Experiment E08-007 measured the proton elastic form factor ratio in the range of by recoil polarimetry. Data were taken in 2008 at the Thomas Jefferson National Accelerator Facility in Virginia, USA. A 1.2 GeV polarized electron beam was scattered off a cryogenic hydrogen target. The recoil proton was detected in the left HRS in coincidence with the elasticly scattered electrons tagged by the BigBite spectrometer. The proton polarization was measured by the focal plane polarimeter (FPP).
In this low region, previous measurement from Jefferson Lab Hall A (LEDEX) along with various fits and calculations indicate substantial deviations of the ratio from unity. For this new measurement, the proposed statistical uncertainty () was achieved. These new results are a few percent lower than expected from previous world data and fits, which indicate a smaller at this region. Beyond the intrinsic interest in nucleon structure, the new results also have implications in determining the proton Zemach radius and the strangeness form factors from parity violation experiments.
High Precision Measurement of the Proton Elastic Form Factor Ratio at Low
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
© Massachusetts Institute of Technology 2010. All rights reserved.
Department of Physics
January 25, 2010
Thomas J. Greytak
Associate Department Head for Education
High Precision Measurement of the Proton Elastic Form Factor Ratio at Low
Submitted to the Department of Physics
on January 25, 2010, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Thesis Supervisor: William Bertozzi
Title: Professor of Physics
Thesis Supervisor: Shalev Gilad
Title: Principle Research Scientist
This work could not have be completed without the people who have supported and helped me along this long journey. I am extremely grateful for their care and considerations along these years for which I don’t have many opportunities to show my gratitude in front of them.
First, I would like to thank my advisor, Prof. William Bertozzi, for giving me the opportunity to start the graduate study at MIT, and for his continuing guidance, attention and support throughout these years. I won’t forget the “hard moments” he gave me during the preparation of the part III exam the same as the encouragement when I was frustrated. He helped me to understand how to become a physicist and at the mean time a happy person in life. I also would like to thank my another advisor Dr. Shalev Gilad for his valuable advices and suggestions during the whole analysis and the encouragement throughout my study and research at Jefferson Lab. Without their support, I would not complete the thesis experiment and finish the degree.
I would like to thank my academic advisor Prof. Bernd Surrow for his careful guidance in my graduate courses and the discussions for the future career. Many thanks to my thesis committee members: Prof. William Donnelly and Prof. Iain Stewart for their valuables comments and suggestions to this thesis.
Although it’s only been less than two years since I join the E08-007 collaboration, I had a wonderful experience and learned a lot in working with the spokespersons, post-docs and former graduate students. Individually, I sincerely appreciate Prof. Ronald Gilman for his guidance and support before and during the running of the experiment, and his helpful suggestions and discussions for the analysis afterwards. I would like to thank Dr. Douglas Higinbotham for being my mentor at Jefferson Lab and giving valuable advices in resolving different problems I encountered along the way. I would like to thank Dr. Guy Ron, for providing the first hand experimental running and analysis experience, the experiment would not run so smoothly without his effort. I would like to thank Dr. John Arrington for the valuable comments and discussions on the analysis and providing the form factors global fits. I also would like to thank Prof. Steffen Strauch, Prof. Eliazer Piasetzky, Prof. Adam Sarty, Dr. Jackie Glister and Dr. Mike Paolone for their guidance and inspiring discussions through the whole analysis process. In addition, I would like to thank the group former post-docs Dr. Nikos Sparveris and Dr. Bryan Moffit for their generous help on the experimental setup and assistance through the experiment. This work could not be done without the contribution from any one of them.
I would like to thank the Hall A staff members and the entire the Hall A collaboration for their commitment and shift efforts for this experiment. I would also like to thank the Jefferson Lab accelerator crew for delivering high quality beam for this experiment.
In the earlier days at Jefferson Lab, I worked with the saGDH/polarized He group. It was very special to me since that’s when I completed my first analysis assignment and learned quite some knowledge about the target system. I would like to thank Dr. Jian-Ping Chen for his supervision and guidance when I started the research in Jefferson Lab without any experience, his passion and rigorous attitude for physics have served as a model for me. I also would like to thank the former graduate students Vince Sulkosky, Jaideep Singh, Ameya Kolarkar, Patricia Solvignon and Aidan Kelleher for their patience and generous support on various things. I would like to thank the He lab/transversity fellow students: Chiranjib Dutta, Joe Katich, and Huan Yao for the great experience we had worked together.
Although I started my graduate life at MIT, I spent the last four years at Jefferson Lab. I am lucky to have friendships at both cities which made my graduate study and research an enjoyable experience. I would like to thank them for their support and encouragement: Bryan Moffit, Vince Sulkosky, Bo Zhao, Kalyan Allada, Lulin Yuan, Fatiha Benmokhtar, Ya Li, Jianxun Yan, Linyan Zhu, Ameya Kolarkar, Xin Qian, Zhihong Ye, Andrew Puckett, Peter Monaghan, Jin Huang, Navaphon Muangma, Kai Pan, Wen Feng, Wei Li, Feng Zhou,. Especially I would like to thank the group post-doc Vince Sulkosky for his effort in reading and correcting my thesis draft.
