Cross Section Measurements of Deuteron Electro-Disintegration at Very High Recoil Momenta and Large 4-Momentum Transfers ( Q 2 )
FFLORIDA INTERNATIONAL UNIVERSITYMiami, FloridaCROSS SECTION MEASUREMENTS OF DEUTERONELECTRO-DISINTEGRATION AT VERY HIGH RECOIL MOMENTA ANDLARGE 4-MOMENTUM TRANSFERS ( Q )A dissertation submitted in partial fulfillment of therequirements for the degree ofDOCTOR OF PHILOSOPHYinPHYSICSbyCarlos Yero Perez2020 a r X i v : . [ nu c l - e x ] S e p o: Dean Michael R. HeithausCollege of Arts, Sciences and EducationThis dissertation, written by Carlos Yero Perez, and entitled Cross Section Mea-surements of Deuteron Electro-Disintegration at Very High Recoil Momenta andLarge 4-Momentum Transfers ( Q ), having been approved in respect to style andintellectual content, is referred to you for judgment.We have read this dissertation and recommend that it be approved.Misak SargsianJoerg ReinholdBrian RaueSteven HudsonWerner Boeglin, Major ProfessorDate of Defense: July 02, 2020The dissertation of Carlos Yero Perez is approved. Dean Michael R. HeithausCollege of Arts, Sciences and EducationAndr´ e s G. GilVice President of Research and Economic Developmentand Dean of the University Graduate SchoolFlorida International University, 2020ii (cid:13) Copyright 2020 by Carlos Yero PerezAll rights reserved.iiiEDICATIONTo my late uncle and nuclear engineer Raul Yero Sosa.ivCKNOWLEDGMENTSFirst, I would like to thank my parents for supporting me during my years as anundergraduate student, specially when I decided to take the important step ofchanging my major from Biology to Physics during the first semester at FloridaInternational University.I would like to thank my advisor, Professor Werner Boeglin, for giving me theopportunity to work on one of Jefferson Lab’s experiments at Hall C which allowedme to gain an unimaginable amount of hands-on experience on both hardware andsoftware related tasks during the initial phases of the 12 GeV experimentalprogram. It is not often that a graduate student has the opportunity to work fromthe ground up on a nuclear physics experiment and be able to have a globalperspective on all the different aspects of what constitutes a nuclear/particlephysics experiment. For this, I considered myself very lucky to have been giventhis opportunity. I am also very thankful to all the Experimental Hall C Staff andUsers for all the useful discussions I had with them on different aspects ofexperimental nuclear physics which allowed me to gain a better perspective onsome of the most difficult (but also the most fun!) topics which I considered to bespectrometer optics and what constitutes the set-up of an electronics trigger.I would also like to thank Dr. Mark Jones from Hall C for his constantsupport and guidance during the analysis of this experiment. I am very grateful toMark as a friend and colleague who has demonstrated infinite patience even whenI ask the most stupid questions one could ever imagine. I cannot really thank himenough for all the help I received.Finally, my gratitude goes to theorists Misak Sargsian, Jean-Marc Laget,Sabine Jeschonnek and J.W. Van Orden for providing the theoretical calculationsas well as helpful discussions on this topic. And from the experimental side, avpecial thanks goes to Dr. Dave Mack from Hall C for diligently (and voluntarily)revising my thesis and making sure to point out any inconsistencies, typos andsilly mistakes.This work was supported in part by the U.S. Department of Energy (DOE),Office of Science, Office of Nuclear Physics under grant No. DE-SC0013620 andcontract DE-AC05-06OR23177, the Nuclear Regulatory Commission (NRC)Fellowship under grant No. NRC-HQ-84-14-G-0040 and the Doctoral EvidenceAcquisition (DEA) Fellowship. viBSTRACT OF THE DISSERTATIONCROSS SECTION MEASUREMENTS OF DEUTERONELECTRO-DISINTEGRATION AT VERY HIGH RECOIL MOMENTA ANDLARGE 4-MOMENTUM TRANSFERS ( Q )byCarlos Yero PerezFlorida International University, 2020Miami, FloridaProfessor Werner Boeglin, Major ProfessorThe H( e, e (cid:48) p ) n cross sections have been measured at negative 4-momentum trans-fers of Q = 4 . ± . and Q = 3 . ± . reaching neutron recoil(missing) momenta up to p r ∼ ◦ ≤ θ nq ≤ ◦ with respect to the 3-momentum transfer (cid:126)q .The new data agree well with the previous data which reached p r ∼
550 MeV/c. At θ nq = 35 ◦ and 45 ◦ , final state interactions (FSI), meson exchange currents (MEC)and isobar configurations (IC) are suppressed and the plane wave impulse approx-imation (PWIA) provides the dominant cross section contribution. The new dataare compared to recent theoretical calculations, and a significant disagreement forrecoil momenta p r >
700 MeV/c is observed.The experiment was carried out in experimental Hall C at the Thomas JeffersonNational Accelerator Facility (TJNAF) and formed part of a group of four experi-ments that were used to commission the new Super High Momentum Spectrometer(SHMS). The experiment consisted of a 10.6 GeV electron beam incident on a liq-uid deuterium target which resulted in the break-up of the deuteron into a protonand neutron. The scattered electrons were detected by the SHMS in coincidencewith the knocked-out protons detected in the previously existing High Momentumviipectrometer (HMS) and the recoiling neutrons were reconstructed from energy-momentum conservation laws. To ensure that the H( e, e (cid:48) p ) n reaction channel wasselected, we required the missing energy of the system to be the binding energy ofthe deuteron ( ∼ p r = 80 ,
580 and750 MeV/c, which required the SHMS central angle and momentum to be fixed andthe HMS to be rotated from smaller to larger angles corresponding to the lower andhigher missing momentum settings, respectively. The experiment was carried out ina time period of six days with typical electron beam currents of 45-60 µ A at about50% beam efficiency. viiiABLE OF CONTENTSCHAPTER PAGE1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Deuteron and the Beginning of Nuclear Forces . . . . . . . . . . . . 11.2 From Meson Theory to Phenomenology . . . . . . . . . . . . . . . . . . . 51.3 Historical H( e, e (cid:48) p ) n Experiments . . . . . . . . . . . . . . . . . . . . . . 81.4 First H( e, e (cid:48) p ) n Experiments at Large Q . . . . . . . . . . . . . . . . . 101.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122. THEORETICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . . . 152.1 The H( e, e (cid:48) p ) n Reaction Kinematics . . . . . . . . . . . . . . . . . . . . 152.2 The H( e, e (cid:48) p ) n Cross Section . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Plane Wave Impulse Approximation . . . . . . . . . . . . . . . . . . . . . 222.4 Final State Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Meson Exchange Currents and Isobar Configurations . . . . . . . . . . . 242.6 From Theoretical Potentials to Cross Sections . . . . . . . . . . . . . . . 253. EXPERIMENTAL SETUP . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 CEBAF Accelerator Overview . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Accelerator Upgrade to 12 GeV . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Particle Acceleration at CEBAF . . . . . . . . . . . . . . . . . . . . . 363.2 Hall C 12 GeV Upgrade . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 E12-10-003 Experiment Overview . . . . . . . . . . . . . . . . . . . . . . 393.4 The Hall C Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.1 Beam Energy Measurement . . . . . . . . . . . . . . . . . . . . . . . . 433.4.2 Beamline Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Target Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.5.1 Target Ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.5.2 Cryotarget Loop Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . 613.6 Hall C Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.6.1 Spectrometer Slit System . . . . . . . . . . . . . . . . . . . . . . . . . 643.6.2 Spectrometer Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.6.3 Spectrometer Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . 713.7 The Hall C Electronics Trigger Setup . . . . . . . . . . . . . . . . . . . . 873.7.1 HMS Detector Hut Electronics . . . . . . . . . . . . . . . . . . . . . . 883.7.2 SHMS Detector Hut Electronics . . . . . . . . . . . . . . . . . . . . . . 893.7.3 Hall C Counting Room Electronics . . . . . . . . . . . . . . . . . . . . 903.7.4 HMS Trigger Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.7.5 SHMS Trigger Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.7.6 Electronic Dead Time Monitoring (EDTM) . . . . . . . . . . . . . . . 104ix. GENERAL HALL C ANALYSIS OVERVIEW . . . . . . . . . . . . . . . 1084.1 Reference Time Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2 Detector Time Window Cuts . . . . . . . . . . . . . . . . . . . . . . . . 1154.3 Detector Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3.1 Hodoscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.3.2 Drift Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.3.3 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.3.4 Gas/Aerogel ˇCerenkovs . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4 Hall C Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.5 Optics Checks and Optimization . . . . . . . . . . . . . . . . . . . . . . 1354.5.1 HMS Optics Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.5.2 SHMS Optics Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.5.3 Spectrometer Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.5.4 HMS Momentum Calibration . . . . . . . . . . . . . . . . . . . . . . . 1544.5.5 Spectrometer Acceptance Post-Optimization . . . . . . . . . . . . . . . 1564.6 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.6.1 Missing Energy Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.6.2 Momentum Acceptance Cuts . . . . . . . . . . . . . . . . . . . . . . . 1614.6.3 HMS Collimator Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1624.6.4 Reaction z v -Vertex Difference Cut . . . . . . . . . . . . . . . . . . . . 1634.6.5 SHMS Calorimeter Cut . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.6.6 Coincidence Time Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.6.7 Four-Momentum Transfer ( Q ) Cut . . . . . . . . . . . . . . . . . . . . 1665. DATA CROSS SECTION EXTRACTION . . . . . . . . . . . . . . . . . . 1685.1 Experimental H( e, e (cid:48) p ) n Cross Section . . . . . . . . . . . . . . . . . . . 1685.2 Tracking Efficiencies ( (cid:15) htrk , (cid:15) etrk ) . . . . . . . . . . . . . . . . . . . . . . . 1725.3 DAQ Live Time Efficiency ( (cid:15) tLT ) . . . . . . . . . . . . . . . . . . . . . . 1765.4 Target Density Corrections ( (cid:15) tgt . Boil ) . . . . . . . . . . . . . . . . . . . . . 1785.5 Proton Absorption Corrections ( (cid:15) pTr ) . . . . . . . . . . . . . . . . . . . . 1895.6 Charge Normalization ( Q tot ) . . . . . . . . . . . . . . . . . . . . . . . . . 1985.7 Hydrogen Normalization Check . . . . . . . . . . . . . . . . . . . . . . . 1985.8 Radiative Corrections ( f rad ) . . . . . . . . . . . . . . . . . . . . . . . . . 2005.9 Bin Centering Corrections ( f bc ) . . . . . . . . . . . . . . . . . . . . . . . 2045.10 Systematic Uncertainty Studies . . . . . . . . . . . . . . . . . . . . . . . 2075.11 Normalization Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.12 Kinematical Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2156. RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . 2216.1 Extraction of the H( e, e (cid:48) p ) n Reduced Cross Sections . . . . . . . . . . . 2216.2 H( e, e (cid:48) p ) n Momentum Distributions . . . . . . . . . . . . . . . . . . . . 2226.3 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2336.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235xIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270xiIST OF TABLESTABLE PAGE3.1 The E12-10-003 experiment comprehensive run list. The spectrometercentral momentum and angle were determined based on the dipoleNMR set value and spectrometer camera, respectively. The beamenergy in this table is uncorrected for synchrotron radiation (seeSection 3.4.1). In the column Study, the ( ∗ ) are data taken withSHMS single-arm (electron trigger ONLY) and the ( † ) representsdata taken with SHMS single-arm and Centered Sieve Slit positioned.The remaining runs are taken with SHMS-HMS coincidence triggeronly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Demonstrated performance of the HMS and design specifications for theSHMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Spectrometer apertures at the collimator entrance. . . . . . . . . . . . . 653.4 Spectrometer magnets design parameters [122]. . . . . . . . . . . . . . . 673.5 Summary of hodoscopes paddle dimensions for each plane. . . . . . . . . 774.1 Original H( e, e (cid:48) ) p elastic kinematics in E12-10-003. . . . . . . . . . . . . 1364.2 Corrected H( e, e (cid:48) ) p elastic kinematics in E12-10-003. . . . . . . . . . . . 1364.3 Spectrometer offsets determined from H( e, e (cid:48) ) p elastic run 3288 in E12-10-003. See Section 4.5.3 of this dissertation for more information. . 1364.4 Original H( e, e (cid:48) p ) n kinematics for E12-10-003. . . . . . . . . . . . . . . 1554.5 Corrected H( e, e (cid:48) p ) n kinematics for E12-10-003. . . . . . . . . . . . . . 1555.1 Target density reduction studies run list taken on April 02, 2018. . . . . 1795.2 Target boiling (or density reduction) studies fit results normalized tothe y -intercept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1875.3 Summary of the averaged normalization correction factors (or efficien-cies) in fractional form and the total accumulated charge per dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155.4 Summary of relative systematic error on the measured cross sections dueto the normalization factors (units are in percent). . . . . . . . . . . 2155.5 Kinematic uncertainties corresponding to the diagonal elements of thecorrelation matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2165.6 Kinematic uncertainties corresponding to the off-diagonal elements ofthe correlation matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . 216xii1 θ nq = 5 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . . 255A2 θ nq = 15 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 255A3 θ nq = 25 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 256A4 θ nq = 35 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 256A5 θ nq = 45 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 257A6 θ nq = 55 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 258A7 θ nq = 65 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 259A8 θ nq = 75 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 259A9 θ nq = 85 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 260A10 θ nq = 95 ± ◦ at Q = 3 . ± . . . . . . . . . . . . . . . . . . 260A11 θ nq = 5 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . . 261A12 θ nq = 15 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 261A13 θ nq = 25 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 261A14 θ nq = 35 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 262A15 θ nq = 45 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 263A16 θ nq = 55 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 263A17 θ nq = 65 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 264A18 θ nq = 75 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 264A19 θ nq = 85 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 264A20 θ nq = 95 ± ◦ at Q = 4 . ± . . . . . . . . . . . . . . . . . . 265xiiiIST OF FIGURESFIGURE PAGE1.1 The L = 0 (spherically symmetric) and L = 2 (prolate spheroid) are thecharge distributions of the deuteron, where the spins of the proton(blue) and neutron (red) are aligned. The existence of a positiveelectric quadrupole moment indicates that the deuteron charge dis-tribution is actually elongated about the axis of rotation ( z -axis). . . 31.2 Qualitative N N -potential versus inter-nucleon separation distance. Note:Reprinted from Ref. [15]. . . . . . . . . . . . . . . . . . . . . . . . . 61.3 H( e, e (cid:48) p ) n cross section versus neutron recoil momentum from the MAMI(1998) experiment [48]. Theoretical calculations were performed byH. Arenh¨ovel [49]. Note: Reprinted from Ref. [47]. . . . . . . . . . . 91.4 H( e, e (cid:48) p ) n angular distributions of the cross section ratio, R = σ exp /σ pwia .(Left) Hall A data at Q = 3 . ± .
