mike Snow, all
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of whom I learned very much from during my time at I.U. Thanks to Hendrik
Schatz at Michigan State for giving me the chance to study an exciting nuclear
physics topic during my summer as an REU student at the NSCL.
I cannot speak of my time at IU without mentioning several great friends
from that time who have served to motivate me. I have great memories of sit-
ting around doing quantum mechanics homework over a pitcher at Kilroy’s with
Jonathan Slager, Dave Howell, and many others. Whether it was the many late
night conversations about deep physics topics, or the friendly competition over
who could be the first to figure out the solution to a complicated homework prob-
lem, these guys were always a great source of motivation for me, and they deserve
to be acknowledged here.
Also, I wish I could thank every K-12 teacher who made an impact on me
to get me to this point. I could not go through that entire list, but I would like to
especially thank all the faculty at the Indiana Academy for Science Mathematics
and Humanities. Few, if any, decisions have made as large an impact on my life
as the choice to leave my home high school after my sophomore year and spend
two years as an ”academite”. I would especially like to thank my physics teachers
Don Hey and Dr. Hasan Fakhruddin, who ignited the spark that set the fire for
this entire process. Prior to the academy, the one other teacher who stands out
in my mind above the rest is my 5th grade teacher, Mr. Fisher. He believed in
me, and his encouragement at that time gave me the ambition to reach for a level
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of achievement in my life that I may not have otherwise thought to be possible.
Most importantly, I must thank all of my family. My parents have always
encouraged me to follow my dreams, and the achievement of a Ph.D. in physics
has been my goal for as long as I can remember. At the point when I was first
trying to grasp at an understanding of the universe, I know my dad would read
science magazines so we could have conversations about the topics I was curious
about. I specifically remember going on walks around the neighborhood, probably
with our dog, and asking him complex questions about black holes, quasars, and
parallel universes. The first time I heard of wave-particle duality was actually from
my mom. She got the details wrong, but still triggered my interest, which lead
to me reading about quantum mechanics, and eventually triggered my passion
for particle physics. My sister, Amelia, has always been there for me. She is
one of the most brilliant mathematicians and artists that I have ever known, and
has constantly served as an inspiration to her older brother. I would also like to
acknowledge both of my grandfathers, who each in their own way have served as
role models in my life. My maternal grandfather, Dr. Wesley Kissel, I always saw
as the consummate intellectual, which from a young age I desired to emulate. This
probably was an initial motivation of my desire to reach for the highest degree
achievable. I also owe very much of myself to my Grandpa Goon, who taught me
the importance of living one’s life based on how you influence those around you.
He is the kindest man I have ever known, and he would give his last dollar to help
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someone he thought needed it more. His example is one that I hope to follow with
every action in my life.
Finally, I owe the biggest thanks of all to my wife, Lindsey. As much of a
challenge as this endeavor has been for me, I know it has been harder for her. I
can never express how much I appreciate her encouragement, support, and most of
all patience throughout this entire process. She had to deal with uncountable late
nights in the lab and research trips causing me to be away for sometimes weeks
at a time. While I was away working on an experiment or attending a conference,
she was taking care of everything at home so I knew I never had anything to worry
about when I got back. She always made things so much easier for me, while I
know she was taking on extra stress for herself. I cannot thank her enough for
her patience and tolerance of the entire process. Her love and support has kept
me going like nothing else could, and she is truly the one person without whom,
this entire achievement would not have been possible. Thank you, Lindsey, for
everything.
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TABLE OF CONTENTS
1. Physics Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 TMDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Boer-Mulders Function . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Semi-inclusive DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Moments of SIDIS Cross-section . . . . . . . . . . . . . . . . . . . 10
1.3 Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2. Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 CEBAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 CLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Drift Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Time-of-Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Cerenkov Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.4 Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . . 27
2.3 E1-f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3. Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 MU File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Determination of Good Run List . . . . . . . . . . . . . . . . . . . . 31
3.3 Helicity Determination . . . . . . . . . . . . . . . . . . . . . . . . . . 31
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3.4 Electron Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Number of Photoelectrons in the CC . . . . . . . . . . . . . . . . . 36
3.4.2 CC θ Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3 CC φ Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.4 CC Time Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.5 CC Fiducial Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.6 EC Threshold Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.7 EC Sampling Fraction Cut . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.8 EC Ein vs. Eout Cut . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.9 EC Geometric Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.10 tEC − tSC Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.11 Vertex Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Hadron Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5.1 π+ Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5.2 π− Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.3 π0 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6 DC Fiducial Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7 Kinematic Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8 Kinematic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.8.1 Electron Momentum Corrections . . . . . . . . . . . . . . . . . . . 75
3.8.2 Hadronic Energy Loss Corrections . . . . . . . . . . . . . . . . . . 87
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3.8.3 Photon Energy Correction . . . . . . . . . . . . . . . . . . . . . . . 87
4. Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1 Data Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1.1 Event Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.2 GSim Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.3 GPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Acceptance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5. Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 105
5.0.1 Systematic Uncertainty from Variation of Particle ID cuts . . . . . 105
5.0.2 Pion Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.0.3 Systematic Uncertainty from Variation of Kinematic Cuts . . . . . 112
5.0.4 Systematic Uncertainty from Variation of Fitting Function . . . . . 115
5.0.5 Systematic Uncertainty from Beam Polarization . . . . . . . . . . . 115
5.0.6 Random Helicity Study . . . . . . . . . . . . . . . . . . . . . . . . 119
5.0.7 Comparison to Simulation . . . . . . . . . . . . . . . . . . . . . . . 122
5.0.8 Acceptance Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.0.9 Beam-charge Asymmetry . . . . . . . . . . . . . . . . . . . . . . . 124
5.0.10 Split Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6. Physics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1 Beam-spin Asymmetries and sinφ Moment . . . . . . . . . . . . . . . 128
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6.2 Comparison to other data . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2.1 Comparison to E1-6 for π+ . . . . . . . . . . . . . . . . . . . . . . 137
6.2.2 Comparison to e1-dvcs for π0 . . . . . . . . . . . . . . . . . . . . . 137
6.2.3 Comparison to HERMES for π+, π−, and π0 . . . . . . . . . . . . . 144
6.3 cos 2φ and cosφ Moments . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3.1 Comparison to previous CLAS results . . . . . . . . . . . . . . . . 151
7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Bibliography 156
A. Data Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B. Good Run List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
C. Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
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LIST OF FIGURES
1.1 The GTMD cube. Taking the forward limit in hadron momentum in
a GTMD provides a corresponding TMD, and integrating a TMD
over∫d2k⊥ gives a PDF. . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Diagram of SIDIS process. The incoming lepton with 4-momentum l
scatters off of a proton with 4-momentum P. In the final state the
scattered electron l′ is detected as well as one produced hadron. The
angle φ is the angle between the planes described by the scattering
lepton and that described by the produced hadron. . . . . . . . . 9
1.3 SIDIS scattering process in the QCD factorization approach in the
Bjorken limit; given by the convolution of the parton distribution
function and fragmentation function. . . . . . . . . . . . . . . . . 9
1.4 Leading order diagram for the Boer-Mulders function in the one-gluon
exchange mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Boer-Mulders function as calculated from the light-cone quark model
for up quarks (left) and down quarks (right). . . . . . . . . . . . . 18
1.6 Spin density for transversely polarized quarks in an unpolarized proton
resulting from the Boer-Mulders function in the light cone quark
model. The left panel shows the distribution for up quarks and the
right panel shows the distribution for down quarks. . . . . . . . . 19
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2.1 The continuous electron beam accelerator facility (CEBAF). CLAS is
located in Hall-B. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Vertical slice of the CEBAF Large Acceptance Spectrometer. . . . . . 23
2.3 Diagram of the path of one electron through the CC along with the
light collection system. . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 The optical mirrors for one sector of the CC. . . . . . . . . . . . . . . 27
2.5 One stack of the EC, as shown divided into U, V, and W planes. . . . 28
2.6 E1-f kinematic coverage in relevant variables. AsinφLU is binned in z, x,
PT and Q2. Each column shows a different pion channel . . . . . . 29
3.1 Number of identified particles vs run number normalized by charge.
Most bad runs are cut due to the number of electrons, but one
additional run is cut due to a low number of π+ and π− . . . . . . 32
3.2 Number of electrons normalized by charge vs run number in each sec-
tor. The previous cuts shown for runs with low event rates have
already been applied. . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 The sinφ moment vs. run number for e1f. The transitions in helicity
definition are given by the red lines. The upper plot shows the
full range of run numbers, and the middle plot concentrates on
a central range in which there are numerous flips of the helicity
definition. The bottom plot shows AsinφLU after the helicity helicity
flips correction is applied. . . . . . . . . . . . . . . . . . . . . . . 35
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3.4 Number of photoelectrons in the CC. The solid (black) histogram
shows events passing all electron identification cuts including CC
matching. The dashed (blue) histogram shows events passing all
other electron id cuts, but without CC matching. . . . . . . . . . 37
3.5 θCC vs CC segment in each sector. The black curves denote the cut at
±3σ about the gaussian fits. The plots show all electron candidates
in the CC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 CC φ matching: returns 0 if both PMTs fire, ±1 if track and PMT are
on the same side, and ±2 if there is a mismatch between the track
and PMT. Candidates are automatically kept if φCC < 4o. . . . . 41
3.7 CC time matching for all electron candidates. The 2-dimensional his-
togram shows ∆t = tCC−(tSC−s/c) vs CC segment for each sector.
The crosses denote 3σ cuts on the lower side of ∆t. . . . . . . . . 43
3.8 θCC vs φCC for all electron candidates. . . . . . . . . . . . . . . . . . 44
3.9 Sampling fraction in calorimeter after all other electron identification
plots are applied. Each panel is a different sector of CLAS. The cut
is made by fitting slices in y with a Gaussian and then fitting the
means and widths of each gaussian with polynomials to determine
fit functions as a function of p, µ(p) + 3σ(p)/− 3.5σ(p). . . . . . . 46
xviii
3.10 EC Einner vs Eouter for electron candidates passing the other EC cuts:
A cut is made to keep only candidates with Einner > 55MeV to cut
minimum ionizing particles. . . . . . . . . . . . . . . . . . . . . . 48
3.11 Physical location of hits on the calorimeter. The colored regions denote
candidates that were kept and the black area shows negative tracks
that were eliminated by this cut. . . . . . . . . . . . . . . . . . . 50
3.12 Cut on ∆t to ensure agreement between the time recorded by the EC
and TOF detectors. Each panel represents a different sector of
CLAS. For the histograms shown all other electron id cuts have
already been applied. . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.13 ∆t vs EC position for CLAS Sector 4. Each distribution is fit with a
Gaussian and the widths are printed in red next to each. . . . . . 53
3.14 Cut on interaction vertex for all electron candidates. Each panel dis-
plays a different sector of CLAS. . . . . . . . . . . . . . . . . . . . 54
3.15 Vertex position in each sector for electrons, π+, and π0. . . . . . . . . 55
3.16 ∆t vs p for π+ candidates. ∆t is fit with a gaussian in each momentum
bin. A cut is made around 3σ of the mean as illustrated by the red
lines. Each panel represents a different sector. . . . . . . . . . . . 57
xix
3.17 β vs. momentum for π+ candidates. The colored 2-dimensional his-
tograms show β vs p for each sector before the ∆t cut, which is
overlayed with black 2-dimensional histograms showing β vs p for
each sector after the ∆t cut. . . . . . . . . . . . . . . . . . . . . . 58
3.18 ∆t vs p for π− candidates. ∆t is projected onto the y-axis and fit with
a gaussian. A cut is made around 3σ of the mean as illustrated by
the red lines. Each panel represents a different sector. . . . . . . . 60
3.19 β vs p for π− candidates in sector 1. The top plot shows β vs momen-
tum before the ∆t cut in each of the six sectors. The second plot
shows β vs p after the ∆t cut. And the bottom plot shows β vs p
after the ∆t, EC Einner and number of photoelectron cuts. . . . . 61
3.20 The first six plots illustrate the π− identification cuts on Einner in
the CLAS electromagnetic calorimeter. The second set of six plots
show the π− identification cuts on the number of photoelectrons in
the CLAS Cerenkov counter. . . . . . . . . . . . . . . . . . . . . . 62
3.21 Photon energy vs. invariant mass. A cut is made on Eγ > 0.15 GeV. To
construct this plot the two photons are distinguished by E1 > E2. 65
3.22 θeγ vs invariant mass. A cut is made on θeγ > 12o. . . . . . . . . . . . 67
3.23 XEC vs YEC for all neutral tracks. The part of the plot in color in-
dicates tracks that are kept, while those in black are cut out by
implementing cuts individually in each U, V, and W plane of the EC 68
xx
3.24 Identification of γ’s by fitting the β distribution of neutral tracks in
the EC with a gaussian in each of ten different momentum bins.
Cuts are indicated by the blue lines. The cut is tightened as the
momentum increases to remove the neutron peak, as shown in Ta-
ble. 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.25 Momentum dependent cut on β of neutral tracks to identify photons.
The fits are shown in Fig. 3.24 and the cut values are given in
Table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.26 A comparison of BSAs for π0s using the nominal vs tight cuts on β in
the photon identification. . . . . . . . . . . . . . . . . . . . . . . . 71
3.27 Invariant mass of two photons to reconstruct π0s. The distribution is
fit with a gaussian + polynomial background, and a 3σ cut is made
around the gaussian mean. The red curve is the peak, the gray is
the background, and the black is the sum of the two. . . . . . . . 72
3.28 Electron θ vs φ for one sector with no fiducial cut (left), the EC geo-
metric cuts (center), and the full fiducial cuts (right). . . . . . . . 74
xxi
3.29 MX vs z for each pion channel (top π+, middle π−, and bottom π0).
The cuts on MX and z are illustrated on the graphs (MX > 1.2
and 0.4 < z < 0.7). Both of these cuts help to reject exclusive
events that reside at high-z and low MX . For systematic studies
the cut is varied between 1.1 and 1.3 GeV showing a very small
dependence on the variation. As no strong exclusive peaks are seen
at MX > 1.2 GeV, it is concluded that this is an acceptable value
for the cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.30 AsinφLU vs MX . These results are integrated over x, PT , and 0.4 < z < 0.7. 77
3.31 The above plots compare the kinematics between elastic (first row),
Bethe-Heitler (second row), and SIDIS (third row) events. It is
shown that Bethe-Heitler and SIDIS events share a kinematic phase
space, making the Bethe-Heitler process a strong candidate for com-
puting corrections to the SIDIS electron momentum. In the bottom
left panel SIDIS events are those for which Q2 > 1 GeV and W > 2
GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xxii
3.32 Elastic Events Left: ∆pp
fit with ∆pp
(φ) = A+Bφ+Cφ2 for elastic events.
The top plot is before the correction and shows the quadratic fit
to determine the correction function, and the bottom plot is after
the correction. Right: Missing mass in each θ bin for one sector,
shown before and after the correction. The vertical bar indicates
the known value of the proton’s mass. . . . . . . . . . . . . . . . . 80
3.33 Elastic Events The top (bottom) plot shows the mean (sigma) values
from the Gaussian fits to the missing mass spectrums before (blue
squares) and after (red triangles) the correction is applied. Both
are plotted vs the bin number in θ. . . . . . . . . . . . . . . . . . 81
3.34 Left: Pre-radiative Bethe-Heitler process. Right: Post-radiative Bethe-
Heitler process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.35 Bethe-Heitler event selection. Top: Cut on θγ for one θe bin in each of
the ten W bins. Bottom, 1.5σ cut around ∆φ peak for one θe bin
in each of the ten W bins. The red lines show the cut and the blue
histograms illustrates the given quantity passing the other cut. . . 84
3.36 ∆PP
vs φe is shown as an example for a single bin (2.1 < W < 2.2 and
24o < θ < 29o in sector 2). The correction in other bins is very
similar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
xxiii
3.37 IM(γγ)mπ0
vs Eγ. The first plot shows the fit before the correction for
events in which Eγ1−Eγ2 < 10MeV, which is used to determine the
correction function. The second plot shows the corrected distribution. 89
4.1 φh distributions from simulated data binned in z and PT and weighted
with Cahn effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 GSim output for one simulated event using E1-f kinematics. Red,
curved tracks denote charged particles and grey lines denote neutral
tracks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Width of MX fits for various values of abc and f. The horizontal line
represents the width of MX from the experimental data. Based on
these results a value of abc = 1.3 is chosen for this analysis. Missing
mass is not a useful quantity for determining the best value of f. . 98
4.4 Width of ∆t vs abc and f. The horizontal line represents the width of
∆t in the experimental data. Based on these results a value of f =
1.05 is selected for this data analysis. . . . . . . . . . . . . . . . . 99
4.5 DC occupancies before and after GPP is applied for simulated CLAS
Sector 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6 Comparison of kinematic distributions between simulated data after
reconstruction and experimental data from E1f. The black squares
are E1f data and the blue triangles are from GSim. Plots on the
left show π+ and those on the right show π−. . . . . . . . . . . . . 101
xxiv
4.7 Acceptance for E1f binned in φ, z and PT . . . . . . . . . . . . . . . . 103
5.1 Sources of systematic error vs x. . . . . . . . . . . . . . . . . . . . . . 107
5.2 The above figure shows the EC sampling fraction in one momentum
bin for a single sector. The four colored lines represent the four
cuts used to test the systematic error due to this cut. The cuts
used from right to left are -2.5 to 4 σ, -2.75 to 3.75 σ, -3 to 3.5 σ,
and -3.25 to 3.25 σ. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Variation of vertex cut. Solid red lines show the nominal cut and
dotted blue/green lines show ± 0.5 cm. . . . . . . . . . . . . . . . 109
5.4 Systematic uncertainty due to pion id cuts. The above figure shows
the BSA in each PT bin for π+ using the ∆t cut (blue points, solid
line) and a β cut (red points, dashed). . . . . . . . . . . . . . . . 110
5.5 ALU vs x for an extreme variation of the EC Einner cut to test for pion
contamination in electrons passing the particle identification. . . . 111
5.6 EC Einner passing other electron identification cuts. The ratio of π−
to electron events in the region of Einner > 55 MeV was determined
from the ratio of the integrals of the fit functions in that region. . 112
5.7 AsinφLU vs x for π− in each of the five PT bins using three different missing
mass cuts in the SIDIS event selection. . . . . . . . . . . . . . . . 114
xxv
5.8 AsinφLU is plotted using a very loose MX cut of 0.85 GeV and using
the nominal cut of 1.2 GeV to remove exclusive events. The small
variation in the results show that contamination from exclusive
events is very small. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.9 ALU for π− comparing the nominal missing mass cut on MX > 1.2 GeV
to the data sample with no exclusive events removed (MX > 0.85
GeV) and to a cut above the mass of the ∆++ resonance (MX > 1.4
GeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.10 BSA vs φ for one bin in z for π0 comparing two fitting functions to mea-
sure the moment. The solid line fits the BSA with A sinφ1+B cosφ+C cos 2φ
and the dashed line fits the BSA with A sinφ, where in both the A
coefficient gives the value for AsinφLU in that bin. The p0 in the fit-
parameters box is the A value resulting from the full fit function,
and the ∆A value printed is the difference between the previous
value and that obtained using the sinφ fit. . . . . . . . . . . . . . 118
5.11 Møller measurements of electron beam polarization vs E1-f run num-
ber. The horizontal line shows the average polarization value of
Pe = 0.751 that was used in this analysis. . . . . . . . . . . . . . . 120
xxvi
5.12 BSAs using a randomly generated value for helicity (left). The blue
squares show the BSA from randomly generated helicity and the
open red circles show the normal BSAs. The plot on the right shows
the results of ALU vs. z using randomly generated helicity. It is
expected and shown that the results using random helicity should
be consistent with zero. . . . . . . . . . . . . . . . . . . . . . . . 121
5.13 Simulated SIDIS data weighted with < sinφ >= 0.3. The fits extract
the input value for ALU in every bin. . . . . . . . . . . . . . . . . 123
5.14 The ratio of acceptances of positive and negative helicity binned in z,
PT , and φ. It is seen that the ratio of acceptances is in agreement
with unity in every bin. . . . . . . . . . . . . . . . . . . . . . . . . 125
5.15 Beam-charge asymmetry vs run number for E1-f. . . . . . . . . . . . 126
5.16 ALU from runs with BCA > 0 compared to BCA < 0. . . . . . . . . . 126
5.17 ALU for π+ (left) and π− (right) with the data divided into two samples.
The first uses −26 < z < −24 cm (black), and the second uses z <
−26 cm and z > −24 cm (red). The two samples yield consistent
results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1 p-value vs. χ2 for each fit. Fits resulting with a p-value less than our
significance level of 0.003 do not confirm our hypothesis and are
removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
xxvii
6.2 χ2 distribution for fits to beam-spin asymmetries for fits with a p-value
greater than 0.003. . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 BSAs vs φ, binned in x for π+ (top row), π− (center row), and π0
(bottom row). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.4 Fits to BSAs for π+. Fits with a p-value < 0.003 are ignored (though
all fits shown here for π+ pass this criteria. . . . . . . . . . . . . . 136
6.5 Fits to BSAs for π−. Fits with a p-value < 0.003 are ignored. . . . . 138
6.6 Fits to BSAs for π0. Fits with a p-value < 0.003 are ignored. . . . . . 139
6.7 AsinφLU vs x in different PT bins. The error bars represent statistical
errors and the shaded regions at the bottom represent systematic
errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.8 AsinφLU vs z for each pion channel and integrated over the other variables.