And Finally, I would like to show my deepest appreciation to my parents. I would not be anywhere without them and there is no words could ever match the love and support they gave since I was born. I also want to thank my fiance Yi Qiang, for the endless support over the past 8 years, and loving me for who I am.
- 1.1 Definitions and Formalism
- 1.2 Form Factor Measurements
- 1.3 World Data
- 1.4 Models and Global Fits
- 1.5 Measurements at Low
2 Experimental Setup
- 2.1 The Accelerator and the Polarized Electron Source
- 2.2 Hall A
- 2.3 Beam Line
- 2.4 Target
- 2.5 High Resolution Spectrometers
- 2.6 BigBite Spectrometer
- 2.7 Hall A Data Acquisition System
- 2.8 Trigger Setup
3 Data Analysis I
- 3.1 Analysis Overview
- 3.2 HRS analysis
- 3.3 Events Selection
- 3.4 Recoil Polarization Extraction
- 4 Data Analysis II
5 Discussion and Conclusion
- 5.1 Comparison with World Data
- 5.2 Discussion with Theoretical Models and Fits
- 5.3 Individual Form Factors and Global Fits
- 5.4 Proton RMS Radius
- 5.5 Proton Zemach Radius
- 5.6 Proton Transverse Densities
- 5.7 Strangeness Form Factors
- 5.8 Future Results and Experiment
- 5.9 Conclusion
- A Kinematics in the Breit Frame
- B Algorithm for Chamber Alignment
- C Extraction of Polarization Observables
- D Analyzing Power Parameterizations
- E Neutral Pion Photoproduction Estimation
- F Cross Section Data
List of Figures
- 1-1 The leading order diagram of elastic scattering.
- 1-2 World data of from unpolarized measurements [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], using the Rosenbluth method, normalized to the dipole parameterization.
- 1-3 World data of from unpolarized measurements [1, 3, 14, 15, 5, 7, 9, 10, 11, 12, 13], using the Rosenbluth method, normalized to the dipole parameterization.
- 1-4 World data of the ratio from unpolarized measurements (black symbols) using the Rosenbluth method and from polarization experiments (colored symbols) [16, 17, 18, 19, 20, 21, 22, 23].
- 1-5 Ratio extracted from polarization transfer (filled diamonds) and Rosenbluth method (open circles). The top (bottom) figures show Rosenbluth method data without (with) TPE corrections applied to the cross sections. Figures from .
- 1-6 Pertubative QCD picture for the nucleon EM form factors.
- 1-7 The scaled proton Dirac and Pauli form factor ratio: (upper panel) and (lower panel) as a function of in . The data are from [17, 18]. Shown with statistical uncertainties only. The dash-dotted curve is a new fit based on vector meson dominance model (VMD) by Lomon . The thin long dashed curve is a point-form spectator approximation (PFSA) prediction of the Goldstone boson exchange constituent quark model (CQM) . The solid and the dotted curves are the CQM calculations by Cardarelli and Simula  including SU(6) symmetry breaking with and without constituent quark form factors, repectively. The long dashed curve is a relativistic chiral soliton model calculation . The dashed curve is a relativistic CQM by Frank, Jennings, and Miller . Figure from .
- 1-8 Diagrams illustrating the two topologically different contributions when calculating nucleon EM form factors in lattice QCD .
- 1-9 Lattice QCD results from the Nicosia-MIT group  for the isovector form factors (upper left) and (lower left) as a function of . Both the quenched results () and unquenched lattice results with two dynamical Wilson fermions () are shown for three different pion mass values. The right panels show the results for (upper right) and (lower right), divided by the standard dipole form factor, as a function of in the chiral limit. The filled triangles show the experimental results for the isovector form factors extracted from the experimental data for the proton and neutron form factors. Figure from .
- 1-10 Isovector form factor lattice data with best fit small scale expansion (SSE) at MeV (left panel). The line in the right-hand panel shows the resulting Dirac radii, . Also shown as the data points are the Dirac radii obtained from dipole fits to the form factors at different pion masses. Figure from .
- 1-11 Photon-nucleon coupling in the VMD picture.
- 1-12 The proton form factor ratio from Jefferson Lab Hall A together with calculations from various VMD models.
- 1-13 The proton form factor ratio from Jefferson Lab Hall A together with calculations from dispersion theory fits. Figure from 
- 1-14 The nucleon electromagnetic form factors for space-like momentum transfer with the explicit pQCD continuum. The solid line gives the fit  together with the world data (circles) including the JLab/CLAS data for (triangles), while the dashed lines indicate the error band. Figure from .
- 1-15 Comparison of various relativistic CQM calculations with the data for . Dotted curve: front form calculation of Chung and Coester  with point-like constituent quarks; thick solid curve: front form calculation of Frank et al. ; dashed curve: point form calculation of Boffi et al.  in the Goldstone boson exchange model with point-like constituent quarks; thin solid curve: covariant spectator model of Gross and Agbakpe . Figure from .
- 1-16 Result for the proton form factor ratio computed with four different diquark radii, . Figure from .