25 (GeV/c) and recoil mo-mentum settings (a) p r = 0 . p r = 0 . p r = 0 . Q settings. The green data (withFSI re-scattering peak) correspond to 400 ≤ p r ≤
600 MeV/c, andthe blue data (no FSI re-scattering) correspond to 200 ≤ p r ≤ H( e, e (cid:48) p ) n reaction kinematics andthe respective four momenta of the interacting particles. . . . . . . . 152.2 General H( e, e (cid:48) p ) n reaction kinematics. . . . . . . . . . . . . . . . . . . 162.3 (a) Anti-parallel kinematics. The neutron (blue) recoils in the oppositedirection to (cid:126)p f (red) and (cid:126)q (black). (b) Parallel kinematics. Theneutron recoils in the same direction as (cid:126)p f and (cid:126)q . (c) Perpendicularkinematics. The neutron recoils in a perpendicular direction relativeto (cid:126)q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 H( e, e (cid:48) p ) n Feynman Diagrams. The colors are: (yellow) deuteron tar-get, (red) initial state proton, (blue solid) initial and final stateneutron, (magenta, green) final state proton, (orange oval) FSI re-scattering, (orange rectangle) intermediate resonance state, (navyblue) virtual photon and (black dashed line) meson exchange. . . . . 212.5 Qualitative inclusive deuteron-electron scattering cross section. Note:Reprinted from Ref. [75]. . . . . . . . . . . . . . . . . . . . . . . . . 25xiv.6 Ratio of the theoretical cross section ratios versus neureon recoil angles θ nq (denoted as θ p s q in the figure) calculated within both the GEA(solid) and Glauber approximation (dashed) for varios neutron recoilmomenta p r (denoted as p s in the figure). Note: Reprinted fromRef. [57]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Aerial view of CEBAF at Newport News, Virginia. The service build-ings mark the 7/8-mile (1.4 km) racetrack-shaped accelerator 30 feet(base of tunnel) below the surface. The dome-shaped terrain repre-sent the accelerator end-stations (experimental halls), which are alsounderground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Schematic of CEBAF 12 GeV Upgrade. Note: Reprinted from Ref. [90]. 343.3 A single (C100) 7-cell Niobium cavity. [91]. Note: Reprinted from Ref. [90]. 353.4 A cartoon of electrons being accelerated by a 5-cell cavity. The principleof operation is the same regardless of the SRF cavity design. . . . . . 363.5 Artist’s rendering of Experimental Hall C after the 12 GeV upgrade.Note: Reprinted from Ref. [97]. . . . . . . . . . . . . . . . . . . . . . 383.6 The Hall C arc with the relevant beamline components for the beam en-ergy measurements are shown. The electron (red vector) loses energy(synchrotron radiation shown as yellow wiggly arrows) as it traversesthe arc under a perpendicular magnetic field B ⊥ . Two superharps(wire-scanners) at each end of the arc are used to determine smallvariations in the beam direction. . . . . . . . . . . . . . . . . . . . . 443.7 Hall C beamline from hall entrance to target chamber. Distances tothe relevant beamline components are measured from the origin (thepivot center) and given in meters. The first three colored boxes(green, blue and red) have multiple components with the relevantdistances to the target origin. The codenames used in the Fast Rastermagnets refer to the horizontally (H) and vertically (V) bending air-core magnets. The commonly used names of the other beamlinecomponents are indicated in parentheses. . . . . . . . . . . . . . . . 473.8 Hall C beamline from target chamber to beam dump. . . . . . . . . . . 483.9 Hall C beamline harp diagram. The harp enters (red arrow shows di-rection of motion) at a 45 ◦ angle. The two vertical wires measurethe beam position along the x -axis and a vertical wire measures theposition alng the y -axis. . . . . . . . . . . . . . . . . . . . . . . . . . 493.10 Results from a harp scan of harp IHA3H07A taken at 5-pass on April2018. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.11 Fast raster (X,Y) raw ADC signals measured by the pickup probe duringrun 3289 for the 80 MeV/c setting. The 3D plot (and inset 2Drepresentation) show an approximately uniform XY raster distribution. 51xv.12 Hall C BPM and electronics diagram. In EPICS coordinate system (left-handed), the beam is directed out of the page. The antennae arelocated along the axes of a coordinate system (blue) that is oriented45 ◦ relative to the EPICS coordinate system. Note: Reprinted fromRef. [97]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.13 Hall C BCM electronics diagram [97]. . . . . . . . . . . . . . . . . . . . 553.14 A CAD (computer-aided design) drawing of the Hall C Target Chamberdesign. Note: Reprinted from Ref. [97]. . . . . . . . . . . . . . . . . 573.15 A CAD drawing of the Hall C Target Ladder. The arrow shows thebeam direction. Note: Reprinted from Ref. [97]. . . . . . . . . . . . . 583.16 Hall C Target GUI screen during the E12-10-003 experiment. . . . . . . 593.17 Carbon hole check during the E12-10-003 experiment shows the rasterpattern for FR-A (left) and FR-B (right) raster magents. . . . . . . . 603.18 Hall C cryotarget loop anatomy for the 12 GeV era (not to scale). Figureadaptation from Refs. [118] [119]. . . . . . . . . . . . . . . . . . . . . 613.19 Spectrometer slit system. . . . . . . . . . . . . . . . . . . . . . . . . . . 653.20 High Momentum Spectrometer (HMS) side view. . . . . . . . . . . . . . 683.21 Super High Momentum Spectrometer (SHMS) side view. . . . . . . . . . 703.22 High Momentum Spectrometer (HMS) detector stack. . . . . . . . . . . 723.23 Super High Momentum Spectrometer (SHMS) detector stack. . . . . . . 733.24 Side view of the plane orientation for the DC1 (left) where the coloredplanes represent the wire planes, and DC2 (right) which is identicalin design to DC1 rotated by 180 ◦ about the x -axis (vertical) forminga mirror image along the z -axis. . . . . . . . . . . . . . . . . . . . . 743.25 Front view of the wire (dashed) orientations for each plane, indicatedby representative sense wires of different colors, where the + z -axis(particle direction) is into the page. The wires in each plane aresuperimposed onto a single plane in this figure for convenience andtheir orientation is defined by the vector normal to the wire. . . . . . 753.26 Front view of the SHMS S1X (front) and S1Y (back) hodoscope planes. 783.27 Cartoon of the ˇCerenkov effect. The charged particle (red) traverses amedium faster than the speed of light (blue) in that medium, pro-ducing a conical light wavefront. . . . . . . . . . . . . . . . . . . . . 803.28 CAD rendering of the SHMS Heavy Gas ˇCerenkov detector. Note:Reprinted from Ref. [134]. . . . . . . . . . . . . . . . . . . . . . . . . 81xvi.29 CAD rendering of the SHMS Noble Gas ˇCerenkov detector. . . . . . . . 823.30 CAD rendering of the SHMS aerogel ˇCerenkov detector. The HMSdesign is very similar with slightly different dimensions and an addi-tional PMT at both ends. . . . . . . . . . . . . . . . . . . . . . . . . 833.31 Typical electromagnetic shower cascade in a calorimeter. . . . . . . . . . 853.32 HMS electromagnetic calorimeter. The entire detector is tilted verticallyby 5 ◦ lower relative to the central ray of the spectrometer hut. . . . . 863.33 SHMS electromagnetic calorimeter. The entire detector is tilted ver-tically by 2 ◦ lower relative to the central ray of the spectrometerhut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.34 HMS detector hut electronics rack (left) and patch panels (right). . . . . 883.35 HMS patch diagram from detectors to Counting Room. . . . . . . . . . 893.36 SHMS hut patch panel (left) and electronics racks (right). . . . . . . . . 893.37 SHMS patch diagram from detectors to Counting Room. . . . . . . . . . 903.38 Counting Room patch panels (left 2 racks) and main electronic racks forHMS/SHMS detector signal processing (right 2 racks). . . . . . . . . 913.39 HMS hodoscopes electronics diagram. . . . . . . . . . . . . . . . . . . . 923.40 HMS calorimeter electronics diagram. . . . . . . . . . . . . . . . . . . . 943.41 HMS gas ˇCerenkov electronics diagram. Same electronics diagram ap-plies for SHMS gas ˇCerenkovs. . . . . . . . . . . . . . . . . . . . . . 953.42 HMS single arm pre-trigger electronics diagram. . . . . . . . . . . . . . 973.43 SHMS hodoscopes electronics diagram. It is important to note that only18 of the 21 quartz bars are currently usable. . . . . . . . . . . . . . 983.44 SHMS PreShower and Shower electronics diagram. . . . . . . . . . . . . 1013.45 SHMS gas ˇCerenkovs electronics diagram. . . . . . . . . . . . . . . . . . 1023.46 SHMS single arm trigger electronics diagram. . . . . . . . . . . . . . . . 1033.47 Coincidence trigger electronics diagram. . . . . . . . . . . . . . . . . . . 1043.48 EDTM electronics diagram. . . . . . . . . . . . . . . . . . . . . . . . . . 1053.49 Cartoon representation of EDTM (purple) and physics (magenta) pre-triggers at the TI module front-end. . . . . . . . . . . . . . . . . . . 106xvii.1 Cartoon illustrating the synchronization of the a detector signal withthe internal clocks of a C1190 TDC Module. . . . . . . . . . . . . . . 1094.2 SHMS reference time spectrum for coincidence run 3377 of the E12-10-003 experiment. Background hits are shown in blue and good hitsin red. Inset: Multiplicity histogram corresponding to the referencetime spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3 Cartoon illustrating how reference time logic definition of Eq. 4.4 mightappear on an oscilloscope. This corresponds to a multiplicity of four,or equivalently, 4 good hits in the TDC readout window. . . . . . . . 1144.4 SHMS (top panel) and HMS (bottom panel) reference time cuts forcoincidence run 3289 of the E12-10-003 experiment. The conversionfrom TDC channel to time is ∼ T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.6 Cartoon of individual scintillator paddles to illustrate the various timingcorrections applied. Note: Timewalk Effect illustration reprintedfrom Ref. [148]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.7 Fit correlation between TDC pulse time and fADC pulse amplitude(left). Time-walk corrected pulse time versus fADC pulse amplitudeshows no correlation (right). . . . . . . . . . . . . . . . . . . . . . . . 1194.8 Fit correlation between track position along paddle and time-walk cor-rected (TDC-ADC) time difference used to determined the propaga-tion velocity across the paddle. . . . . . . . . . . . . . . . . . . . . . 1204.9 Illustration of all possible time difference combinations that are consid-ered in this correction. . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.10 HMS/SHMS hodoscope calibration results for the 580 MeV/c settingof E12-10-003. The histograms are plotted using the drift chambertracking information to determine β . . . . . . . . . . . . . . . . . . . 1224.11 Illustration of a single drift cell (top view) in a drift chamber. Thedashed lines represent equipotential surfaces where the electric fieldis perpendicular to the contour. Figure adaptation from G. Niculescu.1234.12 HMS drift time spectrum for plane 1x1. Inset: Fit of the leading edgein a drift time spectrum corresponding to a group of wires from aspecific discriminator card of plane 1x1. . . . . . . . . . . . . . . . . 1254.13 HMS drift times versus wire number for plane 1x1 before t correction.Inset: Same as in Fig. 4.12. . . . . . . . . . . . . . . . . . . . . . . . 125xviii.14 Fit of the leading edge in an HMS drift time spectrum for the wire card t . . . . 1264.15 HMS drift times versus wire number for plane 1x1 after t correction.Inset: Drift time for wire group t correction. . . . . . . . . . . . . . . . . . . . . . . . . 1264.16 HMS drift distance for plane 1x1 before (red) and after (blue) calibrationof the drift maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.17 HMS fit drift residuals for plane 1x1 before (red) and after (blue) cali-bration of the drift maps. The standard deviation ( σ ) from the fit isrepresentative of the spatial resolution. . . . . . . . . . . . . . . . . . 1294.18 SHMS calorimeter calibration plots for the 580 MeV/c setting of E12-10-003. (A) is before calibration and (B), (C) and (D) are aftercalibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.19 Top view of the spectrometer and hall coordinate systems in Hall C. . . 1334.20 Comparison of HMS momentum difference (∆ P ) between data and SIMC.The inset shows the calculated and measured HMS momentum dis-tribution for data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.21 HMS momentum difference, ∆ P diff = ∆ P SIMC − ∆ P data , before and afterapplying the momentum correction to data. . . . . . . . . . . . . . . 1394.22 HMS momentum difference for data, before (blue) and after (green)applying the momentum correction (2 nd iteration) to data. . . . . . . 1394.23 HMS fractional momentum difference vs. focal plane variables for H( e, e (cid:48) ) p elastic run 3288. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.24 HMS fractional momentum difference vs. focal plane variables for H( e, e (cid:48) ) p elastic run 3371. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.25 HMS fractional momentum difference vs. focal plane variables for H( e, e (cid:48) ) p elastic run 3374. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.26 HMS fractional momentum difference vs. focal plane variables for H( e, e (cid:48) ) p elastic runs 3377. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.27 Missing energy spectrum for H( e, e (cid:48) ) p elastic run 3288 before centralmomentum correction. . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.28 SHMS ( δ calc − δ meas ) vs. focal plane variables for H( e, e (cid:48) ) p elastic run3288 before δ -optimization. . . . . . . . . . . . . . . . . . . . . . . . 1474.29 SHMS ( δ calc − δ meas ) vs. focal plane variables for H( e, e (cid:48) ) p elastic run3288 after δ -optimization. . . . . . . . . . . . . . . . . . . . . . . . . 147xix.30 Missing energy spectrum for H( e, e (cid:48) ) p elastic run 3288 after centralmomentum correction and δ -optimization. . . . . . . . . . . . . . . . 1484.31 SHMS δ vs. Y tar for carbon sieve run 3286 before Y tar -optimization. . . . 1494.32 SHMS δ vs. Y tar for carbon sieve run 3286 after Y tar -optimization. . . . . 1504.33 Missing momentum components with no spectrometer offsets applied. . 1524.34 Missing momentum components with out-of-plane central offset applied. 1524.35 HMS X (cid:48) tar for run 3288 before (left) and after (right) applying the offsetcorrection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.36 HMS momentum correction for H( e, e (cid:48) ) p and H( e, e (cid:48) p ) n . . . . . . . . . 1554.37 SHMS target reconstruction after optics optimization of H( e, e (cid:48) ) p elasticrun 3288 for the E12-10-003. . . . . . . . . . . . . . . . . . . . . . . 1564.38 HMS target reconstruction after optics checks of H( e, e (cid:48) ) p elastic run3288 for the E12-10-003. . . . . . . . . . . . . . . . . . . . . . . . . . 1574.39 SHMS target reconstruction of H( e, e (cid:48) p ) n run 3289 (80 MeV/c setting)for the E12-10-003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584.40 HMS target reconstruction of H( e, e (cid:48) p ) n run 3289 (80 MeV/c setting)for the E12-10-003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.41 Missing energy cut on the 80 MeV/c setting of E12-10-003. . . . . . . . 1604.42 Momentum acceptance cuts on the 80 MeV/c setting of E12-10-003. . . 1614.43 Data HMS collimator cut on the 80 MeV/c setting of E12-10-003. Inset:SHMS collimator geometry and reconstructed events. . . . . . . . . . 1624.44 Simulated HMS collimator cut on the 80 MeV/c setting of E12-10-003.Inset: SHMS collimator geometry and reconstructed events. . . . . . 1624.45 Reaction z v -vertex difference cut on the 80 MeV/c setting of E12-10-003. 1634.46 SHMS calorimeter cut on the 80 MeV/c setting of E12-10-003. . . . . . 1644.47 Coincidence time cut on the 80 MeV/c setting of E12-10-003. . . . . . 1654.48 Four-momentum transfer ( Q )cut on the 80 MeV/c setting of E12-10-003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.1 Cartoon representation of a typical coincidence experiment. . . . . . . . 1685.2 Tracking efficiency of HMS (open) and SHMS (full) for the E12-10-003experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175xx.3 Trigger rates for the SHMS (top), HMS (middle) and coincidence trigger(bottom) during the E12-10-003 experiment. . . . . . . . . . . . . . . 1765.4 Total EDTM live time determined for the E12-10-003 experiment. . . . 1775.5 Illustration of scaler and event reads during a typical experimental run. 1805.6 Example of a BCM scaler current cut used to determine the yield. . . . 1815.7 Example of the histograms used to determine the non-tracking (top)and tracking (bottom) yields. Top: x -axis shows the total depositedenergy in the calorimeter normalized by the central spectrometermomentum, E DEP /P c . Bottom: x -axis shows the HMS momentumacceptance, δ , in percent. . . . . . . . . . . . . . . . . . . . . . . . . 1825.8 Normalized tracking yields using BCM4A (red) and BCM4B (black)beam current cuts on carbon-12 (top), LH (middle) and LD (bot-tom) targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.9 Linear fit of the charge normalized yields for carbon-12. . . . . . . . . . 1845.10 Linear fit of the charge normalized yields for LH . . . . . . . . . . . . . 1855.11 Linear fit of the charge normalized yields for LD . . . . . . . . . . . . . 1865.12 Linear fit function normalized to the y -intercept, Y for each of the threetargets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1875.13 Target density correction of deuterium determined for the E12-10-003experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.14 Averaged beam currents (BCM4A) determined for the E12-10-003 ex-periment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.15 Cartoon to illustrate how the proton absorption coefficient is determinedexperimentally by selecting the ep elastics acceptance region. . . . . 1905.16 Momentum acceptance correlation between SHMS and HMS for coinci-dence H( e, e (cid:48) ) p data run 3248. Inset: Missing energy spectrum cutbelow 30 MeV to select true elastic events. . . . . . . . . . . . . . . . 1915.17 SHMS electron angular acceptance for coincidence H( e, e (cid:48) ) p data run3248. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1925.18 HMS proton angular acceptance for coincidence H( e, e (cid:48) ) p data run 3248. 1925.19 Reaction z -vertex cut determination from aluminum dummy run 3254taken with SHMS singles trigger. . . . . . . . . . . . . . . . . . . . . 193xxi.20 SHMS angular acceptance from electron singles run 3259. The redsquare is the elastic acceptance region determined from coincidenceelastics data at the same kinematics. . . . . . . . . . . . . . . . . . . 1945.21 Proton invariant mass ( W ) determined from SHMS electron singles run3259. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1945.22 Ratio of X (cid:48) tar from SHMS electron singles run 3259. Inset: X (cid:48) tar his-tograms before taking the ratio, where did is in red and should is inblue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.23 Ratio of Y (cid:48) tar from SHMS electron singles run 3259. Inset: Y (cid:48) tar histogramsbefore taking the ratio, where did is in red and should is in blue. . . 1965.24 Invariant mass W determined from SHMS elastic electron singles run3259. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1975.25 Fully corrected experimental data to SIMC yield ratio for four H( e, e (cid:48) ) p elastic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1995.26 Examples of internal bremsstrahlung photons (red) for a general N ( e, e (cid:48) ) N .In the top diagrams, the electron emits a real bremsstrahlung photon(a) before and (b) after scattering off a nucleus N . In the bottomdiagrams, the electron exchanges a virtual bremsstrahlung photonwith (c) the nucleus N and (d) between its own initial and final state.2015.27 2D histogram of radiative correction factor, f rad , binned in ( p r , θ nq ) forthe 80 MeV/c setting of E12-10-003. Inset: Y-projection of θ nq be-tween 30 and 40 degrees. . . . . . . . . . . . . . . . . . . . . . . . . 2035.28 Missing momentum yield for θ nq = 35 ± ◦ before and after radiativecorrections for the 80 MeV/c setting of E12-10-003. . . . . . . . . . . 2035.29 Cartoon illustrating bin-centering calculation for this experiment. . . . . 2055.30 Bin-centering correction factor at θ nq = 35 ± ◦ for each p r setting ofE12-10-003. The dashed reference lines indicate ±
10% (black) or ±
20% (red) deviation from unity. The correction factor was calcu-lated by taking the ratio of cross sections (see Eq. 5.28) either withinthe PWIA (full data points) or by including FSI (empty data points)for each dataset (see Section 3.3). The theoretical cross section cal-culations were by J.M. Laget [60] using the Paris potential [41]. . . . 2065.31 Bin-centering correction factor at θ nq = 45 ± ◦ for each p r setting ofE12-10-003. See Fig. 5.30 for definition of lines and data points. . . . 2065.32 Systematic effects of the missing energy cut. The inner dashed and outersolid lines represent the ∆ = ± σ ∆ and ± σ ∆ boundaries, respectively.2085.33 Systematic effects of the Z tar difference cut. The lines are described inFig. 5.32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209xxii.34 Systematic effects of the HMS collimator cut. The lines are describedin Fig. 5.32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.35 Systematic effects of the coincidence time cut. The lines are describedin Fig. 5.32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.36 Systematic effects of the SHMS calorimeter cut. The lines are describedin Fig. 5.32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.37 Systematic effects of the model dependency of radiative corrections. Thelines are described in Fig. 5.32. . . . . . . . . . . . . . . . . . . . . . 2125.38 Systematic effects of the model dependency of bin-centering corrections.The lines are described in Fig. 5.32. . . . . . . . . . . . . . . . . . . 2125.39 Laget FSI model cross section derivatives at θ nq = 35 ± ◦ for the 80MeV/c setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2185.40 Kinematic systematics relative error contributions to the experimentalcross sections at θ nq = 35 ± ◦ for the 80 MeV/c setting. The overlallrelative error on the cross section is shown in red. . . . . . . . . . . . 2196.1 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 5 ± ◦ . Top panel: The bluelines represent the theoretical calculations by J.M. Laget [60] us-ing the Paris potential [41] denoted by JML and the green/magentalines are calculations from M. Sargsian [59] using either the AV18(green) [42] or CD-Bonn (magenta) [43] potentials denoted by MS.The dashed lines are calculations within the PWIA and the solidlines are calculations including FSI. Bottom panel: The dashed ref-erence (magenta) line refers to MS CD-Bonn PWIA calculation (ormomentum distribution) by which the data and all models are di-vided. Inset (bottom panel): Close-up plot of the reduced crosssection ratio shown in the bottom panel. . . . . . . . . . . . . . . . 2236.2 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 15 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2246.3 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 25 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2256.4 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 35 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.5 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 45 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.6 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 55 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2286.7 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 65 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229xxiii.8 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 75 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2306.9 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 85 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.10 H( e, e (cid:48) p ) n reduced cross sections at θ nq = 95 ± ◦ . The lines are de-scribed in Fig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232B.1 Cartoon showing how the fast-raster system works. The beam bunchesfeel a kick (time-varying magnetic force) along the ( x, y ) coordinatesdue to a time-varying magnetic field and form a rectangular patternat the target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266xxivHAPTER 1 INTRODUCTION
In this introductory chapter I will first give a historical overview of the deuteron.Then I will briefly discuss the transition from meson theory to phenomenology aswell as historical electron-scattering experiments on the deuteron that were done inan effort to understand how the nucleon-nucleon (
N N ) interaction works. Finally,I will give the motivation for doing this experiment.
