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 + , π 0 and π - ) were measured simultaneously over a large range of kinematics (Q 2 ≈ 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, P T , and Q 2 , from which A sinφ LU 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.
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Probing the Proton’s Quark Dynamics in
Semi-inclusive Pion Electroproduction
Wesley P. Gohn, Ph.D.
University of Connecticut, 2012
Measurements of pion electro-production in semi-inclusive deep inelastic scatter-
ing (SIDIS) have been performed. Data were taken with the CEBAF Large Ac-
ceptance Spectrometer (CLAS) at Jefferson Lab using a 5.498 GeV longitudinally
polarized electron beam and an unpolarized liquid hydrogen target during the
E1-f run period in 2003. All three pion channels (π+, π0 and π−) were measured
simultaneously over a large range of kinematics (Q2 ≈ 1-4 GeV 2 and x ≈ 0.1-0.5).
Single-spin azimuthal asymmetries from all three pion channels were measured as
functions of x, z, PT , and Q2, from which AsinφLU were extracted. This new high
statistical data could provide access to transverse-momentum dependent parton
distribution functions (TMD’s), which are thought to be important in understand-
ing of the physics underlying the spin structure of the nucleon.
Probing the Proton’s Quark Dynamics in
Semi-inclusive Pion Electroproduction
Wesley P. Gohn
B.S., Indiana University, Bloomington, IN, 2004
M.S., University of Connecticut, 2007
A Dissertation
Submitted in Partial Fullfilment of the
Requirements for the Degree of
Doctor of Philosophy
at the
University of Connecticut
2012
Copyright by
Wesley P. Gohn
2012
APPROVAL PAGE
Doctor of Philosophy Dissertation
Probing the Proton’s Quark Dynamics in
Semi-inclusive Pion Electroproduction
Presented by
Wesley P. Gohn,
Major Advisor
Kyungseon Joo
Associate Advisor
Harut Avakian
Associate Advisor
Peter Schweitzer
University of Connecticut
2012
ii
This dissertation is dedicated to my wife, Lindsey,
for your infinite patience.
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
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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