- 1-17 The proton form factors in the relativistic baryon PT of  (IR scheme) and  (EOMS scheme). The results of  including vector mesons are shown to third (dashed curves) and fourth (solid curves) orders. The results of  to fourth order are displayed both without vector mesons (dotted curves) and when including vector mesons (dashed-dotted curves). Figure from .
- 1-18 Comparison between charge and magnetization densities for the proton and neutron. Figure from 
- 1-19 Kelly’s fits  to nucleon electromagnetic form factors. The error bands were of the fits. Figure from .
- 1-20 The parametrization of Bradford et al. compared with Kelly’s, together with world data. Figure is from .
- 1-21 Extracted values of and from the global analysis. The open circles are the results of the combined analysis of the cross section data and polarization measurements. The solid lines are the fits to TPE-corrected cross section and polarization data. The dotted curves show the results of taking and from a fit to the TPE-uncorrected reduced cross section. Figure from .
- 1-22 The difference between the measure nucleon form factors and the 2-components phenomenological fit of  for all four form factors.
- 1-23 The world data from polarization measurements. Data plotted are from [23, 45, 46, 21, 22, 16, 19, 20]
- 1-24 Recent world high precision polarization data [16, 19, 20] compared to several fits [47, 24, 48, 44] and parameterizations [49, 36, 50, 51].
- 2-1 Layout of the CEBAF facility. The electron beam is produced at the injector and further accelerated in each of two superconduction linacs. The beam can be extracted simultaneously to each of the three experimental halls.
- 2-2 Hall A floor plan during E08-007.
- 2-3 Schematic of beam current monitors.
- 2-4 Beam spot at target.
- 2-5 Layout of the Møller polarimeter.
- 2-6 Beam helicity sequence used during experiment E08-007.
- 2-7 Target ladder.
- 2-8 Schematic of Hall A High Resolution Spectrometer and the detector hut.
- 2-9 Left HRS detector stack during E08007.
- 2-10 Schematic diagram and side view of VDCs.
- 2-11 Configuration of wire chambers.
- 2-12 Layout of scintillator counters.
- 2-13 Layout of the Focal Plane Polarimeter.
- 2-14 The simulated FPP figure of merit with different carbon door thicknesses .
- 2-15 FPP coordinate system.
- 2-16 Straws in two different planes of a FPP straw chamber.
- 2-17 Block diagram for the logic of the FPP signal. (l.e. = leading edge, t.e. = trailing edge).
- 2-18 A side view (left) and top view (right) of the BigBite magnet showing the magnetic field boundary and the large pole face gap.
- 2-19 A side view (left) and top view (right) of the BigBite spectrometer during this experiment.
- 2-20 A side view of the BigBite detector package during this experiment.
- 2-21 The BigBite shower counter hit pattern for kinematics K8, = -2. The hot region corresponds to the elastic electrons. For production data taking, only the shower blocks inside the ellipse were on.
- 2-22 Left HRS single arm triggers diagram during E08-007.
- 2-23 The BigBite trigger diagram during E08-007.
- 2-24 Coincidence trigger diagram during E08-007.
- 3-1 The flow-chart of the E08-007 analysis procedure.
- 3-2 Hall coordinate System (top view).
- 3-3 Target coordinate system (top and side views).
- 3-4 Detector coordinate system (top and side views).
- 3-5 Transport coordinate system.
- 3-6 Rotated focal plane coordinate system.
- 3-7 The TDC width of the u1 wire group and the demultiplexing cut.
- 3-8 Illustration of the procedure to find clusters in a FPP chamber. The three layers represent the three planes, and the circles are cross-sectional cuts of the straws. The filled circles represent the fired straws.
- 3-9 4 possible tracks for two given fired straws with given drift distances and . The good track is the one with the lowest when taking into account all planes of all chambers.
- 3-10 The difference between the VDC track and the FPP front track before (in black) and after (in red) the chamber alignment. The difference is centered at 0 after the alignment.
- 3-11 versus before and after the FPP chamber alignment.
- 3-12 Cartesian angles for tracks in the transport coordinates system.
- 3-13 Spherical angles of the scattering in the FPP.
- 3-14 Left HRS VDC track number distribution.
- 3-15 HRS acceptance cuts for kinematic setting K5 .
- 3-16 Elastic cut on dpkin (left), and the corresponding 2D cut on the proton angle versus momentum .
- 3-17 The BigBite pre-shower ADC sum versus shower ADC sum with (right panel) and without (left panel) the coincidence trigger cut (T5). The low energy background were highly suppressed with the coincidence configuration.
- 3-18 BigBite shower counter hit pattern in the upper panel and the profiles on (vertical) and (horizontal) in the left and right panels, respectively.
- 3-19 Proton acceptance (angle versus momentum) with BigBite shower and .
- 3-20 The distribution of the FPP polar scattering angle and the applied cut.
- 3-21 Cut applied to after the manual correction for setting K2 .
- 3-22 distribution and cut applied to it for setting K2 .
- 3-23 The cone-test in the FPP. The cone of angle around track 1 is entirely within the rear chambers acceptance, while the one around track 2 is not. Track 2 fails the cone-test and is rejected.
- 3-24 Polarimetry principle: via a spin-orbit coupling, a left-right asymmetry is observed if the proton is vertically polarized.