The deuteron (originally called by various names such as “deuton,” “diplon” or“diplogen”) was discovered in 1931 by H. Urey [1] while spectroscopically examiningthe residue of a distillation of liquid hydrogen. It was not until the discovery of theneutron by J. Chadwick [2] a few months later that the deuteron mass could beexplained. Within a few months, the first attempt to describe the nuclear forcebetween the proton and neutron using a quantum-mechanical approach was madeby W. Heisenberg [3–5] under the faulty assumption that the neutron was a boundsystem of a proton and an electron, as this was the existing view of the nucleusat the time. In 1934, H. Bethe and R. Peierls introduced for the first time theHamiltonian of the deuteron [6] (“diplon” at the time) treating it as a two-bodysystem with a nucleon-nucleon (
N N ) interactive potential, even though the detailsof the interaction were unkown at the time. The approach to describe the
N N potential via a Hamiltonian would become a basis for the successful description ofnuclear systems and reactions in the future [7]. In that same year, the first semi-successful attempt at explaining how the nuclear force worked was presented by H.Yukawa using the idea of particle exchange introduced in the Quantum Field Theory1QFT) of electromagnetic interactions known as Quantum Electrodynamics (QED)and developed by P.M. Dirac in the late 1920s. In a simplified version of Yukawa’stheory, the
N N potential is expressed as: V Yukawa = − C Y e − r/R r , R ≡ (cid:126) cm π c (1.1)where the overall “-” sign means that the force is attractive, C Y is related to thecoupling strength between the nucleons, r is the distance between the nucleons, R isthe range of interaction, (cid:126) c = 197 . · fm, and m π is the mass of the exchangedparticle. The attractive force between two nucleons is mediated by the exchangeof a single massive boson (meson) that Yukawa estimated to be m π ∼
200 timesthe mass of an electron. The particle was later discovered in a cosmic ray experi-ment in 1947 [8], and became known as the charged pion, which earned Yukawa thePhysics Nobel Prize in 1949. Since the exchanged particle has a finite mass (unlikethe virtual photon in QED), the strong nuclear force operates at small distanceswhere the mass of the exchanged mesons mediating the
N N interaction is inverselyproportional to the interaction range. Using Heisenberg’s uncertainty principle andthe known pion mass, the
N N interaction range is estimated to be R ∼ . N N interaction.Before additional discussion of nuclear interactions, it is worth mentioning theimplications that an important experimental discovery had on the nature of the nu-clear force. In 1939, Rabbi et al. [9, 10] measured the deuteron’s electric quadrupolemoment ( Q zz = +2 . e fm ). The implications of a static quadrupole moment werethat the nuclear potential did not only have a central (spherical symmetric) part,but also a complicated non-central component that needed to be accounted for. Tounderstand the non-central component, consider the multipole expansion of a charge2istribution in the presence of an external electric field, (cid:126)E = − (cid:126) ∇ V , which can beexpressed as: E int = V (0) q + ∂V∂z (cid:12)(cid:12)(cid:12) p z − ∂ V∂z (cid:12)(cid:12)(cid:12) Q zz + ... (1.2)where E int is the interaction energy, V is the electric potential and q, p z and Q zz arethe monopole ( L = 0), dipole ( L = 1) and quadrupole ( L = 2) terms, respectively.Figure 1.1: The L = 0 (spherically symmetric) and L = 2 (prolate spheroid) arethe charge distributions of the deuteron, where the spins of the proton (blue) andneutron (red) are aligned. The existence of a positive electric quadrupole momentindicates that the deuteron charge distribution is actually elongated about the axisof rotation ( z -axis).The monopole term conserves angular momentum ( L = 0), which is a propertyof central forces and corresponds to a spherically symmetric charge distribution withradius (cid:104) r (cid:105) = (cid:104) x + y + z (cid:105) ≡ (cid:104) z (cid:105) assuming that the expectation values of thesquare of the distance from the center to the surface, (cid:104) x (cid:105) = (cid:104) y (cid:105) = (cid:104) z (cid:105) , are equal.The existence of a quadrupole term ( Q zz = e (3 z − r )) in the deuteron, however,indicates that the nuclear force has a tensor component that arises from the spin-orbit interaction between the angular momentum ( L = 2) and the intrinsic nuclear3pins. There also exist an interaction between the intrinsic particle spins known asthe spin-spin interaction that contributes to the tensor component of the nuclearforce, however, this interaction arises from a purely quantum-mechanical effect andhas no classical analog. The quadrupole term measures the lowest order departurefrom a spherical charge distribution in a nucleus (see Fig. 1.1). To understand theadditional tensor component, consider the following: J = L + S, (1.3)where S = s p + s n , ( s p , s n ) are the proton and neutron intrinsic spins, L is therelative angular momentum between the two nucleons and J is the total angularmomentum. The range of possible angular momentum states is given by | L − S | ≤ J ≤ L + S. (1.4)From the experimental fact that J = 1 (cid:126) [11] for the deuteron, the possible combi-nations are: L = 0 , S = 1 ( s p , s n ) parallel (1.5a) L = 1 , S = 0 ( s p , s n ) anti-parallel (1.5b) L = 1 , S = 1 ( s p , s n ) parallel (1.5c) L = 2 , S = 1 ( s p , s n ) parallel (1.5d)From the observation that the deuteron parity is even or “+,” only even valuesof relative angular momentum are allowed, which implies that the deuteron wavefunction is not in a pure L = 0 state, but rather a superposition of L = 0 and L = 2states. Using the spectroscopic notation ( S +1 L J ), the L = 0 and L = 2 are referred Parity refers to the eigenvalue of the angular wave function under the trasnformation:ˆ
P Y ( θ, φ ) = Y ( π − θ, φ + π ) = P Y ( θ, φ ), where P = ( − L
4o as “sharp” (or S -wave) and “diffuse” (or D -wave) components, respectively. Thedeuteron wave function can then be expressed as a linear combination of two possiblestates | Ψ d (cid:105) = p S (cid:12)(cid:12) S (cid:11) + p D (cid:12)(cid:12) D (cid:11) , (1.6)where p S + p D = 1 and the normalization coefficiencts, ( p S , p D ), represent the proba-bility of finding the deuteron in either an S -state ( p S ) or D -state ( p D ). The relativecontribution from the S - or D -state are sensitive to the radial part of the deuteronwave function, which is determined phenomenologically. A summary of the D -stateprobability for different N N potentials can be found in Ref. [12], with typical ranges p D ∼ − After the discovery of the pion in cosmic rays in 1947 and its artificial productionin the lab at the Berkeley Cyclotron in 1948 [13], a great deal of effort was devotedto the development of meson theory as the fundamental theory of nuclear forces.In 1951, Taketani, Nakamura and Sasaki [14] proposed that the nuclear potentialshould be divided into different regions that should be treated separately (see Fig.1.2). It was suggested that the long-range part of the potential should be treatedusing meson theory, while the intermediate and short-range parts should be ap-proached phenomenologically as additional complications because of heavy mesons,higher order perturbations, coupling strengths and relativistic effects become diffi-cult to solve. Nevertheless, in the early 1950s there were various attempts to developa fundamental theory of strong interactions (meson theory) [16–19] that ultimatelyfailed when multi-pion exchanges where included in the theory. Only the long-range5igure 1.2: Qualitative
N N -potential versus inter-nucleon separation distance.Note: Reprinted from Ref. [15].part of the
N N potential—or the One-Pion Exchange Potential (OPEP)—was foundto describe the
N N scattering data at the time. A more general overview of thedevelopment of pion theory can be found in Refs. [20, 21].In the early 1960s, the possibility of the existence of heavier mesons started toemerge theoretically [22] and experimentally [23]. These ideas led to the develop-ment of the One-Boson Exchange Potential (OBEP) [24], where the idea of a singlepion exchange between two nucleons was generalized to a single boson exchange, inwhich heavier mesons were also included in the model and would account for shorterdistances in the
N N potential. Soon afterwards, several heavier mesons were dis-covered experimentally, most notably the ρ (770) [25] and ω (783) [26] mesons. Withthe discovery of heavier mesons, increased efforts were devoted to the developmentof the OBEP [27, 28] and soon afterwards, the first N N potential models emergedthat seemed to describe the
N N scattering data better than any previous models6o date. Some of the best known potentials of the 1960s were by Hamada-Johnston(HJ) [29] and Reid68 [30].In the 1970s and 1980s efforts continued towards the development of improvednuclear phenomenological models using the OBEP. Of particular importance was thedevelopment of the relativistic OBEP [31–34] in the 1970s where the full relativisticscattering amplitudes were used in the calculations. The inclusion of relativisticamplitudes produced a significant improvement in the agreement between
N N scat-tering data and phenomenological models using the relativistic OBEP as comparedto previous non-relativistic models (see Fig. 2 of Ref. [35]). During the 1970s, therewas also an effort devoted to derive the 2 π − exchange contributions to the nuclearpotential, which accounted for the intermediate range of the nuclear force. The rea-son was that during its initial years, the OBEP models had to introduce the σ mesonin order to describe the intermediate-range nuclear force, however, no experimentalevidence for the σ meson had been found.Well known examples of potentials that included 2 π exchange contributions werethe Stony-Brook [36] and Paris [37, 38] group potentials. In the 1980s, more sophis-ticated potentials based on the OBE approach were constructed, particularly by theArgonne and Bonn groups with N N potentials that included 2 π exchange contri-butions such as the Argonne V14 and V28 (AV14 and AV28) [39] potentials andthe full Bonn [40] potential, which included both 2 π exchange contributions as wellas relativistic effects on the OBEP. It is important to note that some of the poten-tials mentioned above were improved further in later years. As an example, in thisexperiment we used the parametrized Paris [41], AV18 [42] and charge-dependentBonn (CD-Bonn) [43] potentials to compare with experimental data.7 .3 Historical H ( e, e (cid:48) p ) n Experiments
In order to probe the internal structure and dynamics of nuclei, electron-nucleonscattering serves as the most valuable tool since the interaction is described by thewell established theory of QED, which is capable of making accurate predictions.Electron scattering experiments can be separated into inclusive or exclusive types.In the former, only the electron is detected in the final state (single-arm experi-ments), and one studies the nucleus in question by integrating over all possible finalstates [44]. In the latter, one or more particles are detected in coincidence with thescattered electron, which allows one to investigate properties unique to the specificreaction in question. In deuteron electro-disintegration ( H( e, e (cid:48) p ) n ), for example,one detects the scattered electron in coincidence with the proton and the missingneutron is reconstructed from momentum conservation laws. The H( e, e (cid:48) p ) n re-action proves to be the most direct way of probing the internal structure of thedeuteron since it is possible to deduce the internal momentum of the nucleons fromthe neutron recoil (“missing”) momenta.Historical H( e, e (cid:48) p ) n experiments were started in 1962 at the Stanford Mark IIILinear Accelerator (linac) at a very low Q = 0 .
085 (GeV/c) [45] and shortly afterin 1965 at the Orsay linac at Q = 0 .
264 (GeV/c) [46]. At the time, the smallestcross sections measured were limited by the duty factor of the particle acceleratorsat the time ( f duty ∼ − ) [47]. In the 1970s and 1980s, the duty factor of accelera- The duty factor is defined by the ratio f duty = δT pulse /δT rep , where δT pulse is the pulselength and δT rep is the pulse repetition period of the electron beam. The small dutyfactor of the accelerators leads to high instantaneous particle rates and therefore highaccidental coincidence rates, or equivalently, low signal-to-noise ratio. As a result, theamount of beam time required to measure smaller cross sections is not feasible due tothe high accidentals rate [44]. H( e, e (cid:48) p ) n cross sections,corresponding to larger missing momenta.Figure 1.3: H( e, e (cid:48) p ) n cross section versus neutron recoil momentum from theMAMI (1998) experiment [48]. Theoretical calculations were performed by H.Arenh¨ovel [49]. Note: Reprinted from Ref. [47].For example, the Kharkov Institute [50] 2 GeV linac extended the missing mo-mentum range to ∼
300 MeV/c and experiments done at SACLAY [51,52] measured H( e, e (cid:48) p ) n cross sections up to missing momenta ∼
500 MeV/c. In the 1990s, theduty factor of electron accelerators increased further ( f duty ∼
1) such as the Amster-dan Pulse Stretcher (AmPS) at the National Institute for Nuclear and High EnergyPhysics (NIKHEF) in the Netherlands, the Mainz Microtron (MAMI) in Germany9nd Thomas Jefferson National Accelerator Facility (TJNAF) in the United States.The dramatic improvement in the electron accelerator duty factor allowed for thefirst time measurements of very small cross sections in relatively short periods oftime. For example, H( e, e (cid:48) p ) n cross sections were measured at missing momentaup to ∼
700 MeV/c ( Q = 0 .
28 (GeV/c) ) at NIKHEF [53] and ∼
928 MeV/c( Q = 0 .
33 (GeV/c) ) at MAMI [48]. While at TJNAF [54], the unique combi-nation of high energy, duty factor and beam current allowed the measurements tobe carried out for the first time at relatively high missing momnetum up to ∼ Q = 0 .
