## A Strong-QCD Regime Measurement of the Proton’s Spin Structure

BY

DAVID RUTH

B.A. in Physics, McDaniel College, 2015

DISSERTATION

Submitted to the University of New Hampshire

in Partial Fulfillment of

the Requirements for the Degree of

Doctor of Philosophy

in

Physics

December, 2022

ALL RIGHTS RESERVED

©2022

David Ruth

This dissertation has been examined and approved in partial fulfillment of the
requirements for the degree of Doctor of Philosophy in Physics by:

Dissertation Director, Karl Slifer

Professor of Physics

Mark McConnell

Professor of Physics

William Brooks

Associate Professor of Physics

Universidad Tecnica Federico Santa Maria

David Mattingly

Associate Professor of Physics

Elena Long

Assistant Professor of Physics

On November 16th, 2022

Original approval signatures are on file with the University of New Hampshire
Graduate School.

## DEDICATION

This thesis is dedicated to my granddad, Ray Somerlock, who helped me to
become interested in science very early with his tireless curiosity and
enthusiasm for engineering. I know he would have loved to read this thesis and
I strive to always carry his passion with me.

## ACKNOWLEDGEMENTS

Above all else, I want to thank my amazing wife Michelle. You make this all
feel worthwhile, and I couldn’t ask for a better partner to be on this journey
with. Thank you sunshine, for all of your support and encouragement. I
couldn’t have gotten through this without you.

I want to thank my advisor Karl Slifer, who I am honored to say is both a
mentor and a true friend. Karl has helped me to become a better researcher and
created one of the best environments to learn in that I could imagine. Thank
you for sharing your experience with me over the last seven years.

I’m also eternally grateful for my parents, Nick and Sheila, who have
supported me on every step of this journey. I couldn’t ask for a more caring
family and I will never forget that you’ve been there for me every time I
needed you, in every way possible. Thank you for always coming through for us,
and for setting such an amazing example with your own lives that I can aspire
to reach.

Thank you also to all my friends in the polarized target group, especially
Elena Long and Nathaly Santiesteban, who have both helped me to mature as a
scientist and been some of the best friends I could ask for at UNH. Some of my
fondest grad school memories are working on cooldowns with all of you, despite
the hard work and long hours!

Thank you to my family who has always supported and encouraged me, my cousins
Rachel, Jessie, Nicole, and Kasey, my grandparents Dolores, John, Edna, and
Ray, my aunts and uncles Ann, George, Joan, Bill, Robin, Dave, Carl, and
Christy, and my lovely in-laws Robin, Frank, and Matt.

Thank you also to Jian-Ping Chen, whose experience and keen eye for detail
have helped me learn a lot and become a more rigorous scientist, and without
whom the E08-027 analysis would never have been completed.

Thank you as well to my best man and friend Sam, who has helped me stay sane
and avoid becoming completely antisocial, and the rest of the 203 crew at
McDaniel, who I hope I will stay lifelong friends with.

###### TABLE OF CONTENTS

  1. DEDICATION
  2. ACKNOWLEDGEMENTS
  3. ABSTRACT
  4. 1 Background
  5. 2 Theory
    1. 2.1 Inclusive Electron Scattering
    2. 2.2 Interaction Rates & Cross Sections
    3. 2.3 Lepton and Hadron Tensors
    4. 2.4 Structure Functions
    5. 2.5 Moments and Sum Rules
    6. 2.6 Chiral Perturbation Theory
  6. 3 Existing Data
  7. 4 Experimental Setup
    1. 4.1 Electron Beam
    2. 4.2 Beam Current Monitors (BCMs)
    3. 4.3 Fast and Slow Raster
    4. 4.4 Møller Polarimeter
    5. 4.5 Chicane Magnets
    6. 4.6 Beam Position Monitors (BPMs)
    7. 4.7 Polarized Target
    8. 4.8 Beam Dumps
    9. 4.9 Septa Magnets
    10. 4.10 Spectrometers
    11. 4.11 Data Acquisition System
  8. 5 UNH Polarized Target
    1. 5.1 Superconducting Magnet
    2. 5.2 Vacuum and Cryogenics
    3. 5.3 NMR
    4. 5.4 DAQ
    5. 5.5 Cooldowns & Results
  9. 6 Analysis
    1. 6.1 Experimental Cross Section
    2. 6.2 Packing Fractions
    3. 6.3 Dilutions
    4. 6.4 Asymmetries
    5. 6.5 Model Cross Section
    6. 6.6 Scattering Angle and Out of Plane Polarization Angle
    7. 6.7 Polarized Cross Section Differences & Radiative Corrections
    8. 6.8 Extracting the Structure Functions
    9. 6.9 Systematic Contributions
  10. 7 Results
    1. 7.1 Burkhardt-Cottingham Sum Rule ($\Gamma_{2}$)
    2. 7.2 Transverse-Longitudinal Spin Polarizability ($\delta_{LT}$)
    3. 7.3 Twist-3 Local Operator ($d_{2}$)
    4. 7.4 First Moment of $g_{1}$ ($\Gamma_{1}$)
    5. 7.5 Generalized Forward Spin Polarizability ($\gamma_{0}$)
    6. 7.6 Extended GDH Sum ($I_{A}$)
    7. 7.7 Hyperfine Structure
    8. 7.8 P-N and P+N Channels
  11. 8 Physics Conclusions

###### LIST OF TABLES

  1. 3.1 Existing proton structure function data as of August 2022
  2. 4.1 Electron Beam Polarization for the E08-027 Experiment
  3. 4.2 Chicane configurations for each transverse kinematic setting. Values reproduced from Zielinski (2017).
  4. 6.1 Final Packing Fractions for E08-027.
  5. 6.2 Final systematics for E08-027.

###### LIST OF FIGURES

  1. 1.1 Model of the Proton over time. Adapted from a figure in pro
  2. 1.2 The “$\delta_{LT}$ Puzzle”: (Left) The results of Jefferson Lab E94-010 showing the initial disagreement Amarian et al. (2004). (Right) The results of Jefferson Lab E97-110 showing a new disagreement in 2021 Sulkosky et al. (2021).
  3. 2.1 $ep$ Scattering interaction
  4. 2.2 ep $\rightarrow$ eX scattering cross section as a function of the invariant mass W. Figure edited from Halzen and Martin (1984) and produced with SLAC data.
  5. 2.3 Virtual-Virtual Compton Scattering Interaction
  6. 2.4 Asymptotic freedom and confinement. Adapted from a figure in asy
  7. 2.5 Definitions of handedness in helicity. Adapted from a figure in hel
  8. 3.1 The $g_{1}$ structure function of the proton measured by the EG4 experiment. The data (red circles) are compared to the CLAS Hall B phenomenological model Fersch et al. (2017). The solid and hatched bands refer to the systematic uncertainty of the data and CLAS model parametrization respectively. The black crosses indicate earlier data from the EG1b Fersch et al. (2017) experiment. Reproduced from Zheng et al. (2021).
  9. 4.1 The experimental setup of the g2p Experiment. Reproduced from Ryan Zielinski’s thesis Zielinski (2017)
  10. 4.2 The Jefferson Lab Electron Accelerator. Reproduced from Alcorn et al. (2004)
  11. 4.3 The polarized electron generation technique. Reproduced from Hernandez-Garcia et al. (2008).
  12. (a) Regular GaAs Cathode
  13. (b) Strained GaAsP Cathode
  14. 4.4 Raster swept beam profiles. Plots are a function of current in arbitrary units. Reproduced from Zielinski (2017)
  15. (a) Fast Raster
  16. (b) Slow Raster
  17. 4.5 Møller Polarimeter Schematic. (a) Side View. (b) Top View. Reproduced from Alcorn et al. (2004)
  18. 4.6 Chicane magnet configuration for the g2p experiment. Reproduced from Zielinski (2017).
  19. 4.7 Beam position monitor cavity from the side (left) and front (right). Reproduced from Gu (2016).
  20. 4.8 Superharp wire scanner used to calibrate the BPMs. Reproduced from Gu (2016).
  21. 4.9 Polarized target for the g2p experiment. Reproduced from Pierce et al. (2014).
  22. 4.10 Energy levels of the dynamic nuclear polarization process. Reproduced from Crabb and Meyer (1997).
  23. 4.11 NMR Signal examples from the g2p experiment. Reproduced from Gu (2016).
  24. (a) Raw NMR Signal on Real Q-Meter Impedance
  25. (b) Background-subtracted NMR Signal
  26. 4.12 Target polarizations for the 2.5 T (top) and 5 T (bottom) settings. Reproduced from Badman (2016).
  27. 4.13 Target insert for E08-027 showing Carbon (First), Dummy (Second), and Ammonia (Third and Fourth) targets. Reproduced from Zielinski (2017).
  28. 4.14 Hall A main beam dump schematic. Reproduced from TEC (1997).
  29. 4.15 Local beam dump for E08-027. The septa magnets are visible in the background. Holes on the left and right of the blue beam dump support structure allow scattered electrons to pass through to the detectors. In the center, copper and tungsten plates absorb the beam energy. In this photo, the insert is not inserted into the beam dump. Reproduced from Zielinski (2017).
  30. 4.16 Septa magnets for E08-027. Reproduced from Zielinski (2017).
  31. 4.17 Detector stack for the Hall A High-Resolution Spectrometers. Reproduced from Zielinski (2017).
  32. 4.18 VDC Diagram. Reproduced from Alcorn et al. (2004).
  33. 4.19 Gas Cerenkov diagram. Reproduced from Iodice et al. (1998); Zielinski (2017).
  34. 4.20 Lead calorimeter for HRS detectors. Reproduced from Alcorn et al. (2004).
  35. 4.21 Overall Data Acquisition setup for E08-027. Reproduced from Zielinski (2017).
  36. 5.1 Example of a deuteron signal and high-accuracy fit. Reproduced from Dulya et al. (1997).
  37. 5.2 UNH solenoid and insulating cryostat.
  38. 5.3 Inner vacuum shell. Outer shell is similar but with larger diameter, designed to join with the bottom of the large T-piece via a Copper Flange (CF) seal.
  39. 5.4 Indium seal in progress, showing the folded knot structure
  40. 5.5 LHe Evaporation Refrigerator for the UNH System. Reproduced from Aliaga et al. (2020).
  41. 5.6 Liverpool Q-Meter System. Reproduced from Crabb and Meyer (1997).
  42. 5.7 The LANL VME-Based NMR System.
  43. (a) LANL System Analog Board
  44. (b) LANL System in VME Crate
  45. 5.8 The Cold Board for the Deuteron NMR System
  46. (a) Cold Board Photo
  47. (b) Cold Board Circuit Diagram. Reproduced from Long (2020)
  48. 5.9 DAQ System designed for the UNH Polarized Target Lab.
  49. 5.10 Photos from the UNH Polarized Target Cooldowns.
  50. (a) Author loading a target insert into the cryostat
  51. (b) Main UNH Cryostat containing superconducting magnet and LHe refrigerator
  52. (c) UNH Group monitoring the DAQ system during a cooldown
  53. 6.1 Kinematic coverage of the E08-027 experiment, in terms of invariant mass and momentum transfer. Reproduced from Zielinski (2017).
  54. 6.2 2.5T 1.7 GeV Transverse Carbon Comparison.
  55. (a) Carbon Simulation/Data Ratio
  56. (b) Scaled Carbon Simulation & Data Comparison
  57. 6.3 2.5T 2.2 GeV Transverse Carbon Comparison.
  58. (a) Carbon Simulation/Data Ratio
  59. (b) Scaled Carbon Simulation & Data Comparison
  60. 6.4 5T 2.2 GeV Transverse Carbon Comparison.
  61. (a) Carbon Simulation/Data Ratio
  62. (b) Scaled Carbon Simulation & Data Comparison
  63. 6.5 5T 2.2 GeV Longitudinal Carbon Comparison.
  64. (a) Carbon Simulation/Data Ratio
  65. (b) Scaled Carbon Simulation & Data Comparison
  66. 6.6 2.5T 1.7 GeV Transverse Fits.
  67. 6.7 2.5T 2.2 GeV Transverse Fits.
  68. 6.8 5T 2.2 GeV Transverse Fits.
  69. 6.9 5T 2.2 GeV Longitudinal Fits.
  70. 6.10 2.5T 1.7 GeV Transverse Simulation & Yields.
  71. 6.11 2.5T 2.2 GeV Transverse Simulation & Yields.
  72. 6.12 5T 2.2 GeV Transverse Simulation & Yields.
  73. 6.13 5T 2.2 GeV Longitudinal Simulation & Yields.
  74. 6.14 2.5T 1.7 GeV Transverse Linear pf Fit.
  75. 6.15 2.5T 2.2 GeV Transverse Linear pf Fit.
  76. 6.16 5T 2.2 GeV Transverse Linear pf Fit.
  77. 6.17 5T 2.2 GeV Longitudinal Linear pf Fit.
  78. 6.18 2.5T 1.7 GeV Transverse Dilution Yields.
  79. (a) Carbon
  80. (b) Empty
  81. (c) Dummy
  82. 6.19 2.5T 1.7 GeV Transverse Production Yields.
  83. (a) Material 7
  84. (b) Material 8
  85. 6.20 2.5T 2.2 GeV Transverse Dilution Yields.
  86. (a) Carbon
  87. (b) Empty
  88. (c) Dummy
  89. 6.21 2.5T 2.2 GeV Transverse Production Yields.
  90. (a) Material 7
  91. (b) Material 8
  92. 6.22 5T 2.2 GeV Longitudinal Dilution Yields.
  93. (a) Carbon
  94. (b) Empty
  95. (c) Dummy
  96. 6.23 5T 2.2 GeV Longitudinal Production Yields.
  97. (a) Material 17
  98. (b) Material 18
  99. 6.24 5T 2.2 GeV Transverse Dilution Yields.
  100. (a) Carbon
  101. (b) Empty
  102. (c) Dummy
  103. 6.25 5T 2.2 GeV Transverse Production Yields.
  104. (a) Material 19
  105. (b) Material 20
  106. 6.26 5T 3.3 GeV Transverse Dilution Yields.
  107. (a) Carbon
  108. (b) Empty
  109. (c) Dummy
  110. 6.27 5T 3.3 GeV Transverse Production Yields.
  111. (a) Material 19
  112. (b) Material 20
  113. 6.28 2.5T 1.7 GeV Transverse Dilutions.
  114. (a) Material 7
  115. (b) Material 8
  116. 6.29 2.5T 2.2 GeV Transverse Dilutions.
  117. (a) Material 7
  118. (b) Material 8
  119. 6.30 5T 2.2 GeV Longitudinal Dilutions.
  120. (a) Material 17
  121. (b) Material 18
  122. 6.31 5T 2.2 GeV Transverse Dilutions.
  123. (a) Material 19
  124. (b) Material 20
  125. 6.32 5T 3.3 GeV Transverse Dilutions.
  126. (a) Material 19
  127. (b) Material 20
  128. 6.33 2.5T 1.7 GeV Transverse Asymmetry.
  129. 6.34 2.5T 2.2 GeV Transverse Asymmetry.
  130. 6.35 5T 2.2 GeV Longitudinal Asymmetry.
  131. 6.36 5T 2.2 GeV Transverse Asymmetry.
  132. 6.37 5T 3.3 GeV Transverse Asymmetry.
  133. 6.38 5T 2.2 GeV Longitudinal Unpolarized Cross Section.
  134. 6.39 5T 2.2 GeV Longitudinal Cross Section Comparison to Bosted-Christy Model Christy and Bosted (2010).
  135. 6.40 5T 2.2 GeV Longitudinal Polarized Cross Section Difference Data Comparison to Bosted-Christy Model Christy and Bosted (2010).
  136. 6.41 Ratio of data to Bosted-Christy Model Christy and Bosted (2010) for experiments in the low $Q^{2}$ region. The shaded region shows the approximate coverage of the transverse results of the E08-027 experiment.
  137. 6.42 Born Process and Bremstrahhlung corrections. Reproduced from Zielinski (2017).
  138. 6.43 2.5T 1.7 GeV Transverse Polarized Cross Section Difference.
  139. 6.44 2.5T 2.2 GeV Transverse Polarized Cross Section Difference.
  140. 6.45 5T 2.2 GeV Longitudinal Polarized Cross Section Difference.
  141. 6.46 5T 2.2 GeV Transverse Polarized Cross Section Difference.
  142. 6.47 5T 3.3 GeV Transverse Polarized Cross Section Difference.
  143. 6.48 Final structure functions results for all kinematic settings of the g2p experiment. The solid lines indicate the Clas Hall B Model Fersch et al. (2017). Red points indicate $g_{1}$ results, while blue points show the $g_{2}$ results. The shaded regions show the respective systematic errors of each structure function.
  144. 6.49 Comparison between the $g_{1}$ data of the g2p Ruth et al. (2022) and EG4 Zheng et al. (2021) experiments at Jefferson Lab.
  145. 7.1 The $\Gamma_{2}$ moment. Light blue points represent an integral of the full measured region of E08-027, while dark blue points are limited to a maximum W of 1725. The MAID phenomenological model Drechsel et al. (2007) is also run over this limited W range, as are the other experiments’ results, to provide an even comparison. Results of RSS Wesselmann et al. (2007) are shown in red, and results of SANE Armstrong et al. (2019) are shown in yellow. The open symbols represent the sum of the measured and elastic parts of the integral, excluding only the unmeasured high-W part.
  146. 7.2 The Transverse-Longitudinal Spin Polarizability $\delta_{LT}$. The 8-pointed marker indicates the $Q^{2}$ where both transverse and longitudinal data from E08-027 are available, while the other three points are formed with $g_{1}$ from the CLAS Hall B Model Fersch et al. (2017). The points are compared to the MAID Drechsel et al. (2007) and Hall B Fersch et al. (2017) phenomenological models, as well as $\chi$PT calculations from Alarcon et al. Alarcón et al. (2020) and Bernard et al Bernard et al. (2013). The blue shaded region indicates the systematic uncertainty.
  147. 7.3 The Twist-3 Local Operator $d_{2}$. The 8-pointed marker indicates the $Q^{2}$ where both transverse and longitudinal data from E08-027 are available, while the other three points are formed with $g_{1}$ from the CLAS Hall B Model Fersch et al. (2017). The points are compared to the MAID Drechsel et al. (2007) phenomenological model, as well as a $\chi$PT calculations from Alarcon et al. Alarcón et al. (2020) and a higher $Q^{2}$ lattice QCD calculation Gockeler et al. (2001). Also shown are results from the RSS Wesselmann et al. (2007), SANE Armstrong et al. (2019) and SLAC E155 Anthony et al. (2003a) experiments.
  148. 7.4 The First Moment of $g_{1}$, $\Gamma_{1}$. The points are compared to the MAID Drechsel et al. (2007) and Hall B Fersch et al. (2017) phenomenological model, as well as a $\chi$PT calculations from Alarcon et al. Alarcón et al. (2020) and Bernard et al Bernard et al. (2013). Also shown are results from the EG1B Fersch et al. (2017) experiment.
  149. 7.5 The Generalized Forward Spin Polarizability, $\gamma_{0}$. The points are compared to the MAID Drechsel et al. (2007) and Hall B Fersch et al. (2017) phenomenological model, as well as a $\chi$PT calculations from Alarcon et al. Alarcón et al. (2020) and Bernard et al Bernard et al. (2013). Also shown are results from the EG1B Fersch et al. (2017) and EG4 Zheng et al. (2021) experiments.
  150. 7.6 The Extended GDH Sum, $I_{A}$. The points are compared to the MAID Drechsel et al. (2007) and Hall B Fersch et al. (2017) phenomenological model, as well as a $\chi$PT calculations from Alarcon et al. Alarcón et al. (2020) and Bernard et al Bernard et al. (2013). Also shown are results from the EG1B Fersch et al. (2017) and EG4 Zheng et al. (2021) experiments.
  151. 7.7 The $B_{1}(Q^{2})$ part of the $\Delta_{1}$ integral. The points are compared to the MAID Drechsel et al. (2007) and Hall B Fersch et al. (2017) phenomenological model. Also shown are calculated results from internal data of the EG1B Fersch et al. (2017) experiment.
  152. 7.8 The $B_{2}(Q^{2})$ part of the $\Delta_{2}$ integral. The points are compared to the MAID Drechsel et al. (2007) and Hall B Fersch et al. (2017) phenomenological model.
  153. 7.9 The Transverse Longitudinal Spin Polarizability $\delta_{LT}$ in the P-N Channel. Chiral Perturbation Theory calculations are shown from Alarcon et al. Alarcón et al. (2020) and Bernard et al. Bernard et al. (2013). Data is combined from the g2p Ruth et al. (2022) proton results and Small-Angle GDH Sulkosky et al. (2021) neutron results.
  154. 7.10 The Transverse Longitudinal Spin Polarizability $\delta_{LT}$ in the P+N Channel. Chiral Perturbation Theory calculations are shown from Alarcon et al. Alarcón et al. (2020) and Bernard et al. Bernard et al. (2013). Data is combined from the g2p Ruth et al. (2022) proton results and Small-Angle GDH Sulkosky et al. (2021) neutron results.
  155. 7.11 The Generalized Forward Spin Polarizability $\gamma_{0}$ in the P-N Channel. Chiral Perturbation Theory calculations are shown from Alarcon et al. Alarcón et al. (2020) and Bernard et al. Bernard et al. (2013). Data is combined from the g2p Ruth et al. (2022) proton results and Small-Angle GDH Sulkosky et al. (2021) neutron results.
  156. 7.12 The Generalized Forward Spin Polarizability $\gamma_{0}$ in the P+N Channel. Chiral Perturbation Theory calculations are shown from Alarcon et al. Alarcón et al. (2020) and Bernard et al. Bernard et al. (2013). Data is combined from the g2p Ruth et al. (2022) proton results and Small-Angle GDH Sulkosky et al. (2021) neutron results.
  157. 7.13 The $\Gamma_{1}$ moment in the P-N Channel. Chiral Perturbation Theory calculations are shown from Alarcon et al. Alarcón et al. (2020) and Bernard et al. Bernard et al. (2013). Data is combined from the g2p Ruth et al. (2022) proton results and Small-Angle GDH Sulkosky et al. (2021) neutron results.
  158. 7.14 The $\Gamma_{1}$ moment in the P+N Channel. Chiral Perturbation Theory calculations are shown from Alarcon et al. Alarcón et al. (2020) and Bernard et al. Bernard et al. (2013). Data is combined from the g2p Ruth et al. (2022) proton results and Small-Angle GDH Sulkosky et al. (2021) neutron results.