- 3-25 The dipole approximation of the spin transport in the spectrometer: only a perfect dipole with sharp edges and a uniform field. The proton spin only processes along the out-of-plane direction.
- 3-26 Asymmetry difference distribution along the azimuthal scattering angle at kinematics K6 (). The black solid curve represents the sinusoidal fit to the data (). The dashed light blue curve corresponds to a hypothetical distribution assuming in dipole approximation.
- 3-27 Close up view of Fig. 3-26. The black solid curve represents the sinusoidal fit to the data, while the dashed light blue curve corresponds to a hypothetical distribution assuming in dipole approximation. There is shift between these two curves at the zero crossing.
- 3-28 The target scattering coordinate system (solid lines) is the frame where the polarization is expressed while the TCS (dashed lines) is the one in which COSY does the calculation.
- 3-29 Histograms of the four spin transport matrix elements, (upper left), (upper right), (lower left) and (lower right) at GeV for the elastic events. The ones plotted in black are from dipole approximation, and the ones in red are from the full spin transport matrix generated by COSY. For the dipole approximation, and are exactly zero, and by ignoring the transverse components of the field. The full spin precession matrix gives broad distributions for these elements which represent the effect from the quadrupoles and the dipole fringe field.
- 3-30 Analyzing power fit part 1: plotted with different parameterization in the low energy region ( MeV). The error bars shown are statistical only. The dashed lines are from the LEDEX  parameterization, the dashed dotted lines are from the “low energy” McNaughton parameterization , and the solid lines are from the new parameterization for experiment E08-007.
- 3-31 Analyzing power fit part 2: plotted with different parameterization in the high energy region ( MeV). The error bars shown are statistical only. The dashed lines are from the LEDEX  parameterization, the dashed dotted lines are from the “low energy” McNaughton parameterization , and the solid lines are from the new parameterization for experiment E08-007.
- 3-32 Weighted average analyzing power with respect to for scattering angles .
- 4-1 The spectrum for LH and Al dummy data with the cut shown by the vertical solid lines.
- 4-2 The normalized dpkin spectrum for LH and Al dummy at setting K2 . The unfilled and filled spectra are with and without the proton dpkin cut respectively.
- 4-3 Different elastic cuts on proton dpkin.
- 4-4 The ratio difference with different elastic cuts. The -axis is the difference between the results, the -axis which was manually shifted for different cuts for a better view.
- 4-5 Coordinates for electrons scattering from a thin foil target. is the distance from Hall center to the floor mark, and is the horizontal displacement of the spectrometer axis from its ideal position. The spectrometer set angle is and the true angle is denoted by when the spectrometer offset is considered.
- 4-6 Carbon pointing for kinematics K8 ( GeV).
- 4-7 NMR reading with probe D versus the central momentum setting (left panel), and the deviation between the value from the linear fit function and the set value.
- 4-8 Proton scattering angle versus the momentum for kinematics K8 . The anticipated elastic peak position is plotted as the black dash line.
- 4-9 Dependence of on the proton target quantities for kinematics K7 ( GeV). The full precession matrix calculated by COSY (solid quare) is compared to the dipole approximation (open square) and a constant fit. The data points are shown with statistical error bars only.
- 4-10 Alternative ways to calculate the spin precession matrix .
- 4-11 Fit of the out-of-plane angle difference between the target and the focal plane. (K6 ). The peak at zero corresponds to a bending angle in the spectrometer.
- 4-12 FPP chamber rotation along and the shift of the azimuthal angle .
- 4-13 FPP chamber rotation along and the change of distribution.
- 4-14 The non-zero component in the rotated frame.
- 4-15 The track difference in versus before and after the software alignment.
- 4-16 The track difference () and its profile versus after the software alignment. The solid line is a linear fit to the profile with a slope of .
- 4-17 The form factor ratio binning on the FPP polar scattering angle for kinematic setting K6 ( GeV) and K7 ( GeV).
- 4-18 Comparison of the major contributions to the systematic uncertainties and the statistical uncertainty for each kinematics.
- 4-19 Radiative corrections to the recoil polarization. The solid and dashed lines correspond to the longitudinal and transverse components with GeV. Figure from .
- 4-20 Radiative corrections to the ratio of the recoil proton polarization in the region where the invariant mass of the unobserved state is close to the pion mass and GeV. Figure from .
- 4-21 The 2 exchange correction to the recoil proton longitudinal polarization components and the ratio of the transverse to longitudinal component for elastic scattering at GeV. Figure from .
- 4-22 The relative correction to the proton form factor ratio from 2 exchange as a function of for 5 different [57, 58].
- 5-1 The proton form factor ratio as a function of with world high precision data [16, 19, 20] (). For the new data, the inner error bars are statistical, and the outer ones are total errors. For the world data sets, the total errors are plotted. The dashed lines are fits [42, 24, 48, 44].
- 5-2 The proton form factor ratio as a function of shown with world high precision data [16, 19, 20] (). For the new data, the inner error bars are statistical, and the outer ones are total errors. For the world data sets, the total errors are plotted. The solid lines are from vector-meson dominance calculations [50, 59], a light-front cloudy-bag model calculation , a light-front quark model calculation , and a point-form chiral constituent quark model calculation .