The expected range in z for SIDIS kinematics 0.4 < z < 0.7. The
shaded regions denote systematic errors. . . . . . . . . . . . . . . 141
6.9 AsinφLU vs x for each pion channel and integrated over the other variables.
The integrated range in z for SIDIS kinematics 0.4 < z < 0.7. The
shaded regions denote systematic errors. . . . . . . . . . . . . . . 141
6.10 AsinφLU vs PT for each pion channel and integrated over the other vari-
ables. The integrated range in z for SIDIS kinematics 0.4 < z < 0.7.
The shaded regions denote systematic errors. . . . . . . . . . . . . 142
xxviii
6.11 AsinφLU vs Q2 for each pion channel and integrated over the other vari-
ables. The integrated range in z for SIDIS kinematics 0.4 < z < 0.7.
The shaded regions denote systematic errors. . . . . . . . . . . . . 142
6.12 AsinφLU vs x using binning to match the E1-6 data. The black square
points indicate data from the current analysis of E1-f. The blue
circles are the most recent published CLAS data from E1-e, and
the red triangles show CLAS data from E1-6. . . . . . . . . . . . 143
6.13 Comparison of AsinφLU vs x in five bins in PT for π0s between the E1-f
and e1-dvcs datasets. The black squares represent the measurement
from E1-f and the red triangles represent the points from e1-dvcs.
The large discrepancy in the first PT bin is due to the fact that e1-
dvcs has significantly better coverage in low-PT due to the addition
of the EC, so the fits in that region are much more accurate. . . . 145
6.14 Comparison of AsinφLU vs x between several datasets, each scaled by a
factor of < Q > /f(y) where f(y) is given by Eq. 6.10. . . . . . . 147
6.15 Comparison of measurement to a theoretical model taking into account
only the contributions due to the e(x)⊗H⊥1 term. . . . . . . . . . 148
xxix
6.16 Momentum dependent cut on θ for electrons and pions in order to
precisely match the phase space of the experimental and simulated
data samples. Electrons are shown in the left two plots and pions
on the right. The upper plots show experimental data and the lower
plots show GSim data. The cuts are denoted by the red curves in
each plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.17 Acceptance-corrected φ distributions are fit with the function A(1 +
B cosφ+C cos 2φ), where for each bin B is extracted as AcosφUU and
C is taken as Acos 2φUU . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.18 Comparison of 1κAcos 2φUU for π+ for a single bin in PT between e1f and
e16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
xxx
LIST OF TABLES
1.1 TMDs for leading twist (left) and subleading twist (right). The labels in
the top row denote the polarization of the quark and the labels in the
first column denote the polarization of the nucleon. For the twist-3
terms it is impossible to define a quark polarization because each term
inherently contains a gluon term in addition to the quark, so they can
not be described in the parton model in the same manner as the twist-2
TMDs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 Cut on θCC . Each segment in the CC is fit with a gaussian, and the
means of those gaussian fits are then fit with a polynomial. The
cut is made at ±3σ around the mean given by the equation, and σ
is retained individually for each segment. The table above gives the
coefficients for the mean, given by mean = a0 + a1segm+ a2segm2. 39
3.2 EC sampling fraction cut for electron id. Cut is µ(p) ± 3σ(p), where
the mean and width are given by µ(p) = µ0 + µ1p + µ2p2 + µ3p
3
and σ(p) = σ0 + σ1p+ σ2p2 + σ3p
3 . . . . . . . . . . . . . . . . . 47
3.3 Vertex cuts in centimeters. The liquid hydrogen target was centered
at -25.0 cm during the E1-f run. . . . . . . . . . . . . . . . . . . 55
xxxi
3.4 Minimum β cut on neutral particles to identify photons. As momentum
increases a tighter cut must be used to remove neutron contamination. 66
3.5 Elastic missing mass before and after correction. . . . . . . . . . . . . 79
3.6 Binning for Bethe-Heitler events. Binning is performed in each sector. 83
3.7 Bethe-Heitler missing mass before and after the electron correction is
applied. Both sets have the energy loss correction applied to the
protons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1 Control options for clasDIS . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 FFREAD card for E1-f. . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Input parameters for GPP. . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1 Sources of systematic uncertainty. The second column gives the av-
erage relative uncertainty from each source. For comparison, the
average statistical uncertainty is given. . . . . . . . . . . . . . . . 106
6.1 Kinematic binning of E1-f data. Data is binned five-dimensionally in
z, PT , x, Q2 and φ. . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 E1-f data binned in x using binning to match E1-6 and integrated over
all other variables. The table shows the average value of several
kinematic variables in each bin. . . . . . . . . . . . . . . . . . . . 143
A.1 AsinφLU in one dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . 160
A.2 AsinφLU binned in x and PT . . . . . . . . . . . . . . . . . . . . . . . . . 161
xxxii
C.1 Systematic errors for x vs. PT binning of π+. . . . . . . . . . . . . . 166
C.2 Systematic errors for x vs. PT binning of π−. . . . . . . . . . . . . . 167
C.3 Systematic errors for x vs. PT binning of π0. . . . . . . . . . . . . . 168
xxxiii
Chapter 1
Physics Motivation
1.0.1 Introduction
The three-dimensional structure of the proton is an exciting and rapidly expand-
ing topic in hadronic physics, containing many unanswered questions for both
experiment and theory. It has been known for many years that baryons such as
the proton are composed primary of three quarks, as well as gluons, the force
carriers of the strong nuclear force, and a sea of quark-antiquark pairs that are
constantly created and annihilated inside the hadron. Early particle physics ex-
periments at labs such as SLAC used deep-inelastic scattering (DIS) to map the
monodimensional momentum distributions of the individual partons by measuring
parton distribution functions (PDFs). While these PDFs afforded a rich increase
in understanding of physics inside the nucleon, questions soon arose that could
not be answered in this one-dimensional picture. In 1987, the EMC experiment
at CERN measured the contribution of the quark spin to the total spin of the
proton [1], finding that the quarks contribute only about 30% of the total, which
1
2
was at odds with the best QCD predictions at the time. The three-dimensional
structure of the proton plays an important role in understanding this phenomenon.
The surprising result from the EMC experiment has been dubbed the ”pro-
ton spin crisis,” and has been the subject of significant theoretical and experi-
mental work over the past two decades. Three possible contributors to the proton
spin were identified as the most likely candidates. The first was the spin carried
by the gluons in the proton, the second was the orbital angular momentum of the
quarks, and the third is the orbital angular momentum of the gluons. Multiple
sum rules to achieve the total proton spin have been proposed, one of which is
the Jaffe and Manohar spin sum rule [2] as shown in Eq. 1.1.
1
2=
1
2∆Σ +
∑q
Lq + ∆G+ Lg (1.1)
where ∆Σ is the contribution from quark spin, ∆G is the contribution from gluon
spin, and Lq and Lg are the contributions from quark and gluon orbital angular
momentum respectively. ∆Σ has been measured in DIS and ∆G is measurable in
deeply inelastic reactions. So far we do not know a method to measure Lq and
Lg.
The contribution from the gluon spin, ∆G, has been measured extensively
in STAR, PHENIX, and COMPASS, but so far it appears to be in agreement with
zero within error bars over the kinematic region that has been observed (0.05 <
x < 0.2 at RHIC). This leaves the orbital angular momentum of the quarks as the
3
most likely candidate. In order to understand quark orbital angular momentum,
we must measure how quarks move inside the proton in three dimensions.
A most general object associated with the description of the nucleon struc-
ture is given in terms the position and momentum distributions of quarks in a
nucleon formalized as Wigner functions [3] as shown in Eq. 1.2.
W (p, x) =∫d4ηeipηψ∗(x + η/2)ψ(x− η/2) (1.2)
The Heissenburg uncertainty principle tells us that it is impossible to simul-
taneously know the position and momentum of a quark to an arbitrary precision.
Hence the Wigner functions themselves cannot be measured, but their integrals
can. Measurements of the position and momentum distributions must be made
independently. Measurements of the partons’ positions in the two-dimensional
transverse plane, with a third dimension measured in terms of the longitudinal
momentum fraction, are performed by accessing generalized parton distributions
(GPDs) by processes such as deeply virtual Compton scattering (DVCS), and
measurement of the partons’ three dimensional structure in momentum space
are carried out by accessing transverse momentum dependent parton distribution
functions (TMDs) using processes such as semi-inclusive deep inelastic scattering
(SIDIS) or Drell-Yan. The TMDs and GPDs can be related to Generalized Trans-
verse Momentum Distributions (GTMDs) [4], [5], as shown in Fig. 1.0.1, which
in turn are related by Fourier transform to Wigner functions. The current work
4
Fig. 1.1: The GTMD cube. Taking the forward limit in hadron momentum in
a GTMD provides a corresponding TMD, and integrating a TMD over
∫d2k⊥ gives a PDF.
focuses on measurements of observables associated with some TMDs using the
semi-inclusive production of pions.
1.1 TMDs
Transverse momentum dependent parton distributions are an important tool for
studying the three dimensional structure of the nucleon. TMDs describe the
distributions of quarks in a nucleon in momentum space in three dimensions,
which is imperative to understanding of quark orbital angular momentum. There
is no known relation to measure quark orbital angular momentum from a TMD,
but it is known that orbital angular momentum is a necessity for certain TMDs
5
twist-2 twist-3
N / q U L T
U f1 h⊥1
L g1L h⊥1L
T f⊥1T g1T h1, h⊥1T
f⊥, g⊥, h,e
f⊥L , g⊥L , hL, eL
fT , f⊥T , gT , g⊥T , h⊥,eT h⊥T , e⊥T
Table 1.1: TMDs for leading twist (left) and subleading twist (right). The labels in
the top row denote the polarization of the quark and the labels in the first
column denote the polarization of the nucleon. For the twist-3 terms it
is impossible to define a quark polarization because each term inherently
contains a gluon term in addition to the quark, so they can not be described
in the parton model in the same manner as the twist-2 TMDs.
to exist [6]. TMDs enter certain factorizable processes in which more than one
hadron is involved, such as SIDIS or Drell-Yan. There are a total of eight TMDs
at leading twist1, as shown in Table. 1.1.
In Table 1.1, the second and third row require a nucleon with longitudinal
and transverse polarization respectively, hence it is impossible to measure these
terms with an unpolarized target. This leaves only the first row, which at leading
twist consists of f1, which describes unpolarized quarks in an unpolarized nucleon,
1 The technical definition of twist is twist = dimension - spin of the operator. For convenience,
this has been redefined to a working definition of twist = 2 + power of M/Q [7]. Hence twist-2
is called leading twist and twist-3 is subleading twist because these terms are suppressed by
O(M/Q).
6
and h⊥1 , the Boer-Mulders function, which describes transversely polarized quarks
in an unpolarized proton [8].
To gain an intuitive view of the relevant physics distributions, it is useful to
describe each twist-2 function in the naive parton model, which neglects gluonic
interactions. This is not possible for the higher twist terms because they contain
an intrinsic gluonic interaction, and thus must be equal to zero in the parton
model as shown in Eq. 1.3. The twist-2 term describes a parton density, and the
twist-3 term is the correlation of quark and gluon fields.
twist2 = 〈qq〉 (1.3)
twist3 = 〈qGq〉
At twist-3, the number of TMDs increases to 16, but only those in the first
row can be measured by SIDIS with an unpolarized target. These TMDs include
g⊥, the twist-3 T-odd TMD [9], that has been described as the higher-twist analog
to the Sivers function, and e(x), which is a chiral-odd, twist-3 PDF [10]. The x2
moment of e(x) has been suggested to describe the transverse force acting on
transversely polarized quarks in an unpolarized nucleon [11].
These higher-twist effects can partly be related to the average transverse
force acting on quarks just after interaction with the virtual photon in the colli-
sion [12]. The LU term in particular is due to the imaginary part of the interference
between longitudinal and transverse photon amplitudes.
7
1.1.1 Boer-Mulders Function
The Boer-Mulders function, h⊥1 , can be described in the parton model as the
distribution of transversely polarized quarks in an unpolarized nucleon. It is of
particular interest because it couples only to the Collins fragmentation function
in the cos 2φ moment, making its extraction easier than many of the other TMDs.
It is also measurable using an unpolarized target, as it does not take into account
the polarization of the target hadron.
The Boer-Mulders function is one of two leading order TMDs that are T-Odd
(along with the Sivers function, the measurement of which requires a transversely
polarized nucleon), meaning that it will change sign under time reversal. The
T-Odd functions are expected to have opposite sign when measured by SIDIS
and by Drell-Yan (hh→ `¯X), many measurements of which are currently being
planned at hadron colliders including Fermilab, RHIC, GSI, and CERN.
A non-zero measurement of the Boer-Mulders function shows that the orbital
angular momentum of the quarks in the nucleon is also non-zero. There is so far
no quantitative relationship between TMDs and orbital-angular momentum, but
the Boer-Mulders function has been interpreted as being due to the interference
between nucleon wave functions of different orbital angular momentum with ∆L =
1. This would confirm previous evidence of orbital angular momentum in the
nucleon due to the anomalous magnetic moment, which would also be zero unless
∆L = 1 [13].
8
1.2 Semi-inclusive DIS
Semi-inclusive deep inelastic scattering is one useful tool for measuring TMDs.
SIDIS describes the process in which a lepton collides with a hadron, and the
scattered electron and one hadron are detected in the final state, ignoring all other
particles produced in the fragmentation of the target hadron. For this analysis,
the reactions studied are those of the type ep→ eπ±,0X.
Deep inelastic scattering (DIS) describes a scattering process in which the
incoming lepton has a momentum great enough to break the target hadron into
multiple parts. Detection of such events require W > 2 GeV and Q2 > 1 GeV2,
where W is the invariant mass of the final state and Q2 is the virtuality, or
momentum transfer, given by W 2 = (P + q)2 and Q2 = −q2 = −(k−k′)2. Here P
is the 4-vector of the target and k(k′) is the 4-vector of the incoming (outgoing)
lepton.
The additional kinematic variables used to describe SIDIS reactions are x, z,
PT , and φ. Here x is the momentum fraction carried by the quark in the hadron,
which is defined by x = Q2
2P ·q . z is the momentum fraction carried away by the
produced hadron, which is defined manifestly as a Lorentz scaler by z = P ·PhP ·q , and
PT is the transverse momentum of the hadron. The angle φ is the angle between
the leptonic and hadronic planes as defined by the Trento convention [14] and
shown in Fig. 1.2.
9
Fig. 1.2: Diagram of SIDIS process. The incoming lepton with 4-momentum
l scatters off of a proton with 4-momentum P. In the final state the
scattered electron l′ is detected as well as one produced hadron. The
angle φ is the angle between the planes described by the scattering
lepton and that described by the produced hadron.
Fig. 1.3: SIDIS scattering process in the QCD factorization approach in the
Bjorken limit; given by the convolution of the parton distribution func-
tion and fragmentation function.
10
1.2.1 Moments of SIDIS Cross-section
If single-photon exchange is assumed, the scattering cross section for lepton-
hadron scattering can be written in a general way in terms of the leptonic and
hadronic tensors, as shown in Eq. 1.4.
dσ
dxdzdyd2qT=πα2yz
2Q4Lµν2MWµν (1.4)
Here the leptonic tensor, Lµν describes the electron side of the interaction
and the hadronic tensor, Wµν describes the interaction with the quark. The
single-photon approximation is valid because subsequent terms involving multiple
photons are suppressed by powers of αQED.
For a polarized lepton beam, the leptonic tensor is written as:
L(∫)µν = Tr[γµ(/k
′+m)γν(/k +m)
1± γ5/s
2] (1.5)
= 2kµk′ν + 2k′µkν −Q2gµν ± 2imεµνρσq
ρpσ (1.6)
For helicity states λe = ±1, the above equation can be simplified to
Lµν (λe=±1) ≈ Lµν (S) + λeLµν (A) (1.7)
where the superscripts S and A refer to the symmetric and antisymmetric parts of
the tensor. The above relation together with the appropriate parts of the hadronic
tensor give rise to the helicity dependence of the sinφ term of the cross section.
The symmetric and antisymmetric parts of the leptonic tensor are expressed in
11
terms of the vector kµ and a set of Cartesian coordinate vectors t.
Lµν(S) =Q2
y2[−2(1− y +
1
2y2)gµν⊥ + 4(1− y)tµtν
+4(1− y)(kµ⊥kν⊥ +
1
2gµν⊥ ) + 2(2− y)
√1− yt[µkν]
⊥ ]
and
Lµν(A) =Q2
y2[−iy(2− y)εµν⊥ − 2iy
√1− yt[µεν]ρ
⊥ kρ] (1.8)
Because the electron is believed to be a pointlike fundamental particle, and
because quantum electrodynamics (QED) can be solved exactly (in a perturbative
expansion of αQED), Lµν can be computed. If the electron had a finite size, the
numerical coefficients would need to be replaced by functions, as is seen in the
hadronic tensor below.
The hadronic tensor describes the hadronic side of the interaction. In the
Bjorken limit in quantum chromodynamics (QCD) factorization approach, it is
written in terms of two correlators as shown in Eq. 1.9, Φ(x, pT ) is the correlator
describing the quarks inside the nucleon and ∆(z, kT ) is the correlator of the
fragmentation process, describing what happens to the quarks in the final state,
where kT is the transverse momentum of the interacting quark.
2MWµν(q, P, S, Ph) = 2zh
∫d2pTd
2kT δ2(pT−
Ph⊥zh−kT )Tr(Φ(xB, pT , S)γµ∆(zh, kT )γν)
(1.9)
12
where the correlators are written as
Φij(x, pT ;n) =∫ d(ξ−)d2ξT
(2π)3eip·ξ〈P |ψj(0)U
n−(0,+∞)U
n−(+∞,ξ)ψi(ξ)|P 〉|ξ+=0 (1.10)
∆ij(z, kT ;n) =1
2z
∑X
∫ d(ξ−)d2ξT(2π)3
eik·ξ〈0|Un+
(+∞,ξ)ψi(ξ)|Ph, X〉〈Ph, X|ψj(0)Un+
(0,+∞)|0〉|ξ−=0
(1.11)
Here the Un terms are gauge links (or Wilson lines). These terms ensure the
color gauge invariance of the corelators. Without the inclusion of Wilson lines the
quark-quark correllator above is not gauge invariant because it would involve two
quark fields at different space-time points. The quark and remnant are colored,
and thus interact via gluon exchange. The Wilson line restores gauge invariance
by summing over all such gluon interactions. The sign flip of the T-Odd TMDs
such as Boer-Mulders and Sivers is explained by the opposite color flow in the
gauge links due to the interchange of past-pointing and future-pointing Wilson
lines [15].
The information contained in the correlators can be expressed in terms of
various TMDs and fragmentation functions. Here leading twist TMDs are inter-
preted as the probability of measuring a parton with a certain momentum fraction
x, and fragmentation functions give the probability of measuring a hadron with
momentum fraction z in the decay products. For an unpolarized target or frag-
mentation into an unpolarized hadron these are written to leading twist as
Φ[±](x, pT ) =1
2(f1(x, pT )/n+ ± h
⊥1 (x, pT )
i[/pT , /n+]
2M) (1.12)
13
∆(z, kT ) =1
2(D1/n− + iH⊥1
[kT , /n−]
2Mh
) (1.13)
Here the matrices /n are related to the standard Dirac matrices by /n± =
1√2(γ0 ∓ γ3). The Boer-Mulders TMD is not constrained by time-reversal, so in
the correlators for the SIDIS and Drell-Yan processes h⊥1 has the opposite sign.
Because our measurement is sensitive to twist-3 terms, we must add the higher
twist terms as well. The fragmentation correlation function then becomes
∆(z, kT ) = ∆[twist−2] +Mh
2P−h(E +D⊥
/kTMh
+ iH[/n−, /n+]
2+G⊥γ5
ερσT γρkTσMh
) (1.14)
The twist-3 quark-quark correlator is derived in [16]. If we neglect terms
requiring target polarization, this correlator can be written as
Φ(x, PT ) = Φ[twist−2] +M
P+[kiTMf⊥(x, k2
T )− εijT kTjM
g⊥(x, k2T )] (1.15)
where P+ is the target momentum in the light cone coordinates, following the
relation P µ = P+nµ+ + M2
2P+nµ−.
The cross-section for single pion electroproduction in SIDIS may then be
expressed as a set of structure function and trigonometric functions of φ.
dσ = dσ0(1 + AcosφUU cosφ+ Acos 2φ
UU cos 2φ+ λeAsinφLU sinφ) (1.16)
14
where the three moments, AcosφUU , Acos 2φ
UU ,and AsinφLU are ratios of structure functions,
and λe = ±1 is the beam helicity. By measuring the asymmetry in the cross-
section between positive and negative helicity events, the helicity dependent term,
AsinφLU , may be extracted from the beam-spin asymmetry (BSA).