665 (GeV/c) .The comparison of the results (see Fig. 1.3) from the MAMI (1998) experimentwith H. Arenh¨ovel’s calculations [49] demonstrated that only for very specific kine-matics (e.g., SACLAY experiment in Ref. [51]), at missing momenta below ∼ as the dominant contribution to the cross section. However,above ∼
300 MeV/c, the PWBA (dashed blue), FSI (dashed-dotted green), MEC(dotted purple) and IC (solid red) all contribute significantly to the H( e, e (cid:48) p ) n cross section and obscure any possibility of extracting the momentum distributions(PWIA in dashed green). H ( e, e (cid:48) p ) n Experiments at Large Q The first H( e, e (cid:48) p ) n experiments at Q > were carried out at TJNAF inexperimental Halls A [55] and B [56]. Both experiments determined that the cross In the plane wave impulse approximation (PWIA), it is assumed that only the protongets knocked out by the virtual photon whereas in the PWBA, the process in which theneutron is knock-out is also considered. θ nq ∼ ◦ in agreement with the generalized eikonalapproximation (GEA) calculations [57, 58] at high missing momentum and Q > (see Fig. 1.4). ✓ nq (deg)
25 (GeV/c) and recoil momen-tum settings (a) p r = 0 . p r = 0 . p r = 0 . Q settings. The green data (with FSI re-scattering peak)correspond to 400 ≤ p r ≤
600 MeV/c, and the blue data (no FSI re-scattering)correspond to 200 ≤ p r ≤
300 MeV/c. The solid curves are calculations from J.M.Laget [60] and the dashed curves are from M. Sargsian [59]. Note: Reprinted fromRef. [47]. 11n Hall B, the CEBAF Large Acceptance Spectrometer (CLAS) measured an-gular distributions for a range of Q values as well as momentum distributions.However, statistical limitations made it necessary to integrate over a wide angularrange to determine momentum distributions that are therefore dominated by FSI,MEC and IC for recoil momenta above ∼
300 MeV/c.In Hall A, the pair of high resolution spectrometers (HRS) made it possible tomeasure the missing momentum dependence of the cross section for fixed neutron re-coil angles ( θ nq ) reaching missing momenta up to p r = 550 MeV/c at Q = 3 . ± . . For the first time, very different momentum distributions were found for θ nq = 35 ± ◦ and 45 ± ◦ compared to θ nq = 75 ± ◦ . Theoretical models attributedthis difference to the suppression of FSI at the smaller angles ( θ nq = 35 , ◦ ) com-pared to FSI dominance at θ nq = 75 ◦ [55]. Being the most simple neutron-proton ( np ) bound state, the deuteron serves as astarting point to study the strong nuclear force (or N N potential) without additionalcomplications that arise from
A >
N N potentialis sub-divided into three regions with inter-nucleon distance r as follows: • the long range part (LR), where r > • the intermediate or mid-range part (MR), where 1 < r < • the short range part (SR), where r < π exchange, where usually the OPEP is usedby most phenomenological models. The MR part is dominated by 2 π exchange orthe exchange of a heavier mesons. Finally, the SR part is often modeled by a repul-sive hard core and is determined completely phenomenologically. It is this part that12s least known from a theoretical point of view and the most difficult to access exper-imentally. At such small inter-nucleon distances, from a Quantum Chromodynamics(QCD) perspective, a repulsive force is expected. For example, if one considers thethree quarks inside the nucleon, as the proton and neutron start to overlap, thequarks in each nucleon cannot be considered independent of the other. Given thatquarks are fermions, the Pauli exclusion principle prevents any two fermions fromoccupying the same quantum state. As a consequence, any three quarks must go toenergy states above the lowest states occupied by the other three [62]. This processrequires a large amount of energy that shows up as a resistance (repulsive hard core)to bring the two nucleons to sub-Fermi distances.From a nuclear physics perspective, the overlap between the nucleons in thedeuteron is directly related to short-range correlations (SRCs) observed in A > H( e, e (cid:48) p ) n cross section to Q = 4 . ± . at Bjorken scale x Bj > p r ∼ Q and highmissing momenta required a high beam energy and small electron scattering anglesleading to the detection of electrons at ∼ ◦ and 45 ◦ , MEC, IC andFSI are mostly suppressed. This leaves the PWIA as the dominant contribution tothe H( e, e (cid:48) p ) n cross section giving access to the high momentum components of thedeuteron wave function. 14HAPTER 2 THEORETICAL BACKGROUND
In this chapter I will derive the basic formulas for the H( e, e (cid:48) p ) reaction kinematicsusing the Feynmann diagram of Fig. 2.1 (assume natural units for speed of light, c = 1). Then I will briefly discuss the general reaction cross section and the variousreaction mechanisms that can occur. Finally, the theoretical models that are usedto compare with the experimental data will be discussed. H ( e, e (cid:48) p ) n Reaction Kinematics
Figure 2.1: Simplified Feynman diagram of the H( e, e (cid:48) p ) n reaction kinematics andthe respective four momenta of the interacting particles.15he deuteron electro-disintegration reaction kinematics can be described in QED bythe exchange of a single virtual photon between the electron and deuteron assuminga One-Photon Exchange Approximation (OPEA). The Feynman diagram in Fig.2.1 describes a typical deuteron electro-disintegration reaction, where the electroninteracts with the deuteron via the exchange of a virtual photon that breaks thedeuteron up into a proton and a neutron. The scattered electron is detected by theSuper High Momentum Spectrometer (SHMS) in coincidence with the knocked outproton in the High Momentum Spectrometer (HMS). The “missing” (recoil) neutronis reconstructed from momentum conservation laws.Figure 2.2: General H( e, e (cid:48) p ) n reaction kinematics.Figure 2.2 shows a more detailed diagram of the H( e, e (cid:48) p ) n reaction kinematics,where the scattering plane is defined by ˆ y sct = ˆ k × ˆ k (cid:48) where ˆ k and ˆ k (cid:48) are unit vectorsin the direction of the incident and scattered electron, respectively, and ˆ y sct is a unitvector normal to the scattering plane which defines its orientation. Similarly, theorientation of the reaction plane is defined by ˆ y react = ˆ q × ˆ p f , where ˆ q and ˆ p f are unitvectors in the direction of the virtual photon and final state proton, respectively,and ˆ y react is a unit vector normal to the reaction plane. The angle between the twoplanes is defined by ˆ y sct · ˆ y react = cos( φ pq ). In Hall C, the angle φ pq is referred to as16he out-of-plane angle between the two spectrometers, but since the spectrometersare always in the same plane, the two possible values are φ pq = 0 ◦ or 180 ◦ , whichonly apply to the central ray of the spectrometers.The relevant kinematic variables in the H( e, e (cid:48) p ) n reaction can be obtained byapplying energy and momentum conservation at the electron and hadron verticesin Fig. 2.2. At the electron vertex, the initial and final electron four momenta are P µe = ( E, (cid:126)k ) and P (cid:48) µe = ( E (cid:48) , (cid:126)k (cid:48) ), where the final electron scatters at angle θ e relativeto the incident electron direction. The energy and momentum transfer carried bythe virtual photon are defined as q µγ ≡ P µe − P (cid:48) µe = ( E − E (cid:48) , (cid:126)k − (cid:126)k (cid:48) ) = ( ω, (cid:126)q ) . (2.1)By taking the negative square of Eq. 2.1 and assuming E, E (cid:48) ∼ k, k (cid:48) (electron mass m e (cid:28) k , k (cid:48) ), it is convenient to define the four-momentum transfer of the virtualphoton (also known as the virtuality) as Q ≡ − q µ q µ = q − ω ≈ kk (cid:48) sin ( θ e / . (2.2)It is also convenient to define the Bjorken scale, x Bj ≡ Q M p ω , where M p is theproton mass. At the hadron vertex, the deuteron nucleus with mass M D is station-ary with total internal momentum of the proton and neutron, (cid:126)p p, i + (cid:126)p n, i = (cid:126)
0, andfour-momentum, P µ D = ( M D , (cid:126) P µp = ( E f , (cid:126)p f ) and P µn = ( E r , (cid:126)p r ), respectively. Applying energy-momentum conservation at the hadron vertex, q µγ + P µ D = P µp + P µn = ⇒ ( ω, (cid:126)q ) + ( M D , (cid:126)
0) = ( E f , (cid:126)p f ) + ( E r , (cid:126)p r ) . (2.3)From energy conservation of Eq. 2.3, ω + M D = E f + E r = T p + M p + T n + M n = ⇒ E m ≡ E BE = M p + M n − M D = ω − T p − T n , (2.4)17here the missing energy ( E m ) is defined as the binding energy ( E BE ) of the deuteronand ( T p , T n ) are the final kinetic energies of the proton and neutron, respectively.From momentum conservation of Eq. 2.3, (cid:126)p f = (cid:126)q − (cid:126)p r = ⇒ p = q + p − qp r cos( θ nq ) . (2.5)Or equivalenlty, the neutron recoil momentum from Eq. 2.3 can be expressed as, (cid:126)p r = (cid:126)q − (cid:126)p f = ⇒ p = q + p − qp f cos( θ pq ) . (2.6)Substituting Eq. 2.6 into Eq. 2.5 and solving for cos( θ nq ),cos( θ nq ) = q − p f cos( θ pq ) (cid:112) q + p − qp f cos( θ pq ) , (2.7)where ( θ nq , θ pq ) refers to the angle between the virtual photon and the recoiling neu-tron ( θ nq ) or scattered proton ( θ pq ) direction. From Eq. 2.7, under the assumption (cid:126)q > (cid:126)p p, i and that the proton is struck and the neutron is a spectator without furtherinteraction, the limiting cases are shown in Fig. 2.3.From Fig. 2.3, the proton (red) and neutron (blue) are initially inside thedeuteron moving in opposite direction with total internal momentum ( (cid:126)p p, i + (cid:126)p n, i = (cid:126) • Anti-Parallel Kinematics:
The virtual photon knocks out a proton initiallymoving along (cid:126)q , transferring all its momentum to the proton in the final state(solid red vector) such that (cid:126)q < (cid:126)p f . The neutron recoils in opposite directionto (cid:126)q with missing momentum same as its internal momentum in the deuteron. • Parallel Kinematics:
The virtual photon knocks out a proton initially movingopposite to (cid:126)q , transferring all its momentum to the proton causing it to changedirection in the final state such that (cid:126)q > (cid:126)p f . The neutron recoils along the18irection of (cid:126)q with missing momentum same as its internal momentum in thedeuteron. • Perpendicular Kinematics:
The virtual photon knocks out a proton initiallymoving perpendicular to (cid:126)q , transferring all its momentum to the proton caus-ing it to change direction in the final state such that | (cid:126)q | ∼ | (cid:126)p f | and θ pq is at verysmall angles. The neutron recoils perpendicular to (cid:126)q with missing momentumsame as its internal momentum in the deuteron.Figure 2.3: (a) Anti-parallel kinematics. The neutron (blue) recoils in the oppositedirection to (cid:126)p f (red) and (cid:126)q (black). (b) Parallel kinematics. The neutron recoils inthe same direction as (cid:126)p f and (cid:126)q . (c) Perpendicular kinematics. The neutron recoilsin a perpendicular direction relative to (cid:126)q .These are limiting cases, but in general, the vectors do not have to be perfectlyaligned with (cid:126)q when referring to these kinematics. It is sufficient if the final state19ectors are approximately along (cid:126)q . The Parallel and
Anti-Parallel Kinematics aremore directly related to the short-range structure of the deuteron as FSI are ex-pected to be reduced at these kinematics whereas in the
Perpendicular Kinematics ,FSI become dominant at higher missing momentum, which can lead to a larger in-ferred initial momentum than the true internal momentum of the proton [67]. Thisexperiment (E12-10-003) has chosen kinematics ( θ nq at forward angles) that favorthe Parallel Kinematics for short-range structure studies of the deuteron. H ( e, e (cid:48) p ) n Cross Section
Assuming the OPEA, for the general A ( e, e (cid:48) p ) reaction where an electron is detectedin coincidence with a knocked-out proton and the residual ( A −
1) system recoils,the unpolarized 6-fold differential cross section can be expressed as (See Chapter 6of Ref. [72]): d σdE (cid:48) d Ω e d Ω p dT p = σ Mott ( v L W L + v T W T + v LT W LT cos φ pq + v TT W TT cos 2 φ pq ) , (2.8)where the longitudinal ( W L ), transverse ( W T ) and interference ( W LT , W TT ) nuclearresponse functions are determined from matrix elements of the hadronic four-currentoperator and the leptonic kinematic factors ( v L , v T , v LT , v TT ) are determined frommatrix elements of the leptonic four-current operator. The Mott cross section, σ Mott ,describes the scattering of an electron off an infinitely massive and spinless pointcharge and is defined as σ Mott = (cid:16) αk (cid:48) cos( θ e / Q (cid:17) , (2.9)where α ∼ /
137 is referred to as the fine structure constant and characterizes thecoupling strength of the electromagnetic interaction.20he leptonic kinematic factors and nuclear response functions are summarized inTables 2 and 4 of Ref. [72]. For a more detailed discussion of the formalism used toderive the leptonic and hadronic matrix elements from their respective four-currentoperators refer to Chapter 2 and 6 of Ref. [72].The cross section in Eq. 2.8 can include the various nuclear processes such asMEC, IC and FSI, which can significantly alter the nuclei momenta in the finalstate. For the deuteron in particular, the Feynman diagrams in Fig. 2.4 describepossible reaction mechanisms, which are further discussed in the following sections.Figure 2.4: H( e, e (cid:48) p ) n Feynman Diagrams. The colors are: (yellow) deuteron target,(red) initial state proton, (blue solid) initial and final state neutron, (magenta, green)final state proton, (orange oval) FSI re-scattering, (orange rectangle) intermediateresonance state, (navy blue) virtual photon and (black dashed line) meson exchange.21 .3 Plane Wave Impulse Approximation
In the PWIA (Fig. 2.4(a)), the virtual photon couples directly to the bound proton,which is subsequently ejected from the deuteron without any further interactionwith the recoiling neutron. The recoiling neutron carries a momentum equal inmagnitude but opposite in direction to the initial momentum of the bound proton, (cid:126)p r = − (cid:126)p p, i , thus providing information on the momentum of the bound proton andits momentum distribution. Within the PWIA, the general cross section in Eq. 2.8can be factorized as follows: d σdE (cid:48) d Ω e d Ω p dT p = Kσ eN S ( (cid:126)p p, i , E m ) , (2.10)where σ eN describes the elementary cross section for an electron scattering off abound (off-shell) nucleon where the deForest [73] off-shell cross sections, σ cc1 or σ cc2 , are commonly used. The kinematic factor that results from the factorizationis defined as K ≡ E f p f , and S ( (cid:126)p p, i , E m ) is referred to as a spectral function, whichdescribes the probability of finding a bound proton with momentum (cid:126)p p, i and sep-aration energy E m . The separation (binding) energy of the bound state can beintegrated out of Eq. 2.10 to obtain, σ theory ≡ d σ theory dE (cid:48) d Ω e d Ω p = Kf rec σ eN S ( (cid:126)p p, i ) , (2.11)where f rec is the recoil factor that arises from the integration in E m and is definedas [74] f rec ≡ − E f E r q − ( p + p ) p . (2.12)For the deuteron, the spectral function is interpreted as the momentum distribu-tion of the proton inside a nucleus. Experimentally, the reduced cross section isdetermined from the experimental cross section by σ red ≡ σ exp Kf rec σ eN , (2.13)22here σ exp ≡ d σ exp dE (cid:48) d Ω e d Ω p . If the PWIA were completely valid, σ red would be thedeuteron momentum distribution.The inclusion of the process (not shown in Fig. 2.4) in which the virtual photoncouples to the neutron and the proton is a spectator is often defined as the PlaneWave Born Approximation (PWBA) and can be suppressed by choosing the ap-propiate kinematics such that the 3-momentum transfer ( (cid:126)q ) is significantly greaterthan the largest missing momentum ( p m ) studied and approximately on the orderof the momentum of the ejected proton. Both of these conditions are satisfied inthis experiment.In reality, long-range processes such as FSI, MEC and IC always contribute tosome extent to the total H( e, e (cid:48) p ) n cross section, hence the word “Approximation” in PWIA. As will be discussed next, these long-range contributions can significantlyalter the recoiling neutron momentum, thereby obscuring the initial momentum dis-tribution of the bound nucleon reducing the possibility of directly probing the highmomentum component of the deuteron wave function. In direct FSI (Fig. 2.4(b)), the ejected proton and recoiling neutron continue tointeract further causing re-scattering of both nucleons. This situation is unfavor-able for the extraction a momentum distribution as during the interaction of theknocked-out proton with the recoiling neutron, momentum is being exchanged lead-ing to (cid:126)p r (cid:54) = − (cid:126)p p, i . As any possible momentum can be exchanged between the finalstate particles, they are not considered plane waves but rather distorted waves andthe factorization of the cross section breaks down. If the remaining conditions forthe PWIA are still valid, the spectral function in Eq. 2.11 can be replaced by a23istorted spectral function, S D ( (cid:126)p r , (cid:126)p f ), and the approximation is regarded as a Dis-torted Wave Impulse Approximation (DWIA). See Section 6.4 of Ref. [72] for details.At large missing momenta ( p r >
300 MeV/c) and high Q , FSI exhibit a strongangular dependence on θ nq with maximal FSI re-scattering at θ nq ∼ ◦ and aminimal re-scattering at θ nq = 40 ◦ and 120 ◦ as predicted by the GEA [57, 58] andconfirmed by the previous Halls A and B experiments [55, 56]. From these observa-tions, it became clear that FSI dominates the deuteron cross section at the Perpen-dicular Kinematics ( θ nq ∼ ◦ ) whereas in the Parallel/Anti-Parallel Kinematics ( θ nq ∼ ◦ , ◦ ) it is significantly reduced (see Fig. 2.3). In the MEC diagram (Fig. 2.4(c)), the virtual photon couples to the virtual mesonbeing exchanged between the two nucleons, whereas in the IC diagram (Fig. 2.4(d)),the virtual photon excites a bound nucleon into an intermediate isobar resonancestate (∆) that subsequently decays (∆ N → N N ) via FSI to the ground state caus-ing further re-scattering between the final state nucleons via the exchange of a pion.Early H( e, e (cid:48) p ) n experiments [48,52–54] showed that at low Q and high missingmomenta, MEC and IC contribute significantly to the deuteron cross section. Atlarge Q , however, from a theoretical perspective, these contributions are expectedto be significantly reduced.The suppression of MEC can be understood from the fact that the estimatedMEC scattering amplitude ( A MEC ) is proportional to the meson propagator in theelectromagnetic current operator( J µm ( Q )) and the N N -meson form factor (Γ
MNN ( Q ))that have the following Q dependence [57], A MEC ∝ J µm ( Q )Γ MNN ( Q ) ∝ Q /m ) 1(1 + Q / Λ) , (2.14)24here m meson ∼ .