- 5-3 The proton form factor ratio as a function of shown with world high precision polarization data [16, 19, 20, 18, 60].
- 5-4 Rosenbluth separation of and constrained by . For each , the reduced cross section is plotted against . The solid blue line is the standard Rosenluth separation fit without any constraint on . The dotted red line is fit with an exact ratio constraint.
- 5-5 The new extraction of and plotted together with the world unpolarized data.
- 5-6 The global fit for the proton form factor ratio with world high precision data. The red points are the new results (E08-007 I and E03-104), the other points are from previous polarization measurements [16, 19, 20]. The black line is the AMT fit to the world 2 exchange corrected cross section and polarization data. The red line is the new fit by including the new data.
- 5-7 The global fit for the proton electric form factor . The black line is the AMT fit to the world 2 exchange corrected cross section and polarization data. The red line is the new fit by including the new data.
- 5-8 The global fit for the proton magnetic form factor . The black line is the AMT fit to the world 2 exchange corrected cross section and polarization data. The red line is the new fit by including the new data.
- 5-9 The uncertainty of the Zemach radius as a function of . The green band shows the coverage of the new data.
- 5-10 A linear fit to previous world polarization data, shown by the solid (blue) line and error band. The fit was done up to the region of GeV where the linear expansion is valid for the transverse radii difference. The shaded area indicates . The dashed (red) line shows the critical slope when . Figure from 
- 5-11 New fit with the E08-007 data, shown by the solid (blue) line and error band. The shaded area indicates . The dashed (red) line shows the critical slope when .
- 5-12 The accessible kinematic region in space. The black dots represent the chosen settings (centers of the respective acceptance). The dotted curves correspond to constant incident beam energies in steps of 135 MeV (”horizontal” curves) and to constant scattering angles in steps (”vertical” curves). Also shown are the limits of the facility: the red line represents the current accelerator limit of 855 MeV, with the upgrade, it will be possible to measure up to the light green curve. The dark green area is excluded by the minimal beam energy of 180 MeV. The maximum (minimum) spectrometer angle excludes the dark (light) blue area. The gray shaded region is excluded by the upper momentum of spectrometer A (630 MeV/). Figure from .
- 5-13 Spin-dependent elastic scattering in Born appromixation.
- 5-14 The kinematics for the two simultaneous measurements. The scattered electrons and are detected in left and right HRS, respectively. The recoil protons and point in the direction of the q-vector and , respectively. denotes the target spin direction.
- 5-15 The proposed points and projected total uncertainties for the second part of E08-007.
- 5-16 The uncertainty of the Zemach radius as a function of . The green band shows the coverage of the new data from this work, and the yellow band shows the proposed coverage of the second part of E08-007.
- 5-17 Projection of E08-007 part II measurements on the new fit by assuming the same slope as decreases.
- A-1 Elastic scattering in the Breit frame.
- C-1 False asymmetry Fourier series coefficients vs. for kinematics K6 .
- C-2 Histograms of the extracted ratio by weighted-sum method with no false asymmetry () in the simulation. is the sample size of each trial in the simulation. At large statistics, the extracted ratio is in good agreement with the set ratio in the simulation.
- C-3 Extracted ratio mean value by weighted-sum method vs. different sample size with false asymmetry (left) and (right). There is no noticeable difference between the two. Upper panel with set polarization , lower panel with set polarization , showing that the results of the tests do not depend on the value of the set ratio .
- C-4 Extracted ratio mean value deviation from the set value divided by the sample standard deviation (RMS) vs. different sample size with false asymmetry (left) and (right). There is no noticeable difference between the two. Upper panel is with set polarization , lower panel is with set polarization .
- C-5 Proton induced polarization component, as a function of the electron scattering angle for different beam energies. The dash (solid) line shows the total (elastic only) 2 exchange effect. The y-axis is actually for the convention used here.
- C-6 Extracted ratio mean value and relative deviation vs. different sample size with false asymmetry , and different combinations of set polarization .
- C-7 Extracted ratio mean value and relative offset from the set value vs. different sample size with difference false asymmetries: , and set polarizations: , respectively.
- E-1 Data and simulated spectrum on .
- E-2 Simulated proton kinematics for at MeV and elastic. is the proton momentum and is the scattering angle.
- E-3 Proton elastic cut on spectrum for kinematics K2.
- E-4 Simulated and spectrum for kinematics K2 and K8. The blue lines are the corresponding elastic cut applied to the data.
- E-5 World data and calculations for differential cross section at E = 1185 MeV.
- E-6 Phase space simulation for , and with E = 1190 MeV.
- E-7 The proton singles spectra and the full background simulation from Hall C Super-Rosenbluth experiment with beam energy 849 MeV (left panel) and 985 MeV (right panel). The spectra in red in the proton elastic peak, and the one in magenta is the simulated pion production.
- E-8 Calculations for the polarization observable at E = 1185 MeV.
List of Tables
- 2.1 E08-007 kinematics.
- 2.2 Results of the Møller measurements during E08-007.
- 2.3 Main characteristics of Hall A High Resolution Spectrometers; the resolution values are for the FWHM.