The sinφ moment may be related to its corresponding structure function by
the relation:
AsinφLU =
F sinφLU
FUU,T(1.17)
Here, the denominator FUU,T is well known, and the numerator contains the
interesting physical structure, as described in [17]. The two terms are written in
terms of fragmentation functions and TMDs as:
F sinφLU =
√2ε(1 + ε)
2M
QC[− h · kT
Mh
(xeH⊥1 +Mh
Mf1G⊥
z)+
h · pT
M(xg⊥D1+
Mh
Mh⊥1E
z)]
(1.18)
FUU,T = C[f1D1] (1.19)
where C[] is a shorthand notation for the convolution defined by
C[wfD] = x∑a
e2a
∫d2pTd
2kTδ(2)(pT−kT−Ph⊥/z)w(pT,kT)fa(x, p2
T )Da(z, k2T )
(1.20)
15
where w is a function of pT and kT.
The two azimuthal asymmetries that do not depend on beam helicity can
be measured as well, but because there is no beam-spin asymmetry, these must be
measured from fits to the φ distributions directly. When looking at the distribution
in φ, only the cosφ and cos 2φ terms contribute, because the sinφ term cancels
when averaged over both beam helicities. The two unpolarized structure functions
that can be accessed in this way are given by
F cos 2φUU = C[−2(h · kT)(h · pT)− kT · pT
MMh
h⊥1 H⊥1 ] (1.21)
F cosφUU =
2M
QC[− h · kT
Mh
(xhH⊥1 +Mh
Mf1D⊥z
)− h · pT
M(xf⊥D1 +
Mh
Mh⊥1H
z)] (1.22)
In addition to the TMDs previously discussed, the structure functions also
contain fragmentation functions. Notably H⊥1 is the naive time-reversal odd
Collins fragmentation function [18]. It is interpretable as the left-right asym-
metry in the fragmentation of a transversely polarized quark into a hadron with
z, zkT. Also, E and G⊥ are twist-3 fragmentation functions.
In the above expressions for F sinφLU and F cosφ
UU , the four ”tilde” terms, E, G⊥
D⊥, and H, come from the quark-gluon-quark fragmentation correlator. Some-
times such terms are assumed to be zero (known as the Wandzura-Wilczek ap-
proximation), but because ALU is a completely twist-3 quantity (considering that
16
e = e and g⊥ = g⊥), and all terms are ”small,” this is not a reasonable approx-
imation in this circumstance. These four terms are related to the corresponding
fragmentation functions from the quark-quark correlator by the expressions
E
z=E
z+
m
Mh
D1 (1.23)
G⊥
z=G⊥
z+
m
Mh
H⊥1 (1.24)
D⊥
z=D⊥
z+D1 (1.25)
H
z=H
z+
k2T
M2h
H⊥1 (1.26)
The cos 2φ term is of particular interest because it relates directly to the
convolution of the Boer-Mulders function with the Collins function. The Collins
function has been measured at Belle [19], HERMES, and COMPASS, which makes
the cos 2φ moment a strong candidate for extraction of the Boer-Mulders TMD.
The cosφ term is twist-3, so it is suppressed by a factor of M/Q. It also
includes the Cahn effect, which is a purely kinematic effect that also produces
a modulation in cosφ. The cos 2φ term is sensitive to the Cahn effect as well,
but because the Cahn effect is flavor independent, it is hoped to cancel out for
differences in the azimuthal asymmetries between π+ and π−.
17
Fig. 1.4: Leading order diagram for the Boer-Mulders function in the one-gluon
exchange mechanism.
1.3 Model Predictions
One method of modeling TMDs is in the framework of the light cone quark model
(LCQM) [20] that is used to calculate the T-Odd TMDs using the single gluon
exchange mechanism between the target quark and the nucleon spectators. For
the purpose of this model, the Boer-Mulders function is defined in terms of a
correlation function and the gauge link is expanded up to the next-to leading
order, and is shown in the diagram of Fig. 1.3. Here gluon exchange between two
quarks transfers one unit of orbital angular momentum, so for instance a quark
in the P-wave state switches orbital angular momentum states with the quark in
the D-wave state. The Boer-Mulders function in this case has contributions from
the interference between the S (` = 0) and P (` = 1) wavefunctions as well as
between the P and D (` = 2) wave wavefunctions.
The model computes the Boer-Mulders function individually for up and
18
Fig. 1.5: Boer-Mulders function as calculated from the light-cone quark model
for up quarks (left) and down quarks (right).
down quarks, predicting a larger magnitude for up quarks in the proton than
down quarks, as show in Fig. 1.3. It is important to note that the sign of the
Boer-Mulders function is the same for up and down quarks, which leads to a spin-
density distribution shifted in the same direction for all quarks in the proton, as
shown in Fig. 1.3. The spin-density is related to the Boer-Mulders function by
the expression
ρqh⊥1
(k⊥, s⊥) =∫dx
1
2[f q1 (x, k2
⊥) +siεijkj
Mhq⊥1 (x, k2
⊥)] (1.27)
19
Fig. 1.6: Spin density for transversely polarized quarks in an unpolarized proton
resulting from the Boer-Mulders function in the light cone quark model.
The left panel shows the distribution for up quarks and the right panel
shows the distribution for down quarks.
Chapter 2
Experiment
2.1 CEBAF
The Continuous Electron Beam Accelerator Facility (CEBAF) is a high-luminosity
polarized electron accelerator located at the Thomas Jefferson National Accelera-
tor Facility in Newport News, VA. It is capable of producing a beam of polarized
electrons with energies reaching 6 GeV. After polarized electrons are produced
with an electron gun, they are accelerated by passing five times each through two
linear accelerators (LINACs), before being divided and delivered to the experi-
mental halls. The beam is delivered simultaneously to three experimental halls,
Halls A, B, and C. The CLAS spectrometer is located in Hall-B.
Polarized electrons are produced in the injector using a polarized electron
gun [21]. Here gallium arsenide that has been doped with beryllium is used to
produce electrons with a preferential polarization by optically pumping between
the P3/2 and S1/2 spin states, causing an excess of electrons emitted in one spin
state over the other. These spin-polarized electrons are then directed into the
20
21
Fig. 2.1: The continuous electron beam accelerator facility (CEBAF). CLAS is
located in Hall-B.
accelerator. Using this method CEBAF is able to produce a beam of polarized
electrons with a maximum polarization of ≈ 88%.
The continuous-wave electron beam is emitted from the injector with an
energy of 45 MeV, and is then accelerated using two LINACs and nine recirculation
arcs. Each LINAC is composed of superconducting radiofrequency cavities (SRF)
that give the beam a booste in energy each time it passed through. The higher the
electron energy the less it is bent in a constant magnetic field, so each recirculation
arc is located at a different angle, as shown in Fig. 2.1. A magnet directs the first
pass (lowest energy) beam into the highest arc, and as the beam gains energy
the same magnet will bend it into successively lower arcs until the beam reaches
its desired energy. An RF system then divides the beam into 2 ns bunches that
22
are directed into each hall. The facility has the capability of providing beam
simultaneously to each of the three halls using this method.
2.2 CLAS
The CEBAF Large Acceptance Spectrometer (CLAS) consists of four types of
detectors arranged in layers to cover a large portion of the solid angle in six sec-
tors, as well as a magnet producing a toroidal magnetic field [22]. Three layers of
drift chambers track charged particles as they are bent through the magnetic field,
allowing measurement of the particles’ momentum by studying their radius of cur-
vature. The time-of-flight detector allows measurement of the particles’ velocity,
which coupled with the measured momentum can provide good identification of
charged hadrons. A Cerenkov counter is used in the electron identification to dis-
tinguish between electrons and negative pions. The Electromagnetic Calorimeter
was used in detection of electrons and photons, which were detected to reconstruct
π0s.
2.2.1 Drift Chambers
The CLAS drift chamber (DC) system is used to determine the momentum of
a charged particle by measuring the curvature of its path as the particle travels
through the toroidal magnetic field [23]. The charge of each particle is determined
by the direction of its curvature, and the momentum of each particle is propor-
23
Fig. 2.2: Vertical slice of the CEBAF Large Acceptance Spectrometer.
24
tional to the radius of curvature as shown in Eq. 2.1, where q is the charge of the
track, B is the magnetic field, and ρ is the curvature of the track.
p = qBρ (2.1)
Each chamber is filled with an ionizing gas. As a charged particle passes
through the gas, it leaves a trail of charged ions. An electric field directs the
ions to drift to a wire where they are detected. The information gathered is
used to reconstruct the trajectory of each particle. Using our knowledge of the
CLAS magnetic field we can use extract the radius of curvature from the measured
trajectory and compute the momentum of the particle.
The DC is separated into six sectors, each of which contains three regions.
Region 1 surrounds the target, region 2 are located between the coils in the region
of maximum particle curvature, and region 3 is outside the coils but inside the EC
and time-of-flight. Each region covers the same angular region, so the size of each
DC increases as the distance from the target increases. The sensing wires used are
20-µm diameter gold-plated tungsten with a surface electric field of 280 kV/cm.
Each sensing wire is surrounded by six wires used to produce the electric field
making a hexagonal cell. The gas used in the DC is a mixture of 90% argon and
10% CO2, which was chosen because it provides a high ion drift velocity, which
is needed because fast collection times improve the momentum resolution. These
are arranged in the 18 drift chambers into 35,148 hexagonal drift cells, providing
25
good resolution in both momentum and angular measurements.
2.2.2 Time-of-Flight
The CLAS time-of-flight detector (TOF) is used to measure the velocity of charged
hadrons produced in CLAS [24]. The TOF detector uses plastic scintillators to
precisely measure the time at which it is hit by a particle. Using this time together
with the event start time and the particle path as measured by the DC, the
velocity of each particle can be measured, which is then used as the main source
of identification for charged hadrons.
The TOF was designed to optimize time resolution to give the most precise
possible particle identification. The resolution varies from about 60 ps for the
shorter scintillator paddles to up to 120 ps for the longer paddles. The lengths of
the scintillators vary from 32 cm at low angles to 450 cm at high angles. All are
5.08 cm thick, and either 15 or 22 cm wide, giving a total coverage area of 206
m2. The system is capable of separating pions and kaons up to 2 GeV/c.
2.2.3 Cerenkov Counter
The CLAS Cerenkov counter (CC) [25] is used in conjunction with the calorime-
ter for discrimination between electrons and negative pions. The CC works by
detecting Cerenkov radiation, which is emitted when a particle passes through a
medium with a velocity faster than the speed of light in that medium, given by
26
Fig. 2.3: Diagram of the path of one electron through the CC along with the
light collection system.
(v > c/n, where n the index of refraction in the medium) Photomultipliers are
used to count the number of photons radiated as the particle passes through the
medium.
The medium used in the CLAS CC is C4F10 gas, which was chosen for its
high index of refraction (n = 1.00153) which yields a high photon count and a
pion momentum threshold of pπ ≈ 2.5GeV/c. The Cerenkov radiation is detected
using 216 optical modules, each consisting of three mirrors and a photomultiplier
tube. The path of a typical electron through the CC and its light collection in
one module is shown in Fig. 2.2.3. The optical modules are arranged between the
magnetic coils in each sector in such a way that the photomultiplier tubes are
blocked by the coils, and hence to provide an additional obstacle for the CLAS
acceptance. A schematic of the optical mirrors in one sector is shown in Fig. 2.2.3.
27
Fig. 2.4: The optical mirrors for one sector of the CC.
2.2.4 Electromagnetic Calorimeter
The CLAS electromagnetic calorimeter (EC) [26] is used in this analysis for elec-
tron, photon and π− identification. The EC is composed of six independent spec-
trometers composed lead-scintillator calorimeters arranged in a triangular geom-
etry. The EC covers the forward region (8o < θ < 45o). The energy, position, and
time for each incident track is recorded with high precision. The resolution re-
quirements are e/γ energy resolution of σ/E ≤ 0.1/√E(GeV ), position resolution
of 2 cm at 1 GeV, π/e separation of 99%, and timing resolution of 1 ns [26].
The scintilating strips that make up the calorimeter are arranged in three
planes, labeled U, V and W. Each plane is offset by an angle of 120o to enable
triangulation of the hits. The planes are arranged in 13 layers, each layer being
composed of 36 scintilating strips, which are then divided into an inner and outer
stack as shown in Figure. 2.5. In total the EC is composed of 8424 scintillator
28
Fig. 2.5: One stack of the EC, as shown divided into U, V, and W planes.
strips, requiring 1296 PMTs.
In order to achieve the desired resolution in energy and timing, it was nec-
essary to use fast plastic scintillator with very high transparancy. To meet these
requirements Bicron BC412 was used.The scintillating material was cut into strips
100 mm wide by 10 mm thick, with lengths ranging from 0.15 to 4.2 m (depending
on their location in the UVW plane). The readout end of each scintillator was
diamond milled to achieve the highest possible light transmission.
2.3 E1-f
E1-f was an electron run operating from April through June of 2003 utilizing a
5.498 GeV polarized electron beam on an unpolarized liquid hydrogen target. The
beam polarization was 75.1±0.2%, and the torus was run at 60% of its nominal
29
value to maximize charged pion acceptance. All three pion channels (π+, π0 and
π−) were measured simultaneously over a large range of kinematics(Q2 ≈ 1-4
GeV 2 and x ≈0.1-0.5, as shown in Fig. 2.6).
Fig. 2.6: E1-f kinematic coverage in relevant variables. AsinφLU is binned in z, x,
PT and Q2. Each column shows a different pion channel
Chapter 3
Data Processing
3.1 MU File Format
MU files are data files that are stripped of unnecessary information and written in
a concise and quickly readable binary format [?]. The bos2mu program reads BOS
files (2 Gigabytes each), takes the desired information, and rewrites it into a much
smaller MU files (approximately 200 Megabytes), and are then made much smaller
after the particle id skims are applied (approximately 5 Megabytes). The data is
organized into a tree structure topped with the individual events, each of which
contains a list of particles in the event, as well as the general event information
such as start time, helicity, etc. Each particle contains a four vector structured
by the V4 C++ class, as well as information on that particle from each of the
relevant detectors.
30
31
3.2 Determination of Good Run List
The good run list is determined by plotting the number of electron, π+, π−, and
π0 events per run normalized by the total charge for each run. These quantities
should be very consistent, but for some runs there are fewer events per unit charge,
so these runs are eliminated from the data analysis. Most of the eliminated runs
came from the number of electrons per run, but one additional run, 38339, was
cut by examination of the number of pions per run (this run was deficient in all
three pion channels). These checks were done independently for each sector, and
no discrepancies were observed (Fig. 3.2). The full list of runs passing the above
criteria is given in Appendix B.
3.3 Helicity Determination
Helicity information is recorded for each event using either direct or delayed re-
porting. The program hel bos.C [27] is used to write this helicity information to
the BOS files during the cooking process by replacing the EVTCLASS variable
with a helicity of ±1, or 0 if unknown. This helicity value is then written in each
MU file event as well.
During the data taking, helicity is reported in two ways. One is direct
reporting and the other is delayed reporting. In order to have an accurate helicity
determination it is necessary to know which reporting method was used for every
run.
32
(a) Number of events with a good electron normalized by charge vs run number. Events for which the
torus was run with reversed polarity are denoted by red triangles. All runs for which this quantity falls
below the horizontal line are not used.
(b) Number of events with a good pion normalized by charge vs run number. The left plot
shows π+ and the right plot shows π− and π0. One run is cut for low statistics.
Fig. 3.1: Number of identified particles vs run number normalized by charge.
Most bad runs are cut due to the number of electrons, but one addi-
tional run is cut due to a low number of π+ and π−
33
If helicity was recorded via direct reporting for the run being cooked, the
helicity program is able to easily determine the helicity for each event directly
from the BOS file. The TRGPRS variable is read in from the HEVT bank. If
TRGPRS is positive, the helicity is assigned a value of +1, while if it is negative
helicity gets a value of -1. Additionally, helicity is only allowed to flip if TRGPRS
had changed by more than 1000 from one event to the next and the difference
in the interrupt time between successive helicity flips was less than 2000 ticks off
the microsecond clock. Interrupt time is read from the TGBI bank. This cut was
added to deal with a trigger glitch that was discovered in e1e. For e1-f only 55
events were affected by this trigger glitch.
For runs that used delayed reporting, it is not possible to read the helicity
Fig. 3.2: Number of electrons normalized by charge vs run number in each sector.
The previous cuts shown for runs with low event rates have already been
applied.
34
directly from the BOS file, but it must instead be read in two steps. The first step
reads the BOS file and creates ntuple 13 containing helicity information, and the
second step reads both ntuple 13 and the BOS file to correlate a specific value of
helicity with each event.
Periodically throughout the run, the definition of helicity is reversed by
changing the half-wave plate or a change in the settings of the injector. These
transitions must be mapped out in order to keep a consistent definition of helicity
throughout the run period. To do so, AsinφLU is calculated for each run and plotted
versus run number. The flips in helicity definition can easily be seen as the
locations where AsinφLU changes sign. These transitions can be seen in Fig. 3.3.
3.4 Electron Identification
The electron identification is performed using a series of ten cuts in the Cerenkov
counter and electromagnetic calorimeter as well as a momentum dependent fidu-
cial cut. In the CC, all events with greater than 2.5 photoelectrons are kept, but
if an event has fewer than 2.5 photoelectrons a series of three CC matching cuts
are used with a geometric cut on the CC θ vs φ. The CC matching procedure is
based on previous work for other data sets as described in [28] and [29].
35
Fig. 3.3: The sinφ moment vs. run number for e1f. The transitions in helicity
definition are given by the red lines. The upper plot shows the full
range of run numbers, and the middle plot concentrates on a central
range in which there are numerous flips of the helicity definition. The
bottom plot shows AsinφLU after the helicity helicity flips correction is
applied.
36
3.4.1 Number of Photoelectrons in the CC
A good electron will cause a high number of photoelectrons in the Cerenkov
counter, hence any candidate with more than 2.5 photoelectrons is kept. Candi-
dates with fewer than 2.5 photoelectrons are subjected to the following four CC
matching and fiducial cuts. Fig. 3.4 shows the number of photoelectrons with and
without the CC matching applied.
37
Fig. 3.4: Number of photoelectrons in the CC. The solid (black) histogram shows
events passing all electron identification cuts including CC matching.
The dashed (blue) histogram shows events passing all other electron id
cuts, but without CC matching.
38
3.4.2 CC θ Matching
For good electrons, there should exist a one-to-one correspondence between the
CC θ and segment number. To quantify this relation and determine the values
for the cut, θCC is plotted as a one-dimensional histogram in each of the eighteen
segments in each sector. These histograms are then fit with a gaussian with a
polynomial background, and the means are plotted vs segment number and fit
with a polynomial. The widths of each gaussian are retained independently, and
the cut is then defined as the mean value from the fitted function ±3σ in each
segment (Fig. 3.5). The coefficients measured to calculate the mean as a function
of CC segment are given in Tab. 3.1.
3.4.3 CC φ Matching
φCC is the azimuthal angle of each track relative to the center of each sector as
measured in the Cerenkov counter. CC φ matching is used to check that for each
event the CC PMT fires on the same side as the candidate’s track. Candidates
with φCC < 4o are kept, as is any event for which PMT’s fire on both sides of the
Cerenkov detector. A histogram showing the cut events is shown in Fig. 3.6.
39
Sector a0 a1 a2
1 7.425 1.727 0.02356
2 7.360 1.717 0.02381
3 7.375 1.721 0.02381
4 7.282 1.740 0.02280
5 7.727 1.720 0.02329
6 7.812 1.612 0.02893
Table 3.1: Cut on θCC . Each segment in the CC is fit with a gaussian, and
the means of those gaussian fits are then fit with a polynomial. The
cut is made at ±3σ around the mean given by the equation, and σ
is retained individually for each segment. The table above gives the
coefficients for the mean, given by mean = a0 + a1segm+ a2segm2.
40
Fig. 3.5: θCC vs CC segment in each sector. The black curves denote the cut at
±3σ about the gaussian fits. The plots show all electron candidates in
the CC.
41
Fig. 3.6: CC φ matching: returns 0 if both PMTs fire, ±1 if track and PMT are
on the same side, and ±2 if there is a mismatch between the track and
PMT. Candidates are automatically kept if φCC < 4o.
42
3.4.4 CC Time Matching
CC time matching matches the time recorded for a track in the Cerenkov counter
to the time recorded in the time-of-flight detector. A quantity ∆t = tCC − (tSC −
s/c) is calculated for each particle where s is the pathlength between the Cerenkov
counter and time-of-flight detector and c is the speed of light. ∆t is fit with a
gaussian for each segment in each sector, and a 3σ cut is made on the lower side
only because multiple Cerenkov light reflections could lead to a time delay giving
good tracks on the high side of the fit. The cuts are shown in Fig. 3.7. Because
CC time is not used in the event reconstruction, it is not precisely calibrated.
Hence the cut must be computed individually for each CC segment.
43
Fig. 3.7: CC time matching for all electron candidates. The 2-dimensional his-
togram shows ∆t = tCC − (tSC − s/c) vs CC segment for each sector.
The crosses denote 3σ cuts on the lower side of ∆t.