## ABSTRACT

The theory of the strong force, Quantum Chromodynamics (QCD) remains one of
the most important ways to understand the fundamental properties of ordinary
matter. However, at low momentum transfer $Q^{2}$, in the regime where the
strong force becomes extremely strong, our understanding of QCD for ordinary
nucleons becomes hazy. Several cutting edge theories such as Chiral
Perturbation Theory ($\chi$PT) and Lattice QCD have provided valuable
predictions in this regime, but Lattice QCD has not yet extended predictions
of many important quantities to this kinematic region, and Chiral Perturbation
Theory has faced several important disagreements with experimental data for
the neutron over the last several decades. It is therefore of extreme
importance to have a benchmark of experimental data in the low energy regime
for the proton’s behavior, as a test of leading theories for the behavior of
QCD in this regime.

The E08-027 (g2p) experiment ran at Jefferson Lab in 2012 with the goal of
collecting this valuable data, and though I was still completing my
undergraduate studies at the time, I became involved in the analysis in 2015
and built on the previous work to complete it and analyze the exciting
results. This experiment achieved a high precision measurement of the spin
structure functions $g_{1}$ and $g_{2}$ for the proton, quantities which
describe the internal spin structure of the proton. These measurements were
taken in the valuable low $Q^{2}$ region described above, and used to extract
several moments of these spin structure functions which can be directly
compared to the cutting-edge predictions of Chiral Perturbation Theory.

Though the experiment’s timeline was such that I didn’t have a chance to work
directly on the experimental setup, I had the opportunity to acquire hands-on
experience working on a polarized target at UNH which is very similar to the
crucial polarized target used in the g2p experiment.

Full details of the g2p experiment and my experimental work at the University
of New Hampshire are presented in this thesis, as well as a detailed
description of the analysis process and the exciting benchmark results, which
serve as a direct test of all current and future theories of QCD in the
low-$Q^{2}$ regime.

## CHAPTER 1Background

It is currently estimated that the proton accounts for upwards of 90% of
ordinary matter in the visible universe Grochala (2015). Protons make up the
nuclei of atoms which compose humans and just about everything else we see
every day, and their charge dictates most electromagnetic interactions we can
study. Accordingly, it is surprising that we are still without a robust
understanding of this essential particle. In 1920 when Ernest Rutherford
proposed the proton’s existence, he believed it was a fundamental particle,
one of the most basic building blocks of the universe Rutherford (1920). In
1964, it was theorized that the proton was itself composed of three resonances
Gell-Mann would come to call “quarks” Gell-Mann (1964). Over time, this
picture too has evolved, revealing a tremendously complex internal structure,
where the three valence quarks of the proton are bound with gluons to each
other and to a shifting “quark sea” of quarks and antiquarks, as in Fig. 1.1.

Figure 1.1: Model of the Proton over time. Adapted from a figure in pro

This complex structure comes with a number of ongoing mysteries about the
proton which still require resolution. In 2010, the radius of the proton was
measured to a value about 4% less than was observed for the last century Pohl
et al. (2010), a value well outside of statistical error bars which raised
many ongoing questions about the proton’s true size, a problem which is now
known as the “Proton Radius Puzzle.” Similar to the mystery of its exact size,
and relevant to the topic of this dissertation, the proton’s spin is also not
yet well understood, and the source of several puzzles. Spin is a fundamental
property of particles which acts as an intrinsic angular momentum that exceeds
the size which relativistic physics would allow in a real angular momentum,
and which has great impact on the particle’s electromagnetic behavior
Griffiths (2005). Most famous among the mysteries about the proton’s spin is
the fact that though the proton is known to be a composite particle, adding
the spins of its known constituents fails to produce the measured spin of the
proton. This is known as the “Proton Spin Puzzle”, and current research
focuses on investigating the idea that the proton’s spin may in part derive
from the real internal angular momentum of its constituentsChen et al. (2018).

Figure 1.2: The “$\delta_{LT}$ Puzzle”: (Left) The results of Jefferson Lab
E94-010 showing the initial disagreement Amarian et al. (2004). (Right) The
results of Jefferson Lab E97-110 showing a new disagreement in 2021 Sulkosky
et al. (2021).

Additional difficulties arise when trying to determine how the proton’s spin
is distributed within this complex particle. The proton’s spin distribution
can be studied with a quantity known as a structure function, the moments of
these quantities can likewise be directly compared to theoretical predictions
to test the theory of the strong nuclear force, quantum chromodynamics (QCD).
It was expected that one of these moments, $\delta_{LT}$, would be a reliable
test due to its insensitivity to the contribution of the $\Delta$-resonance
Camsonne et al. . However, initial measurements of this quantity showed
sizable disagreement with the predictions of chiral perturbation theory
($\chi$PT) for the neutron Amarian et al. (2004). Chiral perturbation theory
is to date one of the best theoretical frameworks for predicting the behavior
of QCD in low momentum collisions. Newer predictions of $\chi$PT seem to
resolve this initial disagreement, but recent data for the neutron show a new
disagreement at even lower $Q^{2}$, as is shown in Fig. 1.2. The theoretical
difficulty in reproducing data at low momentum transfer for the neutron
implies the dire need for proton data in this region.

The low momentum transfer ($Q^{2}$) region is important, as it is in this
regime that the strong nuclear force becomes “truly strong”; while high
$Q^{2}$ is useful for mapping out the partonic constituents of the nucleon, at
low $Q^{2}$ it is possible to study the behavior of the nucleon as a whole and
how its properties can derive continuously from those of its component
particles. There is some important theoretical work done in this region, most
notably chiral perturbation theory and lattice QCD, which can both reproduce
some quantities at low-to-medium $Q^{2}$. However, there is a severe lack of
experimental data, as taking low $Q^{2}$ data, especially for the proton, is
very difficult. This thesis will detail the results of the Jefferson Lab
E08-027 Experiment (g2p) which attempts to remedy this with a new high
precision measurement of the proton’s spin structure in this low $Q^{2}$,
strong-QCD regime.

## CHAPTER 2Theory

### 2.1 Inclusive Electron Scattering

To study the structure of the nucleon, modern experiments rely on the
experimental technique of particle scattering, a technique which helped to
pioneer the field of nuclear physics when Ernest Rutherford used it to
discover the existence of the nucleus by scattering helium nuclei
($\alpha$-particles) through a gold foil. Rutherford and his students Hans
Geiger and Ernest Marsden discovered that most of these positively charged
particles passed straight through the foil, while a small number deflected at
dramatic angles. The study of the deflected particles led Rutherford to
theorize that the mass of the atom is compressed into a tiny volume known as a
nucleus, with most of the atom composed of empty space, and likewise, modern
scattering experiments can glean ever more detailed information about the
nucleus by studying the scattered particles Rutherford (1911).

The quantity that is usually measured and studied in these experiments is
known as a scattering cross section. This quantity describes the normalized
interaction rate over a solid angle area of the target, and varies with the
energy and momentum of the measured particles. The nucleon’s structure can be
probed by measuring the cross sections which result from scattering a beam of
electrons off of a fixed proton target ($ep$ scattering). Depending on the
energy of the electron beam, a number of different particles may result from
this as the proton’s constituents may be knocked out and form new hadrons, but
here we are concerned with inclusive electron scattering, in which only the
scattered electron is measured, ignoring other products which may result.

feyngraph (320,160) ip,il x1,x2,K,x3,x4,o2,o3,o4,ol arrow_len10
fermionil,vl,ol style_def marrowea expr p = drawarrow subpath (1/4, 3/4) of p
shifted 6 up withpen pencircle scaled 0.4; label.top(btex $k^{\mu}$ etex,
point 0.5 of p shifted 6 up ); enddef; marrowea,tension=0il,vl style_def
marroweb expr p = drawarrow subpath (1/4, 3/4) of p shifted 6 up withpen
pencircle scaled 0.4; label.top(btex $k^{\prime\mu}$ etex, point 0.5 of p
shifted 6 up ); enddef; marroweb,tension=0vl,ol $e^{-}$il $e^{-}$ol
photon,tension=1,label=$\gamma^{*}$vl,vp style_def marrowec expr p = drawarrow
subpath (1/4, 3/4) of p shifted 6 right withpen pencircle scaled 0.4; label.
lrt (btex $q^{\mu}$ etex, point 0.5 of p shifted 6 right ); enddef;
marrowec,tension=0vl,vp phantomip,vp,x1 phantomip,vp fermionvpath (__ip,__vp)
scaled 1.01 shifted (-1.8, 6) style_def marrowed expr p = drawarrow subpath
(1/4, 3/4) of p shifted 12 up withpen pencircle scaled 0.4; label.top(btex
$p^{\mu}$ etex, point 0.5 of p shifted 12 up ); enddef;
marrowed,tension=0ip,vp fermionvpath (__ip,__vp) scaled 1.01 fermionvpath
(__ip,__vp) scaled 1.01 shifted ( 1.8,-6) 15 leftx1,x2,x3,x4 phantomvp,x1
phantomvp,x2 phantomvp,x3 phantomvp,x4 fermionvpath (__vp,__x1) scaled 1.02
shifted ( 0.0,-6.0) fermionvpath (__vp,__x2) scaled 1.00 shifted ( 0.0,-2.0)
fermionvpath (__vp,__x3) scaled 0.98 shifted ( 0.0, 2.0) fermionvpath
(__vp,__x4) scaled 0.96 shifted ( 0.0, 6.0) l.d=0,l.a=0,l=[]. } XK $p$ip 50vp

Figure 2.1: $ep$ Scattering interaction

A Feynman diagram for the electron-proton scattering interaction is shown in
Figure 2.1. In this process, an electron enters with four-momentum (in natural
units) $k^{\mu}$ = (E,k), and exchanges a virtual photon with a proton in the
target, which has initial four-momentum $p^{\mu}$ = ($\epsilon$,p). This
virtual photon carries a four momentum of $q^{\mu}$ = ($\nu$,q), where $\nu$
is the energy transfer from the electron to the struck proton, and q is the
momentum exchanged in the interaction. The electron then scatters at an angle
$\theta$ from its initial path, with a four-momentum of $k^{\prime\mu}$ =
(E′,k′), and some unknown products X are produced by the proton Povh et al.
(1995). Since the electron is comparatively well understood, and we are only
measuring the scattered electrons in inclusive scattering, we define
everything in the rest frame of the proton where $\epsilon$ = $M_{p}$, and in
terms of the initial and final state of the electron. The energy transfer
$\nu$ is then defined simply as:

$\nu=E-E^{\prime}$ (2.1)

with E and E′ the initial and final energies of the electron, respectively. We
can also use these quantities in combination with the angle at which the
scattered electron is measured to define the momentum transfer $Q^{2}$ of the
interaction:

$Q^{2}=-\textbf{q}^{2}=4EE^{\prime}\sin{\theta}$ (2.2)

This is an especially important kinematic variable because it defines the
distance scale which the scattering interaction is probing. The de Broglie
wavelength of a particle is considered to be De Broglie (1924)

$\lambda=\frac{h}{\textbf{q}}$ (2.3)

where h is the Planck constant. So it is plain to see that the distance scale
associated with our interaction is inversely proportional to the momentum
transfer. A high $Q^{2}$ probes the small-distance region of the proton where
we are looking at the nucleon’s individual constituents, while a low $Q^{2}$
conversely probes the large distance region and reveals information about how
the proton’s constituents work together to give the proton its observed
properties. If we wish to test the aforementioned theories of QCD such as
chiral perturbation theory, and to understand how the proton obtains its spin
from the quarks and gluons that make it up, the latter is our kinematic region
of interest Slifer (2004).