71 GeV and Λ ∼ . − . Therefore, at Q > ,MEC are expected to be suppressed by an overall factor of ∼ /Q as compared tothe PWIA.Figure 2.5: Qualitative inclusive deuteron-electron scattering cross section. Note:Reprinted from Ref. [75].The suppression of IC arises in part due to the kinematics chosen. At large Q ,one is able to select x Bj >
1, which corresponds to probing the lower energy ( ω ) partof the deuteron quasi-elastic peak, which is maximally far away from the inelasticresonance electroproduction threshold. From Fig. 2.5, the inclusive H( e, e (cid:48) ) showsqualitatively that at the left end of the quasi-elastic peak ( x Bj >
1) one is max-imally away from the inelastic ∆ and N ∗ resonance electroproduction region andcorresponds to the kinematics where this experiment was done. In the E12-10-003 experiment, the theoretical cross sections used to compare to datawere determined using the following phenomenological
N N -potentials:25
Parametrized Paris (1980) [41] • Argonne V18 (AV18) (1995) [42] • Charge-Dependent Bonn (CD-Bonn) (2001) [43]Each of these potentials are improved versions of the original potentials and weredeveloped by Paris, Argonne and Bonn theoretical groups, respectively. The groupshave employed different techniques used in their approach to describe the interme-diate and short range parts of the
N N potential, whereas for the long-range part,all have used the well-known One Pion Exchange Potential.In general, the construction of
N N potentials is largely based on parametersthat the model must fit to either neutron-neutron ( nn ), proton-proton ( pp ) orneutron-proton ( np ) scattering data and the results are usually presented in textsas χ / datum to determine the success of the model in describing the experimentaldata.The calculations to determine the theoretical cross sections from an N N poten-tial are based on solving the Schrodinger equation,ˆ H D ψ D ( (cid:126)r ) = E D ψ D ( (cid:126)r ) = ⇒ ( ˆ T p + ˆ T n + ˆ V NN ) ψ D ( (cid:126)r ) = E D ψ D ( (cid:126)r ) , (2.15)where ˆ H D is the Hamiltonian operator that acts on the deuteron wave function( ψ D ( (cid:126)r )) and is expressed in terms of the proton and neutron kinetic energy opera-tors ( ˆ T p , ˆ T n ) and the interactive N N potential ( ˆ V NN ), which is determined by thetheory groups. By solving Eq. 2.15, the deuteron wave function as well as the scat-tering amplitude (and theoretical cross section) can be determined. In reality, Eq.2.15 is restricted to the non-relativistic description of the wave function as it usesa classical definition of the kinetic energies in the Hamiltonian. In this situation,a generalized form of the Schrodinger wave equation can be used to describe thesystem relativistically. Alternatively, the Bethe-Salpeter equation [76], which uses26 relativistically covariant formalism (Feynman S-matrix formalism), can also usedto describe a 2-body bound state including relativistic effects.Different theoretical calculations [49, 59–61, 77, 78] have been developed to de-scribe the deuteron wave function within the PWIA as well as to account for addi-tional processes such as FSI, MEC or IC that are not described by theoretical poten-tials. In addition, some of the most recent theoretical calculations also account foroff-shell effects , which become important at higher missing momenta [59,61]. Sometheoretical potentials may also include off-shell effects in their models, however,there is no way of knowing a priori whether they are correct since these potentialswere derived from N N scattering data where the interacting particles are by defini-tion on their energy shell (free interacting particles). See Chapter 2 of Ref. [62] fora detailed discussion.In this experiment, the theoretical calculations used to determine the H( e, e (cid:48) p ) n cross sections from the AV18 and CD-Bonn potentials were performed by M. Sargsian[59], while those for the Paris potential were by J.M. Laget [60]. In the former, aneffective Feynman diagrammatic approach described in Ref. [57] is used to calculatethe scattering amplitudes within the virtual nucleon approximation. This approx-imation has three main assumptions described in Ref. [59], which also defines itsrange of validity. The first two assumptions are satisfied by requiring the neutronrecoil momenta to be p r ≤
700 MeV/c, while the third assumption made is that atlarge Q ( > ), MEC are considered to be a sufficiently small correction (seeSection 2.5) such that they can be ignored. The assumptions of the virtual nucleon The off-shell effects arise from the fact that for a bound system, the energy-momentumconservation applies to the nucleus as a whole, but the momentum of a pair of nucleonswithin the nucleus is no longer restricted and the individual particles are considered tobe “off the energy shell” (off-shell). Whereas for a pair of free interacting nucleons, theenergy-momentum conservation applies and the particles are said to be “on the energy-shell” (on-shell). (not shown) and IC (Fig. 2.4(d)) wherethe IC can be suppressed kinematically by choosing x Bj > θ nq ∼ ◦ . This prediction was confirmed by thefirst high Q deuteron electro-disintegration experiments carried out at Halls A [55]and B [56] of Jefferson Lab (see Fig. 1.4). Additionally, it was also found that atvery forward and backward neutron recoil angles, FSI were significantly reduced andcomparable to the PWIA. The reduction in FSI can be understood from the fact inthe high energy limit ( Q > ) of the GEA, the pn re-scattering amplitude ismostly imaginary: A = A PWIA + iA FSI , (2.16)with A FSI ≈ i | A FSI | , where the total scattering amplitude A is expressed as thesum of the PWIA ( A PWIA ) and the imaginary part of the FSI ( A FSI ) scatteringamplitudes. The total theoretical cross section can then be obtained by taking themodulus square of the total scattering amplitude and can be expressed as σ PWIA+FSI ∼ | A | = | A PWIA | − | A PWIA || A FSI | (cid:124) (cid:123)(cid:122) (cid:125) “Screening” orinterference term + | A FSI | (cid:124) (cid:123)(cid:122) (cid:125) re-scattering term . (2.17) This process corresponds to the scenario in which the virtual photon strikes a neutronthat re-interacts with the spectator proton in the final state via np → pn charge-exchangere-scattering. R = σ PWIA+FSI σ PWIA = 1 − | A PWIA || A FSI || A PWIA | + | A FSI | | A PWIA | . (2.18)From the ratio of cross sections the interference term enters with an opposite sign ascompared to the re-scattering term, which provides an opportunity for an approxi-mate cancellation at certain neutron recoil angles as shown in Fig. 1.4 of Ref. [47].This cancellation is also approximately independent of the neutron recoil momenta,which opens a kinematic window at θ nq ∼ ◦ where one can probe the short-rangestructure of the deuteron beyond p r ∼
500 MeV/c and is the main focus of thisexperiment.In contrast to GEA approach used by M. Sargsian, J.M. Laget uses a diagram-matic approach described in Ref. [60], which he first introduced in Ref. [79] and isused to calculate the theoretical cross sections including the IC, MEC and FSI con-tributions. The kinematics of this experiment suppress IC and MEC contributionsand therefore we only used the PWIA and FSI contributions to the theoretical crosssections to comparare with the data. The PWIA and FSI scattering amplitudes forthe H( e, e (cid:48) p ) n reaction have been reproduced by Laget in Ref. [60] and utilize therelativistic expressions of the proton and neutron on-shell current density operators,( J p ( q ), J n ( q )), in both amplitudes. The current densities use the conventional dipoleexpression for the magnetic form factors of the proton and neutron, while for theneutron and proton electric form factor, the Galster parametrization [80] and theresults from the Hall A experiment described in Ref. [81] were used, respectively.Similar to the predictions from the GEA, the FSI calculations from J.M. Lagetalso show that the FSI peak at θ nq ∼ ◦ for p r ∼
500 MeV/c, whereas for lower recoilmomenta, the peak shifts towards larger recoil angles with a dip at about θ nq ∼ ◦ for the smallest recoil momenta. This can be understood from the fact that as the29ncident electron scatters from a proton at rest, it transfers most of its energy andmomentum to the struck proton while the neutron (also at rest) recoils at θ nq ∼ ◦ ,which is predicted by a non-relativistic eikonal approximation known as the Glauberapproximation [82].The Glauber approximation assumes the bound nucleons are stationary withinthe nucleus and predicts an FSI re-scattering peak at θ nq ∼ ◦ corresponding to thetransverse re-scattering of the stationary neutron relative to the exchanged virtualphoton direction. For configurations in which the internal momenta of the nucle-ons increases, this approximation is valid up to a certain extent for nucleon recoilmomenta up to about the fermi momentum, k F ∼
250 MeV/c. Beyond the fermimomentum, however, the relativistic effects within the nucleus cannot be ignoredand must be accounted for in the theoretical calculations. In the classical Glauberapproximation, these relativistic effects are ignored since the nucleon propagator islinearized and the FSI peak stays at θ nq ∼ ◦ for recoil momenta p r > k F . Whenrelativistic effects are accounted for, as it is done for both the GEA and Laget’sdiagrammatic approach, the FSI re-scattering peak shifts towards θ nq ∼ ◦ withincreasing recoil momenta. The agreement of the FSI peak location between M.Sargsian’s and J.M. Laget’s approach can be understood from the fact that in theGEA, the higher order recoil terms in the nucleon propagator are accounted forwhile in the calculations by Laget, the full kinematics of the reaction are taken intoaccount from the beginning of the calculations as stated in Ref. [60].To illustrate the results from this discussion, Fig. 2.6 shows the ratio of thetheoretical cross sections using the PWIA+FSI calculations to cross sections calcu-lated within the PWIA plotted versus neutron recoil angles. At the lowest missingmomenta ( p r = 100 MeV/c), the GEA and Glauber calculations are within almost aperfect agreement, which validates the GEA approach, which reduces to the Glauber30pproximation at very small recoil momenta well within the fermi momentum of thenucleons. At p r = 200 MeV/c, however, a shift in the FSI peak can already beobserved towards ∼ ◦ whereas at p r = 400 MeV/c, a significant shift of ∼ ◦ from θ nq ∼ ◦ to θ nq ∼ ◦ can be observed. While for the Glauber approximation,the FSI peak stays “fixed” at θ nq ∼ ◦ .Figure 2.6: Ratio of the theoretical cross section ratios versus neureon recoil angles θ nq (denoted as θ p s q in the figure) calculated within both the GEA (solid) andGlauber approximation (dashed) for varios neutron recoil momenta p r (denoted as p s in the figure). Note: Reprinted from Ref. [57].31HAPTER 3 EXPERIMENTAL SETUP
In this chapter I will discuss the experimental equipment used to carry out the 12GeV Hall C commissioning experiments at the Continuous Electron Beam Accel-erator Facility (CEBAF). First I will give a brief overview of the accelerator andthen discuss the Hall C 12 GeV upgrade and components required for experiments:beamline, target, spectrometer systems (magnets and detectors) and the triggerelectronics setup used to collect data.
With the discovery of quarks inside the proton in a series of ep scattering exper-iments at Stanford Linear Accelerator (SLAC) [83, 84] in the late 1960s and thedevelopment of a new theory of strong interactions (QCD) in the early 1970s, manyquestions regarding the role of quarks in nuclear forces arose. For example, “Whyweren’t the effects of the underlying quark structure immediately visible?” or “Couldnew phenomena be discovered that were a direct consequence of QCD and our newunderstanding of nuclear theory?” [85]. To answer these questions, electron-hadroncoincidence experiments would have to be carried out at high energies in relativelyshort periods of time—a task that could not be done by the accelerators at the timedue to the low duty factors and high accidental rates (see Section 1.3). As a possiblesolution to this issue it was recommended by both the Friedlander panel (1976) andthe Livingston panel (1977) that a new high energy, continuous wave (CW) beam,electron accelerator should be built for nuclear physics research [85].In 1985, the United States Department of Energy (DOE) approved the conceptof CEBAF based on superconducting radio-frequency (SRF) technology that would32llow for a high energy and high duty factor machine to be built and in February1987, the construction project of CEBAF along with three experimental end stations(Halls A, B and C) officially began [85]. In 1994, the first beam was successfullyFigure 3.1: Aerial view of CEBAF at Newport News, Virginia. The service buildingsmark the 7/8-mile (1.4 km) racetrack-shaped accelerator 30 feet (base of tunnel)below the surface. The dome-shaped terrain represent the accelerator end-stations(experimental halls), which are also underground.delivered to experimental Hall C and the following year CEBAF reached the designenergy of 4 GeV. Finally, in June 1998, beam was successfully delivered simultane-ously to all three experimental halls [86, 87].Although the CEBAF was initially designed to operate at 4 GeV, the researchand development work on SRF technology at Jefferson Lab allowed the acceleratorto be upgraded to beam energies of nearly 6 GeV and total beam currents of upto 200 µ A combined for all experimental halls starting in the year 2000 [86, 88].CEBAF operations at 6 GeV concluded in Spring 2012 by completing its 178thexperiment since 1994. 33 .1.1 Accelerator Upgrade to 12 GeV
The idea of a 12 GeV upgrade at CEBAF had already started in the late 1990s withthe purpose of probing the nuclear structure at even smaller scales (larger Q ) thatwould enable new insights into the structure of the nucleon, the transition betweenhadronic and quark/gluons degrees of freedom and the nature of confinement. In2004, the U.S. DOE approved the development of the 12 GeV conceptual designand approved start of construction in September 2008 [89].Figure 3.2: Schematic of CEBAF 12 GeV Upgrade. Note: Reprinted from Ref. [90].Figure 3.2 shows a schematic of CEBAF with the 12 GeV upgrade components. Theracetrack-shaped accelerator site consists of an injector, 2 ( ∼ / • [90]: The C100 cryomodules are an im-proved design of the original C20 and refurbished C50 cryomodules of the 6GeV era [88]. A single C100 cryomodule (see Fig. 3.3) consists of 8 7-cell 1497MHz Niobium SRF cavities as compared to the previous 5-cell cavities andcan accelerate electrons up to 100 MeV/cryomodule, which yields 0.5 GeVacceleration per linac. The existing cryomodules accelerate electrons to 0.6GeV/linac for a total acceleration of 1.1 GeV/linac or 2.2 GeV per pass.Figure 3.3: A single (C100) 7-cell Niobium cavity. [91]. Note: Reprinted fromRef. [90]. • upgrade recirculating arc magnets [90, 91]: The arcs dipole magnets were up-graded in order to accomodate the higher beam energies. In addition, a 5thpass separator and 10th arc were added in order to extract and steer the beamto the new experimental Hall D, which receives an extra half-pass for a totalof 5.5 passes (12.1 GeV beam), whereas the other halls, at a maximum of 5passes, receive beam energies only up to 11 GeV . Beam energies are actually smaller by a few 100 MeV mostly due to the inability of thecryomodules to maintain sufficiently high gradients at acceptable trip rates and partlydue to energy loss due to synchrotron radiation in the arcs. double cryogenic capacity [90]: The upgraded SRF linacs (adding the newcryomodules) required doubling the cryogenics supply for the cryomodules tooperate at 2 K temperatures. This was done by constructing a 2nd heliumrefrigerator building (CHL-2) to meet the demands. • upgrade injector to 123 MeV [90]: The injector energy was upgraded by addinga new C100 cryomodule towards the final acceleration portion of the injectorto increase the electrons’ acceleration from 67 to 123 MeV before entering thenorth linac. An additional 4th laser was added for the new Hall D operation. • upgrade experimental halls : To meet the demands of the higher beam energiesand the new experimental programs [92], the three existing experimental hallswere upgraded as well [89]. In addition, a new experimental hall (Hall D)was built to carry out the GlueX physics program which requires a 9 GeVpolarized photon beam from a 12 GeV electron beam.