44
3.4.5 CC Fiducial Cut
A purely geometric cut is used in the Cerenkov counter, keeping events for which
θCC > 44.5o − 35
√1− φ2CC
575(Fig. 3.8). The equation was obtained empirically
by looking at the function compared to all data, data passing the number-of-
photoelectrons cut, and data failing the number-of-photoelectrons cut. It was
observed that most of the events that fail this cut also have fewer than 2.5 pho-
toelectrons per event. This cut is shown in Fig. 3.8.
Fig. 3.8: θCC vs φCC for all electron candidates.
45
3.4.6 EC Threshold Cut
In electron identification, the purpose of the calorimeter is to differentiate electrons
from minimum ionizing particles, namely π−. The first cut used is an EC threshold
cut to determine the minimum momentum of electrons that can be detected in
CLAS.
A minimum value for momentum is calculated from the threshold energy of
the calorimeter’s trigger discriminator. The value is calculated from:
pmin(MeV ) = 214 + 2.47× ECthreshold(mV ) (3.1)
For E1-f, the EC threshold was 172 mV, giving a minimum momentum of
0.639 GeV.
3.4.7 EC Sampling Fraction Cut
Electrons passing through CLAS will have a distribution of Etotp
that is nearly
constant in p. These good electrons can be distinguished from minimum ionizing
particles by isolating and cutting around this band. To determine the cut the Etotp
distribution is fit with a gaussian in each momentum bin. The means and widths
of these fits are then fit with polynomials, and the two functions are combined to
determine functions for the upper and lower cuts, µ(p)+3.5σ(p) and µ(p)−3σ(p).
This cut is shown in Fig. 3.9 with the coefficients on the cut functions given in
Table 3.2.
46
Fig. 3.9: Sampling fraction in calorimeter after all other electron identification
plots are applied. Each panel is a different sector of CLAS. The cut is
made by fitting slices in y with a Gaussian and then fitting the means
and widths of each gaussian with polynomials to determine fit functions
as a function of p, µ(p) + 3σ(p)/− 3.5σ(p).
47
Sector µ0 µ1 µ2 µ3 σ0 σ1 σ2 σ3
1 0.28 0.0051 0.0062 -0.00094 0.049 -0.021 0.0063 -0.00067
2 0.31 0.017 0.0026 -0.00073 0.055 -0.019 0.0043 -0.00027
3 0.30 0.0013 0.013 -0.0024 0.047 -0.016 0.0037 -0.00026
4 0.30 -0.0067 0.012 -0.0018 0.046 -0.012 0.0033 -0.00039
5 0.27 0.016 0.00085 -0.00041 0.055 -0.022 0.0061 -0.00058
6 0.28 0.0043 0.0067 -0.0012 0.050 -0.019 0.0051 -0.00051
Table 3.2: EC sampling fraction cut for electron id. Cut is µ(p)± 3σ(p), where
the mean and width are given by µ(p) = µ0 + µ1p+ µ2p2 + µ3p
3 and
σ(p) = σ0 + σ1p+ σ2p2 + σ3p
3
48
3.4.8 EC Ein vs. Eout Cut
Electrons shower and deposit a great deal of energy in the EC, but pions are
minimum ionizing particles that deposit very little energy. The energy deposited
by minimum ionizing particles and good electrons in the inner calorimeter can
be easily separated. A cut is made to keep only particles with Ein > 55MeV to
remove minimum ionizing particles, as is shown in Fig. 3.10.
Fig. 3.10: EC Einner vs Eouter for electron candidates passing the other EC cuts:
A cut is made to keep only candidates with Einner > 55MeV to cut
minimum ionizing particles.
49
3.4.9 EC Geometric Cut
Near the edges of the calorimeter, the energy cannot be properly reconstructed
because the shower could occur only partially in the detector. To correct for this,
cuts are made around the edges of the calorimeter in each sector. The scintillating
strips in the EC are laid out in three planes, labeled U, V, and W. Cuts are made
individually in each of these three planes of the detector, the values of which are
determined by looking at one dimensional histograms of hits in each plane, and
visually determining the value at which the expected trend terminates. he cuts
imposed are:
70cm ≤ U ≤ 400cm
V ≤ 362cm
W ≤ 395cm
, the results of which can be seen in Fig. 3.11 once applied to the two-
dimensional geometry of the EC.
3.4.10 tEC − tSC Cut
The final EC cut used for electron id checks the agreement between the time
recorded in the calorimeter and the time-of-flight detector. A quantity ∆t is
calculated as the difference between the EC time and the SC time modified by
the distance between the two detectors divided by the speed of light, as shown in
50
Fig. 3.2. The distribution is fit with a gaussian and a 3σ cut is used around the
mean of the fit (see Fig. 3.12).
Fig. 3.11: Physical location of hits on the calorimeter. The colored regions de-
note candidates that were kept and the black area shows negative
tracks that were eliminated by this cut.
51
∆t = tEC − tSC − 0.7ns (3.2)
It can be seen in Fig. 3.12 that sectors 3, 4, and 5 have widths larger than
do the other three. These widths are generally consistent with the ∆t width
measured in the EC time calibration of about 0.4 ns. To investigate further the
∆t distributions were looked at with regards to position in the EC, and it was
found that those tracks nearer the edge of the calorimeter typically had a slightly
larger width than those tracks hitting closer to the center. An example of one
sector is displayed in Fig. 3.13.
52
Fig. 3.12: Cut on ∆t to ensure agreement between the time recorded by the EC
and TOF detectors. Each panel represents a different sector of CLAS.
For the histograms shown all other electron id cuts have already been
applied.
53
Fig. 3.13: ∆t vs EC position for CLAS Sector 4. Each distribution is fit with a
Gaussian and the widths are printed in red next to each.
54
Fig. 3.14: Cut on interaction vertex for all electron candidates. Each panel dis-
plays a different sector of CLAS.
3.4.11 Vertex Cuts
A cut is made on the interaction vertex to remove events that do not originate
within the target. The cut is set independently for each sector to include the full
target region. Fig. 3.14 shows this cut in each sector, and Table 3.3 gives the
exact values. The sector-dependent cut is necessary because a vertex correction
is not implemented. It cannot be assumed that each sector is precisely the same
distance from the vertex, as is shown in Fig. 3.15.
55
Fig. 3.15: Vertex position in each sector for electrons, π+, and π0.
Sector zmin (cm) zmax (cm)
1 -28.0 -22.9
2 -27.5 -22.0
3 -26.9 -21.9
4 -27.5 -22.0
5 -28.0 -22.8
6 -28.5 -23.25
Table 3.3: Vertex cuts in centimeters. The liquid hydrogen target was centered
at -25.0 cm during the E1-f run.
56
3.5 Hadron Identification
3.5.1 π+ Identification
Positive tracks are identified by CLAS based on their direction of curvature in the
toroidal magnetic field. Using the normal magnetic field direction, positive tracks
will bend away from the beamline. Positive tracks seen in CLAS while running
at 6 GeV are mostly protons, π+, and K+. Pions are separated from the others
by looking at the time differences required for particles of different mass to reach
the CLAS time-of-flight detector.
Positive pions are identified by comparing the time measured in the time-
of-ight detector to a flight time calculated using the momentum measured by the
drift chamber, p, the particle’s path length, L, and the known pion mass. [30].
∆t = tmeasured− tcalculated is fit with a gaussian function in each of ten momentum
bins. The mean of each fit ±3σ are then fit to determine the cuts in order to select
good pions as shown in Figure 3.16. To obtain the p-dependence the positive side
of ∆t the fit function is a0(a1 + a2p + a3p2 + a4p
3)e−p/a5 and the negative side is
fit with a0 + a1√a2+a3p
.
tcalc =Lβ
c(3.3)
β =p√
p2 +m2π
For comparison, a cut is computed on β vs momentum, and the results are
57
compared to those using the ∆t cuts and included in the computed systematic
uncertainty. Because pions have the lowest mass of the charged particles produced
in ep collisions, they will have the highest velocity as measured by the CLAS TOF
detector. In Fig. 3.17, the pions form the top band with β ∼ 1 except for at very
low values of momentum. Proton bands are also visible before the pion selection
criteria on ∆t are applied.
Fig. 3.16: ∆t vs p for π+ candidates. ∆t is fit with a gaussian in each momentum
bin. A cut is made around 3σ of the mean as illustrated by the red
lines. Each panel represents a different sector.
58
3.5.2 π− Identification
The detection of negative pions utilized three cuts. First, the ∆t cut is used as
shown in Figure 3.18, just as in the π+ case. By examining the β vs momen-
tum spectra after this cut is applied, it can be seen that there is still a large
contamination by electrons with β ≈ 1 (see Figure 3.19). These electrons can
be eliminated by adding cuts opposite to those used in the electron identification
Fig. 3.17: β vs. momentum for π+ candidates. The colored 2-dimensional his-
tograms show β vs p for each sector before the ∆t cut, which is over-
layed with black 2-dimensional histograms showing β vs p for each
sector after the ∆t cut.
59
(Figure 3.20). Candidates are kept if they have Einner < 60MeV in the electro-
magnetic calorimeter and the number of photoelectrons less than ∼2.5 (depending
on 3σ cut to gaussian fits in each sector) in the Cerenkov Counter. A similar π−
identification is performed in [31].
The Einner cut for pions is not at exactly the same value as for electrons.
The electron cut was done by eye by looking at E¡sub¿in¡/sub¿ for all negative
tracks to visibly separate the electron and π− distributions. The value for the pion
cut was determined by looking at a plot of negative tracks that did not include
the trigger electrons. 60 MeV is near the 3σ limit of the pion distribution. The
nominal cut for electrons is tested to 60 MeV in the systematic error studies, and
it is seen that this causes a very small change in the measured asymmetries.
60
Fig. 3.18: ∆t vs p for π− candidates. ∆t is projected onto the y-axis and fit
with a gaussian. A cut is made around 3σ of the mean as illustrated
by the red lines. Each panel represents a different sector.
61
Fig. 3.19: β vs p for π− candidates in sector 1. The top plot shows β vs mo-
mentum before the ∆t cut in each of the six sectors. The second plot
shows β vs p after the ∆t cut. And the bottom plot shows β vs p
after the ∆t, EC Einner and number of photoelectron cuts.
62
Fig. 3.20: The first six plots illustrate the π− identification cuts on Einner in
the CLAS electromagnetic calorimeter. The second set of six plots
show the π− identification cuts on the number of photoelectrons in
the CLAS Cerenkov counter.
63
3.5.3 π0 Identification
Neutral pions have a decay time on the order of 10−17 seconds, so they do not travel
far enough to be detected in the CLAS spectrometer. Instead, their primary decay
products must be detected. The decay with the highest probability is π0 → γγ
with a decay probability of 98.8% [32]. CLAS is capable of detecting both photons
in the EC, and the π0 4-vector can be reconstructed by adding the 4-vectors of
the two photons. A cut is imposed on the two-photon invariant mass spectrum
around the π0 mass peak.
Photon Identification
Photons are detected by measuring the velocity of neutral particles in the electro-
magnetic calorimeter. Any event containing two or more neutral particles with
energy Eγ > 0.15 GeV (Fig. 3.21), θeγ > 12o (Fig. 3.22), and passing the EC
geometric cuts (Fig. 3.23) is analyzed by fitting the β distribution in ten different
momentum bins with a gaussian + polynomial background and placing a lower
bound on β in each bin in order to remove heavy neutral particles (see Figures 3.24
and 3.25). Here β for the neutral tracks is defined as
β =L
c(tEC − tstart)(3.4)
Slight variations in L between the sectors introduce very small offsets in β.
The EC timing calibration minimized the difference between the EC time and the
64
SC time, but photon β is also checked. Our values of β are in agreement with
what is expected from the EC timing calibration for E1-f. The cut is checked for
each sector individually, so a sector-dependent offset is not necessary.
The gaussian mean and σ of the β distributions do not change greatly with
increasing momentum, but at higher momentum the distribution is encroached
upon by the neutron peak, necessitating a momentum dependent cut. A 3σ cut
is utilized at the lower momentum bins, but as momentum increases it becomes
necessary to tighten the cut as shown in Table 3.4. The β cut is performed
separately in each sector to account for any small sector-dependencies that may
occur. Each neutral particle passing this photon cut is considered as a π0 decay
candidate (if Nγ ≥ 2).
In order to test the effect of neutron contamination at high momentum,
the analysis was completed using an extremely tight cut on β to compare against
the nominal pion identification. A total of 750423 π0s are identified using the
nominal β cut, and 723534 are identified using the tighter cuts. The beam spin
asymmetries resulting from each cut are displayed in Fig. 3.5.3, showing that the
effect due to neutron contamination is very small.
π0 Invariant Mass
After the photon detection cuts and energy corrections (see Section 10.3), the π0
invariant mass is reconstructed from two photons, binned in z, x, PT and φ, fit with
65
Fig. 3.21: Photon energy vs. invariant mass. A cut is made on Eγ > 0.15 GeV.
To construct this plot the two photons are distinguished by E1 > E2.
66
pγ (GeV) Cut Width βmin
0.1 - 0.3 3σ 0.90
0.3 - 0.6 3σ 0.92
0.6 - 0.9 3σ 0.92
0.9 - 1.2 3σ 0.92
1.2 - 1.5 3σ 0.92
1.5 - 1.8 3σ 0.92
1.8 - 2.1 2σ 0.94
2.1 - 2.4 1.5σ 0.95
2.4 - 2.7 0.9σ 0.97
>2.7 0.9σ 0.97
Table 3.4: Minimum β cut on neutral particles to identify photons. As momen-
tum increases a tighter cut must be used to remove neutron contam-
ination.
67
Fig. 3.22: θeγ vs invariant mass. A cut is made on θeγ > 12o.
a gaussian function plus polynomial background, and a 3σ cut is applied in each
bin. Every pair of photons is tested. For instance for the case of three photons,
three invariant masses are tested corresponding to each possible pair (IMγ1γ2 ,
IMγ2γ3 , and IMγ1γ3). A comparison was made between events with Nγ ≥ 2 and
Nγ = 2. 750,423 π0s were found with Nγ ≥ 2 and 590,877 π0s were found with
Nγ = 2.
Three techniques for background subtraction were tested. The most reli-
able method is thought to be a subtraction of the background function, which
requires the number of events be calculated by integrating the signal function af-
ter background subtraction between ±3σ using Eq. 3.5, where f(φ) is the function
resulting from the Gaussian fit. Each bin was fit using a linear, quadratic, and
68
Fig. 3.23: XEC vs YEC for all neutral tracks. The part of the plot in color
indicates tracks that are kept, while those in black are cut out by
implementing cuts individually in each U, V, and W plane of the EC
69
Fig. 3.24: Identification of γ’s by fitting the β distribution of neutral tracks in
the EC with a gaussian in each of ten different momentum bins. Cuts
are indicated by the blue lines. The cut is tightened as the momentum
increases to remove the neutron peak, as shown in Table. 3.4.
70
Fig. 3.25: Momentum dependent cut on β of neutral tracks to identify photons.
The fits are shown in Fig. 3.24 and the cut values are given in Ta-
ble 3.4.
71
Fig. 3.26: A comparison of BSAs for π0s using the nominal vs tight cuts on β in
the photon identification.
cubic background function, and that giving the best fit was chosen. This was
done by observation of each fit. The χ2 was considered, but we also took care to
insure that the fit curve was below the data so as to not overestimate the number
of π0s. Some bins were also tested using a double-Gaussian, but this method
was not used because the concavity of the Gaussian background could lead to an
over-estimation of the number of π0s.
The range of the fit was also specified for each individual bin. The second
method tested was the sideband method, which entails counting the number of
events between −6σ and −3σ and adding to that the number of events between
3σ and 6σ. The background is taken as this sum and subtracted from the total
number of events within ±3σ of the Gaussian fit. The third method tested was
to count all events within ±3σ of the Gaussian mean. Before the photon energy
correction there is a significant variation in the invariant mass peak for each bin,
72
Fig. 3.27: Invariant mass of two photons to reconstruct π0s. The distribution is
fit with a gaussian + polynomial background, and a 3σ cut is made
around the gaussian mean. The red curve is the peak, the gray is the
background, and the black is the sum of the two.
but after the correction the peaks are very consistent. These fits are shown in
Fig. 3.27.
An additional cut is implemented on θγγ, the angle between the two photons
used to reconstruct each π0. If the angle is too small, the resolution of the EC is not
fine enough to give a good reconstruction, resulting in an innacurate measurement
of the invariant mass. To remove these photon pairs a cut removing candidates
with θγγ < 5o is included.
Nevents =1
binsize
∫ +3σ
−3σf(φ) dφ (3.5)
73
3.6 DC Fiducial Cuts
Momentum dependent fiducial cuts on θ and φ are made for electrons and charged
pions [33]. This cut is important to constrain the data to a region in CLAS that
can be accurately recreated by simulation. The formula for the cuts is given by
eq. 3.6 for electrons and eq. 3.7 for pions, with the E1-f torus current set to 2250.0
A. There is a great deal of overlap between events removed by this cut and the
geometric cuts in the EC and CC, but the momentum dependent nature of this
cut causes some additional tracks to be removed. For E1-f, I/Imax = 0.6. An
example of the fiducial cut for electrons is shown in Figure 3.28.
θemin = 11.5 +26.0
(pe + 0.5)Imax/I(3.6)
φemax = 22.0 sin(θ − θmin)0.01( ImaxI
pe)1.2
θπ±
min = 8.0 + 20.0(1− pπ8.0
ImaxI
)15 (3.7)
φπ±
max = 28.0 sin(θ − θmin)0.22( ImaxI
pπ)0.15
3.7 Kinematic Cuts
After particles are identified and we are left with only events containing an elec-
tron and at least one good pion, kinematic checks must be imposed to determine
whether each event falls into the SIDIS kinematic region. It is important to
74
Fig. 3.28: Electron θ vs φ for one sector with no fiducial cut (left), the EC
geometric cuts (center), and the full fiducial cuts (right).
eliminate exclusive events as well as any events that do not fall in the deeply-
inelastic region. To select deeply inelastic events cuts are made to keep only
events with W > 2 GeV and Q2 > 1 GeV2. A cut is made to keep only events
with 0.4 < z < 0.7 to exclude events in the current fragmentation region on the
lower side and the exclusive region on the upper side. Exclusive events will have a
missing mass equal to the mass of the exclusive particle produced in the reaction.
The removal of exclusive events is necessary for looking at a semi-inclusive process
in which we do not know the full final state. We cut to keep events with missing
mass greater than 1.2 GeV to exclude exclusive events from processes such as
ep → eπ+n, ep → eπ−∆++ and ep → epπ0 (See Fig. 3.29). Also, charged pions
are kept only if they have energy greater than 1 GeV and photons are kept for
reconstruction of π0 only if they have energy greater than 0.15 GeV. The cut on
photon energy was varied between 0 GeV and 0.3 GeV, and it was found that the
75
cut at 0.15 GeV adequately reduced the number of background events.
In order to determine the correct value of the missing mass cut, AsinφLU was
binned in terms of MX to see at what value a significant deviation was seen,
which would indicate a contamination from exclusive events (see Figure 3.30).
After careful consideration, it was determined that a cut on MX > 1.2 GeV,
combined with the cut on z, provides an adequate separation between exclusive
and semi-inclusive events.
3.8 Kinematic Corrections
3.8.1 Electron Momentum Corrections
Momentum corrections must be implemented to deal with inconsistent momentum
measurements in CLAS due to imperfections in the magnetic field map and drift
chamber misalignments. Momentum corrections for electrons with W > 2 GeV
in E1-f were calculated using Bethe-Heitler events, ep → epγ, which share a
kinematic phase space with the SIDIS events. These events provide a mechanism
for precisely calculating the electron’s momentum as a function of the electron and
proton θ angles, which can be compared to the measured value. The calculated
value of momentum is given by 3.9, and the difference between the measured and
calculated values are fit as functions of φlab to determine the correction in each
bin in W, θe, and φe.
In order for the computed correction to be valid for SIDIS events, it is
76
Fig. 3.29: MX vs z for each pion channel (top π+, middle π−, and bottom π0).
The cuts on MX and z are illustrated on the graphs (MX > 1.2 and
0.4 < z < 0.7). Both of these cuts help to reject exclusive events that
reside at high-z and low MX . For systematic studies the cut is varied
between 1.1 and 1.3 GeV showing a very small dependence on the
variation. As no strong exclusive peaks are seen at MX > 1.2 GeV, it
is concluded that this is an acceptable value for the cut.
77
Fig. 3.30: AsinφLU vs MX . These results are integrated over x, PT , and 0.4 < z <
0.7.
necessary that the kinematic regions of the two processes overlap as much as pos-
sible. Fig. 3.31 shows the kinematic regions covered by Bethe-Heitler, elastic, and
SIDIS events. Bethe-Heitler and SIDIS events share very much of the kinematic
phase-space.
Testing Technique Using Elastic Events
To show that this momentum correction technique works, it was first tested using
the well known elastic collision process, ep → ep. Elastic events are selected
by putting a cut around the proton mass in the W spectrum. The correction is
performed in much the same way as for Bethe-Heitler events. A theoretical value of
momentum is calculated based on the angle θe of the electron, using equation 3.8.