We must also define the invariant mass of the hadron, a frame-independent
kinematic variable which quantifies the total mass-energy of the proton and
its unknown products after scattering:

$W^{2}=(p^{\mu}+q^{\mu})^{2}=M_{p}^{2}+2M_{p}\nu-Q^{2}$ (2.4)

Wherein $M_{p}$ $\approx 938.3$ MeV is the mass of the proton. Using the
kinematic variables already defined, we can also define the Bjorken-x, a
quantity which can be identified with the fraction of the proton’s momentum
carried by the struck quark:

$x_{bj}=\frac{Q^{2}}{2M_{p}\nu}$ (2.5)

This quantity can also be considered to track the elasticity of the
interaction, as discussed in the next section. Finally, we define the
fractional energy loss of the electron simply as:

$y=\frac{\nu}{E}$ (2.6)

These invariant quantities will be used to map out the response of the nucleon
over the tested kinematic range.

### 2.2 Interaction Rates & Cross Sections

To define the measurable cross section mentioned in the previous section, we
will first need to discuss the interaction rate of the $ep$ scattering
process. The interaction rate $W_{fi}$ describes the number of transitions
from the initial state of $e+p$ to the final state of $e^{\prime}+X$ per unit
time. Fermi’s Golden Rule states Halzen and Martin (1984):

$W_{fi}=2\pi|V_{fi}|^{2}\delta(k_{f}-k_{i})$ (2.7)

where $V_{fi}$ is the interaction potential and the delta function enforces
energy and momentum conservation. By adopting a covariant normalization, we
can rewrite per unit volume in four dimensions for a generic scattering
interaction A+B $\rightarrow$ C+D Halzen and Martin (1984):

$W_{fi}=\bigg{(}\frac{2\pi}{V}\bigg{)}^{4}|\mathcal{T}|^{2}\delta^{(4)}\big{(}[p_{C}+p_{D}]-[p_{A}+p_{B}]\big{)}$
(2.8)

Here, $\mathcal{T}$ is the probability amplitude of the interaction process, V
is the volume over which the interactions take place, and $p_{A,B,C,D}$ are
the four-momenta of the particles involved in scattering. The probability
amplitude will be a combination of the electron part, which is easy to compute
using the Feynman rules because Quantum Electrodynamics (QED) is well
understood, and the hadronic part, which comprises both the initial proton
state and the final unobserved products X Srednicki (2012). Using the Feynman
rules for spinor electrodynamics, we have one incoming and outgoing electron,
one internal photon, and two vertices. Though the exact vertex on the hadronic
side of the virtual photon is not known, we know it must be a QED vertex, and
so add the associated vertex factor. We then find:

$\mathcal{T}_{Lepton}=ie^{2}\gamma^{\mu}\gamma^{\nu}\frac{g^{\mu\nu}}{\textbf{q}^{2}-i\epsilon}\overline{u}_{s^{\prime}}(\textbf{k${}^{\prime}$})u_{s}(\textbf{k})$
(2.9)

Where $\gamma$ is the gamma-matrix, $u_{s}$ and $\overline{u}_{s}$ are the
spinors, and e is the fundamental charge. Because we don’t know the exact
structure of the hadronic part of the process, we must instead write it
generally as a transition amplitude from the initial state of a proton with
momentum P and spin S to a final unknown state X Zielinski (2017):

$\mathcal{T}_{Hadron}=<P,S|J^{\mu}|X>$ (2.10)

Plugging into Equation 2.8 and simplifying by making use of gamma matrix
identities to contract the metric with two of our gamma matrices we obtain:

$\begin{split}W_{fi}=\bigg{(}\frac{2\pi}{V}\bigg{)}^{4}\bigg{(}\frac{e^{4}}{Q^{4}}\bigg{)}\overline{u}_{s^{\prime}}(\textbf{k${}^{\prime}$})u_{s}(\textbf{k})\gamma^{\mu}\overline{u}_{s}(\textbf{k})u_{s^{\prime}}(\textbf{k${}^{\prime}$})\gamma^{\nu}<P,S|J^{\mu}|X>\\\
<X|J^{\nu}|P,S>\delta^{(4)}\big{(}[p+p_{x}]-[k+k^{\prime}]\big{)}\end{split}$
(2.11)

We can define the cross section which can be experimentally measured in terms
of the interaction rate Halzen and Martin (1984):

$d\sigma=\frac{W_{fi}}{\phi}N_{f}$ (2.12)

Here $\phi$ is the initial flux of particles, and $N_{f}$ is the number of
available final states. We will later associate these quantities with the
number of incident electrons and the density of scattering centers in the
target, respectively, to form an experimental cross section Povh et al.
(1995).

For now, the initial flux will be a product of the number of beam particles
passing through unit area per unit time, $\frac{2E_{e}}{V}$ and the number of
target particles per unit volume, $\frac{2E_{p}}{V}$ Halzen and Martin (1984).
Since the electron mass $m_{e}$ is much smaller than $M_{p}$, we neglect it,
and in our lab frame, the protons are at rest, so we can simplify the initial
flux as:

$\phi=\frac{4EM_{p}}{V^{2}}$ (2.13)

The number of available final states for a given particle A in a volume V is
limited by quantum theory to be Halzen and Martin (1984):

$N_{f}=\frac{Vd^{3}p_{A}}{(2\pi)^{3}2E_{A}}$ (2.14)

Our final states are the scattered electron with momentum $k^{\prime}$ energy
$E^{\prime}$ and some unknown products with momenta $p_{x}$ and energy
$E_{x}$. We will need to sum over all possible X states to obtain the full
cross section, so combining equations 2.14, 2.13 and 2.12 we obtain the
differential cross section:

$\begin{split}d\sigma=\frac{2\pi}{8EE^{\prime}M_{p}}\bigg{(}\frac{e^{4}}{Q^{4}}\bigg{)}d^{3}k^{\prime}\overline{u}_{s^{\prime}}(\textbf{k${}^{\prime}$})u_{s}(\textbf{k})\gamma^{\mu}\overline{u}_{s}(\textbf{k})u_{s^{\prime}}(\textbf{k${}^{\prime}$})\gamma^{\nu}\times\\\
\displaystyle\sum_{X}\frac{d^{3}p_{x}}{(2\pi)^{3}2E_{x}}<P,S|J^{\mu}|X><X|J^{\nu}|P,S>\delta^{(4)}\big{(}[p+p_{x}]-[k+k^{\prime}]\big{)}\end{split}$
(2.15)

To simplify this, in natural units the fine structure constant $\alpha$ =
$\frac{e^{2}}{4\pi}$, and we can define the lepton tensor:

$L^{\mu\nu}=\overline{u}_{s^{\prime}}(\textbf{k${}^{\prime}$})u_{s}(\textbf{k})\gamma^{\mu}\overline{u}_{s}(\textbf{k})u_{s^{\prime}}(\textbf{k${}^{\prime}$})\gamma^{\nu}$
(2.16)

and the hadron tensor:

$W_{\mu\nu}=\frac{1}{2M_{p}}\displaystyle\sum_{X}\frac{d^{3}p_{x}}{2E_{x}}<P,S|J^{\mu}|X><X|J^{\nu}|P,S>\delta^{(4)}\big{(}[p+p_{x}]-[k+k^{\prime}]\big{)}$
(2.17)

This tensor can also be written in a simpler form by employing completeness:

$W_{\mu\nu}=i\int e^{iqx}<P,S|J^{\mu}(x)J^{\nu}(0)|P,S>d^{4}x$ (2.18)

The meaning of these two quantities will be elaborated on in the following
section, but for now we use them to write our cross section as:

$d\sigma=\frac{\alpha^{2}}{Q^{4}}\frac{d^{3}k^{\prime}}{E^{\prime}}\frac{1}{E}L^{\mu\nu}W_{\mu\nu}$
(2.19)

Finally, we will transform into spherical coordinates Srednicki (2012):

$d^{3}k^{\prime}=k^{\prime 2}dk^{\prime}d\Omega$ (2.20)

Where $d\Omega$ = $\sin\theta d\theta d\phi$ is the differential solid angle
which will be measured in the detector. Since the mass of the electron is
small enough to be negligible, k′ $\simeq$ E′. This gives us a doubly
differential cross section definition of:

$\frac{d^{2}\sigma}{dE^{\prime}d\Omega}=\frac{\alpha^{2}}{Q^{4}}\frac{E^{\prime}}{E}L^{\mu\nu}W_{\mu\nu}$
(2.21)

How this quantity is derived experimentally is detailed in Chapter 6. But we
can interpret it most readily with equation 2.12 as being the fraction of
incoming electrons which interact with a proton per unit time, scaled by the
number of possible scattering centers in the target.

For the coming sections, it will be useful to make a distinction for an
elastic measurement of this quantity as compared to an inelastic measurement.
In an elastic measurement, the energy transfer is absorbed entirely by the
recoil proton, such that the final invariant mass of the hadronic system is
simply the rest mass of the proton: $W_{Elastic}$ = $M_{p}$. For elastic
reactions, the proton behaves as a point particle, corresponding to a
Bjorken-x of $x_{bj}$ = 1, and the final energy of the scattered electron can
be easily calculated. For reactions with a higher W, the final hadronic system
is more complex, and we are probing more the partonic behavior of the proton’s
constituents, corresponding with lower values of x. If we go to a very high W,
we reach the Deep Inelastic Scattering (DIS) region where we can probe the
individual quarks and gluons, but between this region and the elastic response
is known as the Resonance Region, this is our area of interest because it
concerns how the proton’s constituents contribute to its behavior on a
hadronic level.

Figure 2.2: ep $\rightarrow$ eX scattering cross section as a function of the
invariant mass W. Figure edited from Halzen and Martin (1984) and produced
with SLAC data.

The behavior of the cross section as a function of W is shown in figure 2.2.
The elastic response is shown in the peak at the proton mass of $M_{p}$
$\approx 938.3$ MeV. The resonance region then begins at the pion production
(pp) threshold, which is the minimum energy transfer necessary to create a
pion, W $\approx 1038.2$ MeV. Several excited states of the proton form
resonances in the following region, but the most important resonance is the
first one, corresponding to the $\Delta$-baryon at W $\approx 1232$ MeV. Our
experiment will focus on measurements which begin at the pion production
threshold, and extend to the highest W which is able to be measured at the
relevant kinematics. These inelastic measurements require a slightly more
complex treatment than an elastic measurement would.

### 2.3 Lepton and Hadron Tensors

The leptonic and hadronic tensors of the previous section encode the
nonperturbative behavior of the electron and hadronic system involved in the
scattering, respectively. What we have defined so far can be considered a
generic form used in unpolarized scattering. But for polarized scattering in
the inelastic regime, we will need to define both tensors a bit differently.
Since the leptonic tensor is a bit simpler, we will start by looking at the
unpolarized version, which is averaged over the initial and final spin of the
scattering electron. To average the T-Matrix we write Thomas and Weise (2001):

$<|\mathcal{T}|^{2}>=\frac{1}{2}\displaystyle\sum_{s,s^{\prime}}|\mathcal{T}_{Lepton}|^{2}$
(2.22)

For an unpolarized initial electron, we can adopt the normalization condition
Thomas and Weise (2001):

$\displaystyle\sum_{s=\pm\frac{1}{2}}u_{s}(k)\overline{u}_{s}(k)=\gamma^{\mu}k_{\mu}+m_{e}$
(2.23)

To average the unpolarized version of the lepton tensor of 2.16 we must take
the trace of the T-matrix, so using this normalization condition:

$L^{\mu\nu}_{U}=\frac{1}{2}Tr\bigg{(}(\gamma^{\mu}k_{\mu}+m_{e})\gamma^{\mu}(\gamma^{\nu}k^{\prime}_{\nu}+m_{e})\gamma^{\nu}\bigg{)}$
(2.24)

Multiplying out these terms and performing the trace we find:

$L^{\mu\nu}_{U}=2\big{[}k^{\mu}k^{\prime\nu}+k^{\prime\mu}k^{\nu}-g^{\mu\nu}(\textbf{k}\cdot\textbf{k${}^{\prime}$}-m_{e}^{2})\big{]}$
(2.25)

In the case that the electron in question has an initial polarization, we will
need to pick a different normalization, yielding an extra term Zielinski
(2017); Thomas and Weise (2001):

$L^{\mu\nu}_{P}=2\big{[}k^{\mu}k^{\prime\nu}+k^{\prime\mu}k^{\nu}-g^{\mu\nu}(\textbf{k}\cdot\textbf{k${}^{\prime}$}-m_{e}^{2})+i\epsilon^{\mu\nu\alpha\beta}s_{\alpha}q_{\beta}\big{]}$
(2.26)

Here $\epsilon$ is the Levi-Civita tensor, which enforces
$\epsilon_{0123}=+1$, $q_{\beta}$ is the momentum of the virtual photon, and
$s_{\alpha}$ is the lepton spin vector, which points along the spin
quantization vector z with magnitude
$s_{\mu}=\overline{u}\gamma_{\mu}\gamma_{5}u$ and sign determined by whether
the electron helicity is forward (along z) or backward (antiparallel to z).
Note that the polarization dependent term makes the lepton tensor asymmetric,
but the asymmetric term cancels when averaged over the initial spins.

The lepton tensor characterizes the scattering electron part of the diagram in
Fig 2.1. However, we are far more interested in the inelastic behavior of the
other part, since it contains information on our proton target. This part is
characterized by the hadronic tensor of (2.17). This can be split into a
symmetric unpolarized part and an antisymmetric polarized part, much like the
lepton tensor. We can first get the most general form of the unpolarized part
by expanding its previous form considering Lorentz and gauge invariance Thomas
and Weise (2001):

$W_{\mu\nu}^{U}=W_{1}(\nu,Q^{2})\bigg{(}\frac{q_{\mu}q_{\nu}}{Q^{2}}-g_{\mu\nu}\bigg{)}+W_{2}(\nu,Q^{2})\frac{1}{M_{p}^{2}}\bigg{(}p_{\mu}-\frac{\textbf{p}\cdot\textbf{q}}{Q^{2}}q_{\mu}\bigg{)}\bigg{(}p_{\nu}-\frac{\textbf{p}\cdot\textbf{q}}{Q^{2}}q_{\nu}\bigg{)}$
(2.27)

Here, $W_{1}$ and $W_{2}$ are inelastic unpolarized structure functions which
will be discussed in the next section. For now, as in the lepton tensor, we
add an antisymmetric term to get the polarized variant of the hadronic tensor
Thomas and Weise (2001):

$W_{\mu\nu}^{P}=W_{\mu\nu}^{U}+i\epsilon_{\mu\nu\alpha\beta}q^{\alpha}\bigg{(}G_{1}(\nu,Q^{2})M_{p}S^{\beta}+G_{2}(\nu,Q^{2})\frac{1}{M_{p}}(S^{\beta}\textbf{p}\cdot\textbf{q}-p^{\beta}\textbf{S}\cdot\textbf{q})\bigg{)}$
(2.28)

Where $S^{\beta}$ corresponds to the proton’s spin vector, similar to the
electron’s in the prior discussion. $G_{1}$ and $G_{2}$ are polarized
structure functions which will be discussed fully in the next section. To
continue our understanding of an unpolarized part with additional terms that
represent the polarized physics, let us first contract the unpolarized
leptonic tensor of (2.25) and the unpolarized hadronic tensor of (2.27) to get
a fully symmetric, unpolarized result:

$L^{\mu\nu}W_{\mu\nu}=4W_{1}(\nu,Q^{2})(\textbf{k}\cdot\textbf{k${}^{\prime}$})+2W_{2}(\nu,Q^{2})\bigg{[}\frac{2}{M_{p}^{2}}(\textbf{p}\cdot\textbf{k})(\textbf{p}\cdot\textbf{k${}^{\prime}$})-(\textbf{k}\cdot\textbf{k${}^{\prime}$})\bigg{]}$
(2.29)

In the lab reference frame, the proton is at rest, and the electron only has
energy in the direction of propagation along the z-axis. It scatters from the
central axis by an angle $\theta$ along a vector with azimuthal angle $\phi$.
This allows us to write in full the relevant four-vectors Zielinski (2017):

$k_{\mu}=(E,\textbf{k})=(E,0,0,E)$ (2.30)
$k^{\prime}_{\mu}=(E^{\prime},\textbf{k${}^{\prime}$})=(E^{\prime},E^{\prime}\sin\theta\cos\phi,E^{\prime}\sin\theta\sin\phi,E^{\prime}\cos\theta)$
(2.31) $p_{\mu}=(M_{p},\textbf{p})=(M_{p},0,0,0)$ (2.32)

Plugging these definitions and (2.29) into our equation for the cross section
in (2.21) we can write our unpolarized cross section as:

$\frac{d^{2}\sigma}{dE^{\prime}d\Omega}=\frac{\alpha^{2}E^{\prime}}{4E^{3}}\bigg{(}2W_{1}(\nu,Q^{2})\frac{1}{\sin^{2}\frac{\theta}{2}}+W_{2}(\nu,Q^{2})\frac{\cos^{2}\frac{\theta}{2}}{\sin^{4}\frac{\theta}{2}}\bigg{)}$
(2.33)

To simplify our understanding of this quantity, it is helpful to factor out
the Mott cross section, which describes scattering off of a simple point
particle Halzen and Martin (1984):

$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}=\frac{\alpha^{2}\cos^{2}\frac{\theta}{2}}{4E^{2}\sin^{4}\frac{\theta}{2}}\frac{E^{\prime}}{E}$
(2.34)
$\frac{d^{2}\sigma}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\bigg{(}2W_{1}(\nu,Q^{2})\tan^{2}\frac{\theta}{2}+W_{2}(\nu,Q^{2})\bigg{)}$
(2.35)

This allows us to interpret the structure functions as quantities which
characterize the difference in behavior from a pointlike particle. If we add
in the asymmetric terms from the unpolarized lepton and hadron tensors of
(2.26) and (2.28) to our contraction, we get several additional terms in this
cross section expansion:

$\begin{split}\frac{d^{2}\sigma}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\frac{E^{2}}{Q^{2}}\tan^{2}\frac{\theta}{2}\sin^{2}\frac{\theta}{2}\bigg{(}2W_{1}(\nu,Q^{2})\tan^{2}\frac{\theta}{2}+W_{2}(\nu,Q^{2})\frac{1}{\tan^{2}\frac{\theta}{2}}+\\\
4M_{p}G_{1}(\nu,Q^{2})\big{[}(\textbf{s}\cdot\textbf{S})+\frac{1}{Q^{2}}(\textbf{s}\cdot\textbf{q})(\textbf{q}\cdot\textbf{S})\big{]}+\\\
\frac{4}{M_{p}}G_{2}(\nu,Q^{2})\big{[}(\textbf{p}\cdot\textbf{q})(\textbf{s}\cdot\textbf{S})-(\textbf{S}\cdot\textbf{q})(\textbf{p}\cdot\textbf{s})\big{]}\bigg{)}\end{split}$
(2.36)

To simplify this, we will define the spin vectors as we did the others in the
lab frame. The electron’s spin vector s points along the initial momentum k
and thus differs only by a factor of the electron mass. The proton’s spin
vector is situated at some angle $\beta$ from the scattering plane at an
azimuthal angle of $\rho$:

$s_{\mu}=(\frac{E}{m_{e}},\textbf{s})=\frac{1}{m_{e}}(E,0,0,E)$ (2.37)
$S_{\mu}=(0,\textbf{S})=(0,\sin\beta\cos\rho,\sin\beta\sin\rho,\cos\beta)$
(2.38)

Using these and our earlier four vector definitions we can write, making use a
few trigonometric identities:

$\begin{split}\frac{d^{2}\sigma}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\frac{E^{2}}{Q^{2}}\tan^{2}\frac{\theta}{2}\sin^{2}\frac{\theta}{2}\bigg{(}2W_{1}(\nu,Q^{2})\tan^{2}\frac{\theta}{2}+W_{2}(\nu,Q^{2})\frac{1}{\tan^{2}\frac{\theta}{2}}+\\\
4M_{p}G_{1}(\nu,Q^{2})\big{[}(E\cos\beta+E^{\prime}\sin\theta\sin\beta\cos(\phi-\rho)+E^{\prime}\cos\theta\cos\beta\big{]}+\\\
4G_{2}(\nu,Q^{2})\big{[}2E\sin\theta\sin\beta\cos(\phi-\rho)-Q^{2}\cos\beta\big{]}\bigg{)}\end{split}$
(2.39)

Though this seems messy, this construction of the cross section will be our
basic template for extracting the structure functions, and will make
significantly more sense with smart choices of angle for the proton
polarization vector $S_{\mu}$.

### 2.4 Structure Functions

We have already identified four structure functions $W_{1}$, $W_{2}$, $G_{1}$,
and $G_{2}$, though we have not attempted to explain them properly. Before we
try to do so, it is necessary to consider how we can simplify the cross
section above to extract the structure functions we care about. Firstly, we
consider the cross section where the electron has forward helicity, to one
where it has backward helicity. The latter will flip the sign of the latter
two terms by inverting the sign of $s_{\mu}$, so if we subtract one from the
other we will cancel the unpolarized terms and double the magnitude of the
polarized terms, leaving us with:

$\begin{split}\frac{d^{2}\sigma^{\uparrow}}{dE^{\prime}d\Omega}-\frac{d^{2}\sigma^{\downarrow}}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\frac{8E^{2}}{Q^{2}}\tan^{2}\frac{\theta}{2}\sin^{2}\frac{\theta}{2}\times\\\
\bigg{(}M_{p}G_{1}(\nu,Q^{2})\big{[}(E\cos\beta+E^{\prime}\sin\theta\sin\beta\cos(\phi-\rho)+E^{\prime}\cos\theta\cos\beta\big{]}+\\\
G_{2}(\nu,Q^{2})\big{[}2EE^{\prime}\sin\theta\sin\beta\cos(\phi-\rho)-Q^{2}\cos\beta\big{]}\bigg{)}\end{split}$
(2.40)

Let us now consider the proton polarization. We will look at two specific
cases, one where the proton is polarized longitudinally, or in the same
direction as k, in which case $\beta$ = 0 or $\pi$.

$\begin{split}\Delta\sigma_{\parallel}=\frac{d^{2}\sigma^{\uparrow\Uparrow}}{dE^{\prime}d\Omega}-\frac{d^{2}\sigma^{\downarrow\Uparrow}}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\frac{4E^{2}}{Q^{2}}\tan^{2}\frac{\theta}{2}\sin^{2}\frac{\theta}{2}\times\\\
\bigg{(}M_{p}G_{1}(\nu,Q^{2})\big{[}E+E^{\prime}\cos\theta\big{]}-Q^{2}G_{2}(\nu,Q^{2})\bigg{)}\end{split}$
(2.41)

This is known as the longitudinal cross section difference. We can also define
a transverse cross section difference where $\beta$ = $\frac{\pi}{2}$ or
$\frac{3\pi}{2}$:

$\begin{split}\Delta\sigma_{\perp}=\frac{d^{2}\sigma^{\uparrow\Rightarrow}}{dE^{\prime}d\Omega}-\frac{d^{2}\sigma^{\downarrow\Rightarrow}}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\frac{4E^{2}}{Q^{2}}\tan^{2}\frac{\theta}{2}\sin^{2}\frac{\theta}{2}\times\\\
E^{\prime}\sin\theta\cos(\phi-\rho)\bigg{(}M_{p}G_{1}(\nu,Q^{2})+2EG_{2}(\nu,Q^{2})\bigg{)}\end{split}$
(2.42)

Notably, these structure functions are dimensionful quantities. So, there is a
convention to redefine several dimensionless structure functions as a function
of the Bjorken-x defined above Thomas and Weise (2001):

$F_{1}(x,Q^{2})=M_{p}W_{1}(\nu,Q^{2})$ (2.43) $F_{2}(x,Q^{2})=\nu
W_{2}(\nu,Q^{2})$ (2.44) $g_{1}(x,Q^{2})=\nu M_{p}^{2}G_{1}(\nu,Q^{2})$ (2.45)
$g_{2}(x,Q^{2})=\nu^{2}M_{p}G_{2}(\nu,Q^{2})$ (2.46)

We can define the angle difference $\phi-\rho$ as the out-of-plane
polarization angle, the angle between the polarization and scattering planes,
$\theta_{OoP}$. This gives us a final definition of our polarized cross
section differences:

$\begin{split}\Delta\sigma_{\parallel}=\frac{d^{2}\sigma^{\uparrow\Uparrow}}{dE^{\prime}d\Omega}-\frac{d^{2}\sigma^{\downarrow\Uparrow}}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\frac{4E^{2}}{Q^{2}}\tan^{2}\frac{\theta}{2}\sin^{2}\frac{\theta}{2}\times\\\
\frac{1}{M_{p}}\bigg{(}\frac{1}{\nu}g_{1}(x,Q^{2})\big{[}E+E^{\prime}\cos\theta\big{]}-\frac{Q^{2}}{\nu^{2}}g_{2}(x,Q^{2})\bigg{)}\end{split}$
(2.47)
$\begin{split}\Delta\sigma_{\perp}=\frac{d^{2}\sigma^{\uparrow\Rightarrow}}{dE^{\prime}d\Omega}-\frac{d^{2}\sigma^{\downarrow\Rightarrow}}{dE^{\prime}d\Omega}=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}\frac{4E^{2}}{Q^{2}}\tan^{2}\frac{\theta}{2}\sin^{2}\frac{\theta}{2}\times\\\
\frac{E^{\prime}}{M_{p}}\sin\theta\cos\theta_{OoP}\bigg{(}\frac{1}{\nu}g_{1}(x,Q^{2})+2\frac{E}{\nu^{2}}g_{2}(x,Q^{2})\bigg{)}\end{split}$
(2.48)

The former equation is dominated by the $g_{1}$ term, while the latter is
dominated by the $g_{2}$ term. For this reason, we refer to $g_{1}$ as a
longitudinal structure function, which can be measured with a longitudinally
polarized target, and $g_{2}$ as a transverse structure function, which can be
measured with a transverse polarized target. At this point we have two
equations and two unknowns, so a measurement of both polarized cross section
differences would let us solve for our structure functions $g_{1}$ and
$g_{2}$. In practice, these are rarely both measured at the same kinematics,
so we will need to solve for them slightly differently, a discussion which we
will pick up in the Analysis chapter.

Let us now consider the meaning of these structure functions. It is clear from
the equations above that $g_{1}$ and $g_{2}$ characterize the spin-dependent
part of the scattering event which was introduced by our asymmetric tensors,
and that they each act as a type of scaling on the basic Mott cross section.
So, it is simplest to interpret that for a fixed momentum transfer $Q^{2}$,
these structure functions represent the deviation of the proton’s internal
spin from point-like behavior, with measurements over the Bjorken-x at varying
$Q^{2}$ allowing us to map out this spin structure over a range of distance
scales.

### 2.5 Moments and Sum Rules

The natural question now is how we can relate these structure functions back
to theoretical predictions of QCD. The structure functions themselves are
difficult to calculate directly from the known theory of the standard model
because they encompass all possible unknown states X for the final state of
the hadronic system. Instead, we can investigate a class of kinematically
weighted integrals of the spin structure functions known as moments or sum
rules Zielinski (2017).

To handle the unknown states X in a way that we can calculate we can employ
the optical theorem Jackson (1999). The optical theorem states that the total
scattering cross section is proportional to the imaginary part of the forward
scattering amplitude:

$\sigma_{tot}=\frac{4\pi}{K}\operatorname{Im}\big{(}f(0)\big{)}$ (2.49)

Here K is the virtual photon flux, the definition of which is solely based in
convention. Commonly, and in this analysis, we will employ the Hand convention
Slifer (2004):

$K=\nu-\frac{Q^{2}}{2M_{p}}$ (2.50)

In Compton scattering, a high frequency photon scatters off of a stationary
target. Specifically, we are interested in virtual-virtual Compton scattering
(VVCS), which concerns the scattering of a virtual photon as both the initial
and final state. By using completeness and writing out the amplitude of Figure
2.3 in terms of the probability currents, much as we did for the general form
of the hadron tensor, we can write the Compton tensor as Drechsel et al.
(2003):

$T^{\mu\nu}=i\mathcal{T}\int e^{iqx}<P,S|J^{\mu}(x)J^{\nu}(0)|P,S>d^{4}x$
(2.51)

We note that this is nearly identical to the generic form of our hadron tensor
in (2.18), with the addition of a time-ordered product $\mathcal{T}$ which
demands the chronological ordering of the probability current operators.
Consequently, we can relate the hadron and Compton tensors with Slifer (2004):

$W_{\mu\nu}=\frac{1}{2\pi M_{p}}\operatorname{Im}\big{(}T_{\mu\nu}\big{)}$
(2.52)

This means instead of the unknown states of Figure 2.1, we can investigate the
amplitude of the simpler diagram in Figure 2.3 to derive the measurable
quantities of interest.

feyngraph2 (280,160) ip,il op,x1,x2,K,x3,x4,o2,o3,o4,ol arrow_len10

$\gamma^{*}$il $\gamma^{*}$ol photonil,vp,ol style_def marrowpa expr p =
drawarrow subpath (1/4, 3/4) of p shifted 6 up withpen pencircle scaled 0.4;
label. top(btex $q^{\mu}$ etex, point 0.5 of p shifted 6 up ); enddef;
marrowpa,tension=0il,vp style_def marrowea expr p = drawarrow subpath (1/4,
3/4) of p shifted 6 up withpen pencircle scaled 0.4; label.top(btex $q^{\mu}$
etex, point 0.5 of p shifted 6 up ); enddef; marrowea,tension=0vp,ol
phantomip,vp,op phantomip,vp fermionvpath (__ip,__vp) scaled 1.01 fermionvpath
(__ip,__vp) scaled 1.01 shifted (-1.7, 6) fermionvpath (__ip,__vp) scaled 1.01
shifted ( 1.7,-6) style_def marrowed expr p = drawarrow subpath (1/4, 3/4) of
p shifted 12 up withpen pencircle scaled 0.4; label.top(btex $p^{\mu}$ etex,
point 0.5 of p shifted 12 up ); enddef; marrowed,tension=0ip,vp style_def
marrowex expr p = drawarrow subpath (1/4, 3/4) of p shifted 18 up withpen
pencircle scaled 0.4; label.top(btex $p^{\mu}$ etex, point 0.5 of p shifted 18
up ); enddef; marrowex,tension=0vp,op phantomvp,op fermionvpath (__vp,__op)
scaled 1.05 shifted (-5.1, 6) fermionvpath (__vp,__op) scaled 1.05 shifted
(-7.0, 0) fermionvpath (__vp,__op) scaled 1.05 shifted (-8.9,-6) $p$ip $p$op
70vp

Figure 2.3: Virtual-Virtual Compton Scattering Interaction

To begin, we can write down the decomposition of the VVCS forward scattering
amplitude, which takes the form Drechsel (2009):

$T=\vec{\epsilon}^{\prime}\cdot\vec{\epsilon}f_{T}+f_{L}+i\vec{S}\cdot(\vec{\epsilon}^{\prime}\times\vec{\epsilon})f_{TT}+-i\vec{S}\cdot\big{[}(\vec{\epsilon}^{\prime}-\vec{\epsilon})\times\hat{q}\big{]}f_{LT}$
(2.53)

Here the cross section is expanded in terms of the four amplitudes
$f_{L,T,TT,LT}$ which describe the scattering amplitude of the photon’s
longitudinal polarization states and transverse polarization states
respectively, as well as the transverse-transverse and longitudinal-transverse
interference amplitudes. These interference amplitudes are produced by the
interaction with the proton’s spin, as without the additional spin vector the
only nonvanishing terms would be $f_{T}$ and $f_{L}$. $\epsilon^{\prime}$ and
$\epsilon$ represent the transverse polarization vectors of the virtual
photon, and $\hat{q}$ represents the longitudinal polarization vector of the
virtual photon. These amplitudes are related by the optical theorem back to
four partial virtual photon cross sections Drechsel (2009):

$\operatorname{Im}\\{f_{L},f_{T},f_{TT},f_{LT}\\}=\frac{K}{4\pi}\\{\sigma_{L},\sigma_{T},\sigma_{TT},\sigma_{LT}\\}$
(2.54)

These virtual photon cross sections are related back to the structure
functions of the previous section by Drechsel (2009):

$\\{\sigma_{L},\sigma_{T},\sigma_{TT},\sigma_{LT}\\}=\frac{4\pi^{2}\alpha^{2}}{M_{p}K}\\{F_{1},\frac{M_{p}(1+\frac{Q^{2}}{\nu^{2}})}{\frac{Q^{2}}{\nu}}F_{2}-F_{1},g_{1}-\frac{Q^{2}}{\nu^{2}}g_{2},\frac{Q}{\nu}(g_{1}+g_{2})\\}$
(2.55)

Because understanding the imaginary part of the amplitude will be difficult,
it will be better to relate this back to the real part of the amplitude. We
can do this by writing a dispersion relation with the Cauchy integral formula
Zielinski (2017):

$f_{i}(\nu,Q^{2})=\frac{2}{\pi}\int_{\nu_{pp}}^{\infty}\frac{\nu^{\prime}\operatorname{Im}\big{[}f_{i}(\nu,Q^{2})\big{]}}{\nu^{\prime
2}-\nu^{2}}d\nu^{\prime}$ (2.56)

Here, $\nu_{pp}$ is the energy transfer of the pion production threshold
discussed earlier. $F_{1}$ and $F_{2}$ are interesting for the insight they
provide into the quark-gluon distribution of the nucleon, but here we are
interested in the polarized structure functions, so we will focus on the
interference amplitudes $f_{LT}$ and $f_{TT}$. By plugging (2.54) into (2.56)
we obtain:

$f_{TT}(\nu,Q^{2})=\frac{1}{2\pi^{2}}\int_{\nu_{pp}}^{\infty}\frac{K\nu\sigma_{TT}(\nu,Q^{2})}{\nu^{\prime
2}-\nu^{2}}d\nu^{\prime}$ (2.57)
$f_{LT}(\nu,Q^{2})=\frac{1}{2\pi^{2}}\int_{\nu_{pp}}^{\infty}\frac{K\nu^{\prime}\sigma_{LT}(\nu,Q^{2})}{\nu^{\prime
2}-\nu^{2}}d\nu^{\prime}$ (2.58)

To study this integral, we perform a low-energy power series expansion on the
real part of the amplitude. For $\nu<\nu_{pp}$ we obtain Drechsel (2009):

$\operatorname{Re}f_{TT}(\nu,Q^{2})=\frac{2\alpha}{M_{p}^{2}}I_{A}(Q^{2})\nu+\gamma_{0}(Q^{2})\nu^{3}+\mathcal{O}(\nu^{5})...$
(2.59)
$\operatorname{Re}f_{LT}(\nu,Q^{2})=\frac{2\alpha}{M_{p}^{2}}QI_{LT}(Q^{2})+Q\delta_{LT}(Q^{2})\nu^{2}+\mathcal{O}(\nu^{4})...$
(2.60)

The terms here are described as moments of the spin structure functions,
comparison to the integral and the use of (2.55) yields a series of sum rules
defining them. The first moment in the $f_{TT}$ low-energy expansion (LEX) is:

$I_{A}(Q^{2})=\frac{2M_{p}^{2}}{Q^{2}}\int_{0}^{x_{pp}}\bigg{(}g_{1}(x,Q^{2})-\frac{4M^{2}x^{2}}{Q^{2}}g_{2}(x,Q^{2})\bigg{)}dx$
(2.61)

This is known as the GDH Sum Rule. At the real photon point, $Q^{2}$ = 0, the
GDH Sum Rule can be identified with the magnetic moment $\kappa$ of the
proton, such that:

$I_{A}(0)=-\frac{2\pi^{2}\alpha\kappa^{2}}{M_{p}^{2}}$ (2.62)

The second term in $f_{TT}$’s expansion yields the generalized forward spin
polarizability:

$\gamma_{0}(Q^{2})=\frac{16\alpha
M_{p}^{2}}{Q^{6}}\int_{0}^{x_{pp}}x^{2}\bigg{(}g_{1}(x,Q^{2})-\frac{4M^{2}x^{2}}{Q^{2}}g_{2}(x,Q^{2})\bigg{)}dx$
(2.63)

As will be discussed later, this and other polarizabilities describe the
nucleon’s response to an external field, with spin polarizabilities
correlating to a spin-dependent response. In both $I_{A}$ and $\gamma_{0}$,
the integral is dominated by the $g_{1}$ term due to the strong kinematic
weighting on the $g_{2}$ term, and thus is measured primarily with
longitudinally polarized nucleon data.

For the $f_{LT}$ expansion, the first term yields the moment:

$I_{LT}(Q^{2})=\frac{2M_{p}^{2}}{Q^{2}}\int_{0}^{x_{pp}}\bigg{(}g_{1}(x,Q^{2})+g_{2}(x,Q^{2})\bigg{)}dx$
(2.64)

While the second term gives us the longitudinal-transverse spin
polarizability:

$\delta_{LT}(Q^{2})=\frac{16\alpha
M_{p}^{2}}{Q^{6}}\int_{0}^{x_{pp}}x^{2}\bigg{(}g_{1}(x,Q^{2})+g_{2}(x,Q^{2})\bigg{)}dx$
(2.65)

Unlike our first two moments, $g_{1}$ and $g_{2}$ contribute on a similar
scale to these integrals, so a transverse measurement is vital to compute the
moment. Further, it is worth noting that in the second moments $\gamma_{0}$
and $\delta_{LT}$, the strong $x^{2}$ weighting means that the part of the
integral closest to the pion production threshold will dominate, and a
measurement that goes out to x = 0 or W = $\infty$ is unnecessary. For the
first moments it is necessary to calculate this low-x, DIS contribution, but
for the higher moments the integral is completely dominated by the resonance
region shown in Figure 2.2.

There are several other moments of interest for which this is not the case.
These moments are obtained by attempting to construct the VVCS amplitudes
which correspond directly to the structure functions, which is done by casting
(2.66) into a covariant form Drechsel et al. (2003):

$\begin{split}T=\epsilon_{\mu}^{\prime}\epsilon_{\nu}\Bigg{\\{}\bigg{(}-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{Q^{2}}\bigg{)}T_{1}(\nu,Q^{2})+\frac{1}{\textbf{p}\cdot\textbf{q}}\bigg{(}p^{\mu}-\frac{\textbf{p}\cdot\textbf{q}}{Q^{2}}q^{\mu}\bigg{)}\bigg{(}p^{\nu}-\frac{\textbf{p}\cdot\textbf{q}}{Q^{2}}q^{\nu}\bigg{)}T_{2}(\nu,Q^{2})\\\
+\frac{i}{M_{p}}\epsilon^{\mu\nu\alpha\beta}q_{\alpha}S_{\beta}S_{1}(\nu,Q^{2})+\frac{i}{M_{p}^{3}}\epsilon^{\mu\nu\alpha\beta}q_{\alpha}\bigg{(}\textbf{p}\cdot\textbf{q}S_{\beta}-\textbf{S}\cdot\textbf{q}p_{\beta}\bigg{)}S_{2}(\nu,Q^{2})\end{split}$
(2.66)

This form introduces four covariant VVCS amplitudes, which are related to the
previously introduced VVCS amplitudes by:

$T_{1}(\nu,Q^{2})=f_{T}(\nu,Q^{2})$ (2.67) $T_{2}(\nu,Q^{2})=\frac{\nu
Q^{2}}{M_{p}(\nu^{2}+Q^{2})}\big{(}f_{T}(\nu,Q^{2})+f_{L}(\nu,Q^{2})\big{)}$
(2.68) $S_{1}(\nu,Q^{2})=\frac{\nu
M_{p}}{\nu^{2}+Q^{2}}\big{(}f_{TT}(\nu,Q^{2})+\frac{Q}{\nu}f_{LT}(\nu,Q^{2})\big{)}$
(2.69)
$S_{2}(\nu,Q^{2})=-\frac{M_{p}^{2}}{\nu^{2}+Q^{2}}\big{(}f_{TT}(\nu,Q^{2})-\frac{\nu}{Q}f_{LT}(\nu,Q^{2})\big{)}$
(2.70)

If we employ our previous application of the optical theorem and take the
imaginary part of these definitions, we can relate these amplitudes back to
the structure functions:

$\operatorname{Im}T_{1}(\nu,Q^{2})=\frac{2\pi}{M_{p}}F_{1}(x,Q^{2})$ (2.71)
$\operatorname{Im}T_{2}(\nu,Q^{2})=\frac{2\pi}{\nu}F_{2}(x,Q^{2})$ (2.72)
$\operatorname{Im}S_{1}(\nu,Q^{2})=\frac{2\pi}{\nu M_{p}^{2}}g_{1}(x,Q^{2})$
(2.73)
$\operatorname{Im}S_{2}(\nu,Q^{2})=\frac{2\pi}{\nu^{2}M_{p}}g_{2}(x,Q^{2})$
(2.74)

Again, we will focus on the latter two amplitudes, as we can see they are
directly related to our spin-dependent structure functions. Writing a
dispersion relation for each with the Cauchy theorem:

$S_{1}(\nu,Q^{2})=\frac{2}{\pi}\int_{\nu_{pp}}^{\infty}\frac{\nu^{\prime}\operatorname{Im}S_{1}}{\nu^{\prime
2}-\nu^{2}}d\nu^{\prime}$ (2.75)
$S_{2}(\nu,Q^{2})=\frac{2}{\pi}\int_{\nu_{pp}}^{\infty}\frac{\nu\operatorname{Im}S_{2}}{\nu^{\prime
2}-\nu^{2}}d\nu^{\prime}$ (2.76)

It will also be useful to write a dispersion relation for the amplitude $\nu
S_{2}$:

$\nu
S_{2}(\nu,Q^{2})=\frac{2}{\pi}\int_{\nu_{pp}}^{\infty}\frac{\nu^{\prime}\operatorname{Im}S_{2}}{\nu^{\prime
2}-\nu^{2}}d\nu^{\prime}$ (2.77)

For $S_{1}$ we can perform a low-energy expansion just like we did for the
other amplitudes:

$\operatorname{Re}S_{1}(\nu,Q^{2})=\frac{2\alpha}{M_{p}}\Gamma_{1}(Q^{2})+\bigg{[}\frac{2\alpha}{M_{p}Q^{2}}\big{(}I_{A}(Q^{2})-\Gamma_{1}(Q^{2})\big{)}+M_{p}\delta_{LT}(Q^{2})\bigg{]}\nu^{2}+\mathcal{O}(\nu^{4})...$
(2.78)

Here we have several of the moments defined previously, as well as one new
one. Using our definitions of $S_{1}$ in terms of the structure functions:

$\Gamma_{1}(Q^{2})=\int_{0}^{1}g_{1}(x,Q^{2})dx$ (2.79)

It is worth noting that this moment contains the full integral from x of 0 to
1, and thus requires a calculation of the elastic contribution. We can obtain
a similar moment, and with it an important sum rule, by subtracting (2.77)
from (2.76) to obtain Drechsel et al. (2003):

$\Gamma_{2}(Q^{2})=\int_{0}^{1}g_{2}(x,Q^{2})dx=0$ (2.80)

This “Super-Convergence relation” is known as the Burkhardt-Cottingham sum
rule, an identity which has power in equating the elastic and inelastic parts
of the integral, stating that they ought to cancel each other for any value of
$Q^{2}$. Due to the lack of any kinematic weighting, $\Gamma_{1}$ and
$\Gamma_{2}$ require every part of the integral to be accounted for, including
whatever unmeasured low-x part exists.

One final set of moments is of interest to us, and can be produced by
considering the full, general form of the forward Compton scattering amplitude
in (2.51). One way to obtain a useful expansion of this integral is with the
Operator Product Expansion (OPE) Wilson (1969). At small distances, the OPE
allows the following general expansion of two operators Zielinski (2017):

$\lim_{x\rightarrow
0}\mathcal{O}_{a}(x)\mathcal{O}_{b}(0)=\sum_{k}C_{abk}(x)\mathcal{O}_{k}(0)$
(2.81)

Taking the Fourier transform of this equation yields:

$\lim_{x\rightarrow 0}\int
e^{iqx}\mathcal{O}_{a}(x)\mathcal{O}_{b}(0)d^{4}x=\lim_{Q^{2}\rightarrow\infty}\sum_{k}C_{abk}(q)\mathcal{O}_{k}(0)$
(2.82)

This allows us to apply the OPE directly to our forward Compton amplitude, but
as we can see, the method becomes valid only in the high $Q^{2}$ regime. This
yields a Compton amplitude in the high $Q^{2}$ limit of:

$T^{\mu\nu}=\sum_{k}C_{\mu\nu k}(Q^{2})J_{k}(0)$ (2.83)

The exercise of calculating the Wilson coefficients $C_{abk}(q)$ is carried
out in Thomas and Weise (2001), and yields the general form:

$C_{\mu\nu
k}\approx\bigg{(}\frac{1}{x}\bigg{)}^{n}\bigg{(}\frac{M_{p}}{\sqrt{Q^{2}}}\bigg{)}^{\tau-2}$
(2.84)

Here n is defined as the spin of the relevant operator, and with D as
dimension of the operator, we additionally define the twist $\tau$ as
$\tau=D-n$. Twist is a spin-dependent quantity which helps to quantify the
order in $Q^{2}$ that a particular effect contributes for a given interaction
Jaffe (1996). At high $Q^{2}$, the leading twist is twist-2, but at lower
$Q^{2}$, higher twist terms can contribute. If we expand up to twist-3, we can
write our forward Compton amplitude expansion as a function of the matrix
elements $a_{n}$ and $d_{n}$, the twist-2 and twist-3 local operators, such
that:

$\begin{split}T^{\mu\nu}=C^{a}_{n}(Q^{0})a_{n}(Q^{2})+C^{d}_{n}(Q^{1})d_{n}(Q^{2})+\mathcal{O}(Q^{2})+...\\\
\approx\bigg{(}\frac{1}{x}\bigg{)}^{n}a_{n}(Q^{2})+\bigg{(}\frac{1}{x}\bigg{)}^{n}\frac{M_{p}}{\sqrt{Q^{2}}}d_{n}(Q^{2})\end{split}$
(2.85)

We can extract a relation between these matrix elements and our structure
functions by writing a dispersion relation, an exercise performed in Jaffe
(1996). The result gives us the general form:

$\begin{split}\int_{0}^{1}x^{n}g_{1}(x,Q^{2})=\frac{1}{4}a_{n}(Q^{2});n=0,2,4\end{split}$
(2.86)
$\begin{split}\int_{0}^{1}x^{n}g_{2}(x,Q^{2})=\frac{1}{4}\frac{n}{n+1}\big{(}d_{n}-a_{n}\big{)};n=2,4\end{split}$
(2.87)

The n=0 case of the second relation is ignored because the OPE assumes
implicitly that the B.C. sum rule previously introduced holds. Of interest to
us is the matrix element $d_{2}$. For the n=2 case, we can solve the previous
two equations to obtain:

$d_{2}(Q^{2})=\int_{0}^{1}x^{2}\bigg{(}2g_{1}(x,Q^{2})+3g_{2}(x,Q^{2})\bigg{)}dx$
(2.88)

At high $Q^{2}$, this matrix element is identified as a color polarizability,
which quantifies a color-dependent response to an external field. At lower
$Q^{2}$, we can still employ this sum rule to look at the inelastic part:

$\overline{d}_{2}(Q^{2})=\int_{0}^{x_{pp}}x^{2}\bigg{(}2g_{1}(x,Q^{2})+3g_{2}(x,Q^{2})\bigg{)}dx$
(2.89)

As the elastic part does not contribute to this moment, at low $Q^{2}$ we can
still view this as a “pure polarizability” Alarcón et al. (2020). Although the
color polarizability definition fails and we can no longer identify
$\overline{d}_{2}$ with $d_{2}$, the low $Q^{2}$ regime of $\overline{d}_{2}$
contains information about the quark-gluon correlations of the nucleon.

### 2.6 Chiral Perturbation Theory

The moments introduced in the last section show various ways that we can
calculate meaningful quantities from our structure functions, but the question
remains how we can compare these quantities to modern theories of Quantum
Chromodynamics (QCD). QCD is the theory of the strong nuclear force, and
unlike the comparatively simple coupling of the electromagnetic force, it has
been observed to couple quarks and gluons more strongly together in response
to attempts to extract them from the nucleon, a phenomenon known as
confinement. In contrast, at small length scales they are bound very weakly in
what is known as ‘asymptotic freedom’ Scherer (2003). We know that our $Q^{2}$
controls what length scale we are probing, which means that at small $Q^{2}$,
the strong force becomes very powerful, as shown in Figure 2.4.

Figure 2.4: Asymptotic freedom and confinement. Adapted from a figure in asy

At this length scale, rather than characterizing the dynamics of individual
quarks and gluons, QCD handles how they are bound together into a larger
hadron. In the realm of asymptotic freedom, where the strong force is weaker,
QCD is treated perturbatively, but at low $Q^{2}$ such treatments become
impossible. As it is currently impossible to find an analytical solution at
this energy scale, it is most useful to examine low $Q^{2}$ QCD with an
effective field theory. In quantum field theory, an effective field theory is
a subset of a larger theory which details its behavior in a specific energy
regime. Low energy effective theories allow for a perturbative approach in
terms of the momentum, instead of the coupling constant as is used for the
full theory of QCD at high energy. There are a number of effective theories
which present predictions of strong force behavior at low $Q^{2}$, such as
Lattice QCD, but so far the most successful at very low energies is known as
Chiral Perturbation Theory, or $\chi$PT.

To begin, let us consider QCD as a gauge theory, meaning that it is invariant
under local transformations at a given energy scale. QCD is based on the color
gauge group SU(3). In other words, the special unitary group of rank 3
parameterizes many of the symmetries of the strong force. From the gauge
attributes of SU(3) it is possible to obtain the general QCD Lagrangian
Scherer (2003):

$\mathcal{L}_{QCD}=\sum_{f=u,d,s,c,t,b}\overline{q}_{f}(i\gamma^{\mu}D_{\mu}-m_{f})q_{f}-\frac{1}{4}\mathcal{G}_{\mu\nu}\mathcal{G}^{\mu\nu}$
(2.90)

Here we sum over the six known quark flavors, $q_{f}$ and $\overline{q}_{f}$
are the quark fields, $\mathcal{G}_{\mu\nu}$ is the gluon tensor, $m_{f}$ is
the mass of the relevant quark flavor, and $D_{\mu}$ is the covariant
derivative. The heavier quarks (charm, top, and bottom) occur much less
frequently so their contributions may be largely disregarded. For the
remaining quarks (up, down, strange), their mass $m_{f}\ll M_{p}$, so it may
be a natural assumption to treat them as massless.