- 2.4 Carbon thickness along the proton momentum at each kinematics.
- 2.5 Dimensions of the FPP straw chambers.
- 2.6 Trigger summary for E08-007.
- 3.1 FPP performance for E08-007 with . is the proton average kinetic energy at the center of the carbon door.
- 4.1 Aluminum foil thickness.
- 4.2 The upper limit of the Al background fraction for each kinematics. The numbers listed are the average over all settings.
- 4.3 Polarization of LH, Al dummy and corrected values for kinematics K1 ( GeV).
- 4.4 Polarization of LH inside, outside the coincidence timing cut and the corrected values for kinematics K8 ( GeV).
- 4.5 Shifts of the form factor ratio associated with shifts of the individual target quantities for each kinematic setting.
- 4.6 Spectrometer nominal () and real () central angle for each kinematic setting.
- 4.7 Target materials in the beam energy loss calculation.
- 4.8 Converted uncertainty in with MeV.
- 4.9 Recorded magnetic field in kG with probe D for each momentum setting.
- 4.10 Converted uncertainty in from .
- 4.11 Target materials that the proton passed through before entering the spectrometer.
- 4.12 Proton momentum loss [MeV/c] for each kinematics.
- 4.13 Uncertainty of with
- 4.14 Total uncertainty in from the external parameters.
- 4.15 uncertainty for each kinematics.
- 4.16 Systematic uncertainty in for each kinematics associated with left HRS optics.
- 4.17 Systematic error in associated with COSY.
- 4.18 Errors in the FPP scattering angles and the associated systematic error in .
- 4.19 Errors of the VDC angles and associated systematic error in .
- 4.20 Errors of the kinematic factors and the resulting uncertainty in the form factor ratio for kinematics K7 ( GeV).
- 4.21 Final results with statistical and systematic uncertainties for each kinematics.
- 5.1 The extracted values of and , with and without the constraint of from the new measurements. The errors are indicated in parentheses.
- 5.2 Proton charge rms-radius from different parameterizations.
- 5.3 Summary of corrections for electronic hydrogen.
- 5.4 Zemach radii, for different parameterizations.
- 5.5 The absolute asymmetry difference (), the normalized difference by the experimental uncertainty () and the relative asymmetry difference () between using the AMT  parameterization and the new one.
- C.1 Electron scattering angle for each kinematics ().
- C.2 Deviation from the set value with different combinations of and . The set transferred polarization is . Simulation with sample size and number of trial . The standard deviation for extracted values is .
- D.1 Binning on .
- D.2 Binning on .
- D.3 Coefficients of different parameterizations for the analyzing power . The reduced of the new fit is 0.74 with a of 272.5 and 368 degrees of freedom.
- E.1 HRS acceptance.
- E.2 Simulated to phase space ratio at kinematics K2.
- E.3 to phase space ratio for different kinematics with 1180, 1185, and 1190 MeV.
- E.4 Real photon flux at different energies with 1.192 GeV electron beam.
- E.5 and differential cross sections in the lab frame and the ratio for different kinematics.
- E.6 Estimated ratio of to for kinematics K2.
- E.7 and differential cross sections in the lab frame and the ratio for different kinematics.
- E.8 Polarization observable
Chapter 1 Introduction
When the proton and the neutron were discovered in 1919 and 1931 respectively, they were believed to be Dirac particles, just like the electron. They were expected to be point-like and to have a Dirac magnetic moment, expressed by:
where q, m, and s are the electric charge, mass and spin of the particle respectively. However, later measurements of these nucleons magnetic moments revealed the existence of the nucleon substructure. The first direct evidence that the proton has an internal structure came from the measurement of its anomalous magnetic moment 70 years ago by O. Stern ,
Starting from 1950s, electron scattering experiments were used to unravel the nucleon internal structure. Through the measurements of electromagnetic form factors and nucleon structure functions in elastic and deep inelastic lepton-nucleon scattering, it’s commonly accepted that in a simplistic picture, a nucleon is composed of three valence quarks interacting with each other through the strong force. The strong interaction theory, Quantum Chromodynamics (QCD) can make rigorous predictions when the four-momentum transfer squared, , is very large and the quarks become asymptotically free. However, predicting nucleon form factors in the non-perturbative regime is difficult due to the dominance of the soft scattering processes. As a consequence there are several phenomenological models which attempt to explain the data in this domain, and precise measurements of the nucleon form factors are desired to constrain and test these models.
In the one-photon-exchange (OPE) approximation, the elastic scattering cross section formalism is well known and can be parameterized by two form factors, and which are functions of . At low momentum transfer, the form factors can be interpreted as the fourier transform of the nucleon charge and magnetic densities. Earlier experiments measured the cross section of the elastic scattering which contains information about the internal structure responsible for the deviation from the scattering off point-like particles. However, after four decades of effort, there were still large kinematic regions where only very limited measurements of the form factors were possible, since the cross section of the unpolarized electron scattering is only sensitive to a specific combination of the form factors and the lack of a free neutron target.