78
Fig. 3.31: The above plots compare the kinematics between elastic (first row),
Bethe-Heitler (second row), and SIDIS (third row) events. It is shown
that Bethe-Heitler and SIDIS events share a kinematic phase space,
making the Bethe-Heitler process a strong candidate for computing
corrections to the SIDIS electron momentum. In the bottom left panel
SIDIS events are those for which Q2 > 1 GeV and W > 2 GeV.
79
Sector mean W uncorrected σW uncorrected Mean W corrected σW corrected
1 0.974 0.056 0.939 0.042
2 0.957 0.052 0.938 0.045
3 0.959 0.054 0.938 0.043
4 0.955 0.044 0.939 0.041
5 0.941 0.046 0.940 0.040
6 0.917 0.063 0.939 0.052
Table 3.5: Elastic missing mass before and after correction.
∆PP
was then fit with a quadratic function to determine the correction, which was
applied in the same manner as for Bethe-Heitler events. See 3.32 and 3.33 for a
demonstration of the effectiveness of this method.
Pe =P
1 + P (1−cosθe)MP
(3.8)
Bethe-Heitler Event Selection
Bethe-Heitler events are those of the type ep→ epγ in which a photon is radiated
by the electron either before or after the collision, as shown in Figure 3.34. Because
our goal is to correct the final momentum, which is changed by the post-radiative
events, we are using only the pre-radiative type of Bethe-Heitler events for this
study. Two cuts are used to select pre-radiative Bethe-Heitler events. The first
80
is a coplanar cut designed to select only elastic events. For these events, the
difference in φ between the two detected particles, ∆φ = φe − φP , should be very
close to 180o. The ∆φ peak at 180o is fit with a Gaussian, and a 1.5σ cut is used
in each bin. The second cut utilizes the fact that the radiated photon will be
-30 -20 -10 0 10 20 30
-0.04
-0.02
0
0.02
0.04
, elastic events, with correction functionφp/p vs ∆pp∆
φ
=2θSector 6, n
-30 -20 -10 0 10 20 30-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05 , correctedφp/p vs ∆pp∆
φ
0.6 0.7 0.8 0.9 1 1.1 1.20
2000
4000
6000
8000 <20θ15<W, Sector 6
W[GeV]
0.6 0.7 0.8 0.9 1 1.1 1.20
1000
2000
3000
4000
5000<25θ20<
W, Sector 6
W[GeV]
0.6 0.7 0.8 0.9 1 1.1 1.20
500
1000
1500 <30θ25<W, Sector 6
W[GeV]
0.6 0.7 0.8 0.9 1 1.1 1.20
100
200
300
400
500 <35θ30<W, Sector 6
W[GeV]
0.6 0.7 0.8 0.9 1 1.1 1.2020406080
100120
<40θ35<W, Sector 6
W[GeV]
0.6 0.7 0.8 0.9 1 1.1 1.20102030405060 <45θ40<
W, Sector 6
W[GeV]
Fig. 3.32: Elastic Events Left: ∆pp
fit with ∆pp
(φ) = A + Bφ + Cφ2 for elastic
events. The top plot is before the correction and shows the quadratic
fit to determine the correction function, and the bottom plot is after
the correction. Right: Missing mass in each θ bin for one sector,
shown before and after the correction. The vertical bar indicates the
known value of the proton’s mass.
81
traveling in a direction nearly parallel to that of the beam. The four-vector of the
radiated particle is calculated from known kinematics, and an angle θγ, the angle
0 5 10 15 20 25 30
0.91
0.92
0.93
0.94
0.95
0.96
0.97sector 1 sector 2 sector 3 sector 4 sector 5 sector 6
mean W
θn
0 5 10 15 20 25 300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
sector 1 sector 2 sector 3 sector 4 sector 5 sector 6
Wσ
θn
Fig. 3.33: Elastic Events The top (bottom) plot shows the mean (sigma) values
from the Gaussian fits to the missing mass spectrums before (blue
squares) and after (red triangles) the correction is applied. Both are
plotted vs the bin number in θ.
82
between the beam and the radiated photon, is calculated from that four-vector.
If θγ > 0.4o, the event is cut. Cuts ranging from θγ of 0.3-1.0 were studied, and
0.4 was chosen because it provides good statistics in all bins while also giving a
very tight cut on Bethe-Heitler events. Post-radiative Bethe-Heitler events will
have much larger values of θ, so contamination due to these events is expected to
be extremely small.
e
P
e'
P'
e
P
e'
P'
Fig. 3.34: Left: Pre-radiative Bethe-Heitler process. Right: Post-radiative
Bethe-Heitler process.
83
Variable Bin Size Number of Bins Range
W 0.1 GeV 10 2.0GeV < W < 3.0GeV
θe 5o 6 15o < θe < 45o
φe 4o 15 −30o < φe < 30o
Table 3.6: Binning for Bethe-Heitler events. Binning is performed in each sector.
The momentum correction is calculated individually in each bin in W, θe,
and φe, the binning of which is described in Table 3.6. Hence, the Bethe-Heitler
event identification is also performed separately for each bin. An example of the
two cuts is shown in Figure 3.35.
Determination of Correction Function
Use of Bethe-Heitler events allows us to calculate a relation for the final electron
momentum based on angle measurements of the detected electron and proton.
With θe = the measured angle of the electron and θP = the measured angle of the
detected proton, the theoretical value of momentum, Pe is calculated as:
Pe =P ′
1 + P ′(1−cosθe)MP
(3.9)
with
P ′ =MP
1− cosθe(cosθe +
cosθP sinθesinθP − 1
)
In each bin, after the theoretical value of momentum is calculated from
84
0 0.5 1 1.5 20
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Sector 22.0 < W < 2.1
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o < 25θ < o20
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Sector 22.7 < W < 2.8
o < 25θ < o20
170 175 180 185 1900200400600800
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o < 25θ < o20
170 175 180 185 1900
100
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300
400
500
600
φ∆
Sector 22.9 < W < 3.0
o < 25θ < o20
Fig. 3.35: Bethe-Heitler event selection. Top: Cut on θγ for one θe bin in each
of the ten W bins. Bottom, 1.5σ cut around ∆φ peak for one θe bin
in each of the ten W bins. The red lines show the cut and the blue
histograms illustrates the given quantity passing the other cut.
the measured angles, the quantity ∆pp
, where ∆p = pmeasured − pcalculated, a two-
dimensional histogram is plotted in terms of φe. For an ideal detector with perfect
momentum measurement, this quantity would give a flat line at ∆pp
= 0. Any
deviation from this nominal situation must be corrected for. To do so, ∆pp
is
sliced into φe bins, each of which is fit with a Gaussian. The mean values of these
Gaussians are then fit with a linear function, f(φe) = a0 +a1φe, which is then used
for the correction. For this study a second order polynomial was also tested, but
85
Fig. 3.36: ∆PP
vs φe is shown as an example for a single bin (2.1 < W < 2.2
and 24o < θ < 29o in sector 2). The correction in other bins is very
similar.
it was found that the linear function provided a better fit to the data. Once the
correction function is determined, the data is then corrected on an event-by-event
basis using equation 3.10:
Pcorrected = Pmeasured − f(φe)× Pmeasured (3.10)
By examination of the ∆PP
distributions before and after the correction,
it can be seen that the slope is decreased when the correction is applied. See
Figure 3.36 for an example of the fits used to determine the correction.
Evaluation by Bethe-Heitler Missing Mass
To further evaluate the effectiveness of the correction, the missing mass distribu-
tion from ep → epX is fit with a Gaussian in each bin where the correction was
86
Sector Mean, No Correction σ, No Correction Mean, Corrected σ, Corrected
1 0.0022 0.0199 -0.0006 0.0194
2 0.0022 0.0196 0.0002 0.0191
3 0.0016 0.0198 -0.0010 0.0196
4 0.0018 0.0198 -0.0005 0.0195
5 -0.0004 0.0197 -0.0007 0.0195
6 -0.0013 0.0199 -0.0007 0.0198
Table 3.7: Bethe-Heitler missing mass before and after the electron correction
is applied. Both sets have the energy loss correction applied to the
protons.
performed. Because Bethe-Heitler events of ep → epγ have been selected for the
correction, we expect the missing mass spectrum to be distributed around zero,
the mass of the photon. If the correction is working properly, the Gaussian peak
after the correction will have a narrower width and a mean value closer to zero.
Each bin in W, θe, and φe was checked, and the correction was fine tuned until
positive results were seen in every bin, as shown in Table. 3.7.
The effect of applying these electron momentum corrections is very small,
and does not affect the final results in a statistically significant way, but it is
important to demonstrate that the corrections are rigorously computed in order
to demonstrate that they are not needed in this analysis.
87
3.8.2 Hadronic Energy Loss Corrections
Charged particles lose energy as they travel through the target material, walls, and
drift chambers, which is not accounted for in the standard event reconstruction.
Energy corrections for charged hadrons are performed using the eloss pro-
gram, the usage of which is described in [34]. The program takes a four-vector,
along with the particle’s mass, vertex, and some information about the experi-
ment’s configuration, and returns a new four-vector with corrected energy.
The eloss program was modified from its standard usage by replacing the
g11 target with the E1-f target geometry and removing the start counter. The
program is then run while setting icell=7 to specify the new target geometry.
The program is then used to modify the momentum four-vector of every charged
hadron.
This type of correction is not necessary for electrons because it is expected
that the energy lost by electrons passing through the detectors will be much
smaller than that of the hadrons, hence causing a much smaller effect.
3.8.3 Photon Energy Correction
The two photon invariant mass distribution is not consistent as a function of en-
ergy due to inaccurate calibration of the calorimeter, so the calculation of photon
energy must be corrected. The correction is a function of energy,
88
Ecorrected =Emeasured
corr(Emeasured)(3.11)
corr(E) = p0 +p1
E+p2
E2
The two photon invariant mass can be written as:
IM(γγ) = 2√E1E2sin
θ
2(3.12)
Once the correction is applied, the invariant mass should be equal to the
known π0 mass for all values of energy.
mπ0 = 2
√E1
corr(E1
)
√E2
corr(E2)(3.13)
which provides:
IM(γγ)
mπ0
=√
corr(E1)corr(E2) (3.14)
Then, if if we select events with E1 = E2, the correction function can be
determined from IM(γγ)mπ0
= corr(E), so the ratio of invariant mass to pion mass is
plotted against photon energy for events with E1 −E2 < 10MeV , and then fit to
determine the correction function (see Figure 3.37).
This correction is based on similar calculation as described in [35].
89
Fig. 3.37: IM(γγ)mπ0
vs Eγ. The first plot shows the fit before the correction for
events in which Eγ1 − Eγ2 < 10MeV, which is used to determine the
correction function. The second plot shows the corrected distribution.
Chapter 4
Simulation
A comparison of the measured data to a simulation is necessary to calculate the
acceptance of CLAS. Acceptance provides the probability of that CLAS will record
an event occurring in each kinematic bin. To compute the acceptance, a Monte-
Carlo simulation is performed. An event generator produces a random assortment
of simulated data, which is passed through a a simulation of the CLAS detector.
The percentage of events that are detected in each measured kinematic bin is
recorded and used to correct the E1-f data.
4.1 Data Simulation
The following programs are used to perform this calculation.
• clasDIS generates an ideal set of data.
• GSim predicts the portion of data that is seen with the CLAS detector.
• GPP introduces smearing, to make the simulation more realistic.
90
91
• userana and bos2mu are used to reconstruct and format the data.
• The exact same data analysis routines used for real data are then used on
the data that has been reconstructed from GSim.
• The ratio of reconstructed to generated numbers of events is then calculated
in each kinematic bin.
4.1.1 Event Generation
Simulated events are generated with realistic physics distributions using the clas-
DIS event generator. clasDIS is based on the LUND particle generation algo-
rithms, which have been modified to be compatible with CLAS kinematics. The
generator accepts input parameters which are fine-tuned to give as close a match
as possible between the kinematics of simulated and real data. A summary of the
control options used is given in Table 4.1.
An option to introduce the Cahn effect [36] into the generated data (which
provides a modulation in cosφ) was also tested, but it was decided to instead use
a weighting procedure to implement the Cahn effect. To perform this procedure
events were generated using a flat φ distribution, and then weighted with the
function
< cosφ >= −(2p⊥Q
)(2− y)
√1− y
1 + (1− y)2
z2
1 + z2(4.1)
< cos2φ >= (2p2⊥
Q2)
1− y1 + (1− y)2
z4
(1 + z2)2(4.2)
92
Option Description
–trig Gives the total number of events to process.
–datf Tells the generator to output a data file.
–outform 2 Tells the generator to format the output for GSim.
–beam 5498 Inputs the beam energy to match E1-f.
–zpos -250 Sets the z-position at -250 mm.
–t 15 60 Sets the range of acceptable θ in degrees.
–parl3 0.7 Sets mean of kT distribution, which is tuned so
output PT matches E1-f.
–lst37 2 Turns on Cahn effect (Not used).
–lst15 145 Defines set of parton distribution functions used in
the simulation.
–pid 211 or -211 Sets which channel to produce with simulation.
211 gives π+ and -211 gives π−.
–z 0.2 Sets minimum z value to 0.2.
–parj33 0.3 Defines the remaining energy below which the frag-
mentation of a parton system is stopped and two
hadrons are formed.
Table 4.1: Control options for clasDIS
93
Fig. 4.1: φh distributions from simulated data binned in z and PT and weighted
with Cahn effect.
to include the Cahn contributions. Fig. 4.2 shows the φ amplitude after the Cahn
weighting.
One billion events were generated to complete this acceptance calculation.
It is important to use a very large sample to minimize the error that is introduced
into the final result due to the acceptance.
94
4.1.2 GSim Detector Simulation
GSim is a software package used for Monte-Carlo simulation that uses GEANT to
simulate the CLAS detector. Each event generated in clasDIS is passed through
this simulated detector geometry, creating a BOS file containing simulated data
that is exactly analogous to those created while running the experiment. The
output file contains both the generated data from the event generator and the
raw GSim, which is stored as tracks through the simulated detector. Figure 4.2
shows an example of one event being tracked through GSim.
Just as clasDIS must be fine-tuned to match E1-f kinematics, GSim must me
set to match the E1-f CLAS configuration. This is done my setting an ffread card
to configure the GSim input parameters. This configured the CLAS geometry,
magnetic field, and target information. E1-f and E1-e used the same target, so
the E1-e target configuration was used in the E1-f analysis. The ffread card used
for E1-f is described in Table. 4.2.
4.1.3 GPP
GSim Post Processing (GPP) is used after GSim to introduce smearing into the
simulated data and make the results more realistic. The program also introduces
detector inefficiencies in the drift chambers and time-of-flight. The smearing pa-
rameters are fine-tuned to match kinematic distributions between simulated and
experimental data. This is important to insure that kinematic cuts used on the
95
Fig. 4.2: GSim output for one simulated event using E1-f kinematics. Red,
curved tracks denote charged particles and grey lines denote neutral
tracks.
96
Input Description
GEOM ’ALL’ Includes all CLAS geometry
NOGEOM ’PTG’ ’ST’ ’IC’ Lists geometry to be excluded.
MAGTYPE 3 Sets magnetic field type to ’ torus + mini from lookup
table .’
CUTS 5.e-3 5.e-3 5.e-3 5.e-3 Kinetic energy cuts in GeV.
CCCUTS 1.e-3 1.e-3 1.e-3 1.e-3 Cuts for the Cerenkov counter
DCCUTS 1.e-4 1.e-4 1.e-4 1.e-4 Cuts for the drift chamber.
ECCUTS 1.e-4 1.e-4 1.e-4 1.e-4 Cuts for the calorimeter.
SCCUTS 1.e-4 1.e-4 1.e-4 1.e-4 Cuts for the time-of-flight detector.
NTARGET 2 Sets the target type to liquid hydrogen.
MAGSCALE 0.5829 0.7495 Sets the scale of the magnetic fields.
RUNG 10 Default setting.
TARGET ’e1e’ Defines target geometry.
TGMATE ’PROT’ ’ALU’ Sets the target material as protons and aluminum.
TGPOS 0.00 0.00 -25.0 Defines the target position.
NOMCDATA ’ALL’ Default setting to turn off additional GEANT hit infor-
mation.
SAVE ’ALL’ ’LEVL’ 10 Save all secondaries up to cascade level 10.
KINE 5 Setting for LUND event generator.
AUTO 1 Automatic computation of the tracking medium param-
eters.
STOP Geant command to end the ffread file.
Table 4.2: FFREAD card for E1-f.
97
abc f
π+ 1.3 1.05
π− 1.0 1.2
Table 4.3: Input parameters for GPP.
experimental data are compatible with the simulated kinematic variables.
Four input parameters are used to tune the GPP smearing, called a, b,
c, and f. The first three are related to particle momentum information, and f
is used to tune the relative time smearing. Several possible values of each of
these parameters are tested and the results are compared to experimental data to
determine the best values.
To tune the smearing parameters, the missing mass from ep→ eπ+X is fit
with a Gaussian distribution around the proton mass peak for several values of
abc and f, and the results are compared to experimental data to determine the
best value. The widths of this fit are plotted as shown in Fig. 4.3. From these
plots it is possible to select the best value of abc by comparison to experimental
data, but no variation is seen in the width due to changing f. To tune f, the
∆t distribution is fit with a Gaussian, and an equivalent procedure if followed to
determine the best parameter. Plots to illustrate this comparison are shown in
Fig. 4.4. Equivalent analysis is performed for π−, and the results for both pion
channels are shown in Table 4.3.
In addition to tuning the abc and f parameters, it is important to match
98
Fig. 4.3: Width of MX fits for various values of abc and f. The horizontal line
represents the width of MX from the experimental data. Based on these
results a value of abc = 1.3 is chosen for this analysis. Missing mass is
not a useful quantity for determining the best value of f.
99
Fig. 4.4: Width of ∆t vs abc and f. The horizontal line represents the width of
∆t in the experimental data. Based on these results a value of f = 1.05
is selected for this data analysis.
as closely as possible the DC occupancies between the experimental and simu-
lated CLAS. This is input into the simulation using a program called PDU. DC
occupancies are shown for one sector before and after GPP is applied in Fig. 4.5.
4.2 Acceptance Calculation
Acceptance is calculated by taking the ratio of reconstructed to generated simu-
lated data in each bin for which data analysis is performed. It is very important
that the data reconstructed by GSim, gpp, and userana match the experimental
data as closely as possible in order to give an accurate determination of the CLAS
acceptance for the measured processes. This is checked by comparing kinematic
distributions in the relevant variables, as shown in Fig. 4.2. The reconstructed
data is seen to behave in a manner consistent with the experimental data from
100
Fig. 4.5: DC occupancies before and after GPP is applied for simulated CLAS
Sector 4.
E1f.
Care is taken to insure that the reconstructed tracks used to compute accep-
tance are read from the correct generated particle by using momentum matching
between the two. Each of the four components of the four-momentum are required
to be within 0.10 GeV. Reconstructed tracks are seen to differ from their gener-
ated counterparts by as much as 0.02 GeV, so 0.10 GeV is wide enough to avoid
mismatch of good tracks, while still being tight enough to remove any mismatches
when applied to each component of the four-momentum. Momentum matching is
applied for all electrons and pions identified from the reconstructed GSim sample.
It is seen that 2.2% of reconstructed electrons passing the electron identification
fail the momentum matching.
101
Fig. 4.6: Comparison of kinematic distributions between simulated data after
reconstruction and experimental data from E1f. The black squares are
E1f data and the blue triangles are from GSim. Plots on the left show
π+ and those on the right show π−.
102
Acceptance is computed using the exact binning used in the data analysis.
The data is binned in x, z, PT , Q2, and φ. To observe the dependence on one
kinematic variable, the data is integrated over all other variables (except the de-
sired variable and φ), and then fit as a function of φ in each bin of that kinematic
variable. Acceptance is computed in the same way. Both the generated and re-
constructed data are integrated to the exact binning used for the experimental
data, and the ratio of reconstructed events to generated events is computed as
a function of φ and that variable. The experimental data is then corrected for
acceptance by dividing the number of events in each kinematic bin by the accep-
tance value for that bin as determined by the simulation. So for a single bin with
acceptance Ai, the corrected number of events is
N ′i =Ni
Ai(4.3)
When computing acceptances a tighter set of requirements of good events
must be used to insure that the agreement between GSim and experimental data is
as close as possible. First, because our event generator only generates events with
z > 0.2, it is necessary to impose this cut on the experimental data as well. (A cut
on 0.4 < z < 0.7 is already used to select SIDIS events, so this cut only influences
observations of the z-dependence.) A tighter set of fiducial cuts are used on p,
θlab and φlab to insure that we are only looking at regions of the CLAS detector
that can be very accurately reproduced by simulation. A higher cut on missing
mass (MX > 1.5 GeV) is also used because the event generator is designed for DIS
103
Fig. 4.7: Acceptance for E1f binned in φ, z and PT .
104
events only, so it is important to be sure that any exclusive events in the generated
sample are removed. An additional cut is made to keep events only with y < 0.8
to avoid radiative effects, and an additional cut is imposed on electron momentum
to keep only pe > 0.9 GeV because events with lower electron momentum are not
reproduced by the simulated data.
