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.

iii

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

iv

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

v

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

vi

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

vii

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

viii

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

ix

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

x

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.

xi

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

xii

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

xiii

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

xiv

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

xvi

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

xvii

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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γθ

Sector 22.0 < W < 2.1

o < 25eθ < o20

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Sector 22.1 < W < 2.2

o < 25eθ < o20

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100120

γθ

Sector 22.2 < W < 2.3

o < 25eθ < o20

0 0.5 1 1.5 2020406080

100120140160180200220

γθ

Sector 22.3 < W < 2.4

o < 25eθ < o20

0 0.5 1 1.5 2050

100150200250300350

γθ

Sector 22.4 < W < 2.5

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Sector 22.5 < W < 2.6

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200

400600

800

1000

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Sector 22.6 < W < 2.7

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Sector 22.9 < W < 3.0

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Sector 22.0 < W < 2.1

o < 25θ < o20

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300400

500600

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Sector 22.1 < W < 2.2

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Sector 22.2 < W < 2.3

o < 25θ < o20

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Sector 22.3 < W < 2.4

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200400

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Sector 22.4 < W < 2.5

o < 25θ < o20

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10001200140016001800

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Sector 22.5 < W < 2.6

o < 25θ < o20

170 175 180 185 1900200400600800

100012001400160018002000

φ∆

Sector 22.6 < W < 2.7

o < 25θ < o20

170 175 180 185 1900200400600800

1000120014001600

φ∆

Sector 22.7 < W < 2.8

o < 25θ < o20

170 175 180 185 1900200400600800

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φ∆

Sector 22.8 < W < 2.9

o < 25θ < o20

170 175 180 185 1900

100

200

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.