To accelerate the electrons at CEBAF, the SRF resonant cavities (operating at 2K ina He bath) are excited at the fundamental frequency f = 1497 MHz. The resultingoscillating electric field causes the electrons to be accelerated (see Fig. 3.4). E E EE E ~v e
580 and 750 MeV/c. At each of these settings, the electron arm (SHMS)was “fixed” in momentum and angle and the proton arm (HMS) was rotated fromsmaller to larger angles corresponding to the lower and higher missing momentumsettings, respectively. In reality, the spectrometers’ angle and momentum changedback and forth multiple times at each setting which made the reproducibility of theexact setting impossible. As a result, the data collected from the 580 and 750 MeV/csettings were separated into multiple datasets, each corresponding to a change ineither spectrometer. We analyzed separately 2 data sets for the 580 MeV/c setting,and 3 data sets for the 750 MeV/c setting.Hydrogen elastic H( e, e (cid:48) ) p data were acquired at kinematic settings close to thedeuteron 80 MeV/c setting for cross-checks with the spectrometer acceptance mod-eled using the Hall C Monte Carlo simulation program, SIMC [104]. Additional H( e, e (cid:48) ) p data were also taken at three other kinematic settings that covered theSHMS momentum acceptance range for the deuteron and were used for spectrometeroptics optimization, momentum calibration, and the determination of spectrometeroffsets and kinematic uncertainties. In addition to elastic data, SHMS data wereobtained using a 3-foil carbon target and a sieve slit to check and fix a problem en-countered with one of the spectrometer magnets. A complete list of the kinematicsettings is given in Table 3.1.From Table 3.1, only the data taken after the SHMS Optics studies (with the ex-ception of Proton Absorption measurements) were analyzed since during data takingit was found by the experts that the SHMS Q3 and HB magnets had a saturation40 ate Study Runs Target E b P SHMS θ SHMS P HMS θ HMS (April) [GeV] [GeV] [deg] [GeV] [deg]03 Carbon Hole 3242 C 10.60314 8.7 12.2 2.938 37.29 H( e, e (cid:48) ) p H 10.60314 8.7 12.2 2.938 37.29Proton Absorption ∗ H 10.60314 8.7 12.2 2.938 37.2904 Aluminum Dummy 3252-3258 Al 10.60314 8.7 12.2 2.938 37.29Proton Absorption ∗ H 10.60314 8.7 12.2 2.938 37.2980 MeV/c (set0) 3264-3268 H 10.60314 8.7 12.2 2.8438 38.8904-05 580 MeV/c (set0) 3269-3282 H 10.60314 8.7 12.2 2.194 54.94505 SHMS Optics † C 10.60314 8.7 8.938 2.194 54.9453286 C 10.60314 8.7 8.938 2.765 37.3383287 C 10.60314 8.7 12.06 2.938 37.338 H( e, e (cid:48) ) p H 10.60314 8.7 12.194 2.938 37.33880 MeV/c (set1) 3289 H 10.60314 8.7 12.194 2.843 38.89605-06 580 MeV/c (set1) 3290-3305 H 10.60314 8.7 12.194 2.194 54.99206-08 750 MeV/c (set1) 3306-3340 H 10.60314 8.7 12.194 2.091 58.39108 580 MeV/c (set2) 3341-3356 H 10.60314 8.7 12.194 2.194 55.008-09 750 MeV/c (set2) 3357-3368 H 10.60314 8.7 12.194 2.091 58.39409 H( e, e (cid:48) ) p H 10.60314 8.7 13.93 3.48 33.545 H( e, e (cid:48) ) p ∗ H 10.60314 8.7 9.928 3.48 33.545 H( e, e (cid:48) ) p ∗ H 10.60314 8.7 9.928 3.017 42.9 H( e, e (cid:48) ) p H 10.60314 8.7 9.928 2.31 42.9 H( e, e (cid:48) ) p ∗ H 10.60314 8.7 8.495 1.8904 42.9 H( e, e (cid:48) ) p H 10.60314 8.7 8.495 1.8899 47.605 H( e, e (cid:48) ) p H 10.60314 8.7 8.495 1.8899 47.605 H( e, e (cid:48) ) p H 10.60314 8.7 8.495 1.8898 47.60509 750 MeV/c (set3) 3380-3387 H 10.60314 8.7 12.21 2.091 58.391
Table 3.1: The E12-10-003 experiment comprehensive run list. The spectrometercentral momentum and angle were determined based on the dipole NMR set valueand spectrometer camera, respectively. The beam energy in this table is uncorrectedfor synchrotron radiation (see Section 3.4.1). In the column Study, the ( ∗ ) are datataken with SHMS single-arm (electron trigger ONLY) and the ( † ) represents datataken with SHMS single-arm and Centered Sieve Slit positioned. The remainingruns are taken with SHMS-HMS coincidence trigger only.41orrection that was not needed. This correction was removed from the magnetcontrols software and the experiment resumed data-taking starting at run 3288without the HB/Q3 correction. The analysis of Proton Absorption studies was notimpacted as it involved taking yield ratios. When the electron beam exits the south linac, it is steered at the beam switchyardto any one of the three experimental hall beamlines (A, B or C). In Hall C, thebeam is sent through a transport line with an entrance channel of 63.5 mm innerdiameter stainless steel tubing connected with conflat flanges which reduces theinner diameter to 25.4 mm when passing through the steering magnets (dipoles,quadrupoles, hexapoles and beam correctors) [97]. To reach the hall entrance, thebeam is bent by a series of 8 dipole magnets located at the hall arc (see Fig. 3.6).The beam is then transported through the Hall C alcove into the scattering (target)chamber and the beam dump at the other end of the hall. Several beam diagnosticscomponents are placed throughout multiple locations in the accelerator and hallbeamlines. The relevant ones used to monitor the beam in Hall C are the harps (wirescanners), beam position monitors (BPMs) and beam current monitors (BCMs).The beamline is also equipped with two permanent beam raster systems with thepossibility to add a third raster. See HCLOG entries below1. https://logbooks.jlab.org/entry/3555385 https://logbooks.jlab.org/entry/3555428 https://logbooks.jlab.org/entry/3555436 https://logbooks.jlab.org/entry/3555447 Transport line between green shield wall and hall entrance where the Compton and MøllerPolarimeters are located. .4.1 Beam Energy Measurement The accurate determination of the absolute beam energy is important as its uncer-tainty is directly related to the uncertainty in the cross section. Various methodshave been proposed to determine the beam energy at CEBAF [105]. In Hall C,the beam energy is determined by using the beamline arc as a spectrometer (firstproposed in Ref. [106]) and is the method used in this experiment. The method isbased on the equations of motion of a charged particle in a magnetic field. For anelectron the magnetic force is given by | (cid:126)F B | = e | (cid:126)v e × (cid:126)B | = ev e B ⊥ = γm e v e ρ c , (3.1)where e, v e , B ⊥ and ρ c are the elementary charge, electron velocity, magnetic field(perpendicular to the velocity) and the local radius of curvature, respectively, and γ ≡ (1 − v e /c ), is the Lorentz factor to account for relativistic effects. From Eq.3.1, the electron momentum is given by p e = γm e v e and the radius of curvature canbe expressed as ρ c = dL (cid:107) /dθ arc where dL (cid:107) and dθ arc are the infinitesimal arc lenthand arc angle, respectively. Using these definitions, Eq. 3.1 may be expressed as p e = eB ⊥ dL (cid:107) dθ arc . (3.2)In reality, as the electron beam traverses through the Hall C arc, the dipole magneticfields are not homogeneous and need to be integrated over infinitesimal ( dL (cid:107) ) arcelements along the beam trajectory. Eq. 3.2 can then be expressed as p e = C k (cid:82) L B ⊥ dL (cid:107) (cid:82) θ arc dθ arc , (3.3)where C k is a constant determined from dimensional analysis and using the conver-sion factor 1 [C][T][m] ≡ × GeV/c and e = 1 . × − C: C k ≡ . × − C × . × GeV / c1[C][T][m] = 0 . / c][T − ][m − ] , B ⊥ . Two superharps (wire-scanners) at each end of the arc are usedto determine small variations in the beam direction.The integrated field (cid:82) B ⊥ dL (cid:107) is determined by carefully mapping the magneticfields of the arc dipoles at the corresponding dipole current. The bend angle θ arc is determined from a survey by the relative orientation of the beam at the arc en-trance and exit (see Fig. 3.6). The superharps at both ends of the arc are used Compared to harps, superharps have been more accurately fiducialized and surveyed forabsolute position measurements [107].
44o determine the absolute beam position and direction. See Ref. [108] for technicaldetails of the Hall C superharps.During the beam energy measurement, Machine Control Center (MCC) oper-ators turn off all the arc quadrupole and beam corrector magnets, which wouldotherwise provide an achromatic beam and only use the dipoles to steer the beam.As a result, dispersion (momentum dependent position) builds up across the arc,which provides a very sensitive energy measurement as the beam will be spread outbased on small energy differences. The negative side effect of dispersion is that itbecomes very difficult for operators to guide the beam across the arc due to thisspread. If this is done successfully, the operators use a lookup table determinedfrom the field mapping to convert the dipole current to (cid:82) B ⊥ dL (cid:107) across the arc.To measure the beam direction, a pair of superharps located at the arc entranceand exit (see Fig. 3.6) are used and controlled by MCC as they are invasive to thebeam. During the harp scans, the signals produced by two of the superharps wereunexpectedly wide and it was decided not to use this information. This was not acause of concern as the variations in the beam direction allowed by the beamlinediameter were sufficiently small and were expected to have a small effect on the (cid:82) B ⊥ dL (cid:107) measurements [109]. The measured beam energy at the arc entrance (un-corrected for synchrotron radiation) is shown in Table 3.1. A detailed table withbeam energy measurements performed at different times can be found in Ref. [110].As the electron beam traverses the Hall C arc, it changes direction, which causesthe beam to lose energy due to synchrotron radiation. This loss is not accounted for MCC operators control and steer the beam around the accelerator and into the experi-mental end stations. Quadrupole magnets function as an achromatic (or in this case, momentum independent)lens to the beam by providing the necessary restoring forces to focus the beam andminimize dispersion.
45n the field integral measurements and must be determined separately. The usualformula for energy loss due to synchrotron radiation is given by (see Ref. [111]) δE sync [keV] = 88 . E [GeV] ρ c [m] θ arc ◦ , (3.4)where E meas is the measured beam energy at the arc entrance. Since the originalenergy loss formula is per 360 ◦ , the fractional energy loss in the Hall C arc is θ arc / ◦ , where θ arc = 34 . ◦ from the survey and the arc radius of curvature is ρ c = 40 .
09 m. Substituting the beam energy from Table 3.1 and the geometricalvalues from the arc in Eq. 3.4 and converting keV to GeV, one obtains δE sync = 0 . . (3.5)The corrected beam energy and its relative error at the target are then given by E tgt = E meas − δE sync , (3.6) (cid:16) δE tgt E tgt (cid:17) = (cid:16) δE meas E meas (cid:17) + (cid:16) δE sync E meas (cid:17) , (3.7)where δE meas /E meas is the relative error due to the field integral and δE sync /E meas isthe relative error due to the measured beam energy due to synchrotron radiation.From the beam energy measurement on April 30, 2018 [110]: E meas ± δE meas = 10 . ± . . (3.8)Substituting the numerical values of Eqs.3.5 and 3.8 in Eqs. 3.6 and 3.7 one obtainsthe corrected beam energy and its relative uncertainty E tgt = 10 . , (3.9) δE tgt E tgt = 4 . × − . (3.10)46 .4.2 Beamline Components As the beam enters Hall C, it passes through various beamline components (seeFigs. 3.7 and 3.8) as it is transported to the target chamber and into the beamdump. Upstream of the target chamber, are the fast raster (FR), beam positionand current monitors (BPMs, BCMs), and harps which were briefly mentioned inthe previous section.Figure 3.7: Hall C beamline from hall entrance to target chamber. Distances to therelevant beamline components are measured from the origin (the pivot center) andgiven in meters. The first three colored boxes (green, blue and red) have multiplecomponents with the relevant distances to the target origin. The codenames usedin the Fast Raster magnets refer to the horizontally (H) and vertically (V) bendingair-core magnets. The commonly used names of the other beamline components areindicated in parentheses.Downstream of the target chamber, the entire beam pipe is ∼ . < ◦ ).Figure 3.8: Hall C beamline from target chamber to beam dump. Harps
The harps consist of a fork with three wires (see Fig. 3.9) and a stepper motorattached that enables the entire system to move invasively through the unrasteredbeam. As each of the wires comes in contact with the beam, a current is produced inthe wire due to secondary electron emission. This current signal is amplified beforebeing sent to an Analog-to-Digital Converter (ADC). The ADC spectrum formedfrom the digitized signals is fit to obtain the beam profile (size). To determine theabsolute beam position, as each wire passes through the beam a position encodergenerates the number of pulses equivalent to the number of steps the motor hasmoved, which corresponds to an absolute beam position.Figure 3.10 shows the results of a typical harp scan (with beam currents ∼ µ ACW) where the y -axis represents the ADC value plotted versus the distance the forktravelled (mm) shown in the x -axis. The fit results for each wire (gaussian peak)are shown at the right of the plot. The overall results of the absolute beam position(X Pos (mm), Y Pos (mm)) and beam profile (X Sigma (mm), Y Sigma (mm)) are48igure 3.9: Hall C beamline harp diagram. The harp enters (red arrow showsdirection of motion) at a 45 ◦ angle. The two vertical wires measure the beamposition along the x -axis and a vertical wire measures the position alng the y -axis.Figure 3.10: Results from a harp scan of harp IHA3H07A taken at 5-pass on April2018.shown at the bottom. Since the harps are invasive to the beam, the scan is notperformed during normal experiment operations. Therefore, to monitor the beam49ositions in real time during the experiment, the BPMs must be calibrated usingthe absolute beam positions from the harp scans. Beam Raster Systems
The intrinsic electron beam size of CEBAF in the 12 GeV era is typically 200-700 µ m in diameter (approximately a gaussian full width, 2 σ ). From the harp scanresults in Fig. 3.10 for example, a typical gaussian has a full width of 2 σ ∼ µ m, which is a reasonably good approximation for the diameter of the unrasteredbeam, considering that the beam is often asymmetric. The amount of power perunit area (intensity) deposited by such a small beam size on either the target, targetchamber or the beam dump for extended periods of time can cause damage to thesecomponents by overheating. For cryogenic targets such as liquid hydrogen anddeuterium used in this experiment for example, there are two effects [112]: • At lower beam currents, the target density change is due to warming of thecryogenic fluid with a density variation of the order ∼ • At higher beam currents, bubbles also start to form and break-off at the targetcell windows.These effects have a direct impact on the high-precision cross section measurementsrequired by the Hall C physics program as significant target density changes causethe data yield to be significantly lower. To solve this issue, the intrinsic beam issmeared out (rastered) to reduce the temperature changes over a larger area. In the 6 GeV era, CEBAF delivered average beam spot sizes of 50-200 µ m, which werecomparatively smaller than in the 12 GeV era. The increase in the intrinsic beam sizefrom the 6 to 12 GeV era is attributed to an increase in synchrotron radiation emittedby the beam. ∼
14 m upstream ofthe Target Chamber (see Fig. 3.7). A third raster (Slow Raster) can be added forexperiments that require a polarized target but does not form part of the standardbeamline components [97]. For most experiments (including this experiment), thefast raster is used and is discussed in more detail in Appendix B.Figure 3.11: Fast raster (X,Y) raw ADC signals measured by the pickup probeduring run 3289 for the 80 MeV/c setting. The 3D plot (and inset 2D representation)show an approximately uniform XY raster distribution.Figure 3.11 shows a 3D (and inset 2D projection) of the fast-raster raw ADCsignal distribution measured during the E12-10-003 experiment. The raster was setto 2 × to minimize localized density changes in the 10-cm liquid deuterium51ryogenic target. The raster distribution in Fig. 3.11 shows that the beam isuniformly distributed across the entire 2 × raster, especially at the boundaries,which was a major issue with the original Hall C raster in use from 1996 to 2002 [113]. Beam Position Monitors (BPM)
The BPMs are cylindrical cavities that form part of the beamline and are used tomake continuous, non-invasive measurements of the beam position during normalbeam operations. To measure the beam position in Hall C, three BPMs located up-Figure 3.12: Hall C BPM and electronics diagram. In EPICS coordinate system(left-handed), the beam is directed out of the page. The antennae are located alongthe axes of a coordinate system (blue) that is oriented 45 ◦ relative to the EPICScoordinate system. Note: Reprinted from Ref. [97].stream of the target are used (see Fig. 3.7). Each BPM consists of an enclosure thatforms part of the beam-pipe with four wire-antennae attached to feedthroughs onthe interior wall of the pipe. The antennae (blue) are in a coordinate system orientedat γ =45 ◦ relative to the EPICS [114] coordinate system (see Fig. 3.12). When thebeam (499 MHz sub-harmonic of f ) passes through the BPM cavity it induces a sig-nal in the antennae with an amplitude inversely proportional to the distance betweenthe beam and each of the antennae. This RF (radio-frequency) signal is then con-52erted to a more convenient lower frequency known as IF (intermediate-frequency).This signal is subsequently detected by the S/H (sample-and-hold) section, which asits name indicates, samples the input signal and holds it until it can be further pro-cessed by the ADCs to be analyzed by software. This method, however, requires theuser to retrieve the constants and perform a separate calculation to convert the rawADC (processed antennae signals) to raw beam position values. Alternatively, theantennae signals are interpreted and calibrated by the EPICS readout chain usingthe standard difference-over-sum method to determine the raw beam positions. Thebeam position is averaged over 0.3 seconds and is logged into the EPICS databasewith 1 Hz updating frequency and injected in the data-stream every few seconds,unsynchronized but with a reference timestamp [97]. Using the Hall C analysis soft-ware, the raw BPM positions are retrieved from EPICS and calibrated relative tothe absolute beam position determined from the harp scans. Beam Current Monitors (BCM)
The experimental cross section measurements at Hall C require the data yield tobe normalized by the total charge at the target. To achieve this, multiple BCMsare used for the continuous, non-invasive measurement of the beam current insidethe hall. The primary system consists of two BCMs and an adjacent Unser Monitor(BCM1, Unser, BCM2) located ∼ ∼ (cid:46) . As the electron beam passes through the toroid axis of symmetry,its circular magnetic field magnetizes the strips of permeable material in the toroid.A modulator-demodulator circuit senses this magnetization and sends a current toa compensating coil to cancel out the field established by the beam. This compen-sating current is proportional to the beam current [117].Both the resonant cavities (BCMs) and the Unser are sensitive to temperaturevariations in their surroundings. The variations in temperature cause thermal ex-pansion or contraction of the BCM cavities and the Unser toroid material resultingin the detuning of the cavities and undesirable zero drifts in the output signal ofthe Unser monitor. To minimize the effects due to temperature variations, the HallC BCMs and Unser are thermally insulated in a box and kept at a temperature of110 ◦ C with a tolerance of ± ◦ C, which is monitored periodically. The circular strips used in the Unser toroid consist of (CoFe) (MoSiB) —an amorphousmagnetic alloy that exhibits extreme magnetic permeability, which means the materialinternal dipoles become easily aligned in response to an applied external magnetic field[116]. The “zero drift” refers to the effect where the zero reading of an instrument is modifiedby the ambient conditions. ∼
100 kHz) by an analog down-converter and fed into an RMS-to-DCconverter with a 20 kHz bandpass filter. The output signal is amplified and sent toa Voltage-to-Frequency converter (V-to-F) before being read out by the scalers. Thesupplementary BCMs (BCM4A, BCM4B, BCM4C) RF signals are processed by adigital down-converter with an onboard ADC, FPGA and DAC, which provides ananalog signal that can be sent to a V-to-F and finally to the scalers [97]. The Unser55as a nominal output signal of 4 mV/ µ A and is also sent to a V-to-F before beingread out by a scaler. The BCM/Unser signals from the V-to-F are also read outby EPICS scalers directed by the EPICS controller located in the Hall C CountingRoom. The Unser gain is verified during downtimes by running a precision currentthrough a wire which passes through the toroid head.Even though the Unser has an extremely stable gain, its output signal offset candrift significantly on a time scale of several minutes and cannot be used to contin-uously monitor the beam current. On the other hand, the BCMs in general have astable offset but their gain needs to be calibrated and is not as stable as the Unsergain. Therefore, to use the BCMs as continuous current monitors they must becalibrated relative to the absolute beam current determined by the Unser.