If the acceptance for a bin is less than 2%, that bin is not used (which occurs
for some φ values near 0o or 360o). The accepted values of the CLAS acceptance
fall in the range of 2%−20%, peaking at φ of 180o in each kinematic bin.
Chapter 5
Systematic Uncertainties
Systematic errors are derived by analyzing the shifts in a measured quantity due
to a variation of some analytical technique. For this study cuts used in particle
identification are varied to test their result on the final BSAs. Additionally, the
systematic uncertainty due to the fitting of the BSAs is tested by simplifying the
fit function, and the uncertainty caused by the beam polarization measurement
is calculated. Table 5.1 gives a summary of the sources of systematic error along
with their approximate relative contributions to the total uncertainty. The total
systematic uncertainty is estimated to be 0.5% for π+, 0.7% for π−, and 0.9% for
π0. The x-dependence of the systematic error is shown in Fig. 5. Because the
sources of systematic error are assumed to be independent, the uncertainties are
added in quadrature.
5.0.1 Systematic Uncertainty from Variation of Particle ID cuts
Particle identification cuts were varied for both electron and pion id routines. For
the electron identification in the EC, the cuts on the sampling fraction and EC
105
106
Source of Error Variation Average Uncertainty
π+ π− π0
EC Einner Cut 50, 55, 60 MeV 0.0017 X0.0030 0.0003
EC Sampling Fraction 2.0σ-4.0σ 0.0005 0.0016 0.0020
Electron Fiducial Cut Tight, Medium, Loose 0.0011 0.0029 0.0020
Vertex Cut ±0.5 cm 0.0021 0.0029 0.0036
Pion ID ∆t, β 0.0007 0.0028 -
Pion Fiducial Cut Tight, Medium, Loose 0.0018 0.0040 -
Missing Mass Cut 1.1-1.3 GeV 0.0052 0.0029 0.0064
Background Subtraction None, Fit-function, Sideband - - 0.005
Background Asymmetry - - 0.007
Fitting Function 0.0007 0.0011 0.0010
Beam Polarization 0.0003 0.0005 0.0006
Total: 0.006 0.007 0.009
Statistical Error 0.005 0.014 0.012
Table 5.1: Sources of systematic uncertainty. The second column gives the av-
erage relative uncertainty from each source. For comparison, the
average statistical uncertainty is given.
107
Fig. 5.1: Sources of systematic error vs x.
Einner were varied. The sampling fraction tested four regions; -2.5 to 4 σ, -2.75
to 3.75 σ, -3 to 3.5 σ, and -3.25 to 3.25 σ as shown in Figure 5.2. By shifting the
lower and higher limits of each cut together, the number of events resulting from
each cut is similar, with a total range of variation of 0.75 for both the min and
max, varying the range used between 2.5 and 4.0. The min and max were selected
at the largest limits giving a reasonable cut. Wider cuts were tested but it was
decided not to include them in the systematic studies.
The EC inner energy is varied between 50-60 MeV and the vertex cut was
varied by ±0.5 cm as shown in Fig. 5.0.1.
For identification of the π+ and π− channels, the nominal cut on ∆t were
compared to momentum dependent cuts on the β of each track as measured by
108
Fig. 5.2: The above figure shows the EC sampling fraction in one momentum
bin for a single sector. The four colored lines represent the four cuts
used to test the systematic error due to this cut. The cuts used from
right to left are -2.5 to 4 σ, -2.75 to 3.75 σ, -3 to 3.5 σ, and -3.25 to
3.25 σ.
the time-of-flight detector. Overall, these contributions to the systematic error
were found to be quite small in comparison to the statistical uncertainty.
5.0.2 Pion Contamination
A significant contamination of the electron sample by misidentified pions could
be a source of systematic error. Of the electron identification cuts, the one most
sensitive to pion contamination is that on EC Einner. In order to estimate the
magnitude of the systematic error due to pion contamination, this cut was varied
by a very wide margin. The nominal cut is to keep only events with Einner >
109
Fig. 5.3: Variation of vertex cut. Solid red lines show the nominal cut and dotted
blue/green lines show ± 0.5 cm.
55MeV . To test the effect of pion contamination the data was analyzed with
no cut on Einner, and also with a cut at Einner > 100MeV . The data with
Einner > 0 would have the maximum possible pion contamination, and the data
with Einner > 100 should remove nearly all possible pion contamination events. It
was found that the effect due to pion contamination is significantly smaller than
the other sources of systematic error, as shown in Fig. 5.0.2.
An additional check was performed by examining the EC Einner distributions
to compute the fraction of identified electrons that are actually negative pions.
To perform this study the Einner distribution passing other electron identification
cuts was binned in Q2 and W , and each bin was fit with a combination of functions
to differentiate between the low-energy π− peak and the higher energy electrons.
110
Fig. 5.4: Systematic uncertainty due to pion id cuts. The above figure shows the
BSA in each PT bin for π+ using the ∆t cut (blue points, solid line)
and a β cut (red points, dashed).
111
Fig. 5.5: ALU vs x for an extreme variation of the EC Einner cut to test for pion
contamination in electrons passing the particle identification.
112
Fig. 5.6: EC Einner passing other electron identification cuts. The ratio of π− to
electron events in the region of Einner > 55 MeV was determined from
the ratio of the integrals of the fit functions in that region.
The results were integrated from 0.055 GeV to 1.0 GeV, and the ratio of the two
integrals was taken as the fraction of electrons that are actually pions. The pion
fraction was found to be extremely small in all bins, and example of which is
shown in Fig. 5.0.2.
5.0.3 Systematic Uncertainty from Variation of Kinematic Cuts
The precise values for cuts separating SIDIS events from exclusive events are not
well known, so it is useful to check the analysis under different conditions. For
113
this purpose the cut on missing mass from ep → eπ±,0X were varied between
1.1 and 1.5 GeV to test how they affect the asymmetries. A nominal cut of 1.2
GeV was chosen, so a variation of 1.1-1.3 GeV was used to compute systematic
errors. Our goal for systematic errors was to vary the cut in a way that would not
greatly change the number of exclusive events; in other words all cuts studied in
the variation of the cut should remove nearly all exclusive events.
By comparison to Fig. 3.30, which shows AsinφLU binned in MX for the three
pion channels, it is easy to make the assumption that this systematic error should
be larger for π− when changing the MX cut. It is important however to keep in
mind the difference between these two plots. When the data is binning in MX ,
the MX = 1.3 GeV bin only includes events between 1.25 < MX < 1.35 GeV,
which is a small number of events when compared to the number at MX > 1.35
GeV (This can be easily seen from Fig. 3.29). Therefore, an asymmetry shift in
this range of MX need not lead to an equally large shift when comparing cuts at
1.2 or 1.3 GeV in MX .
To test the affect due to contamination by exclusive events, the asymmetry
was computed using a much wider range of MX cuts than was used for the sys-
tematic studies. A cut at MX > 0.85 GeV was used to include all exclusive events,
and compared to the nominal cut at MX > 1.2 GeV. There is also an exclusive
channel that could provide contamination from the process ep→ eπ−∆++, where
m∆++ = 1.23 GeV. Contamination due to this process is not observed. The results
114
Fig. 5.7: AsinφLU vs x for π− in each of the five PT bins using three different missing
mass cuts in the SIDIS event selection.
115
in both cases are in agreement, as is seen in Fig. 5.0.3 and Fig 5.9 for π+ and π−,
demonstrating that the effect due to contamination by exclusive events is very
small.
5.0.4 Systematic Uncertainty from Variation of Fitting Function
For this analysis, the full fitting function of A sinφ1+B cosφ+C cos 2φ
was used, but it is
possible to simplify the analysis by fitting each BSA with the much simpler func-
tion, A sinφ, and extracting the moment from the ’A’ coefficient as was done
with the full function. By comparing the two methods directly in each bin, it
was found that the variation of the fitting function in this way contributes only a
small portion of the full systematic uncertainty.
Other fit functions were used to determine their influence on the final results.
A constant term was added to both the A sinφ and A sinφ/(1+B cosφ+C cos 2φ)
functions, but with negligible effect. Other functions tested include A sinφ +
B sin 2φ, A sinφ/(1 +B cosφ), and A sinφ/(1 +B cos 2φ). No significant changes
were observed due to any of these variations. It is also relevant to note that the
coefficient on the sin 2φ term was always found to be consistent with zero.
5.0.5 Systematic Uncertainty from Beam Polarization
During the E1-f run period, the beam polarization was measured periodically with
a Møller polarimeter, with an average measurement of Pe = 75.1±0.2%, as shown
116
Fig. 5.8: AsinφLU is plotted using a very loose MX cut of 0.85 GeV and using the
nominal cut of 1.2 GeV to remove exclusive events. The small variation
in the results show that contamination from exclusive events is very
small.
117
x0.1 0.2 0.3 0.4 0.5 0.6
LUφ
sin
A
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
Missing Mass Cut>0.85 GeVXM>1.2 GeVXM>1.4 GeVXM
-π vs x, LUφsin
A
Fig. 5.9: ALU for π− comparing the nominal missing mass cut on MX > 1.2 GeV
to the data sample with no exclusive events removed (MX > 0.85 GeV)
and to a cut above the mass of the ∆++ resonance (MX > 1.4 GeV).
118
Fig. 5.10: BSA vs φ for one bin in z for π0 comparing two fitting functions to
measure the moment. The solid line fits the BSA with A sinφ1+B cosφ+C cos 2φ
and the dashed line fits the BSA with A sinφ, where in both the A
coefficient gives the value for AsinφLU in that bin. The p0 in the fit-
parameters box is the A value resulting from the full fit function, and
the ∆A value printed is the difference between the previous value and
that obtained using the sinφ fit.
119
in Fig. 5.0.5. The quoted uncertainty here is that due to the variation between
measurements, given by:
δP =1
N(∑i
|Pi − P |)1/2 (5.1)
It is well known that the systematic uncertainty of the target polarization
is 1.4% (relative). The combined atomic motion and finite acceptance contribute
another 0.8% (relative) [37]. These sources of error must be accounted for in each
bin and applied to the uncertainty on ALU , which is done with the relation:
∆ALU =∆PePe
ALU (5.2)
Here ∆PP
is computed from the above uncertainties to be ∆PP
= (0.002 +
0.014+0.008)/0.751 = 0.032, so then the uncertainty in each bin is given by δA =
0.032A. These values are averaged over all bins to give the value in Table. 5.1.
5.0.6 Random Helicity Study
The data was tested by assigning a random value for helicity to each event, and
using this to calculate BSAs, which should be consistent with zero. The data set
was randomized by calling a C++ function that returns either ±1 for each event
and assigning that value as the events helicity. The consistency of this function
was checked and it was found that over the entire data sample the function gave
an equal number of positive and negative helicity events with a precision of better
120
Fig. 5.11: Møller measurements of electron beam polarization vs E1-f run num-
ber. The horizontal line shows the average polarization value of
Pe = 0.751 that was used in this analysis.
121
Fig. 5.12: BSAs using a randomly generated value for helicity (left). The blue
squares show the BSA from randomly generated helicity and the open
red circles show the normal BSAs. The plot on the right shows the
results of ALU vs. z using randomly generated helicity. It is expected
and shown that the results using random helicity should be consistent
with zero.
than one tenth of one percent. Using this ”fake” asymmetry data, beam-spin
asymmetries were then calculated and binned in z, x, and two dimensionally in x
and PT . A beam polarization is used that is equal to that measured for the real
data. As can be seen in Figure 5.12, the ”fake” asymmetries are all in agreement
with zero, as is expected. The test was also performed by replacing the randomly
generated helicities with alternating helicity in which the first of each pair was
always picked to be +1 and the second -1. The results were very similar and still
consistent with zero.
122
5.0.7 Comparison to Simulation
The analysis procedure was tested on simulated data produced with the clasDIS
event generator. clasDIS utilized the LUND physics event generation routines to
produce simulated data in the CLAS kinematic regime. This data was given a
random helicity and was weighted with an input value of < sinφ > of 0.3 in every
bin. The analysis and fitting procedures were then applied in exactly the same
manner as for the experimental data in an effort to extract the input value from
the data. The fit results are shown in Fig. 5.13 from which it is seen that the
analysis procedure yields values in agreement with 0.3 in every bin.
5.0.8 Acceptance Effects
CLAS data can be effected by acceptance if detector inefficiencies in a kinematic
bin cause a change the results for that bin. For beam-spin asymmetries, these
effects are expected to be negligible because the acceptance effects for N+ and
N− are very similar as long as the bin-size is sufficiently small. If a bin contains
N events, the bin content after the acceptance correction is N ′ = N/A. Eq. 5.3
shows that if the acceptance for N+ and N− are the same, the BSA remains
unchanged.
BSA′ =N ′+ −N ′−
N ′+ −N ′−=N+/A−N−/AN+/A+N−/A
=N+ −N−
N+ +N−= BSA (5.3)
To show that acceptance does not affect the BSA, it is necessary to show
123
Fig. 5.13: Simulated SIDIS data weighted with < sinφ >= 0.3. The fits extract
the input value for ALU in every bin.
124
that acceptance is not helicity-dependent. A Monte-Carlo simulation was used to
determine the acceptance for each helicity state, and the ratio of the acceptances
is shown in Fig. 5.14. The ratio of acceptances between positive and negative
helicity events is consistent with one in every kinematic bin.
5.0.9 Beam-charge Asymmetry
The beam-charge asymmetry (BCA) could be a source of universal systematic
error to the BSA. The BCA is computed similarly to the BSA, but without con-
sideration for a specific physics process. We take N+ here as the total number of
events with positive helicity and N− as the total number of events with negative
helicity. The BCA is then computed as shown in Eq. 5.4.
BCA =N+ −N−
N+ +N−(5.4)
Ideally the BCA should be zero if exactly the same number of positive and
negative helicity events are sent from the accelerator. If there is a small surplus
of events with one helicity state, that can give a systematic error to the SIDIS
BSAs. Fig. 5.0.9 shows the BCA for E1-f vs run number. It can be seen that it is
consistently compatible with zero. The integrated BCA over the entire run period
is 0.00331±0.00005, which is consistent with the other sources of systematic error.
Since the magnitude of the BCA is on the order of 10% of the BSA, it is
necessary to determine whether or not there is an observable contribution from
125
Fig. 5.14: The ratio of acceptances of positive and negative helicity binned in
z, PT , and φ. It is seen that the ratio of acceptances is in agreement
with unity in every bin.
126
Fig. 5.15: Beam-charge asymmetry vs run number for E1-f.
x0.1 0.2 0.3 0.4 0.5 0.6
LUφ
sin
A
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
BCA
>0
<0
+π vs x, LUφsin
A
Fig. 5.16: ALU from runs with BCA > 0 compared to BCA < 0.
the BCA to the systematic error. This is done by dividing the data into two
regions; one set uses all runs with a positive BCA and the other set uses all runs
where the BCA is negative. Fig. 5.16 shows AsinφLU for runs with positive BCA and
runs with negative BCA. The beam-charge asymmetry is not seen to cause a large
systematic effect.
127
x0.1 0.2 0.3 0.4 0.5 0.6
LUφ
sin
A
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
Vertex Range
-26<z<-24
z<-26 and z>-24
+π vs x, LUφsin
A
x0.1 0.2 0.3 0.4 0.5 0.6
LUφ
sin
A
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
Vertex Range
-26<z<-24
z<-26 and z>-24
-π vs x, LUφsin
A
Fig. 5.17: ALU for π+ (left) and π− (right) with the data divided into two sam-
ples. The first uses −26 < z < −24 cm (black), and the second uses
z < −26 cm and z > −24 cm (red). The two samples yield consistent
results.
5.0.10 Split Data
An estimate of the systematic error was made by splitting the data into two
separate samples; the first with vertex events in −26 < Z < −24 and the second
with Z < −26 and −24 < Z. Each of the two samples contain close to half of the
total statistics. The analysis is carried out independently for each sample, and
the two values are statistically consistent with eachother, as shown in Fig. 5.17.
Chapter 6
Physics Analysis
6.1 Beam-spin Asymmetries and sinφ Moment
Because AsinφLU is helicity dependent, it may be extracted from measurement of the
beam-spin asymmetries (BSA). To measure BSA’s, events are recorded in each
kinematic bin separately for positive and negative helicity events. The BSA is
calculated as in eq. 6.1, where N+ is the number of events with positive helicity,
N− is the number of events with negative helicity, and Pe is the polarization of
the electron beam, 75.1±0.2% for the E1-f dataset. Detector acceptances and
radiative corrections are not expected to significantly affect the BSA’s. Each
N i would be modified by the same acceptance correction in the numerator and
denominator of the BSA, so to first order these correction terms cancel.
BSA =1
Pe
N+ −N−
N+ +N−(6.1)
The statistical uncertainty on the BSA is calculated in a standard way,
starting from the the error on the the number of events in each bin, δN± =√N±,
128
129
Variable Number of Bins Range
z 8 0 - 0.8
PT 5 0.0 - 1.0 GeV
x 5 0.1 - 0.6
Q2 5 1.0 - 4.5 GeV2
φ 12 0o − 360o
Table 6.1: Kinematic binning of E1-f data. Data is binned five-dimensionally in
z, PT , x, Q2 and φ.
the + or - referring to the beam helicity. The error on the asymmetry, δA, is given
by:
δA =
√(∂A
∂N+)2(δN+)2 + (
∂A
∂N−)2(δN−)2 (6.2)
where the two required derivatives are given by:
∂A
∂N±=
±1
N+ +N−− N+ −N−
(N+ +N−)2(6.3)
Then inserting 6.3 into 6.2, the uncertainty for each bin is given by:
δA =
√1− A2
N+ +N−(6.4)
For each kinematic bin in x, z, PT , and Q2 (integrated over other variables),
the BSA distribution is fit with a function derived from the SIDIS cross-section
130
given by eq. 6.5 in terms of φh, where the value of the sinφ moment for this bin is
given by the coefficient on sinφ. Fits with the function AsinφLU sinφ were also tried
as a test of systematic errors, which yielded very similar results. The fitting is
performed in ROOT using the TMinuit class, which utilizes a χ2 minimization to
fit the desired function. The quoted errors are those provided by the Minuit fit,
which are computed using the χ2 of the fit.
AsinφLU sinφ
1 +Bcosφ+ Ccos2φ(6.5)
The beam-spin asymmetries are fit with a χ2 minimization using the MI-
NUIT algorithms. Here the χ2 is defined as
χ2 =∑i,j
(xi − yi(a))Vij(xj − yj(a)) (6.6)
where Vij is the inverse of the error matrix. In the simple case where Vij is
diagonal this simplifies to the usual expression
χ2 =∑i
(xi − yi(a))2
σ2i
(6.7)
where the σ2 are the inverse of the diagonal elements of V , and σ is interpreted
as the error on the corresponding value of x.
Nominally, MINUIT determines the statistical error on fitted parameters
by taking the inverse of the second derivative matrix, assuming parabolic behav-
ior using the HESSE algorithm. This method is strong if the errors on the fit
131
parameters are not correlated and the matrix is diagonal. Since the fit function
is A sinφ1+B cosφ+C cos 2φ
, the errors on the three fit parameters are correlated, leading
to possible non-linearities. It is also necessary to impose limits on the B and C
paramaters in the fitting function to insure they provide physically meaningful
values and prevent divergences in the fit. (Based on physical considerations the
B and C paramaters are expected to be ≈ −0.1 < B,C < 0.1, so limits of ±0.3
are used to allow adequate fluctuation.) These limits cause the error matrix to be
non-diagonal, making the matrix approach less accurate. In this case the HESSE
routine does not provide the best possible method. Instead, the MINOS technique
is used to provide a more accurate description of the error. In general, MINOS
will provide the same or a slightly larger value for the error on each parameter
than will HESSE.
MINOS computes the error using a non-parabolic χ2 method. The error-
matrix approach would use the curvature at the minimum and assume a parabolic
shape, which is not always the case. The MINOS approach determines where the
function crosses the function value by following the function out through the
minimum, leading to a more accurate calculation of the error. It is possible for
this method to yield non-symmetric errors, but for this analysis the errors are
assumed to be symmetric.
Hypothesis testing is used to objectively measure the goodness-of-fit. Our
hypothesis, H0, is defined as the statement: The data is consistent with our fit
132
function. This is evaluated by comparing the χ2 distribution of the fit results
to the χ2 p.d.f. for eight degrees of freedom (given 12 data points and 3 fit
parameters, the number of degrees of freedom is given by ν = 12 − 3 − 1 = 8).
The p.d.f. for ν = 8, normalized to 118 entries and a bin size of 2 is
f(x) =2× 118
96x3e−x/2 (6.8)
which is shown in Fig. 6.2.
The χ2 is used to compute compute a p-value for the hypothesis, where
the p-value is the probability that an observed χ2 exceeds the expected value by
chance. The p-value is computed by
p =∫ ∞χ2
f(x; ν)dx (6.9)
where f(x;nd) is the p.d.f. and ν is the number of degrees of freedom. Fig. 6.1
shows the computed p-values vs χ2. A significance level of 0.003 is set, so any fit
resulting in a p-value less than 0.003 is removed. Our significance level was set
to 0.003 because this value provides adequate separation between the strong and
the poor fits, as well as removing all fits for which the χ2 distribution is less than
1. The impact of significance levels set to 0.01 and 0.05 were also tested, but it
was concluded that these conditions would admonish fits with χ2 values that fit
into the expected distribution and accurately described the data.