$q_{f}$ are color triplets based on the three possible color charges, but we
must also consider left and right handed chirality of the quarks. For massive
particles, chirality refers to the handedness of the group representation, but
for massless particles we can easily understand it as being the same as the
helicity. The helicity of a particle refers to whether its spin is aligned or
antialigned with its momentum, as shown in figure 2.5.

Figure 2.5: Definitions of handedness in helicity. Adapted from a figure in
hel

We define the chiral quark fields in terms of a left or right-handed
projection operator Scherer (2003):

$P^{L,R}=\frac{1}{2}(1\mp\gamma_{5})$ (2.91)
$q^{L,R}_{f}=P^{L,R}q_{f}=P^{L,R}\begin{pmatrix}{\color[rgb]{1,0,0}q_{f}^{red}}\\\
{\color[rgb]{0,1,0}q_{f}^{green}}\\\
{\color[rgb]{0,0,1}q_{f}^{blue}}\end{pmatrix}$ (2.92)
$\overline{q}^{L,R}_{f}=\overline{q}_{f}P^{L,R}=\begin{pmatrix}{\color[rgb]{0,1,1}\overline{q}_{f}^{antired}}\\\
{\color[rgb]{1,0,1}\overline{q}_{f}^{antigreen}}\\\
{\color[rgb]{1,1,0}\overline{q}_{f}^{antiblue}}\end{pmatrix}P^{L,R}$ (2.93)

In the chiral limit of $m_{f}=0$, the left and right handed quark interaction
term vanishes. Disregarding the mass term and plugging these into our general
QCD Lagrangian, we obtain:

$\begin{split}\mathcal{L}_{QCD}^{0}=\sum_{f=u,d,s}\overline{q}_{f}^{L}(i\gamma^{\mu}D_{\mu})q_{f}^{L}+\overline{q}_{f}^{R}(i\gamma^{\mu}D_{\mu})q_{f}^{R}-\frac{1}{4}\mathcal{G}_{\mu\nu}\mathcal{G}^{\mu\nu}\\\
=\sum_{f=u,d,s}\overline{q}_{f}i(\gamma^{\mu}D_{\mu}+\gamma_{5}\gamma^{\mu}D_{\mu}\gamma_{5})q_{f}-\frac{1}{4}\mathcal{G}_{\mu\nu}\mathcal{G}^{\mu\nu}\end{split}$
(2.94)

This chiral Lagrangian is said to have a global $U(3)_{L}\times U(3)_{R}$
symmetry Scherer (2003). The progression of an effective field theory is to
take a symmetric or analytically solvable term and add terms that are
functions of a small parameter to reach the full Lagrangian through
perturbative effects. Here, that is done by considering the missing part of
the Lagrangian, namely, the mass term:

$\mathcal{L}_{QCD}^{I}=-\sum_{f=u,d,s}\overline{q}_{f}^{L}m_{f}q_{f}^{R}+\overline{q}_{f}^{R}m_{f}q_{f}^{L}$
(2.95)

We can see that this part of the Lagrangian contains an explicit breaking of
the chiral symmetry in $\mathcal{L}_{QCD}^{0}$. Because the quarks are weakly
interacting at low energy, in a low energy effective theory it is possible to
treat this term perturbatively with a low energy expansion. This allows us to
consider the total effective $\chi$PT Lagrangian as Bernard et al. (1995):

$\mathcal{L}_{\chi
PT}=\mathcal{L}_{QCD}^{0}+\mathcal{L}_{QCD}^{I}=\mathcal{L}_{QCD}^{0}+\mathcal{L}_{2}+\mathcal{L}_{4}+...$
(2.96)

Where the index on $\mathcal{L}_{2}$, etc refers to the low energy dimension
in terms of derivatives or powers of the quark mass. Each of these terms can
be written explicitly by considering all possible diagrams of a given
dimension, they are constructed by ordering diagrams in powers of momentum. In
this $\chi$PT Lagrangian, corrections are expanded in terms of pion loops,
whereas in the full QCD Lagrangian, quark and gluon loops are used explicitly.
At dimension 2, the tree level diagrams are of order $Q^{2}$ while the one
loop corrections are of order $Q^{4}$, at dimension 4 ($\mathcal{L}_{4}$) the
tree level diagrams are order $Q^{4}$ and so on. This means that to consider a
full Lagrangian up to order $Q^{4}$ we must include tree level and one-loop
corrections of $\mathcal{L}_{2}$, and tree level diagrams of
$\mathcal{L}_{4}$.

To expand these terms in orders of pion loops, the order in $Q^{2}$ also
corresponds to the included order of the pion mass, $m_{\pi}$. The effective
Lagrangian, and all quantities derived from it, is constructed by ordering the
result in powers of the pion mass. The specific way in which this ‘power-
counting’ is done is an important point, and one which may have an effect on
the final calculation, as assumptions must be made to handle the inclusion of
the $\Delta$-resonance. One such method which is important here is the
$\epsilon$-counting scheme, where it is assumed that the pion mass $m_{\pi}$
is of similar scale to the difference between the mass of the
$\Delta$-resonance, $M_{\Delta}-M_{p}=\varDelta\approx m_{\pi}$. This allows
$\varDelta$ and $m_{\pi}$ to be expanded to the same order Bernard et al.
(2013). The other method which will be discussed in the following chapters is
the $\delta$-counting method, in which a different assumption is made; here it
is assumed that the ratio $\frac{m_{\pi}}{\varDelta}$ is of similar scale to
the ratio $\frac{\varDelta}{M_{p}}$, and consequently used to simplify and
expand the symmetry breaking terms to like powers Alarcón et al. (2020).

Once an effective Lagrangian is obtained, it can be used to obtain the
equations of motion, and consequently, calculate the VVCS amplitudes of the
previous section by varying the action such that $\delta\mathcal{Z}=0$:

$\mathcal{Z}=\int d^{4}x\mathcal{L}_{\chi PT}$ (2.97)

The calculation to obtain the Lagrangian expansion and compute the VVCS
amplitudes is an elaborate one far beyond the scope of this thesis, but it is
core to understand that differences such as the choice of power counting cause
the cutting-edge chiral perturbation theory calculations to differ starkly in
their predictions for the proton’s spin structure function moments. These
differences shall be discussed further in the analysis chapter, and the full
details of these calculations can be found in Alarcón et al. (2020); Bernard
et al. (2013). But the fact that such differences exist further illustrates
the importance of experimental measurements in the region where $\chi$PT is
applicable.

## CHAPTER 3Existing Data

Before we begin to discuss the particular measurement which is the topic of
this thesis, let us first consider existing relevant measurements of the
structure functions and their moments. Both structure functions for the
neutron have been measured with high precision, as discussed in chapter 1,
relevant for their revealing of the $\delta_{LT}$ puzzle in Figure 1.2.
Jefferson Lab E94010, published in 2004, ran down to a minimum $Q^{2}$ of 0.1
GeV2, and showed a disagreement with the predictions of $\chi$PT at the time
Amarian et al. (2004). The cutting edge $\chi$PT calculationsBernard et al.
(2013); Alarcón et al. (2020) have since remedied this discrepancy, but are
still in tension with the recently published Small-Angle GDH neutron data from
Jefferson Lab Sulkosky et al. (2021), which revealed a new disagreement at a
lower $Q^{2}$ of 0.05 GeV2. At higher $Q^{2}$, there is even more neutron
structure function data of less relevancy to this thesis.

For the proton, the earliest structure function measurements in the late 1980s
and early 1990s focused on the DIS region and were collected at the Stanford
Linear Accelerator (SLAC) and the European Organization for Nuclear Research
(CERN). These early measurements covered a broad range from medium to high
$Q^{2}$ but were largely focused at higher $Q^{2}$. More recently, a number of
experiments at Jefferson Lab (JLab) have delved to lower $Q^{2}$ and higher
$x$, investigating the resonance region. This list of experiments was
originally compiled by Ryan Zielinski in his thesis Zielinski (2017) and has
been updated for the state of the field in 2022. For reference, $\chi$PT is
valid below a $Q^{2}$ of around 0.3. The complete list of proton structure
function data is summarized and organized by average $Q^{2}$ in Table 3.1.
Though many of these experiments have used various methods to compute both
$g_{1}$ and $g_{2}$, for the purpose of this table, an experiment which
measured a longitudinally polarized target is marked having measured $g_{1}$,
and one which measured a transverse-polarized target is marked as having
measured $g_{2}$.

Experiment | Facility | Avg. $Q^{2}$ (GeV2) |  $x_{bj}$ Range | $g_{1}$ | $g_{2}$
---|---|---|---|---|---
COMPASSAlekseev et al. (2010) | CERN | 15.7 | 0.004-0.7 | ✓ |
EMCAshman et al. (1988) | CERN | 10.7 | 0.01-0.7 | ✓ |
E155Anthony et al. (2000) | SLAC | 10.0 | 0.02-0.8 | ✓ |
E143Abe et al. (1998) | SLAC | 5.0 | 0.03-0.8 | ✓ | ✓
SANEArmstrong et al. (2019) | JLab | 4.5 | 0.3-0.8 |  | ✓
SMCAdeva et al. (1998) | CERN | 4.0 | 0.003-0.7 | ✓ | ✓
E155xAnthony et al. (2003b) | SLAC | 4.0 | 0.02-0.8 |  | ✓
HERMESAirapetian et al. (2012) | HERA | 3.0 | 0.004-0.9 | ✓ | ✓
EG1bFersch et al. (2017) | JLab | 2.1 | 0.02-0.8 | ✓ |
RSSWesselmann et al. (2007) | JLab | 1.3 | 0.03-0.8 | ✓ | ✓
EG4Zheng et al. (2021) | JLab | 0.4 | 0.01-0.9 | ✓ |
g2pRuth et al. (2022) | JLab | 0.07 | 0.01-0.8 | ✓ | ✓
Table 3.1: Existing proton structure function data as of August 2022

The first spin structure function measurement for the proton was an
observation of $g_{1}$ by the European Muon Collaboration (EMC) in 1988 Ashman
et al. (1988). This measurement disagreed with the prediction of the Ellis-
Jaffe sum rule, and showed that the proton’s valence quarks carry only a small
fraction of its spin Ellis and Jaffe (1974). Rather than the electron
scattering described so far, this measurement was performed with polarized
muon-proton scattering. Additional $g_{1}$ and $g_{2}$ measurements were
obtained at CERN with polarized muon scattering in the SMC Adeva et al. (1998)
and COMPASS Alekseev et al. (2010) experiments. Due to the high energy of the
CERN beam, all of these experiments operated at a relatively high $Q^{2}$.

Also at high to medium $Q^{2}$ are several experiments from SLAC in the 1990s
which helped to provide clarity on the structure functions through electron-
proton scattering. SLAC E143 Abe et al. (1998) measured both structure
functions and found at high $Q^{2}$ that the $g_{2}$ results had good
consistency with the leading twist predictions of the Wandzura-Wilczek
relations. SLAC E155 Anthony et al. (2000) and E155x Anthony et al. (2003b)
built on these results by measuring the structure functions with high
precision, with E155x performing one of the highest precision $g_{2}$
measurements to date. These experiments found good agreement with a number of
sum rules as well as the $g_{2}^{WW}$ predictions.

Starting in the early 2000s, Jefferson Lab began to produce a number of modern
structure functions results at much lower average $Q^{2}$, making use of the
high intensity CEBAF electron beam. The RSS Wesselmann et al. (2007); Slifer
(2004) experiment performed an extraction of both structure functions in the
resonance region for both the proton and neutron from a scattering measurement
of Helium-3, a nucleus with 2 protons and 1 neutron. The proton measurement
showed promising agreement with predictions for the Burkhardt-Cottingham and
Gerasimov-Drell-Hearn sum rules, while the neutron result revealed the
surprising result discussed in chapter one, which showed a deviation from the
predictions of chiral perturbation theory. Another more recent measurement of
$g_{2}$ comes from the HERMES experiment with HERA at DESY. HERMES integrated
both polarized electron and positron scattering to measure $g_{2}$, focused in
the DIS regime Airapetian et al. (2012).

At higher $Q^{2}$, the SANE experiment Armstrong et al. (2019), which recently
finished analysis, also measured several kinematic points of $g_{2}$, finding
a potentially surprising result for one of its moments, the $d_{2}$ twist-3
operator. In Hall B, the EG1b experiment likewise measured the $g_{1}$
structure function in the resonance region at medium $Q^{2}$ and extracted the
moments dominated by it. This experiment is notable as it is largely the basis
for the CLAS model discussed in later chapters Fersch et al. (2017).

Most recently, a program at Jefferson Lab encompassing several experiments has
worked to study the low-$Q^{2}$ behavior of the nucleons. The Small-Angle GDH
experiment, discussed in the first chapter, performed a low $Q^{2}$
measurement of the neutron structure functions and moments which revealed the
original $\delta_{LT}$ puzzle is alive and well with a new discrepancy from
$\chi$PT Sulkosky et al. (2021). The reason for the neutron’s disagreement is
still unknown, as the newest calculations agree with the earlier E94-010
neutron data but still disagree with the small-angle GDH results at even lower
$Q^{2}$.

The EG4 experiment Zheng et al. (2021) performed a low $Q^{2}$ measurement of
the $g_{1}$ structure function of the proton with longitudinal scattering
data, in which they directly extracted the polarized cross section differences
instead of relying on the diluted asymmetry method of many prior experiments.
The results show good agreement with calculations of $\chi$PT for several
moments, but also reveal the current lack of precision in the cutting-edge
calculations and the need to improve the uncertainty associated with each.

Figure 3.1: The $g_{1}$ structure function of the proton measured by the EG4
experiment. The data (red circles) are compared to the CLAS Hall B
phenomenological model Fersch et al. (2017). The solid and hatched bands refer
to the systematic uncertainty of the data and CLAS model parametrization
respectively. The black crosses indicate earlier data from the EG1b Fersch et
al. (2017) experiment. Reproduced from Zheng et al. (2021).

Finally, the experiment which is the subject of this thesis, the E08-027 (or
g2p) experiment ran in the spring of 2012 in Hall A at Jefferson Lab. The aim
of this experiment was to fill out the experimental program discussed above
with a very low $Q^{2}$ measurement of the $g_{2}$ structure function of the
proton. This required transverse polarized target scattering data, which
introduced a number of experimental challenges. We have recently finished
analysis on this experiment and its first results were released in the fall of
2022 Ruth et al. (2022).

## CHAPTER 4Experimental Setup

Figure 4.1: The experimental setup of the g2p Experiment. Reproduced from Ryan
Zielinski’s thesis Zielinski (2017)

The g2p experiment was an unprecedented effort in Hall A at Jefferson Lab to
measure the $g_{2}$ structure function of the proton at a very low $Q^{2}$
sufficient to test chiral perturbation theory predictions. The experiment used
the inclusive electron scattering described in Chapter 2, with polarized
electrons scattering off a spin-polarized target before being measured in the
Hall A high resolution spectrometers (HRS). The experimental target was a cell
of frozen ammonia (NH3) beads polarized with the Dynamic Nuclear Polarization
(DNP) process.

Data were taken over two target magnetic fields of 2.5T and 5T, and four beam
energies ranging from 1.1 - 3.3 GeV. The target magnetic field was transverse
to the electron beam for most settings, but longitudinal for one. This gave
the experiment a total of six distinct kinematic settings, five transverse and
one longitudinal. The lowest setting, at a field of 2.5T and a beam energy of
1.1 GeV, was taken primarily for the purpose of radiative corrections and is
not otherwise used in the final results.

I was unfortunately not personally involved in the running of the experiment,
but rather became involved in the intensive analysis process starting in 2015.
However, I was given the opportunity to perform experimental work on a
polarized target setup very similar to that discussed here, which is described
in Chapter 5. Much of the information in this chapter comes originally from my
collaborators who worked on the experiment directly in 2012, Ryan Zielinski
Zielinski (2017), Karl Slifer Camsonne et al. , Jian-Ping Chen Camsonne et al.
, Toby Badman Badman (2017), and Chao Gu Gu (2016). The overall experimental
setup is shown in Figure 4.1, with each component discussed in the following
sections.

### 4.1 Electron Beam

Figure 4.2: The Jefferson Lab Electron Accelerator. Reproduced from Alcorn et
al. (2004)

At Jefferson Lab, experiments are performed with a high intensity electron
beam linear accelerator, known as the Continuous Electron Beam Accelerator
Facility (CEBAF). Currently the beam energy has a maximum of 12 GeV, but at
the time that g2p ran, the maximum energy was 6 GeV. The beam current can be
up to 200 $\mu$A, but was limited to 50 nA for this experiment to prevent
depolarization of the target.