The target chamber is a large evacuated cylindrical aluminum tank in two stackedsections (see Fig. 3.14) that contains the solid and cryotargets in a target ladder.The aluminum chamber is nominally 2 inches thick with an inner diameter of 41inches and outer diameter of 45 inches. The vacuum in the chamber is kept at apressure of a few 10 − torr by a turbomolecular vacuum pump connected through agate valve in the lower cylinder. In the standard configuration, both spectrometersshare a single chamber exit window that covers a horizontal angular range from3.2 ◦ to 77.0 ◦ on the HMS side and 3.2 ◦ to 47.0 ◦ on the SHMS side with a verticalcoverage of ± . ◦ for both sides.Both spectrometers’ entrance window are actually very close to, but not in con-tact (or vacuum coupled) with the chamber window itself. So as the electron beamscatters from the target, the outgoing particles have to exit the target cell, pass56igure 3.14: A CAD (computer-aided design) drawing of the Hall C Target Chamberdesign. Note: Reprinted from Ref. [97].through the exit window of the chamber into the air and then through the spectrom-eter entrance window before entering the spectrometer vacuum. The unscatteredbeam exits the chamber through an opening in the exit window to which the exitbeam pipe is connected via a threaded compression flange. There are also variousopenings in the target chamber through which the beam can enter, two pumpingports, several viewports and some spare ports. The viewports are used with a re-mote TV camera and light to observe the target motion and position in the countingroom [97]. The solid and cryogenic targets are accommodated in a target ladder with a singleaxis (vertical) motion system employed to select the desired target. Figure 3.15shows a CAD representation of the target ladder with the three cryogenic target57ells above the solid targets. The target ladder motion is controlled remotely viaa target GUI in the Counting Room. During this experiment, the cryogenic tar-get cells were filled with liquid helium (Loop 1), hydrogen (Loop 2) and deuterium(Loop 3), respectively.Figure 3.15: A CAD drawing of the Hall C Target Ladder. The arrow shows thebeam direction. Note: Reprinted from Ref. [97].Figure 3.16 shows a typical target GUI screen during this experiment. The cen-tral white screen shows a representation of the targets and an arrow indicating whichtarget was being hit by the beam. The live status of the beam current and the targetchamber vacuum pressure are also visible at the top of that screen. To the extremeleft of the GUI is the panel used to move the target ladder to a specific target.During this operation, it is extremely important for the beam to be taken away toprevent any damage to the target ladder system. To the right of the white screenare two panels with live feedback of the helium coolant supply from the End Station58efrigerator (ESR) as well as the cryogenic targets temperature and pressure andother relevant information which can be displayed on strip charts. See Refs. [97,118]for detailed information.Figure 3.16: Hall C Target GUI screen during the E12-10-003 experiment.The relevant targets used in this experiment were: • Carbon Hole:
A carbon hole target consists of a thin carbon foil with acentral hole of 2 mm in diameter. With a rastered beam of at least 2 × ,this target can be used to check how well is the beam centered at the target.As the beam passes through the hole, the edges of the beam do interact withthe carbon at the edges of the hole causing the electrons to re-scatter and bedetected by either spectrometer. The ( x, y ) raster values are plotted (whenthe spectrometer recorded a particle) forming a raster pattern with a hole (seeFig. 3.17). See HCLOG entry: https://logbooks.jlab.org/entry/3555843 for detailed informa-tion and pictures on the Hall C Target Configuration for Spring 2018 run period. • Aluminum Dummy (10 cm):
The dummy target consists of aluminum foilsmounted on separate frames at the locations Z = ± ∼
10 times thickerthan the actual cryotarget windows which must be accounted for. • Optics-1:
The optics target consists of carbon foils located at Z = -10, 0,10 cm along the beam axis and are used for spectrometer optics optimizationstudies. Even though these foils are beyond the target length used in Hall C,they are necessary for an optics reconstruction along the full target length. • Liquid Hydrogen (10 cm):
The cryogenic liquid hydrogen (LH ) is keptat a temperature of T LH = 19 ± . ∼
25 psia). LH freezing and boilingpoints are T F = 13.8 K and T B = 22.1 K, respectively. A 2 × raster isused to minimize density reduction at high beam currents.60 Liquid Deuterium (10 cm):
The cryogenic liquid deuterium (LD ) is keptat a temperature of T LD = 22 ± . ∼
23 psia). LD freezing and boilingpoints are T F = 18.7 K and T B = 25.3 K, respectively. A 2 × raster isused to minimize density reduction at high beam currents. Cryogenic targets are more complex than the solid targets, which are kept cool bythermal conduction (or direct contact) between the cryotarget and solid target lad-ders. To keep the cryotargets at very low temperatures, the hydrogen and deuteriummust be re-circulated constantly through a heat exchanger. Figure 3.18 shows a sim-plified version of the loop anatomy for a typical target cell used in the Hall C 12GeV era.Figure 3.18: Hall C cryotarget loop anatomy for the 12 GeV era (not to scale).Figure adaptation from Refs. [118] [119].61 gas panel outside the building provides the heat exchanger with a constantsupply of either hydrogen or deuterium, which is cooled to either 19 K (hydrogen)or 22 K (deuterium) by the 15 K He coolant supply from the ESR. The amount ofcoolant sent to the heat exchanger is controlled via a Joule-Thompson (JT) valve.The target fluid is then sent to the target cell and enters the high-performance blockthrough an inlet. The flow diverter inside the target cell then guides the liquid fromthe inlet to the outlet such as to make the flow velocity constant and minimizelocalized density changes caused by the beam. The liquid then leaves the target cellback to the heat exchanger at a higher temperature than which it entered the cell,mostly due to the heat deposited by the electron beam. The liquid is then cooledagain by the 15 K He supply from the ESR completing the loop. This way, thetarget fluid is constantly recirculated between the target cell and heat exchanger tokeep the fluid at operating temperatures.
The main experimental equipment in Hall C consists of a pair of magnetic spectrom-eters designed to perform high-precision cross section measurements at a relativelyhigh luminosity . The spectrometers have bearings at the pivot which permit rapid,remote spectrometer rotation. Each spectrometer consists of a series of optical ele-ments (quadrupoles and dipoles) followed by a series of particle detectors that arehoused in a heavily shielded detector hut. The optical elements are used to trans-port the scattered particles from the target chamber to the particle detectors. Thetracks are then reconstructed at the focal plane and translated back to the target. Luminosity in nuclear physics is defined as ( × ( ), typically expressed in units of cm − s − (see Section 5.1). Parameter HMS SHMSPerformance Specification
Range of Central Momentum 0.4 to 7.4 GeV/c 2 to 11 GeV/cMomentum Acceptance ± −
10% to +22%Momentum Resolution 0 . − .
15% 0 . − . ◦ to 90 ◦ ◦ to 40 ◦ Target Length Accepted at 90 ◦
10 cm 50 cmHorizontal Angle Acceptance ±
32 mrad ±
18 mradVertical Angle Acceptance ±
85 mrad ±
50 mradSolid Angle Acceptance 8.1 msr > ∼ ∼ > > π /K Discrimination 100:1 at 95% efficiency 100:1 at 95% efficiency Table 3.2: Demonstrated performance of the HMS and design specifications for theSHMS.The major component of the Hall C 12 GeV upgrade is the new Super HighMomentum Spectrometer (SHMS) that replaced the orginal companion of the HMSknown as the Short Orbit Spectrometer (SOS). The new SHMS-HMS pair makesHall C the only facility in the world capable of carrying out the rich nuclear physics63rogram detailed in Refs. [99,120]. To be able to carry out this program successfully,the SHMS was designed to achieve a maximum momentum of 11 GeV/c, which iswell matched with the maximum beam energy delivered to Hall C. The SHMS is alsoable to rotate to very small forward central angles down to 5.5 ◦ as well as operateat an unprecedented high luminosity of 10 cm − s − and has a 32% momentumacceptance which measures the percent deviation of the particle momentum relativeto the central momentum of the spectrometer. A detailed description of the HMSperformance parameters and SHMS design specifications is given in Table 3.2. As the electron beam interacts with the target atoms, the final-state particles scat-ter radially outwards in all possible directions, but only a small fraction of these fallwithin the momentum and angular acceptance set by the spectrometer. Each spec-trometer is equipped with a slit system containing collimators defining the angularacceptance, and a sieve slit used for spectrometer optics studies. Table 3.3 gives asummary of the apertures as well as the corresponding solid angles defined by thecollimators in each spectrometer.The slit system in each spectrometer is housed in a vacuum box with a mov-able slit ladder that has space for three separate slits. Each ladder position caneither hold a collimator or an optics sieve slit in accordance to the experimentalrequirements. The ladder motion is controlled remotely from the Hall C CountingRoom. Figure 3.19 shows the slit configuration used for each spectrometer duringthe commissioning experiments. The SHMS was built from 2009 to 2016, and was first commissioned in 2017 as part ofthe Hall C KPP. orizontal (mr) Vertical (mr)
Solid Angle (msr)
Shape
HMS
Large Collimator ± ± Pion Collimator ± ± Collimator ± ± Table 3.3: Spectrometer apertures at the collimator entrance.Figure 3.19: Spectrometer slit system.
The HMS Slits
In the HMS, the slit system is installed at the entrance of the first quadrupolemagnet (Q1) at a distance of Z (HMS)coll =166.37 cm from the center of the target to65he collimator entrance. The slit system consists of two collimators and a sieve slit.Each slit is a rectangular block of a heavy alloy metal (90% W, 10% Cu/Ni) witha density of 17 g/cm . The collimators have a machined octagonal-shaped openingwhereas the sieve slit has several rows of holes drilled into it with the exception oftwo specified locations that are used to determine the orientation of the slit in thereconstruction analysis. During this experiment, there was no need to insert thesieve as the HMS optics is well understood and it was decided to only use the new Large Collimator and not the original
Pion Collimator (6 GeV era) to define theacceptance.
The SHMS Slits
In the SHMS, the slit system is installed between the horizontal bender (HB) andfirst quadrupole (Q1) magnets at a distance of Z (SHMS)coll =253 cm from the center ofthe target to the collimator entrance. The slit system consists of a collimator andtwo sieve slits. The collimator and two sieves are made of Mi-Tech TM TungstenHD-17 (90% W, 6% Ni, 4% Cu) with a density of 17 g/cm . Similar to the HMS,the SHMS collimator has an octagonal-shaped opening and the sieves have severalrows of holes drilled into it (see Fig. 3.19). The centered sieve has 11 columns ofholes with the sixth column at the center, whereas the shifted sieve has 10 columnsand shifted from the central axis. More details about the HMS/SHMS slit systemcan be found in Refs. [97, 121]. The spectrometers’ optical elements consist of a series of superconducting magnetsthat guide the scattered particles towards a detector stack. To keep the magnet66oils at superconducting temperatures, a constant supply of liquid He at a temper-ature of 4.5 K is provided by ESR to Hall C. The cryogenic supply is distributedto each spectrometer via flexible transfer lines emanating from a main distributionbox located over the pivot into each spectrometer cryogenics network. The magnetpower supplies are located on the spectrometer support structure adjacent to themagnets. To remove the excess heat, the power supplies are all water-cooled bya constant flow rate that can be monitored by a water flow meter located on theelectronic boxes on the floor near the pivot [97]. The magnet cryogenics and powersupplies, as with the spectrometer rotation controls, are operated and monitoredremotely via the magnet control screens in the Counting Room.
StoredMagnet Type EFL Aperture Momentum Current Field/Gradient Energy (m) (cm) (GeV/c) (A) (MJ)HMS Q1 Cold Fe 1.867 40 7.4 1012 7.148 T/m 0.335HMS Q2 Cold Fe 2.104 60 7.4 1023 6.167 T/m 1.59HMS Q3 Cold Fe 2.104 60 7.4 1023 6.167 T/m 1.59HMS D Warm Fe 5.122 40 7.4 3000 2.073 T 9.79SHMS d “C” Septum 0.752 14.5 x 18 11 3930 2.56 T 0.2SHMS Q1 Cold Fe 1.86 40 11 2460 7.9 T/m 0.382SHMS Q2 cos(2 θ ) 1.64 60 11 3630 11.8 T/m 7.6SHMS Q3 cos(2 θ ) 1.64 60 11 2480 7.9 T/m 3.4SHMS D cos( θ ) 2.85 60 11 3270 3.9 T 13.7 Table 3.4: Spectrometer magnets design parameters [122].Table 3.4 summarizes the design parameters of the spectrometer magnets. Eachspectrometer is designed to provide point-to-point focusing (Q1, Q2, Q3) and avertical momentum dispersion (D) and can be configured to transport either positiveor negatively charged particles by setting the individual magnets to a “+” or “-”polarity in an alternating pattern. The central momentum is also set individually for EFL refers to the
Effective Field Length of the magnet.
The HMS Magnets
The HMS optics elements consist of three quadrupoles (Q1, Q2, Q3) and a dipole(D) magnet arranged in a (QQQD) configuration that are used to transport thescattered particles into a series of particle detectors located in a detector hut (seeFig. 3.20). The quadrupoles focus the collimated particles into the dipole thatbends the central momentum particles vertically by 25 ◦ into the detector stack.Figure 3.20: High Momentum Spectrometer (HMS) side view.The HMS is capable of detecting particles with central momentum from 0.4 to7.4 GeV/c and can be rotated from 10.5 ◦ to 85 ◦ where the minimum/maximumangles are restricted by administrative, software, and hardware limits. These limitsdepend on the beamline configuration and obstructions in the Hall at the time ofthe experiment [97]. The spectrometer magnets and shield hut (with detectors)68re actually supported by two separate, but firmly attached carriages that keep thedetectors and magnets aligned to each other and to the target.Even though the HMS is a well understood spectrometer from the 6 GeV era, itunderwent minor modifications in preparation for the experimental requirements ofthe 12 GeV era and had to be re-commissioned. First, the NMR probe used forprecise field regulation of the HMS dipole was replaced and second, the old HMS driftchambers were replaced with a new design similar to the SHMS drift chambers. Withthe replacement of the NMR probe, the precise mapping between the NMR probereading and the dipole magnetic field had to be re-done (see Ref. [124]) and, with theinstallation of the new HMS drift chambers, the tracking and optics reconstructionto the target had to be checked as well. Furthermore, some of the 12 GeV eraexperiments required the HMS central momentum to operate above ∼ > The SHMS Magnets
Similar to the HMS, the SHMS optics elements consist of an array of three quadrupoles(Q1, Q2, Q3) and dipole (D) magnet used to guide the scattered particles into aseries of particle detectors in a shielded hut. The SHMS has an additional dipole The NMR probe is used to determine the spectrometer central momentum, which ismostly determined by the dipole. Ideally, the probe is placed at the center of thedipole and picks up a field reading from it and regulates this field by re-adjusting thedipole current to achieve a more precise dipole field and hence a more precise centralmomentum. In reality, more than one probe is used in this procedure (see Ref. [124] fordetails). ◦ away from the beamlineand towards the collimator before entering Q1. To achieve a horizontal bend, theoptical axis of the HB is oriented 3 ◦ lower (towards the beamline) relative to theoptical axis of the rest of the spectrometer (collimator, Q1, Q2, Q3, D, detectors).With the HB equipped, the SHMS is able to detect particles at angles that wouldhave otherwise been impossible to reach due to the obstruction of the quadrupolemagnets and shield house with the beamline.As the particles are bent horizontally towards the collimator and enter Q1, theyare focused through the remaining quadrupoles (Q2, Q3) and into the dipole (D)where the central-momentum particles are vertically bent by 18.4 ◦ into the detectorstack. The SHMS can detect particles with central momentum from 0.2 to 11 GeV/cand can be rotated from a central angle of 8.5 ◦ up to 40 ◦ . In reality, due to the 3 ◦ bend by the HB, the central-ray particles detected at the hut actually scatter from70he target 3 ◦ lower relative to the hut, therefore, the SHMS full angular coverage isfrom 5.5 ◦ to 37 ◦ .Given that the SHMS is a new spectrometer, a significant amount of work duringthe Fall-2017 to Spring-2018 run period has been devoted towards understandingand optimizing the magnetic optics as well as commissioing the particle detectors.See Ref. [125] for details of the optics commissioning work for the SHMS. Detailson the detector calibrations and SHMS optics optimization for this experiment willbe discussed in detail in Chapter 4. Each spectrometer is equipped with a similar set of particle detectors housed ina heavily shielded hut. The detector package consists of a pair of drift chambers(DC1 and DC2) used for track reconstruction, two pairs of hodoscope planes usedfor particle triggering, a calorimeter used for e/π separation, and a gas and aerogelˇCerenkov used for additional particle identification.In the HMS (see Fig. 3.22), the particles enter the detector hut through a cylin-drical vacuum vessel that extends from the dipole exit window (outside the hut)to just upstream of DC1. As the particles exit the vacuum vessel, they first passthrough the drift chamber pair followed by an aerogel ˇCerenkov detector, a firstpair of XY hodoscope planes, a Heavy Gas ˇCerenkov (HGC), a second pair of XYhodoscope planes, and towards the end, a preshower and shower counters that makeup the calorimeter detector. During the commissioning run period, the aerogel de-tector was not installed in the HMS detector stack.71igure 3.22: High Momentum Spectrometer (HMS) detector stack.In the SHMS (see Fig. 3.23), as the particles enter the detector hut, they passthrough a vacuum vessel (similar to HMS) coupled to the dipole, which partiallyprotrudes inside the hut due to space constraints. Depending on the experimentalrequirements for particle identification, the vacuum extension pipe can be replacedwith the Noble Gas ˇCerenkov (NGC) at higher spectrometer momenta, where theeffects of multiple scattering are minimized.As the particles exit the vacuum vessel (or NGC), they pass through a pair of driftchambers, followed by a pair of XY hodoscope planes, a Heavy Gas ˇCerenkov, anaerogel ˇCerenkov, a second pair of XY hodoscope planes and finally, the preshowerand shower counters, which constitute the calorimeter detector.72igure 3.23: Super High Momentum Spectrometer (SHMS) detector stack.During the commissioning run period, the NGC was installed as the spectrometercentral momentum was relatively high in each of the experiments. In this experi-ment (E12-10-003) in particular, both spectrometers used the standard hodoscopoeand drift chamber detectors for event triggering and tracking. Due to the negligiblebackground and low coincidence trigger rates the need of additional particle identi-fication was minimal. Only the SHMS calorimeter was used to select a clean sampleof electrons.