Fits to determine AsinφLU in each x and PT bin are shown in Figures 6.4 - 6.6 in
133
Fig. 6.1: p-value vs. χ2 for each fit. Fits resulting with a p-value less than
our significance level of 0.003 do not confirm our hypothesis and are
removed.
134
Fig. 6.2: χ2 distribution for fits to beam-spin asymmetries for fits with a p-value
greater than 0.003.
the appendix. The results from each bin are then plotted so dependence of AsinφLU
on x, and PT may be measured, and comparisons made between the three pion
channels (see Fig. 6.7). It is also useful to examine the dependence of AsinφLU on x,
PT , z, and Q2 individually by integrating over all other kinematic variables before
computing the beam-spin asymmetries. These results are shown in Figures 6.8
- 6.11.
One advantage of the fitting method over a moment method in determining
AsinφLU is that the fitting method does not require a full coverage in φ. In the
moment method, incomplete φ coverage introduces large uncertainties because it
is not possible to complete the integral in φ over the full 2π. The fitting method
135
is capable of extracting reliable results by fitting the function to only a portion of
the full range in φ.
The following are fits to beam-spin asymmetries. Fig. 6.1 shows the BSAs
binned in x and φ and Figs. 6.5- 6.6 show the BSAs binning in x, PT , and φ. Fits
were also performed using one-dimensional binning in z, PT , and Q2. These results
are very similar. The fit function for all BSA fits displayed is A sinφ1+B cosφ+C cos 2φ
.
Fig. 6.3: BSAs vs φ, binned in x for π+ (top row), π− (center row), and π0
(bottom row).
136
x0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
[GeV
]TP
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 0.8402χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.02
/ndf = 2.6272χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.02
/ndf = 2.4012χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.02
/ndf = 1.9522χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 2.0802χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.02
/ndf = 2.3362χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.03
/ndf = 0.9682χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 0.4762χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.03
/ndf = 1.9842χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 0.6182χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.02
/ndf = 0.2102χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 1.1912χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.03
/ndf = 2.0622χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 0.4592χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.05
/ndf = 0.3812χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 1.1182χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.00±A = 0.04
/ndf = 0.5322χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 0.6572χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.04
/ndf = 0.5062χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 1.1612χ
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.04
/ndf = 1.3742χ
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.02
/ndf = 0.9482χ
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1 0.02±A = 0.02
/ndf = 1.0132χ
φ
, p>0.003+π vs x, TP
Fig. 6.4: Fits to BSAs for π+. Fits with a p-value < 0.003 are ignored (though
all fits shown here for π+ pass this criteria.
137
6.2 Comparison to other data
Previous results for AsinφLU in pion production have been shown by CLAS for
π+ [38], [39] and π0 [40], [41], as well as by HERMES with low statistics in all
three pion channels [42]. The present data is in good agreement with all three
previous measurements, and will provide an improvement in both statistics and
kinematic range.
6.2.1 Comparison to E1-6 for π+
For purpose of comparison, data for π+ was binned in x using the same bin size as
was used for the CLAS E1-6 data, and integrated over all other variables to give
a direct comparison. The E1-6 data was taken using a beam energy of 5.7 GeV
and E1-e used a beam of 4.3 GeV (The beam energy for E1-f was 5.498 GeV).
Figure 6.12 shows the E1-f AsinφLU vs x plotted with the π+ data from E1-6 and
E1-e.
6.2.2 Comparison to e1-dvcs for π0
CLAS has recently published data on AsinφLU for π0 from the e1-dvcs run period.
This data has very good statistics and will provide a solid basis for comparison
to E1-f. The analysis of e1-dvcs utilized multi-dimensional binning in PT and
x that is very similar to the binning used in the present analysis. The primary
experimental difference is that e1-dvcs utilized an inner calorimeter in addition to
138
x0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
[GeV
]TP
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1
φ
, p>0.003-π vs x, TP
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.02
/ndf = 0.6742χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.02
/ndf = 0.1842χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 1.2482χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.04
/ndf = 1.1862χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.02
/ndf = 0.9452χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.02
/ndf = 0.6352χ
φ
0 50 100 150 200250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 1.7612χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 2.8472χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 1.8472χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.02
/ndf = 2.7652χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.01
/ndf = 0.9182χ
φ
0 50 100 150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.03±A = 0.01
/ndf = 0.3982χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.03±A = -0.02
/ndf = 0.4032χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.01
/ndf = 1.7112χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = 0.00
/ndf = 0.4922χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1 0.01±A = -0.03
/ndf = 1.1012χ
φ
0 50 100150 200 250 300 350-0.1
-0.05
0
0.05
0.1
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1 0.03±A = -0.02
/ndf = 0.7102χ
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1 0.02±A = -0.01
/ndf = 0.8312χ
φ
0 50 100 150 200 250300 350-0.1
-0.05
0
0.05
0.1
φ
Fig. 6.5: Fits to BSAs for π−. Fits with a p-value < 0.003 are ignored.
139
x0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
[GeV
]TP
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 100 200 300-0.1
-0.05
0
0.05
0.1
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.04
/ndf = 2.6172χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 Fit Ignored
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.02±A = 0.03
/ndf = 1.2582χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1
φ
0 100 200 300-0.1
-0.05
0
0.05
0.1
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.02
/ndf = 0.9832χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 3.0282χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 Fit Ignored
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1
φ
0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 1.9982χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.00±A = 0.01
/ndf = 1.0642χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.02
/ndf = 2.4192χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.03
/ndf = 2.9552χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1
φ
0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 1.8842χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.00±A = -0.01
/ndf = 2.6702χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 2.2152χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = 0.01
/ndf = 1.3582χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1
φ
0 100 200 300-0.1
-0.05
0
0.05
0.1 Fit Ignored
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.01±A = -0.01
/ndf = 0.5762χ
φ 0 100 200 300-0.1
-0.05
0
0.05
0.1 0.02±A = 0.01
/ndf = 0.2242χ
φ
, p>0.0030π vs x, TP
Fig. 6.6: Fits to BSAs for π0. Fits with a p-value < 0.003 are ignored.
140
Fig. 6.7: AsinφLU vs x in different PT bins. The error bars represent statistical errors
and the shaded regions at the bottom represent systematic errors.
141
Fig. 6.8: AsinφLU vs z for each pion channel and integrated over the other variables.
The expected range in z for SIDIS kinematics 0.4 < z < 0.7. The
shaded regions denote systematic errors.
Fig. 6.9: AsinφLU vs x for each pion channel and integrated over the other variables.
The integrated range in z for SIDIS kinematics 0.4 < z < 0.7. The
shaded regions denote systematic errors.
142
Fig. 6.10: AsinφLU vs PT for each pion channel and integrated over the other vari-
ables. The integrated range in z for SIDIS kinematics 0.4 < z < 0.7.
The shaded regions denote systematic errors.
Fig. 6.11: AsinφLU vs Q2 for each pion channel and integrated over the other vari-
ables. The integrated range in z for SIDIS kinematics 0.4 < z < 0.7.
The shaded regions denote systematic errors.
143
Fig. 6.12: AsinφLU vs x using binning to match the E1-6 data. The black square
points indicate data from the current analysis of E1-f. The blue circles
are the most recent published CLAS data from E1-e, and the red
triangles show CLAS data from E1-6.
x < PT > < Q2 > < z > ALU (×10−2)
0.09-0.15 0.505 1.122 0.506 1.57±0.33
0.15-0.21 0.460 1.375 0.505 2.50±0.27
0.21-0.26 0.406 1.667 0.500 2.68±0.16
0.26-0.32 0.359 1.967 0.491 2.86±0.18
0.32-0.38 0.329 2.350 0.483 2.59±0.22
0.38-0.44 0.303 2.792 0.476 3.21±0.40
0.44-0.49 0.276 3.272 0.468 3.23±0.65
0.49-0.55 0.246 3.773 0.457 3.24±0.85
Table 6.2: E1-f data binned in x using binning to match E1-6 and integrated
over all other variables. The table shows the average value of several
kinematic variables in each bin.
144
the standard CLAS electromagnetic calorimeter, which greatly improves detection
of photons at low angles, while E1-f will miss most of these low-angle photons as
only the EC is used in photon detection. Analysis for the two datasets is very
similar, with the primary exception being on the missing mass cut to exclude
exclusive events. E1-dvcs used a cut on MX > 1.5 GeV and E1-f is using a cut
on MX > 1.2 GeV. In addition to nearly doubling the total statistics for π0s, the
lower value of this cut extends the kinematic range into higher PT . The reason one
would use the higher cut is to exclude exclusive events of the type, ep→ e∆+π0,
though no peak due to these events are not observed in the E1-f data. E1-dvcs
sees a significant peak due to these exclusive events. That dataset has enough
statistics to make this an acceptable loss, but for E1-f it is necessary to keep as
many good events as possible. Since no ∆+ peak is observed, it was preferable to
use the cut at 1.2 GeV. A comparison of AsinφLU for π0 between E1-f and e1-dvcs is
given in Figure 6.13.
6.2.3 Comparison to HERMES for π+, π−, and π0
In 2006 the HERMES collaboration published data for AsinφLU in all three pion
channels [42], which is currently the only published data for π−. Their experiment
utilized a 27.6 GeV polarized positron beam on a gaseous hydrogen target. The
data is shown one-dimensionally vs x, PT , and z. The statistics are much lower
in every pion channel than those for E1-f, and the kinematic range covers a lower
145
Fig. 6.13: Comparison of AsinφLU vs x in five bins in PT for π0s between the E1-f
and e1-dvcs datasets. The black squares represent the measurement
from E1-f and the red triangles represent the points from e1-dvcs. The
large discrepancy in the first PT bin is due to the fact that e1-dvcs
has significantly better coverage in low-PT due to the addition of the
EC, so the fits in that region are much more accurate.
146
region in x (< x >= 0.10). While the results of the two data sets are consistent
with each other, the current analysis will provide a significant extension in data
statistics and kinematic range. In particular the HERMES π− data does not
provide a conclusive measurement of the sign of AsinφLU , but from E1-f it can be
seen to be negative. Figs. ?? and 6.14 show a comparisons between the results of
the two experiments. In fig. 6.14 the results are scaled by a factor of < Q > /f(y),
where f(y) is given by
f(y) =y√
1− y1− y + y2/2
(6.10)
This expression is motivated by the kinematic terms relating AsinφLU to the
structure function F sinφLU .
dσUUdxdydz
≈ (1− y + y2/2)f1(x)D1(z) (6.11)
dσLUdxdydzdφhdP 2
h⊥= λey
√1− y sinφF sinφ
LU (6.12)
AsinφLU =
σLUσUU
≈ f(y)F sinφLU (6.13)
F sinφLU is twist-3, so it goes as 1/Q. Hence weighting by < Q > /f(y)
provides access to a quantity that should be independent of the experimental
parameters [43].
147
Fig. 6.14: Comparison of AsinφLU vs x between several datasets, each scaled by a
factor of < Q > /f(y) where f(y) is given by Eq. 6.10.
The data is compared to a model described in [44], which takes into account
only the contribution of the e(x) ⊗ H⊥1 term to the sinφ moment, as shown
in Fig. 6.15, where [45] is used to model the Collins contribution. The model
prediction is computed for the E1-f kinematics. The opposite sign of the two
charged pion channels is accurately predicted by the model, but the difference in
scale for π+ and π0 in particular suggests that the other three contributions to
the structure function must also play relevant roles.
6.3 cos 2φ and cosφ Moments
Future analysis will include the extraction of the cosφ and cos 2φ moments, AcosφUU
and Acos 2φUU , can be extracted from fits to acceptance-corrected φ distributions.
The two unpolarized moments are highly susceptible to influence from CLAS
acceptance and radiative effects. It is possible to make a cleaner measurement
148
Fig. 6.15: Comparison of measurement to a theoretical model taking into ac-
count only the contributions due to the e(x)⊗H⊥1 term.
of h⊥q by computing a quantity ∆Acos 2φUU = Acos 2φ,π+
UU − Acos 2φ,π−
UU , which removes
much of the contribution from radiative effects. These two moments are measured
only for the charged pion channels because acceptance for photons has not been
computed. Described here are very preliminary results for the two unpolarized
moments.
Because the unpolarized moments are sensitive to acceptance, it is necessary
to impose tighter criteria on event selection in order to insure that the event sample
falls in a region of CLAS that is very accurately reproduced by GSim and that
the physics is very accurately simulated in the event generator. To accomplish
this, events are kept only with z > 0.2 because low z events are not produced
149
by clasDIS, and a cut is made to keep only events with y < 0.8 to minimize
contributions from radiative effects. The cut on electron momentum is increased
from 0.6 GeV as required by the EC threshold cut to 0.9 GeV to insure an overlap
in kinematics with the event generator. It is also necessary to utilize tighter
fiducial cuts in order to insure that the events measured are in a region where the
CLAS detector is very accurately simulated in GSim. The fiducial cuts used for
electrons are
16o +26.0
(pe + 0.5)Imax/I< θe < 68o − 17pe (6.14)
φe < 16.0o sin(θ − θmin)0.01( ImaxI
pe)1.2
For pions, the nominal set of fiducial cuts is adequate, but it is necessary to
also impose a cut on θπ as a function of pπ to remove a region of the phase space
in which the experimental and simulated data do not overlap. This cut is
θπ < 10o +30000
(pπ + 4)4(6.15)
The momentum dependent θ cuts are shown in Fig. 6.3.
The AUU terms are computed by fitting the acceptance corrected φ distri-
butions in each bin using the function
A(1 +B cosφ+ C cos 2φ) (6.16)
as shown in Fig. ??. These fits are performed using a χ2 minimization in Minuit,
exactly as described for the fits to BSAs described in the previous section. Small
150
Fig. 6.16: Momentum dependent cut on θ for electrons and pions in order to
precisely match the phase space of the experimental and simulated
data samples. Electrons are shown in the left two plots and pions
on the right. The upper plots show experimental data and the lower
plots show GSim data. The cuts are denoted by the red curves in each
plot.
151
abnormalities in acceptance can cause large instabilities in these fits, resulting in a
breakdown of the χ2 minimization, so it is necessary for the acceptance calculation
to be very precise in order to get reasonable results.
6.3.1 Comparison to previous CLAS results
The cosφ and cos 2φ moments of pion electroproduction in SIDIS have been pub-
lished by CLAS for π+ [46] and unpublished results exist for π−, both from the
E1-6 run period. These results use an alternative set of definitions for the SIDIS
cross section, which they define by
d5σ
dxdQ2dzdpTdφ=
2πα2
xQ4
Eh|p|||
ζ[εH1 +H2 +(2−y)
√κ
ζcosφH3 +κ cos 2φH4] (6.17)
using κ = 11+γ2
, γ = 2xMP√Q2
, and ζ = 1 − y − 14γ2y2. Based on this definition, the
two unpolarized moments are defined as
< cosφ >= (2− y)
√κ
ζ
H3
H2 + εH1
(6.18)
< cos 2φ >= κH4
H2 + εH1
(6.19)
The values quoted in the e16 paper are actually not the moments directly,
but instead the related structure functions H3
H2+εH1and H4
H2+εH1. By comparing
these to the standard definitions shown in Eq. 1.22 and Eq. 1.21, it is seen that
152
Fig. 6.17: Acceptance-corrected φ distributions are fit with the function A(1 +
B cosφ+C cos 2φ), where for each bin B is extracted as AcosφUU and C
is taken as Acos 2φUU .
153
Fig. 6.18: Comparison of 1κAcos 2φUU for π+ for a single bin in PT between e1f and
e16.
our measurements must first be modified by computing 1κAcos 2φUU and 1
2−y
√ζκAcosφUU
before comparison with these previous results.
Based on preliminary analysis, the values of Acos 2φUU measured for π+ in e1f
are in agreement with the structure functions measured from e16, as shown in
Fig. 6.18. This is shown for one bin in PT in which the measurements overlap.
Further analysis will expand this comparison into the full PT vs z binning, and
make comparisons for π−.
Chapter 7
Conclusion
AsinφLU has been measured with good statistics in all three pion channels by fitting
beam-spin asymmetries in different kinematic bins as a function of φh and extract-
ing the coefficient on the sinφ term. Assuming the Collins mechanism dominates,
it is expected for π+ and π− to be of opposite sign. The π0 results should be the
same sign as π+, and isospin symmetry predicts that the magnitude of π0 will
be roughly a weighted average of those from the two charged pion channels. The
expected flavor separation is clearly seen in these results.
Preliminary results that were shown in [48] have been updated to include
a better particle identification of both electrons and hadrons, fiducial cuts, and
kinematic corrections. The new results are in agreement with previous CLAS re-
sults shown for π+ [38] and for π0 [40], as well as those published by the HERMES
collaboration for all three pion channels [42]. Some work has been done to extract
twist-3 functions from the previously existing data [49], but better statistics are
needed for model-dependent studies of TMDs [50], [51]. E1-f provides a significant
upgrade in statistics over each of [38] and [42]. This will be the first CLAS results
154
155
to show BSA’s for all three pion channels from the same dataset, which minimizes
systematic errors and allows for the opportunity for a better understanding of the
flavor dependence of the effect. This is also the first CLAS result to show AsinφLU
for π−.
Because F sinφLU is entirely twist-3, the commonly used Wandzura-Wilczec
approximation would remand the entire asymmetry to zero. The measurement
of the BSA at the order of 3% leads to the conclusion that quark-gluon-quark
terms are sizeable and should be considered. Because the structure function is
entirely twist-3, it improves our knowledge of quark-gluon-quark correlations in
the nucleon.
Analysis of AcosφUU and Acos 2φ
UU is ongoing. So far the preliminary results for π+
support those previously published by CLAS that show a positive < cos 2φ > at
low z and high PT . Further analysis will be performed to extract these quantities
from the E1-f dataset, and experiments are planned at CLAS12 that will access
these quantities for both pions and kaons [52].
In the coming decades, the study of TMDs will play an important role in our
understanding of hadronic physics. A solid understanding of twist-3 fragmentation
functions and TMDs will be very important for development of the physics case
for future facilities such as the proposed EIC [53], and the measurements discussed
in this dissertation have improved our understanding of these factors.
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Appendix A
Data Tables
159
160
< z > < x > < PT > < Q2 > AsinφLU
, π+ AsinφLU
, π− AsinφLU
, π0
0.05 0.24 0.18 1.80 0.0059 ± 0.0004 ± 0.0004 -0.008 ± 0.001 ± 0.000
0.15 0.26 0.29 1.84 0.0071 ± 0.0009 ± 0.0002 -0.007 ± 0.002 ± 0.001
0.25 0.27 0.37 1.87 0.007 ± 0.001 ± 0.001 -0.010 ± 0.001 ± 0.000 0.019 ± 0.002 ± 0.003
0.35 0.27 0.42 1.90 0.011 ± 0.001 ± 0.001 -0.013 ± 0.002 ± 0.000 0.020 ± 0.002 ± 0.005
0.45 0.28 0.45 1.92 0.015 ± 0.001 ± 0.001 -0.012 ± 0.002± 0.003 0.022 ± 0.002 ± 0.003
0.55 0.28 0.47 1.93 0.029 ± 0.002 ± 0.003 -0.007 ± 0.003 ± 0.001 0.012 ± 0.004 ± 0.006
0.65 0.29 0.48 1.93 0.039 ± 0.005 ± 0.000 -0.017 ± 0.005 ± 0.002 0.019 ± 0.005 ± 0.007
0.75 0.29 0.48 1.95 0.049 ± 0.005 ± 0.002 -0.028 ± 0.007 ± 0.003 0.013 ± 0.011 ± 0.005
0.85 0.30 0.44 1.95 0.052 ± 0.003 ± 0.003 -0.041 ± 0.011± 0.002
0.95 0.30 0.29 1.93 0.066 ± 0.003 ± 0.004 -0.049 ± 0.034 ± 0.008
0.51 0.14 0.52 1.27 0.018 ± 0.003 ± 0.004 -0.014 ± 0.006 ± 0.004 0.012 ± 0.004 ± 0.011
0.51 0.24 0.45 1.69 0.023 ± 0.003 ± 0.001 -0.008 ± 0.003 ± 0.002 0.016 ± 0.003 ± 0.003
0.51 0.34 0.40 2.17 0.026 ± 0.003 ± 0.001 -0.008 ± 0.003 ± 0.003 0.023 ± 0.003 ± 0.006
0.51 0.44 0.38 2.91 0.027 ± 0.002 ± 0.003 -0.010 ± 0.006 ± 0.003 0.026 ± 0.005 ± 0.003
0.51 0.54 0.36 3.78 0.026 ± 0.006 ± 0.002 -0.019 ± 0.015 ± 0.004 -0.015 ± 0.015 ± 0.007
0.52 0.32 0.10 1.94 0.017 ± 0.004 ± 0.003 0.012 ± 0.008 ± 0.004 0.041 ± 0.010 ± 0.011
0.51 0.29 0.30 1.95 0.028 ± 0.002 ± 0.003 0.013 ± 0.006 ± 0.002 0.031 ± 0.005 ± 0.005
0.51 0.28 0.50 1.91 0.024 ± 0.003 ± 0.003 -0.012 ± 0.006 ± 0.003 0.026 ± 0.003 ± 0.002
0.51 0.26 0.70 1.85 0.023 ± 0.004 ± 0.003 -0.016 ± 0.005 ± 0.002 0.003 ± 0.004 ± 0.001
0.52 0.21 0.90 1.69 0.029 ± 0.005 ± 0.003 0.010 ± 0.007 ± 0.005 0.016 ± 0.006 ± 0.002
0.51 0.21 0.46 1.35 0.018 ± 0.002 ± 0.003 -0.011 ± 0.002 ± 0.003
0.51 0.30 0.43 2.05 0.026 ± 0.004 ± 0.001 -0.014 ± 0.003 ± 0.001 0.011 ± 0.004 ± 0.007
0.51 0.37 0.42 2.75 0.030 ± 0.002 ± 0.001 -0.011 ± 0.005 ± 0.003 0.021 ± 0.003 ± 0.001
0.51 0.45 0.40 3.45 0.031 ± 0.004 ± 0.003 -0.016 ± 0.009 ± 0.002 0.023 ± 0.004 ± 0.005
0.51 0.52 0.37 4.15 0.019 ± 0.005 ± 0.004 -0.006 ± 0.019 ± 0.005 0.012 ± 0.007 ± 0.006
Table A.1: AsinφLU in one dimension.