In the last two decades, advances in the technology of intense polarized electron beams, polarized targets and polarimetry have ushered in a new generation of electron scattering experiments which rely on spin degrees of freedom. Compared to the cross section measurement, the polarization techniques have several distinct advantages. First, they have increased sensitivity to a small amplitude of interest by measuring an interference term. Second, spin-dependent experiments involve the measurement of polarizations or helicity asymmetries, and these quantities are independent of the cross section normalization, since most of the helicity independent systematic uncertainties can be canceled by measuring a ratio of polarization observable.
The first experiment to measure the recoil proton polarization observable in elastic scattering was done at SLAC by Alguard et al. , but the impact of the results was severely limited by the low statistics. Followed by that, the proton form factor measurements using recoil polarimetry were carried out at MIT-Bates [45, 67] and MAMI [68, 69]. Due to limited statistics and kinematics coverage, the ratio values were in agreement within uncertainties with the unpolarized measurements. More recent measurements of the proton form factor ratio using recoil polarimetry at Jefferson Lab [16, 17, 18], which have much better precision at high , deviated dramatically from the unpolarized data. This has prompted intense theoretical and experimental activities to resolve the discrepancy. The validity of analyzing data in the OPE approximation has been questioned, and two-photon-exchange (TWE) processes are now considered as an significant correction to the unpolarized data and mostly account for the discrepancy at high .
While extending our knowledge at higher momentum transfer region is an ongoing endeavor, the proton form factor ratio behavior at low has also become the subject of considerable interest, especially, when potential discrepancy was observed from the most recent high precision measurements for . BLAST  did the first proton form factor ratio measurement via beam-target asymmetry at values of 0.15 to 0.65 , and the results are consistent with 1 in this region. LEDEX , which used the recoil polarimetry technique, observed a substantial deviation from unity at . However, the data quality of LEDEX was compromised due to the low beam polarization and background issues . Hence, it was necessary to carry out a new high precision measurement to either confirm to refute the deviations at low momentum transfers. Beyond the intrinsic interest in the nucleon structure, an improved proton form factor ratio also impacts other high precision measurements such as parity violation experiments (HAPPEX) [71, 72], deeply virtual Compton scattering (DVCS) [73, 74], and also determination of other physical quantities such as the proton Zemach radius.
This thesis presents the analysis and results of experiment E08-007, which was conducted in 2008 at Jefferson Lab Hall A. In this experiment, the proton form factor ratio was measured at using recoil polarimetry.
1.1 Definitions and Formalism
1.1.1 Exclusive electron scattering
When scattered off a nuclear target, the electron exchanges virtual photons with the nucleus, which probes the electromagnetic structure of the nucleus. The electromagnetic coupling is small enough that it is valid to only consider the leading order. For the elastic scattering reaction off a proton, , the leading order diagram is shown in Fig. 1-1.
Initial and final electrons have four-momenta and respectively, and the initial and final protons and . The virtual photon has four-momentum , and the Lorentz-invariant four-momentum transfer squared is defined as:
where the last expression is valid in the Lab frame by neglecting the electron mass. The amplitude of is associated with the scale that the electromagnetic probe is sensitive to.
For exclusive elastic scattering, the recoil proton is also detected, so that can be defined from the proton momenta:
In the Lab frame,the initial proton is at rest, and Eq. 1.4 becomes:
where is the proton mass and is the kinetic energy of the final proton in the Lab frame.
One of the advantages of the electromagnetic probe lies in the fact that the leptonic vertex is fully described by the theory of the electromagnetic interaction, Quantum ElectroDynamics (QED), and the information related to the unknown electromagnetic properties of the nucleon are contained by only the hadronic vertex . From the Feynman diagram in Fig. 1-1, the amplitude for elastic scattering can be written as:
where with the 0-th component as the time component, are the Dirac matrices in the chiral representation:
, and is the set of standard Pauli matrices:
and are the Dirac spinors for the initial and final electron, and , are the Dirac four-spinors for the initial and recoil proton respectively. In particular, the proton spinors enter in the plane-wave solution for a spin 1/2 particle which satisfies the Dirac equation:
and one can write:
with and is a normalized two-spinor, such that
While the leptonic current is fully described by QED, the hadronic current involves the factor , which contains the information about the internal electromagnetic structure of the proton. In general is some expression that involves and constants such as the proton mass , the electric charge . Since the hadronic current transforms as a vector, must be a linear combination of these vectors, where the coefficients can only be function of . It is convenient to write the current in the following way:
where , is the proton anomalous magnetic moment and are the proton elastic form factors. They contain the information about the electromagnetic structure of the proton.
1.1.3 Nucleon Form Factors
and are distinguished according to their helicity characteristics, the projection of electron intrinsic spin along its direction of motion . is the Dirac form factor; it represents the helicity-preserving part of the scattering. On the other hand, the Pauli form factor represents the helicity-flipping part. and are defined in a similar way for the neutron. The form factors are normalized according to their static properties at . For the proton:
and for the neutron:
For reasons that will soon become obvious, it is more convenient to use the Sachs form factors : and , which are defined as:
where is a kinematic factor. The Sachs form factors also have particular values at according to the static properties of the corresponding nucleon:
where and in units of nuclear magneton.