Polarized electrons are initially generated by photoemission from a
phosphorous-doped Gallium Arsenide cathode (GaAsP), which is stimulated with a
circularly polarized laser at a frequency of 860 nm, the bandgap energy of the
GaAs crystal Hernandez-Garcia et al. (2008). In pure, undoped Gallium
Arsenide, this photoelectric stimulation introduces a degeneracy in the
valence band states, exciting electrons out of the states $P_{1/2}$ and
$P_{3/2}$. Three electrons are promoted to one spin direction from the
$P_{3/2}$ state for every 1 promoted from $P_{1/2}$ to the other spin state,
yielding a net 50% polarization. The direction of this polarization is based
on whether the polarized laser has left-handed helicity ($\sigma^{-}$) or
right-handed helicity ($\sigma^{+}$). This effect is shown in Figure 4.3(a).

(a) Regular GaAs Cathode

(b) Strained GaAsP Cathode

Figure 4.3: The polarized electron generation technique. Reproduced from
Hernandez-Garcia et al. (2008).

The introduction of the phosphorous doping causes a mechanical strain on the
cathode, lifting the degeneracy and causing the bandgap from the $P_{1/2}$
state to be larger than that from the $P_{3/2}$ state. This means that light
tuned to the bandgap of the latter state can theoretically promote 100% of
electrons to the desired spin state, again based on the helicity of the
stimulating laser, as shown in Figure 4.3(b). In the Jefferson Lab source, a
superlattice structure containing alternating layers of GaAs and GaAsP is used
to create electrons with both a high polarization and high intensity. This
routinely creates a polarization of around 85% Hernandez-Garcia et al. (2008).

The direction of this polarization can be flipped from positive to negative
then by changing the helicity of the stimulating laser. This is done with a
voltage-controlled wave plate, when the sign of the voltage applied to this
plate is flipped the helicity of a photon passing through the plate can also
be flipped. This voltage is flipped in a pseudo-random fashion at a rate of
960.02 Hz, limiting time-dependent systematic uncertainties by frequently
changing the direction of polarization. The beam helicity can also be flipped
with an insertable half-wave plate, which was performed every few hours during
the experiment to study and minimize helicity-dependent systematics Zielinski
(2017).

The electrons generated by the cathode are pulled through an injector by an
applied bias voltage, which gives the electrons an initial energy of 45 MeV
Alcorn et al. (2004). These injected electrons then enter the paired linear
accelerators shown in Figure 4.2. Each linear accelerator consists of 20
cryomodules, units which consist of 8 superconducting niobium cavities each.
These cavities are kept cooled to a temperature of 2K and powered to create an
internal RF field, which accelerates the electrons as they pass through, with
each cavity achieving an average gradient of 7 MeV/m. After their trip through
the first linear accelerator, electrons achieve an energy of around 400 MeV,
before entering a recirculating arc and being transferred to a second,
identical linear accelerator. From here, the electrons can either be
distributed to the experimental halls A, B, and C, or sent back to the first
linear accelerator to achieve higher energies. By controlling the number of
loops, it is possible to control the beam energy which reaches the hall Alcorn
et al. (2004).

The beam energy is measured by observing the bend angle through the
recirculation arc, in what is known as the “Arc Method” Gu (2016). In
principle, the momentum of the electrons traveling through a magnetic field is
directly related to the bend angle of the electrons and the strength of the
magnetic field:

$E=k_{c}\frac{\int\vec{B}\times d\vec{l}}{\theta}$ (4.1)

Where $k_{c}$ = 0.299792 GeV$\cdot$rad$\cdot$T$\cdot$m corresponds to the
speed of light, $\theta$ is the measured deflection as the electrons travel
through the recirculation arc, and the integral runs over the measured
magnetic fields of the dipoles in the arc Gu (2016). For the g2p experiment, a
total of four different beam energies were used: 1.1 GeV, 1.7 GeV, 2.2 GeV,
and 3.3 GeV. These differing beam energies combined with the direction and
magnitude of the target field generate the different kinematic settings of the
experiment.

### 4.2 Beam Current Monitors (BCMs)

Once the beam enters the hall, the first component in the experimental setup
is the Beam Current Monitors (BCMs) used to measure the incident flux of
electrons. The BCMs are located 23 m upstream of the target, and consist of a
pair of stainless steel, cylindrical TM010 mode waveguides tuned to the
frequency of the beam, 1497 MHz Zielinski (2017); Gu (2016). A receiver
antenna inside each cavity is excited by the passing electrons, emitting a
signal proportional to the beam current, or total flux of electrons flowing
through the waveguide.

Traditionally this signal is processed with a RMS-to-DC converter, but the
previously used receiver did not work with the low beam current necessary for
the g2p experiment. Instead, a new receiver was designed by the Jefferson Lab
instrumentation group specifically for the experiment Gu (2016). The new
receiver employed an analog part, which used several analog amplifiers and
filters to boost the strength of the signal, converting it from a 1497 MHz
radiofrequency (RF) signal to a 45 MHz intermediate frequency (IF) signal. The
system then uses an Analog-Digital-Converter (ADC) to digitize the signal and
further boost the Signal-to-Noise ratio by using a digital filter tuned to cut
off at 10.4 kHz. Once the signal has been fully processed, it is converted
back to an analog 0-10V DC signal and sent to the Hall A Data Acquisition
(DAQ) system Zielinski (2017).

To calibrate the BCMs, a tungsten calorimeter was employed to measure the
effective current based on heating effects of the beam Bevins (2005). A
tungsten slug is placed in the path of the beam, absorbing the full beam
current and heating as a result. The number of electrons $N_{e}$ to hit the
slug is obtained with:

$N_{e}=\frac{K_{w}\Delta T}{E}$ (4.2)

Where $K_{w}$ = 8555.5 J/K is the specific heat capacity of the tungsten,
$\Delta T$ is the temperature change of the slug, and E is the beam energy
measured with the arc method before entering the hall. The scaling factor on
the signal conversion of the BCMs was fixed to normalize the BCM measured
current to that obtained with the tungsten calorimeter Zielinski (2017); Gu
(2016).

By performing the current measurement with two independent cavities, it is
possible to estimate the uncertainty of the BCM measurement by comparing the
difference between the measurement of the upstream and downstream BCM. For a
vast majority of the runs, the difference between the two BCMs was less than
0.7% Gu (2016).

### 4.3 Fast and Slow Raster

After passing through the BCMs, the beam encounters a series of beam rasters.
As was mentioned earlier, it was necessary to limit the beam current to 50 nA
to prevent depolarization of the target, as the high intensity of of the
Jefferson Lab beam would quickly destroy the spin polarization of the target
and cause radiation damage to the material. On top of this current limiting,
it is necessary to sweep the beam around the target so as to avoid
concentrating it on a single spot for too long, ensuring the target can
maintain its polarization and avoiding radiation effects.

The fast raster increases the initial 200 $\mu$m beam to a 4mm $\times$ 4mm
square profile. This is a standard piece of Hall A equipment, and employs two
perpendicular dipole magnets to move the beam. Both dipoles are driven by the
same 25 kHz triangle wave current, sweeping out the square area shown in
Figure 4.4(a).

(a) Fast Raster

(b) Slow Raster

Figure 4.4: Raster swept beam profiles. Plots are a function of current in
arbitrary units. Reproduced from Zielinski (2017)

The target cell for the g2p experiment however was a much larger cylinder with
a diameter of 25 mm, requiring an additional raster to sweep out the full area
of the target. A slow raster was added specially for the experiment,
consisting of a second pair of dipoles and sweeping out the area shown in
Figure 4.4(b). However, to sweep out the circular profile of the target cell,
a simple triangle wave could not be used as the driving current of the dipole
magnets. Instead, the $\hat{x}$ and $\hat{y}$ dipoles were driven by a dual
channel function generator, with respective forms of Gu (2016):

$I_{x}(t)=A_{x}\sqrt{t}\sin(\omega t)$ (4.3)
$I_{y}(t)=A_{y}\sqrt{t+t_{0}}\cos(\omega t)$ (4.4)

Here $A_{x}$ and $A_{y}$ are the amplitudes of the respective signals and t
represents the time-based amplitude modulation. The frequency $\omega$ of the
sin wave is equal to 99.412 Hz. This pattern allows the beam to sweep out a
uniform 2.5 cm circular raster pattern to match the target cell.

### 4.4 Møller Polarimeter

The polarization of the electron beam was measured using a Møller Polarimeter,
which employs the Møller process of electron-electron ($e^{-}+e^{-}\rightarrow
e^{-}+e^{-}$) scattering to measure the beam polarization. The incident
electrons scatter off polarized atomic electrons in a magnetized iron and
copper foil, and are focused through three quadrupole magnets used to set the
acceptance and one dipole magnet used to select for the electron momentum
Zielinski (2017). Scattered electrons are detected by a lead-glass
calorimeter.

Figure 4.5: Møller Polarimeter Schematic. (a) Side View. (b) Top View.
Reproduced from Alcorn et al. (2004)

We can write the Møller cross section as a function of the beam polarization
$P_{b}$ and the target polarization $P_{t}$ Alcorn et al. (2004):

$\sigma_{M\o{}ller}=\sigma_{0}\bigg{[}\sum_{i,j=x,y,z}P^{b}_{i}A_{ij}P^{t}_{j}\bigg{]}$
(4.5)

Where $A_{ij}$ represents the asymmetry components and $\sigma_{0}$ is the
unpolarized cross section. With the z-axis along the direction of the electron
beam and the y-axis perpendicular to the scattering plane, we can write the
longitudinal and transverse asymmetries as Alcorn et al. (2004):

$A_{zz}=\frac{\sin^{2}\theta_{CM}\big{(}7+\cos^{2}\theta_{CM}\big{)}}{\big{(}3+\cos^{2}\theta_{CM}\big{)}^{2}}$
(4.6)
$A_{xx}=-A_{yy}=-\frac{\sin^{4}\theta_{CM}}{\big{(}3+\cos^{2}\theta_{CM}\big{)}^{2}}$
(4.7)

Here $\theta_{CM}$ is the center of mass scattering angle for the Møller
interaction. $A_{z}z$ is our component of interest, as it tracks the
longitudinal polarization. There is some transverse polarization, but it is
small and can be corrected for by averaging one transverse direction with the
other Gu (2016).

A schematic of the polarimeter is available in Figure 4.5. To measure the
polarization, an asymmetry is measured at two center-of-mass angles of +20∘
and -20∘, the asymmetry between these two angles is averaged to cancel the
transverse polarization effects. With the asymmetry A quantifying the
difference between left ($N^{+}$) and right ($N^{-}$) handed helicity
electrons counts, we can write the beam polarization as:

$P_{b}=\frac{A}{P_{t}\cos\theta_{t}<A_{zz}>}=\frac{1}{P_{t}\cos\theta_{t}<A_{zz}>}\frac{N^{+}-N^{-}}{N^{+}+N^{-}}$
(4.8)

The average $A_{zz}$ is calculated with a Monte-Carlo simulation, and the
target polarization $P_{t}$ and target scattering angle $\theta_{t}$ are
extracted from the polarized target setup as described in the following
sections. Nine measurements of the beam polarization were taken throughout the
experiment, as is shown in Table 4.1 Zielinski (2017):

Date |  $P_{b}$ (%) | Stat. Error (%) | Syst. Error (%)
---|---|---|---
03/03/2012 | 79.91 | 0.2 | 1.7
03/30/2012 | 80.43 | 0.46 | 1.7
03/30/2012 | 79.89 | 0.58 | 1.7
04/10/2012 | 88.52 | 0.3 | 1.7
04/23/2012 | 89.72 | 0.29 | 1.7
05/04/2012 | 83.47 | 0.57 | 1.7
05/04/2012 | 81.82 | 0.59 | 1.7
05/04/2012 | 80.40 | 0.45 | 1.7
05/15/2012 | 83.59 | 0.31 | 1.7
Table 4.1: Electron Beam Polarization for the E08-027 Experiment

### 4.5 Chicane Magnets

One of the most major experimental difficulties in the experiment was the beam
deflection caused by the polarized target. To measure $g_{2}$, a transverse
polarization is needed, and consequently, a transverse magnetic field which
can cause a severe bending, especially at low beam energies. To compensate for
this, a pair of chicane magnets were used ahead of the beam reaching the
target, to pre-bend the beam and compensate for the deflection of the target
field such that the beam strikes the center of the target as intended. The FZ1
dipole magnet, located 5.92 m before the target, bends the beam downwards out
of its initial straight path Gu (2016). The FZ2 magnet then bends the beam
back up 2.66 m before the target field Zielinski (2017). The FZ2 magnet was
mounted on a hydraulic support which could be used to move it up and down for
optimization. The operation of the chicane magnets is shown in Figure 4.6.

Figure 4.6: Chicane magnet configuration for the g2p experiment. Reproduced
from Zielinski (2017).

This configuration is only necessary for the transverse settings, for the
longitudinally polarized setting the field is along the direction of the beam
and subsequently does not deflect it significantly. The chicanes are operated
in a ’straight-through’ configuration for this setting Zielinski (2017).

The chicane configurations for each transverse momentum setting are shown in
Table 4.2. $\theta_{in}$ and $\theta_{out}$ represent the angle of the beam
when entering and leaving the target field, while $\theta_{target}$ is the
angle of the beam when it strikes the target. For the 2.5T settings, the
chicanes were calibrated to strike the target at an angle and leave along the
normal beam path. For the 5T settings, the bending was significant enough that
it was impossible to reach the main hall beam dump, so instead the
configurations cause the beam to strike the target straight-on and then
deflect into a local beam dump.

E (GeV) |  $B_{t}$ (T) | $\theta_{in}$ | $\theta_{target}$ | $\theta_{out}$
---|---|---|---|---
1.16 | 2.5 | 11.96∘ | 5.98∘ | 0.0∘
1.72 | 2.5 | 8.06∘ | 4.03∘ | 0.0∘
2.25 | 2.5 | 6.09∘ | 3.05∘ | 0.0∘
2.25 | 5.0 | 6.09∘ | 0.0∘ | -6.09∘
3.35 | 5.0 | 4.09∘ | 0.0∘ | -4.09∘
Table 4.2: Chicane configurations for each transverse kinematic setting.
Values reproduced from Zielinski (2017).

### 4.6 Beam Position Monitors (BPMs)

The Beam Position Monitors (BPM) are used to properly determine the beam’s
position and angle after the deflection from the chicane magnets. The beam
passes through a cavity which contains four antennas similar to the one used
in the BCM. These antennas are open-ended and lie coaxial to the beam, and are
excited by the passage of the beam, stimulating a signal proportional to the
distance of the beam from that particular antenna. By using these four
antennas at right angles around the cavity, the beam angle and position can be
measured, as shown in Figure 4.7.

Figure 4.7: Beam position monitor cavity from the side (left) and front
(right). Reproduced from Gu (2016).

As with the BCM receiver, the low current necessary for E08-027 necessitated a
new BPM cavity designed by the Jefferson Lab instrumentation group Gu (2016).
These BPM cavities were calibrated with the use of a superharp, shown in
Figure 4.8. One of these superharps was inserted between the two BPM cavities,
while the other was placed directly before the chicane magnets. These
superharps contain three 50 $\mu$m wires angled at -45∘, 0∘, and 45∘
respectively. When these wires are passed through the beam by a motor control
on the harp, they produce a shower of radiation measured in nearby
photomultiplier tubes (PMTs). This radiation can be used to calculate an
absolute measurement of the beam position and angle, which is used to
calibrate the BPM cavities.

Figure 4.8: Superharp wire scanner used to calibrate the BPMs. Reproduced from
Gu (2016).

### 4.7 Polarized Target

The polarized target is the central feature of the g2p experiment, being
necessary to produce the structure functions introduced in Chapter 2. The
target was a spin polarized solid ammonia target, doped with irradiation and
polarization enhanced with the Dynamic Nuclear Polarization (DNP) process. The
overall experimental setup is shown in Figure 4.9.

Figure 4.9: Polarized target for the g2p experiment. Reproduced from Pierce et
al. (2014).

The target itself consists of solid ammonia (NH3) beads, contained within a
cup made out of Kel-F plastic, which is used primarily for its absence of free
protons so as not to contribute any background signal. Ammonia is an ideal
target material in this case because it has a short time-constant, allowing
maximum polarization in a relatively short amount of time, and because it is
relatively durable against the depolarizing effect of the electron beam. The
target ammonia is frozen initially using liquid nitrogen, (LN2) and then pre-
irradiated at a linear accelerator at the National Institute of Standards and
Technology (NIST) Pierce et al. (2014). This irradiation produces an excess of
paramagnetic centers in the ammonia, which is necessary for the polarization
enhancement process. The target is sealed off with thin aluminum end-caps to
hold the material in place.

Initial polarization is created by cooling the target to 4 K by submerging it
in a bath of Liquid Helium (LHe), and applying a strong magnetic field with a
superconducting magnet. The superconducting magnet is a split-pair Helmholtz
configuration coil, to allow the beam to pass unimpeded through it, and is
composed of niobium-titanium wire, which becomes superconducting below 10 K
and is also cooled to 4 K with LHe. For this experiment, the magnet was run at
strengths of 2.5 and 5.0 T, and was able to be swapped between transverse and
longitudinal polarization by way of a rotating vacuum seal beneath the coils.