161
< x > < PT > < z > < Q2 > AsinφLU
, π+ AsinφLU
, π− AsinφLU
, π0
0.19 0.19 0.54 1.20 0.030 ± 0.009 ± 0.009
0.17 0.34 0.50 1.25 0.024 ± 0.004 ± 0.005 0.049 ± 0.012 ± 0.014
0.16 0.51 0.49 1.26 0.018 ± 0.004 ± 0.007 -0.020 ± 0.009 ± 0.005 0.042 ± 0.011 ± 0.008
0.16 0.69 0.49 1.27 0.025 ± 0.004 ± 0.004 -0.020 ± 0.008 ± 0.005 0.019 ± 0.017 ± 0.011
0.16 0.88 0.50 1.27 0.015 ± 0.005 ± 0.006 0.014 ± 0.011 ± 0.010
0.26 0.16 0.54 1.49 0.025 ± 0.004 ± 0.003 0.000 ± 0.000 ± 0.008
0.25 0.32 0.51 1.58 0.031 ± 0.002 ± 0.006 0.049 ± 0.011 ± 0.003 0.044 ± 0.007 ± 0.007
0.24 0.50 0.50 1.63 0.029 ± 0.002 ± 0.005 -0.022 ± 0.005 ± 0.005 0.049 ± 0.007 ± 0.005
0.24 0.69 0.50 1.66 0.026 ± 0.003 ± 0.004 -0.017 ± 0.005 ± 0.005 0.014 ± 0.010 ± 0.011
0.23 0.87 0.50 1.75 0.033 ± 0.004 ± 0.009 -0.023 ± 0.010 ± 0.013 0.000 ± 0.000 ± 0.001
0.34 0.14 0.54 1.87 0.012 ± 0.003 ± 0.003 0.019 ± 0.014 ± 0.008 0.037 ± 0.013 ± 0.008
0.33 0.31 0.51 2.00 0.031 ± 0.004 ± 0.004 0.011 ± 0.006 ± 0.004 0.026 ± 0.006 ± 0.004
0.33 0.50 0.50 2.05 0.032 ± 0.003 ± 0.004 -0.022 ± 0.006 ± 0.003 0.017 ± 0.004 ± 0.004
0.33 0.67 0.49 2.16 0.027 ± 0.003 ± 0.007 -0.036 ± 0.007 ± 0.011 0.035 ± 0.010 ± 0.006
0.32 0.85 0.49 2.51 0.030 ± 0.006 ± 0.006 -0.002 ± 0.013 ± 0.018
0.43 0.14 0.53 2.61 0.020 ± 0.005 ± 0.004 -0.018 ± 0.027 ± 0.007 -0.008 ± 0.008 ± 0.007
0.43 0.31 0.51 2.77 0.035 ± 0.004 ± 0.009 -0.000 ± 0.012 ± 0.010 -0.004 ± 0.004 ± 0.010
0.43 0.50 0.50 2.80 0.037 ± 0.006 ± 0.008 -0.005 ± 0.008 ± 0.006 0.014 ± 0.008 ± 0.008
0.43 0.66 0.48 2.95 0.039 ± 0.006 ± 0.012 -0.044 ± 0.010 ± 0.019 0.033 ± 0.013 ± 0.011
0.41 0.83 0.47 3.31 0.028 ± 0.010 ± 0.021 -0.016 ± 0.025 ± 0.018
0.52 0.15 0.53 3.59 0.040 ± 0.013 ± 0.008 0.007 ± 0.011 ± 0.009
0.52 0.31 0.51 3.70 0.039 ± 0.010 ± 0.013 -0.020 ± 0.029 ± 0.033 0.015 ± 0.011 ± 0.009
0.52 0.50 0.49 3.73 0.026 ± 0.008 ± 0.008 -0.009 ± 0.021 ± 0.028 0.031 ± 0.026 ± 0.010
0.52 0.64 0.47 3.78 0.042 ± 0.013 ± 0.043 -0.054 ± 0.026 ± 0.011
Table A.2: AsinφLU binned in x and PT .
Appendix B
Good Run List
37658 37659 37661 37662 37664 37665 37666 37667 37670 37672 37673 37674 37675
37677 37678 37679 37680 37681 37683 37684 37685 37686 37687 37688 37689 37690
37691 37692 37693 37694 37698 37699 37700 37701 37702 37703 37704 37705 37706
37707 37708 37709 37710 37711 37712 37713 37714 37715 37716 37717 37719 37721
37722 37723 37724 37725 37726 37740 37744 37745 37746 37747 37748 37750 37753
37762 37763 37766 37767 37769 37770 37772 37773 37775 37776 37778 37780 37781
37782 37783 37784 37785 37788 37789 37790 37801 37802 37803 37804 37805 37806
37807 37808 37809 37810 37811 37812 37813 37814 37815 37816 37817 37818 37819
37820 37822 37823 37824 37825 37828 37831 37832 37833 37844 37845 37846 37847
37848 37849 37850 37851 37852 38046 38047 38048 38049 38050 38051 38052 38053
38070 38071 38072 38074 38075 38076 38077 38078 38079 38080 38081 38082 38083
38084 38085 38086 38089 38090 38091 38092 38093 38094 38095 38096 38097 38098
38099 38100 38114 38117 38118 38119 38120 38121 38122 38131 38132 38133 38134
38135 38136 38137 38138 38139 38140 38141 38142 38143 38144 38146 38172 38173
38174 38175 38176 38177 38182 38183 38184 38185 38186 38187 38188 38189 38190
162
163
38191 38192 38194 38195 38196 38197 38198 38199 38200 38201 38203 38204 38205
38206 38207 38208 38209 38210 38211 38212 38213 38214 38215 38216 38217 38218
38219 38220 38221 38222 38223 38225 38226 38265 38266 38268 38271 38272 38273
38274 38275 38276 38277 38278 38283 38284 38285 38286 38288 38289 38290 38300
38301 38302 38304 38305 38306 38307 38309 38310 38312 38313 38314 38315 38317
38318 38320 38322 38328 38331 38337 38338 38341 38342 38344 38346 38347 38350
38351 38353 38354 38355 38356 38359 38360 38364 38365 38378 38379 38380 38381
38382 38383 38384 38385 38387 38388 38389 38390 38391 38392 38393 38394 38395
38396 38397 38398 38399 38400 38401 38402 38403 38404 38405 38408 38409 38410
38411 38412 38415 38417 38418 38419 38420 38421 38422 38423 38430 38431 38432
38433 38434 38435 38436 38437 38438 38440 38441 38443 38446 38447 38449 38450
38451 38452 38453 38454 38455 38456 38457 38458 38459 38460 38461 38462 38463
38464 38465 38466 38467 38468 38469 38470 38471 38472 38473 38474 38475 38476
38477 38479 38480 38483 38484 38485 38486 38487 38488 38489 38490 38491 38492
38493 38494 38495 38497 38498 38499 38500 38501 38507 38508 38509 38510 38511
38512 38513 38514 38515 38516 38517 38518 38519 38520 38521 38522 38523 38524
38525 38526 38527 38528 38529 38530 38531 38534 38536 38537 38538 38539 38540
38541 38542 38543 38544 38545 38548 38549 38550 38551 38552 38553 38554 38558
38559 38560 38561 38562 38563 38568 38571 38572 38573 38574 38575 38576 38577
38578 38579 38580 38581 38582 38583 38584 38585 38596 38597 38598 38600 38601
38602 38603 38604 38606 38607 38608 38609 38610 38611 38612 38613 38614 38616
164
38617 38618 38619 38620 38621 38622 38623 38624 38625 38626 38627 38628 38629
38630 38631 38632 38633 38634 38635 38636 38637 38638 38639 38640 38641 38642
38645 38646 38647 38648 38649 38650 38651 38652 38653 38654 38655 38656 38670
38671 38672 38673 38674 38675 38676 38677 38681 38682 38684 38685 38686 38687
38689 38690 38691 38692 38693 38694 38696 38697 38698 38699 38700 38701 38702
38703 38704 38705 38706 38707 38708 38709 38710 38711 38712 38713 38714 38715
38716 38717 38719 38720 38721 38722 38723 38724 38725 38727 38728 38729 38730
38731 38732 38733 38734 38735 38737 38738 38739 38740 38743 38744 38745 38746
38748 38749 38750 38751
Appendix C
Systematic Errors
165
166
xPT
AδASF
δAEin
δAefid
δA
∆t
δApifid
δAfit
δApol
δAstat
δAsys
0.15
0.1
0.0135019
0.000618893
0.00340803
0.00430103
0.000706188
0.00644828
0.000301961
0.000310544
0.0095169
0.00904525
0.15
0.3
0.00953054
0.000499232
5.84325e-0
50.0015687
0.000351291
0.00111581
0.00125425
0.000219202
0.00392123
0.00496592
0.15
0.5
0.00821448
0.00100764
7.37221e-0
50.00169833
0.000310986
0.00055591
0.00165357
0.000188933
0.00253892
0.00659749
0.15
0.7
0.0126281
0.000307466
0.00060503
0.000535333
1.28693e-0
50.0018
0.00168557
0.000290446
0.0033665
0.00419125
0.15
0.9
0.00651272
0.000495588
0.000326566
0.000942478
0.00142348
0.00404819
0.000122796
0.000149793
0.00507697
0.00600021
0.25
0.1
0.00495387
0.000335872
9.38896e-0
60.00112693
0.000952119
0.000457787
0.000500998
0.000113939
0.00353757
0.00254128
0.25
0.3
0.0138446
0.000164097
3.28826e-0
50.00173572
4.03051e-0
60.000814624
0.000235532
0.000318426
0.00202655
0.00608573
0.25
0.5
0.0185823
0.000334639
0.000115255
0.000976207
9.94192e-0
50.000771826
0.00219829
0.000427393
0.00281077
0.00493537
0.25
0.7
0.00867379
0.000313384
3.0267e-0
50.000794925
5.56493e-0
50.000890506
3.73875e-1
00.000199497
0.00224301
0.00384552
0.25
0.9
0.0129223
0.000472191
0.00274397
0.00211566
0.000942432
0.000698675
3.78114e-1
00.000297213
0.00472036
0.00919425
0.35
0.1
0.00718759
0.000217032
4.56594e-0
50.000334025
0.000555432
0.00235915
0.000452355
0.000165315
0.00336227
0.00283777
0.35
0.3
0.0199537
0.000124082
0.000142285
3.57022e-0
50.000389828
0.000395139
0.0025542
0.000458935
0.00261593
0.0036742
0.35
0.5
0.0179742
0.00028174
0.00192375
0.000213547
0.000249467
0.000529401
0.00239362
0.000413407
0.00217682
0.00383339
0.35
0.7
0.0107417
0.000192659
0.00166799
0.000408142
0.00102417
0.00114509
4.56198e-1
10.000247059
0.00299895
0.00708554
0.35
0.9
0.0115823
3.99495e-0
50.00341368
0.000411535
0.000777605
0.00208474
0.000379052
0.000266393
0.00919253
0.00638459
0.45
0.1
0.0176492
0.000556295
0.00202262
0.000212354
0.0014053
0.00105001
0.0022981
0.000405932
0.00552146
0.00371982
0.45
0.3
0.0208705
0.000485677
4.74494e-0
50.000226328
0.000647922
0.00320598
0.00297438
0.000480021
0.00385028
0.0088964
0.45
0.5
0.00766314
0.000500941
0.000180264
0.000196313
0.00101894
0.00110908
0.00143394
0.000176252
0.00364335
0.00840696
0.45
0.7
0.0272832
6.78284e-0
50.00981661
0.000499262
0.000330077
0.00170315
0.000795295
0.000627514
0.00585087
0.0119312
0.45
0.9
0.0285481
0.000916309
0.00493195
0.00134977
0.000701379
0.000627633
1.05824e-0
90.000656606
0.0348727
0.021449
0.55
0.1
0.0349296
0.000706535
0.00249407
0.000353114
0.00169197
0.00542326
0.00451346
0.000803381
0.0106321
0.00827451
0.55
0.3
0.00618981
0.00130682
0.00065652
0.00252087
0.000942856
0.00209383
4.53835e-1
00.000142366
0.00930323
0.0129966
0.55
0.5
-0.00481736
0.000986044
0.00495716
0.00146476
0.000619991
0.000921813
0.00103366
-0.000110799
0.00846512
0.00808301
0.55
0.7
00.000622407
00.00112729
0.000335451
0.003761
3.79067e-1
00
00.0431923
0.55
0.9
00.00178117
00.00282575
0.00144893
00
00
0.00364099
Table
C.1
:Syst
emat
icer
rors
for
xvs.PT
bin
nin
gofπ
+.
167
xPT
AδASF
δAEin
δAefid
δA
∆t
δApifid
δAfit
δApol
δAstat
δAsys
0.15
0.1
00
00
00
00
00
0.15
0.3
0.0465613
00.0002486
00
00
0.00107091
0.041439
0.00109939
0.15
0.5
-0.015702
0.00158096
7.98888e-0
50.00205275
0.00153591
0.00286257
0.00260147
-0.000361146
0.00979398
0.0049164
0.15
0.7
-0.00525771
0.000582456
0.000168271
0.00117207
0.00234074
0.00435331
0.000296139
-0.000120927
0.00968196
0.00516492
0.15
0.9
-0.00400529
0.0013492
0.00507715
0.00441639
0.00480084
0.00369572
1.30119e-0
9-9
.21217e-0
50.0104727
0.00967815
0.25
0.1
00.00412549
00
00.0125464
0.000998516
00
0.013245
0.25
0.3
0.0180768
0.000782309
2.15585e-0
50.00194449
0.00185877
0.000489645
0.000336241
0.000415766
0.0104683
0.00290349
0.25
0.5
-0.0128931
0.000571894
5.40227e-0
50.00102462
0.00145855
0.00216225
0.00340561
-0.000296541
0.00549322
0.00488742
0.25
0.7
-0.00857787
0.000613864
0.000316658
0.00136379
0.00312449
0.00319721
0.00172332
-0.000197291
0.00543004
0.0053258
0.25
0.9
-0.00402875
0.000297251
0.00901806
0.00185142
0.00302097
0.00119245
0.00318493
-9.26613e-0
50.0113471
0.0131212
0.35
0.1
0.00304459
0.00295238
0.000516208
0.00163473
0.0074324
0.000513793
0.000164692
7.00256e-0
50.0142156
0.00821009
0.35
0.3
0.00754881
0.000572216
0.000103916
0.00181528
0.00236531
0.00123939
0.000428052
0.000173623
0.00703091
0.00352483
0.35
0.5
0.0007702
0.000547767
0.00030937
0.00183278
0.00166587
0.00119885
2.20375e-1
11.77146e-0
50.00538915
0.00310575
0.35
0.7
-0.0097008
0.000609788
0.00455771
0.000583052
2.38295e-0
50.000851383
3.8988e-1
1-0
.000223118
0.00566371
0.0111717
0.35
0.9
0.0166411
0.000967961
0.00706312
0.00636334
0.00477649
0.00815901
0.000834706
0.000382745
0.0182012
0.0183253
0.45
0.1
-0.0202932
0.00142908
0.0038684
0.00216926
0.00335239
0.00207827
0.00353284
-0.000466744
0.0229702
0.00721176
0.45
0.3
-0.0111303
0.00306161
3.18234e-0
50.00385036
0.000330268
0.00825868
0.00100203
-0.000255997
0.0121374
0.00982464
0.45
0.5
-0.00252204
0.000818455
0.00275967
0.000954858
0.0018977
0.00346813
0.00028256
-5.80069e-0
50.00873466
0.00624887
0.45
0.7
-0.00631458
0.000746703
0.0106528
0.00161787
0.00180326
0.000524788
4.76828e-1
0-0
.000145235
0.0122083
0.0193443
0.45
0.9
00.000854716
00.00702544
0.00349905
0.00852129
00
00.0176283
0.55
0.1
00.00277757
00
00
00
00.00277757
0.55
0.3
-0.0398339
0.0015562
0.00396485
0.00808655
0.00470049
0.0112937
0.00876651
-0.00091618
0.0282643
0.0329386
0.55
0.5
0.0296353
0.00460982
0.00561904
0.00240679
0.00444137
0.00454175
0.00685039
0.000681612
0.0217801
0.0280216
0.55
0.7
00.00308347
00.00644732
0.00203428
0.00621074
0.00417749
00
0.010547
0.55
0.9
00
00
00
00
00
Table
C.2
:Syst
emat
icer
rors
for
xvs.PT
bin
nin
gofπ−
.
168
xPT
AδASF
δAEin
δAefid
δABackground
δAfit
δApol
δAstat
δAsys
0.15
0.1
00
00
00
00
0
0.15
0.3
0.0485237
0.000839667
0.000204674
0.0028642
0.0132004
00.00111605
0.011506
0.0135811
0.15
0.5
0.0418857
0.000294557
0.000494851
0.00127228
0.00782938
0.00164385
0.000963371
0.0107529
0.0081784
0.15
0.7
0.0194034
0.00437973
0.000300379
0.00420399
0.00837271
0.000984116
0.000446278
0.016953
0.0107186
0.15
0.9
00
00
00.00458037
00
0.00458037
0.25
0.1
00
00
00.00764922
00
0.00764922
0.25
0.3
0.043949
0.0013704
0.000331865
0.00432163
0.00215008
0.00439724
0.00101083
0.00727309
0.00676691
0.25
0.5
0.0490626
0.000585643
7.45543e-0
50.000616772
0.00387501
0.00263832
0.00112844
0.00700583
0.00492751
0.25
0.7
0.013861
0.00221052
6.82887e-0
50.00221074
0.00925061
0.00350462
0.000318803
0.0103454
0.0114124
0.25
0.9
00
00
00.00126218
00
0.00126218
0.35
0.1
0.0374302
0.00287853
0.000204522
8.95215e-0
50.00503445
0.00555226
0.000860895
0.0127151
0.00807776
0.35
0.3
0.0263536
0.000586921
1.80624e-0
50.00122442
0.0038822
2.11877e-0
60.000606133
0.00640769
0.00420216
0.35
0.5
0.0174142
0.000399453
4.89073e-0
50.00133253
0.000603135
0.00115919
0.000400527
0.00398777
0.0037275
0.35
0.7
0.0349682
0.00071466
0.00037704
0.00313468
0.00475604
0.000683074
0.000804269
0.010002
0.00636472
0.35
0.9
00
00
00.00331108
00
0.00331108
0.45
0.1
-0.0081264
0.00132104
0.00184511
0.00460363
0.00408696
0.00279158
-0.000186907
0.00756967
0.00713854
0.45
0.3
-0.00405361
0.000434106
0.000144229
0.00181441
0.00288588
0.00960513
-9.3233e-0
50.00365646
0.010448
0.45
0.5
0.0139414
0.00145101
0.000127384
0.00121243
0.00130733
0.00497157
0.000320652
0.00774076
0.00802356
0.45
0.7
0.0325581
0.000761272
8.93783e-0
50.00558755
0.00382793
0.000481063
0.000748836
0.0133686
0.0109778
0.45
0.9
00
00
00
00
0
0.55
0.1
0.00723277
0.00756197
0.000530515
0.00134282
0.00463123
00.000166354
0.0111079
0.00911856
0.55
0.3
0.0150606
0.00302467
0.000812129
0.000680112
0.00353789
00.000346394
0.0111
0.00941343
0.55
0.5
0.0305583
0.00916254
8.66913e-0
50.00263129
0.00197203
00.000702841
0.0258549
0.00976044
0.55
0.7
00
00
00
00
0
0.55
0.9
00
00
00
00
0
Table
C.3
:Syst
emat
icer
rors
for
xvs.PT
bin
nin
gofπ
0.