1.1.4 Hadronic Current in the Breit Frame
In the Breit frame, which is defined as the frame where the initial and final nucleon momenta are equal and opposite, the hadronic current has a simplified interpretation. A definition of variables in the Breit frame, which are noted with a subscript , is elaborated in Appendix A. Using the Gordon identity 
similarly, we can write:
In the Breit frame, the explicit expression of the hadronic current is simplified:
where is the Using , the time component can be expressed by:
Then, by the expressions:
we can finally get the simple relation:
The vector current can also be expressed in a similar way in the Breit frame:
Therefore, in the Breit frame, the electric form factor is directly related to the electric part of the interaction between the virtual photon and the nucleon, and the magnetic form factor is related to the magnetic part of this interaction. The electric and magnetic form factors can be associated with the Fourier transforms of the charge and magnetic current densities in this frame in the non-relativistic limit. The Fourier transforms can be expanded in powers of :
Notice that the first integral gives the total electric or magnetic charge, and the second integral defines the RMS electric or magnetic radii of the nucleon. However, the Breit frame is a mathematical abstraction, and for different value, the Breit frame experiences relativistic effect which is essentially a Lorentz contraction of the nucleon along the direction of motion. This results in a non-spherical distribution for the charge densities, and complicates the Fourier transform interpretation of the form factors in the rest frame.
1.2 Form Factor Measurements
1.2.1 Rosenbluth Cross Section
The differential cross section for scattering in the lab frame can be written as:
where we have neglected the electron mass, and is the amplitude defined in Eq. 1.7. Integrating over and gives:
where is the solid angle in which the electron is scattered, and has the form:
The hadronic and leptonic tensors are defined respectively as:
For unpolarized cross section, and are averaged over the incident particle spin states, and summed over the final particle spin states. Since the contraction of these two tensors is a Lorentz invariant, they can be calculated in any frame, as long as they are both calculated in the same frame.
In the single-photon exchange (Born) approximation, the formula for the differential cross section of electron scattering off nucleons is given by :
is the Mott cross section for the scattering of a spin-1/2 electron from a spinless, point-like target, and is the recoil factor. This is the most general form for the electron elastic scattering cross section. Using Eq. 1.16, we can rewrite Eq. 1.37 without the interference term:
where is the electromagnetic fine structure constant, and this expression is known as the Rosenbluth formula.
The Rosenbluth cross section has two contributions: the electric term and the magnetic term . As noted earlier, there is no interference term, so that the two contributions can be separated. We define the reduced cross section as:
where is the virtual photon polarization parameter. The quantity can be changed at a given , by changing the incident electron beam energy and the scattering angle. Therefore, at a fixed by varying , one can measure the elastic scattering cross section and separate the two form factors using a linear fit to the cross section. The slope is equal to and the intercept is equal to .
This method has been extensively used in the past 40 years to measure the elastic form factors and proved to be a very powerful method to measure the proton and the neutron magnetic form factors up to large . However, there are practical limitations. First, the neutron electric form factor is normalized to the static electric charge of the neutron, which is 0, and the cross section is completely dominated by the magnetic form factor. For the proton, the magnetic term will also dominate at large , since the factor increases quadratically as increases. As an example, at the magnetic term contributes about 95 of the total cross section. On the other hand, in the low region, the magnetic term is suppressed for the same reason and the electric term becomes dominant. Besides the difficulties from the physics side, the precision of Rosenbluth separation is also limited by the cross section measurements due to a widely different kinematic settings in order to cover a wide range of . Systematic errors are introduced by the inconsistent acceptance, luminosity, detector efficiency between different kinematics.
1.2.2 Polarization Transfer Measurements
In 1974, Akhiezer and Rekalo  discussed the interest of measuring an interference term of the form by measuring the transverse component of the recoiling proton polarization in the reaction . Thus, one could obtain in the presence of a dominating at large . Instead of directly measuring the separate form factors, the ratio can be accessed by measuring the polarization of the recoil nucleon. The virtue of the polarization transfer technique is that it is sensitive only to the ratio and does not suffer from the dramatically reduced sensitivity to a small component. Another way of measuring the interference term would be to measure the asymmetries in the scattering of a polarized beam off a polarized target.
By measuring the polarization of the recoil nucleon along a unit vector , we measure a preferential orientation of the spin along . In this case, when we average over initial proton spin states and sum over final proton spin states, the completeness relation
no longer holds. Instead we have to use:
so that the hadronic tensor becomes:
where is the unpolarized hadronic tensor and is the polarized one:
For recoil proton polarization measurements, a longitudinally polarized beam is required. The polarization of the beam is defined as:
where and are the number of electrons with their spin parallel and anti-parallel to their momentum respectively. Therefore, with a polarized electron beam, the leptonic tensor is modified and a new matrix is introduced:
projects the spin along the momentum in a preferential direction. If the beam polarization is , and by further neglecting the electron mass, the leptonic tensor can be written as:
where is the Levi-Civita symbol. It is 0 if any two indices are identical, -1 under an even number of permutations and +1 under an odd number of permutations. Note that is anti-symmetrical.
In order to get the polarized amplitude, we contract the leptonic and the hadronic tensors:
is the amplitude squared of the unpolarized process.