Due to the strong magnetic field and cold temperature, magnetically
interacting particles in the target align themselves with the direction of the
magnetic field. The Zeeman Effect introduces a spin-dependent energy level
splitting for these target particles, dividing the spin-$\frac{1}{2}$ protons
and electrons in the target into spin up ($\uparrow$) and spin down
($\downarrow$) states, designating particles which are aligned or anti-aligned
with the B-field. The vector polarization of these target particles is defined
as the fraction of nuclei which are spin-polarized in a particular direction
Crabb and Meyer (1997):

$P_{\frac{1}{2}}=\frac{N^{\uparrow}-N^{\downarrow}}{N^{\uparrow}+N^{\downarrow}}$
(4.9)

Where $N^{\uparrow}$ and $N^{\downarrow}$ are the number of nuclei aligned or
anti-aligned with the magnetic field. The polarization induced by the magnetic
field alone can be determined at thermal equilibrium through the use of
Boltzmann statistics to be:

$P^{TE}_{\frac{1}{2}}=\tanh{\bigg{(}\frac{\mu B}{k_{B}T}\bigg{)}}$ (4.10)

Where $\mu$ is the magnetic moment of the relevant particle, $k_{B}$ is the
Boltzmann constant, B is the magnetic field, and T is the temperature. The
electron has a small mass and consequently, a large magnetic moment of 9.3e-24
J/T, so at 5 T and 4 K, we get a very large electron spin polarization
exceeding 99%. The proton, with its much smaller magnetic moment of 1.4e-26
J/T, does not polarize nearly so easily, so we only get a proton polarization
of around 0.5% in this fashion.

The DNP process takes advantage of the high electron polarization induced in
this process. Internal to the target, unpaired electrons and protons interact,
creating a hyperfine splitting into four distinct energy levels comprising all
possible combinations of proton and electron electron spins, as shown in
Figure 4.10. The irradiation doping process described above ensures there is a
large number of unpaired electrons to interact in this fashion in the target.

Figure 4.10: Energy levels of the dynamic nuclear polarization process.
Reproduced from Crabb and Meyer (1997).

To enhance the proton polarization, microwaves are applied tuned to the energy
level gaps shown in the figure. This causes the electrons and protons to
couple, such that a polarizing electron will also cause a coupled proton to
polarize. Since the electron has a much smaller relaxation time, a single
electron can repeatedly relax and couple with new protons before the
relaxation time of a single proton elapses. This allows the proton
polarization to be enhanced to very high values, upwards of 70% or more at 5
T.

The polarization is also enhanced from thermal equilibrium with the use of a
LHe evaporation refrigerator, shown in Figure 4.9. By pumping on the bath of
liquid helium immersing the target with high flow roots pumps, the temperature
can be lowered from the base of 4 K, while fresh liquid helium is fed to the
target by way of a separator, which allows liquid helium to drip through a
fine mesh while hot gaseous helium is pumped out Crabb and Meyer (1997);
Pierce et al. (2014). The helium which drips through then must pass through a
heat exchanger before reaching the target, which serves to create a
temperature gradient such that the coldest helium submerges the target cell
without allowing warm 4 K helium to reach the bottom of the refrigerator. This
process serves to cool the target to a temperature of 1 K, enhancing the
polarization greatly.

The polarization is measured by way of a Nuclear Magnetic Resonance (NMR)
system. In this system, polarizing protons either emit or absorb a photon at
the Larmor frequency of the proton, depending on the direction of
polarization. This can be observed with a Q-Meter, a high precision RLC
(Resistance-Inductance-Capacitance) circuit, which has its inductance coupled
to the photons emitted by the target through a coil inductor placed inside the
cup surrounding the target material. This coupling can be seen as a signal in
the real part of the circuit’s impedance, which is centered at the 2.5/5T
proton Larmor frequency, and the area of which is directly proportional to the
proton polarization of the target Crabb and Meyer (1997).

(a) Raw NMR Signal on Real Q-Meter Impedance

(b) Background-subtracted NMR Signal

Figure 4.11: NMR Signal examples from the g2p experiment. Reproduced from Gu
(2016).

To extract the signal, an example of which is shown in Figure 4.11(a), a
baseline of the circuit background near the correct frequency is taken at a
lower magnetic field and subtracted from the final result. Due to temperature
shifts in the target and other effects, this baseline is rarely perfect, so it
is necessary to also fit a cubic polynomial to the wings of the signal, and
subtract it to obtain a raw signal as is shown in Figure 4.11(b). Since the
polarization is only known explicitly at thermal equilibrium, we calibrate our
signal with a long measurement of the thermal equilibrium polarization in
equation 4.11. We can then use the area of the thermal equilibrium signal to
find the enhanced polarization based on the measured area A:

$P=\frac{P_{TE}}{A_{TE}}A$ (4.11)

This method works for both positive and negative spin polarization. Both
directions of polarization are employed in the experiment to try and remove
systematic effects dependent on the direction. Further discussion of the
underlying principles of NMR can be found in Chapter 5, where the UNH
Polarized Target is discussed.

Figure 4.12: Target polarizations for the 2.5 T (top) and 5 T (bottom)
settings. Reproduced from Badman (2016).

Measured polarizations for the experiment are shown in Figure 4.12.
Polarizations for the 5 T settings averaged between 60% and 80%, acceptable
values but not always reaching the full potential of the material as measured
by other groups Crabb and Meyer (1997). For the 2.5 T settings, the target
polarization was between 40% and 50%, but a year after the experiment’s run,
it was discovered that the T.E. calibration constant used was still set to 5 T
for all settings, meaning the 2.5 T settings were less by a factor of 2. This
results in a larger statistical error on the results of the 2.5 T kinematic
settings Badman (2016).

The target cup must be held in the region of the superconducting magnet with
the highest field uniformity, the more uniform the magnetic field, the sharper
the measured NMR signal will be. The target ladder which holds the Kel-F cup
is made of aluminum and suspended from a carbon fiber target insert which can
be removed to replace the target material or moved up and down to change to
another target cup. For the purposes of the dilution analysis, several non-
ammonia targets were featured on the target ladder as shown in Figure 4.13
Zielinski (2017). The right two targets are filled with frozen ammonia, the
clear beads of which are turned a deep purple by the irradiation process at
NIST. The second target from the left is the ‘Dummy’ target, left empty to
measure the yield which results from the aluminum end-caps and an NMR coil
inside. The leftmost target is a carbon disk, which is used to estimate
indirectly the nitrogen contribution to the scattering yields, as discussed in
Chapter 6. The full target cup is a cylinder with diameter of 2.72 cm and
length of 2.83 cm, though a shorter cup of length 1.3 cm was used for the 1.2
GeV beam energy runs to lessen radiation lengths Zielinski (2017). An
additional ‘Empty’ target was employed for the dilution analysis of Chapter 6,
for which a production cup was simply left empty and filled with LHe to
measure the helium contribution.

Figure 4.13: Target insert for E08-027 showing Carbon (First), Dummy (Second),
and Ammonia (Third and Fourth) targets. Reproduced from Zielinski (2017).

### 4.8 Beam Dumps

Though many electrons in the beam will interact within the target and scatter
into the detectors, a large portion of the beam intensity will pass through
the target without significant interaction and must reach a final destination
which can safely dissipate the beam energy. For standard experiments in Hall
A, a hall beam dump is located at the end of the hall. This beam dump is
composed of an aluminum plate heat exchanger inside an aluminum pressure
vessel. The heat exchanger is cooled by a water recirculator, which feeds a
bath of water that helps to dissipate beam energy and heat TEC (1997). A cross
sectional schematic of the Hall A beam dump is shown in Figure 4.14. The beam
dump is optimized such that 70% of the power is dissipated directly in the
aluminum, minimizing the production of tritium and other elements which might
serve to activate the cooling water.

Figure 4.14: Hall A main beam dump schematic. Reproduced from TEC (1997).

For the 2.5 T settings, the chicane magnets were optimized such that the beam
exits the target field at a 0∘ angle, to reach the beam dump. At the 5 T
settings, the high field combined with the low beam energy made it impossible
to configure the chicanes to both hit the target in the center and reach the
main hall beam dump, so a local beam dump had to be commissioned to dissipate
the beam energy for the 5 T transverse settings. A photo of the local beam
dump is shown in Figure 4.15.

Figure 4.15: Local beam dump for E08-027. The septa magnets are visible in the
background. Holes on the left and right of the blue beam dump support
structure allow scattered electrons to pass through to the detectors. In the
center, copper and tungsten plates absorb the beam energy. In this photo, the
insert is not inserted into the beam dump. Reproduced from Zielinski (2017).

The local beam dump is composed of a series of copper and tungsten plates,
sufficient to diffuse the beam energy for this experiment due to the low beam
current. It contained a removable insert which could be used to control
radiation levels in the hall, and was located 0.64 m downstream of the
polarized target Gu (2016).

### 4.9 Septa Magnets

To obtain the desired low $Q^{2}$ of the experiment, it is necessary to
measure scattered electrons at an angle of around 6∘. However, due to space
constraints in the hall, the Hall A spectrometers could not be placed closer
than 12.5∘ from the center. It was therefore necessary to employ a pair of
dipole septa magnets after the polarized target, to bend scattered electrons
at 6∘ into the angular acceptance of the Hall A detectors. These magnets were
composed of a pair of coils above and below the beamline designed to bend it
the requisite amount, as shown in Figure 4.16.

Figure 4.16: Septa magnets for E08-027. Reproduced from Zielinski (2017).

The top set of coils for the right spectrometer septa became damaged during
the operation of the experiment several times, requiring multiple
reconfigurations of the three top coils for the RHRS. This required changing
the number of loops of wire in the coils. The ideal configuration is 48-48-16,
but on 3/18/2012 and 4/11/2012, the top right coils became damaged and were
instead changed to 40-32-16 and then 40-00-16 loops for the latter sections of
the run Zielinski (2017).

The septa magnets were employed for all settings with a transverse target
field, but for the longitudinal setting, counts were instead measured at an
acceptance of 11∘, for which the septa were not used.

### 4.10 Spectrometers

Electrons and other particles which scatter from the target at the appropriate
angle are directed into the Hall A high resolution spectrometers, used to
identify the measured particles and track their momentum and position. There
are two identical detectors, referred to as the Left (LHRS) and Right (RHRS)
arms. These spectrometers consist of three superconducting quadrupoles, which
focus the scattered electrons, and one superconducting dipole, which is used
to select the electron momentum which reaches the detector stack. After
passing through the focusing magnets the scattered particles reach the
detector stack shown in Figure 4.17.

Figure 4.17: Detector stack for the Hall A High-Resolution Spectrometers.
Reproduced from Zielinski (2017).

Particles which enter the HRS first pass through two vertical-drift wire-
chambers (VDCs). These chambers consist of cells filled with a gas mixture of
argon and ethane, as well as an array of wires used to sense the approach of
nearby electrons Zielinski (2017). The cell is operated under an electric
field produced by a gold-plated Mylar foil connected to a voltage differential
of -4 kV. There are two VDCs in each spectrometer, each composed of two wire
planes in the UV configuration. Each plane consists of 368 sense wires spaced
4.2 mm from each other Gu (2016).

When an electron passes through the VDC cell, it ionizes the gas around it,
creating a shower of radiation. The argon is used as a quenching agent to
suppress secondary emissions and limit the produced radiation to that which
comes from scattered electrons. The produced radiation, stimulated by the
electric field, couples with the nearest sense wire, creating a signal that
can be measured by the Data-Acquisition system (DAQ). A single scattered
electron triggers signals on multiple sense wires as it passes, with the
shortest drift time associated showing what wire corresponds to the electron’s
closest approach. From the measured drift distances and the known position of
the wires, the electron’s trajectory can be reconstructed back to the target
Zielinski (2017); Charpak (1977). This, in combination with the HRS dipole,
allows the spectrometers to measure and select the particle’s momentum and
exact scattering angle.

A diagram of the VDCs is shown in Figure 4.18. The upper and lower chambers
for each VDC have the sensing wires oriented 90∘ from each other. The U and V
planes are oriented at a 45∘ angle from the direction of the travelling
electrons.

Figure 4.18: VDC Diagram. Reproduced from Alcorn et al. (2004).

After passing through the VDCs, scattered particles pass through the first of
two plastic scintillators. These plastic scintillators function to trigger
photomultiplier tubes (PMTs) on each side. These photomultiplier tubes are
calibrated with cosmic rays and used to form the Data Acquisition trigger.
When both the left and right PMTs are fired for each scintillator, the DAQ
trigger registers a scattering event.

Between the two plastic scintillators is a gas Čerenkov detector, used for
particle identification. Because we are only interested in the scattered
electron, we will need a way to eliminate events which come from other
negatively charged particles that may result from scattering, such as
$\pi^{-}$. The detector is filled with CO2 gas, and operates on the
measurement of Čerenkov radiation, which is produced when a particle travels
faster than the phase velocity of light in a given medium Iodice et al.
(1998). Čerenkov radiation can only be produced by a particle with a minimum
momentum of Iodice et al. (1998):

$p=\frac{mc}{\sqrt{n^{2}-1}}$ (4.12)

Where p is the particle’s momentum, m is its rest mass, c is the speed of
light, and n is the index of refraction for the medium, in this case n =
1.00041 for CO2. Because electrons are very light, they have a low momentum
threshold of 0.017 GeV. Pions are much heavier and have a momentum threshold
of 4.8 GeV. Because of the momentum selected by the dipole magnet in the HRS,
any pions or heavier particles which reach the Čerenkov detector will have
insufficient momentum to produce this radiation, so a signal will only be
produced by a travelling electron.

Figure 4.19: Gas Cerenkov diagram. Reproduced from Iodice et al. (1998);
Zielinski (2017).

The produced radiation is measured by way of ten overlapping mirrors at the
end of the detector, which focus produced radiation into a corresponding
series of photomultiplier tubes. The signal from each PMT is summed to get the
total radiation measured in the detector. A diagram of the Čerenkov detector
is shown in Figure 4.19. Though heavier particles do not produce Čerenkov
radiation directly, they may produce some electron radiation due to ionization
of the CO2 gas, which may be mistaken for an electron scattering event.

To isolate and remove these events, a lead-glass calorimeter is used after the
second plastic scintillator. This calorimeter is a series of lead glass blocks
at the end of the detector stack which are stimulated by energetic particles
to produce e--e+ pairs and bremsstrahlung radiation photons. The radiation
produced is measured in a series of photomultiplier tubes, and is directly
proportional to the energy deposited by the particle, which can be used in
combination with the Čerenkov to definitively rule out heavier negatively
charged particles Gu (2016).

Figure 4.20: Lead calorimeter for HRS detectors. Reproduced from Alcorn et al.
(2004).

The number of lead-glass blocks differs slightly between the left and right
HRS. For the left HRS, the calorimeter is composed of two layers each with 34
blocks perpendicular to the incoming particle trajectory, each with a
dimension of 14.5$\times$14.5$\times$30 cm for the first layer and
14.5$\times$14.5$\times$35 cm for the second. The right HRS instead has on the
first layer 48 10$\times$10$\times$35 cm perpendicular blocks, and on the
second, 80 14.5$\times$14.5$\times$35 cm blocks oriented parallel to the
incoming particle. The increased number of blocks on the right HRS means that
it completely absorbs the energy of the incoming particle, while this is not
true of the left HRS. The calorimeter setup is shown in Figure 4.20.

### 4.11 Data Acquisition System

Scattering events which activate the trigger by tripping a PMT in both s1 and
s2m scintillators and which fall within the particle identification and
momentum acceptance of the detector are recorded in the data acquisition
system. The overall DAQ system is shown in Figure 4.21.

Figure 4.21: Overall Data Acquisition setup for E08-027. Reproduced from
Zielinski (2017).

Signals from the detectors are directly received by trigger modules formed by
several NIM and CAMAC electronics crates Zielinski (2017). These determine the
primary trigger, which is whether both scintillators were tripped as discussed
above, as well as a secondary trigger used to measure the trigger efficiency,
which trips when only one scintillator plane measures an event instead of
both, but the event also registered in the Čerenkov detector. These triggers
are sent to the trigger supervisor, which determines whether or not to accept
an event based on if the upstream data acquisition (DAQ) system is busy
processing a previous event, and the prescale factor, chosen to reduce the
missed events due to busy time as much as possible. The trigger supervisor
maintains the maximum efficiency by keeping the DAQ busy for the maximum time
possible.

Accepted events are passed to a FASTBUS crate containing analog-to-digital
(ADC) and time-to-digital (TDC) converter modules. These modules serve to
digitize the signals and pass them to the CEBAF Online Data Acquisition (CODA)
software on a connected Unix computer.

In parallel, signals from the BCMs and BPMs, as well as the beam helicity
measured at the source are recorded in ADC modules re-used from the HAPPEX
experiment, before being paired with matching events based on time
synchronization in the CODA system.