Drift Chambers
The drift chambers in both the HMS and SHMS are of similar design (see Refs.[126, 127]). In each spectrometer, the two drift chambers are mounted on an alu-minum frame and are separated by about 80 cm as measured from their middleplane. Each chamber consists of 6 anode (wire) planes and 8 cathode planes con-fined between two cathode windows. The middle plane is used for mounting (onboth sides of the plane) the 16-channel amplifier discriminator cards required for73ense wire readout. The middle plane also divides the chamber in half, each con-sisting of three wire planes and four cathode planes (see Fig. 3.24).Figure 3.24: Side view of the plane orientation for the DC1 (left) where the coloredplanes represent the wire planes, and DC2 (right) which is identical in design toDC1 rotated by 180 ◦ about the x -axis (vertical) forming a mirror image along the z -axis.DC1 and DC2 are separated by an “imaginary” plane referred to as the focalplane, which is chosen such that the focal point of the spectrometer optics coincideswith the origin of the focal plane. This means that the particles transported tothe hut are focused at the focal plane and those with a momentum equal to thecentral spectrometer momentum are focused at the origin. This assumes that thespectrometer is also positioned at the central angles corresponding to the centralmomentum, otherwise, the focal point will be shifted.The wire planes for each chamber were designed such that a 180 ◦ rotation of theunprimed wire planes about the z -axis produce the primed planes with wires at thesame orientation, but slightly shifted, which allows the resolution of the left/rightambiguities . For each wire plane, the wire orientation is defined by a vector The left/right ambiguities refers to our ignorance of whether a particle passed to theleft or right side of the sense wire that detected it ± ◦ relative tothe (X,X’) wires as illustrated in Fig. 3.25.Figure 3.25: Front view of the wire (dashed) orientations for each plane, indicated byrepresentative sense wires of different colors, where the + z -axis (particle direction)is into the page. The wires in each plane are superimposed onto a single plane inthis figure for convenience and their orientation is defined by the vector normal tothe wire.A wire plane consists of alternating field and sense wires. In both the HMS andSHMS, the sense wires are made of 20 µ m gold-plated tungsten and the field wiresare made of copper plated beryllium with thickness of 100 µ m in HMS and 80 µ m inSHMS. The numbering scheme of the sense wires is determined by the direction ofthe perpendicular vector to the wire (see Fig. 3.25) which points towards increasingwire numbers. For the HMS drift chambers, the (U, U’, V, V’) consist of 96 sensewires per plane and the (X, X’) consist of 102 sense wires per plane. In contrast, theSHMS (U, U’, V, V’) consist of 107 sense wires per plane and the (X, X’) consist of79 sense wires per plane. 75uring operations, each chamber was filled with a carefully chosen gas mixture(50:50 argon/ethane) such that the charged particles that pass through the chamberionize the surrounding argon gas atoms producing an avalanche of electrons wherethe ethane served as the quenching element. In addition, the cathode planes andfield wires were kept at a negative potential ( ∼ -1940 V) relative to the sense wires,which were kept grounded at zero potential. The potential difference between thefield wires and cathode planes relative to each sense wire established an electric fieldwith field lines pointing away from the sense wires and towards the adjacent fieldwires and cathode planes. The calibration procedures will be discussed in detail inSection 4.3.2. Hodoscopes
Each spectrometer is equipped with a series of four scintillator arrays (hodoscopeplanes) grouped into two pairs separated by a distance of about 2.2 m. Each pairis segmented along the dispersive ( x -axis) and non-dispersive ( y -axis) direction byan array of long rectangular elements that can be either a plastic scintillator paddle or quartz bar with a photomultiplier tube (PMT) coupled at each end. The plasticscintillating materials used in the HMS and the first three planes of the SHMS are theBC-404 [128] from Saint-Gobain Crystals and RP-408 [129] from Rexon Corporarion,respectively. The last plane in the SHMS, known as the quartz plane , is composedof Corning HPFS 7980 Fused Silica (or quartz) [130] bars. To eliminate the possiblegaps and avoid dead spots between adjacent elements where a particle could passundetected, the paddles/bars are slightly overlapped by a few millimeters in everyplane. Table 3.4 summarizes the dimensions of each paddle for every hodoscopeplane in both spectrometers. 76 lane Thickness (mm) Width (cm)
Length (cm) of Elements
HMS1X 2.12 8.0 75.5 161Y 2.12 8.0 75.5 102X 2.12 8.0 75.5 162Y 2.12 8.0 75.5 10SHMS1X 5 8.0 100 131Y 5 8.0 100 132X 5 10.0 110 142Y 25 5.5 125 21
Table 3.5: Summary of hodoscopes paddle dimensions for each plane.The fast timing properties of the plastic scintillators makes the hodoscope de-tector ideal for particle triggering specially after the 12 GeV energy upgrade whereparticle rates become significantly higher. The addition of the quartz plane in theSHMS detector package provides a clean detection of charged particles while main-taining a high level of background rejection that optimizes the hodoscope trackingefficiency at higher rates where the background is also expected to be larger.Even though the plastic scintillators and the quartz both emit light due tocharged particle interactions, the process by which the light is produced is dif-ferent. In a scintillator, as a charged particle traverses the medium, it excites themolecules in the scintillator material that decay back into the ground state via theemission of scintillation photons in (or near) the visible light range, a process knownas fluorescence. The photons propagate towards the end of the scintillator paddleswhere they are detected by the PMT. As the photons interact with the PMT photo-cathode, a certain number of photoelectrons will be produced via the photoelectriceffect. These electrons are accelerated towards a series of dynodes creating an elec-tron avalanche towards the end of the PMT at the anode creating a measurableanalog signal that is sent via a signal cable to the Counting Room for further signalprocessing (see Fig. 3.26). 77s the light propagates through the scintillator material, when it reaches theFigure 3.26: Front view of the SHMS S1X (front) and S1Y (back) hodoscope planes.boundaries, the light may not be completely reflected and can be lost due to re-fraction. If the light ray incident on the boundary exceeds the critical angle , itis reflected via total internal reflection where no losses occur at the boundary. Tomaximize the scintillator light output in case of partial refraction, the scintillatorsare wrapped in a layer of a highly reflective material (aluminum foil) and multiplelayers of Tedlar (HMS) or electrical tape (SHMS) to ensure light tightness.In contrast to the plastic scintillators, the radiation produced in the quartz planeis based on the ˇCerenkov effect, which will be discussed in more detail in the next The critical angle required for total internal reflection depends on the index of refractionbetween the two media at the boundary.
Threshold ˇCerenkovs
At high particle momenta ( ≥ t ∼ /p , where ∆ t is the hodoscope time differ-ence between the first and second pair of planes and p is the particle momentum.This means that at higher particle momenta, it becomes very difficult to identifyeach particle due to the small and indistinguishable time difference between differ-ent particle masses. Therefore, the use of additional particle identification detectorsbecomes a necessity.The Hall C spectrometers are equipped with various threshold ˇCerenkov par-ticle detectors. These detectors depend on ˇCerenkov effect, which occurs when acharged particle traverses a transparent medium faster than the speed of light in themedium. As a result, the charged particle creates an electromagnetic disturbancein the medium that causes ˇCerenkov radiation to be emitted and distributed in aconical shape about the tracjectory of the particle (see Fig. 3.27).From Fig. 3.27, the electromagnetic disturbance (light) created by the passageof the charge particle is analogous to the sound waves emitted by a supersonic jet.Since the charged particle moves faster than the spherical waves it emits, a conicalshape is formed given by the relationcos( θ c ) = 1 nβ , (3.11)where β = v/c is the ratio of the velocity ( v ) of the charged particle to the speedof light in vacuum ( c ) and n = c/u is the index of refraction of the medium, where79 is the speed of light in the medium. Alternatively, β = p/ (cid:112) m + p where m isthe particle’s mass. Since the charged particle must travel faster that light in themedium for ˇCerenkov light to be emitted, one requires v > c/n = ⇒ n > /β whichcan be expressed in terms of momentum as n > (cid:112) m + p p . (3.12)From this inequality, the index of refraction of the medium can be adjusted ac-cordingly such that at a fixed momentum, the mass of the particle will determinewhether or not ˇCerenkov radiation will be produced.Figure 3.27: Cartoon of the ˇCerenkov effect. The charged particle (red) traverses amedium faster than the speed of light (blue) in that medium, producing a conicallight wavefront.Both the gas and aerogel threshold ˇCerenkov detectors in Hall C utilize thisbasic inequality for particle identification. In addition, for the gas ˇCerenkovs, thequantity ( n −
1) is proportional to the gas pressure, which can be adjusted to changethe index of refraction and hence select the desired particle mass that will triggerthe ˇCerenkov effect. Below is a brief description of each ˇCerenkov detector.80 eavy Gas ˇCerenkov (HGC)
Each spectrometer is equipped with an HGCdetector located between the front and rear hodoscope planes. The detector is filledwith a gas that is kept at a specific pressure depending on the experimental require-ments. The detector is also equipped with several mirrors mounted and oriented soas to reflect and focus the ˇCerenkov light towards the PMTs.Figure 3.28: CAD rendering of the SHMS Heavy Gas ˇCerenkov detector. Note:Reprinted from Ref. [134].The HMS ˇCerenkov [97, 135] consists of a 1.5 meter-long cylindrical tank with2 spherical mirrors installed that focus the ˇCerenkov light onto 2 PMTs. The de-tector can operate as an e/π or π/p discriminator, depending on whether the tankis filled with C F or N gas for the former, or Freon-12 gas for the latter, at arange of operating pressures depending on the gas used. In contrast, the SHMSˇCerenkov [97, 134] (see Fig. 3.28) consists of a 1.3 meter-long cylindrical tank (1.8881 in diameter) and has 4 mirrors installed that focus the light onto 4 PMTs. Thedetector is filled with C F or C F O gas, which are functionally equivalent, and isable to operate as an e/π or π/K discriminator depending on the gas pressure used. Noble Gas ˇCerenkov (NGC)
The SHMS is equipped with an additional gasˇCerenkov detector [97, 136] (see Fig. 3.29) located in front of the drift chambers dueto space constraints in the hut. The detector consists of a rectangular tank 2.5Figure 3.29: CAD rendering of the SHMS Noble Gas ˇCerenkov detector.meters in length (along z -axis) and 0.8 meters wide with 4 spherical mirrors used toreflect and focus the ˇCerenkov light onto 4 PMTs. The tank is designed to operateat a gas pressure of 1 atm of either argon, neon or a mixture of the two gases, whichprovides e/π discrimination at momenta above 6 GeV/c. At lower central momenta,the NGC is replaced with an extended vacuum pipe of the same length to minimizethe effects of multiple scattering on the particle trajectory.82 erogel ˇCerenkov The strange physics part of the program at Hall C requiresthe capability to carry out a clean p/π/K separation. To achieve a clean particleidentification, both spectrometers are equipped with an aerogel ˇCerenkov detector.Figure 3.30: CAD rendering of the SHMS aerogel ˇCerenkov detector. The HMSdesign is very similar with slightly different dimensions and an additional PMT atboth ends.In contrast to the gas ˇCerenkovs, this detector uses an aerogel, which is a trans-parent, highly porous material with a refractive index typically between those ofgases and liquids. The aerogel detector in each spectrometer are of similar designconsisting of an aerogel tray followed by a light diffusion box with a highly reflectivematerial at the inner boundaries. As the charged particle passes through the aero-gel material, depending on the refractive index, specific particles will emit ˇCerenkovlight that travels into the diffusion box where it is reflected at the boundaries ofthe box and into an array of PMTs mounted on each side. The analog signal from83he light collected by the PMTs is sent to the Counting Room for further signalprocessing. A detailed description of the aerogel detector including the dimensionsand material specifications can be found in Refs. [137, 138].
Calorimeters
The electromagnetic (EM) calorimeter in each spectrometer is primarily used for e/π discrimination and to complement the gas ˇCerenkov detectors for a more ro-bust electron identification and pion suppression. The calorimeters provide a de-structive measurement of the projectile particle energy and are therefore located atthe end of the detector stack. The projectile energy is measured by the detection ofˇCerenkov radiation primarily from EM showers produced mainly via bremsstrahlungand pair production processes. As the electrons traverse the calorimeter, they areslowed down (decelerated) by the calorimeter radiator and emit bremsstrahlungphotons that decay to e + e − pairs via pair production. These pairs further radiatebremsstrahlung photons triggering an EM shower cascade reaction until most or allthe initial electron energy has been deposited in the calorimeter.From Fig. 3.31, a single electron with initial energy, E , enters the calorimeterradiator material and produces a particle cascade in a chain reaction that becomesless energetic as it traverses the radiator. The horizontal axis (along particle trajec-tory) is defined as a multiple of one radiation length ( X [g/cm ]), which is definedas the mean distance over which a high-energy electron loses all but 1 /e of its initialenergy ( E ) due to bremsstrahlung radiation [139]. The radiation length can alsobe expressed in [cm] by dividing X by the density of the material in [g/cm ].To ensure that all (or most) of the incident projectile energy is deposited in theradiator material, the Hall C calorimeters are made of several stacked layers of thicklead glass blocks (or modules) that are tilted a few degrees lower relative to the84igure 3.31: Typical electromagnetic shower cascade in a calorimeter.spectrometer central ray. The original HMS calorimeter [140] (see Fig. 3.32) wascommissioned in 1994 as one of the first detectors to operate in Hall C and remainsin the stack since no significant deterioration in performance has been observed.The detector consists of 52 TF-1 lead glass modules (refractive index 1.65, density3.86 g/cm ) stacked in four layers of 13 blocks/layer with dimensions of 10 × × per block. A single layer measures 3.65 radiation lengths along the particletrajectory (+ z ) for a total of ∼ new SHMS calorimeter [140] (see Fig. 3.33) consists of TF-1 and F-101type lead glass modules (refractive index 1.65, density 3.86 g/cm ) assembled sepa-85igure 3.32: HMS electromagnetic calorimeter. The entire detector is tilted verti-cally by 5 ◦ lower relative to the central ray of the spectrometer hut.rately into a preshower (TF-1) and shower (F-101) counter. The preshower blocksused in the SHMS were removed from the decommissioned SOS calorimeter andconsist of 28 modules stacked in two adjacent columns. Each module has dimen-sions of 10 × ×
70 cm and is coupled to a PMT at one end. In contrast to thepreshower, the shower counter blocks were obtained from the decommissioned HER-MES calorimeter detector and consist of 224 modules of dimensions 8 . × . ×
50 cm per module with a coupled PMT towards the long end of the block. The moduleswere stacked in a fly’s eye configuration behind the preshower plane. This configura-tion, which is ∼
18 radiation lengths deep, guarantees that the EM showers from thehighest energetic projectiles ( ∼
10 GeV) will be mostly absorbed. The preshowercounter, in contrast, is only 3 . ◦ lower relative to the central ray of the spectrometer hut. The majority of the detector electronics in the HMS/SHMS detector huts are readout by
Read-Out Controllers (ROCs) crates located in the Counting Room exceptfor the HMS/SHMS drift chambers and the SHMS shower counter signals, whichare read in their respective ROCs in the detector huts. In the HMS/SHMS huts,the drift chamber signals are transmitted by 20-25 foot-long ribbon cables that areread out in the hut electronics rack (see Figs. 3.34 and 3.37). On the SHMS side,the shower counter consists of 224 signal cables read directly in the hut electronicsrack. 87 .7.1 HMS Detector Hut Electronics
The HMS drift chambers are read out through a VXS Crate (
ROC 03 ) in thedetector hut electronics rack (see Fig. 3.34). The signals are carried through 16-channel ribbon cables fed into various CAEN1190 (C1190) [141] TDC modules.The Trigger Interface (TI) [142] module at the front end of the crate distributes thereadout trigger throughout all modules in the crate and initiates data readout.The rest of the HMS detector signals (gas ˇCerenkov, hodoscopes, calorimeter)
HMS Drift Chamber Rack (HMS Hut)
ROC03 HMS VXS CRATE
4 5
7 8 . . . 17 19 20
NIM CRATE C T D C C T D C C T D C C T D C C T D C TI MODULE
HMS Hut Patch Panel 1 HMS Hut Patch Panel 2 P A T C H P A N E L P A T C H P A N E L Hodoscope Planes h1X h1Y h2X h2Y Calorimeter Layers hA hB hC hD Gas Cherenkov Hodoscope Planes h1X h1Y h2X h2Y Calorimeter Layers hA hB hC hD
Figure 3.34: HMS detector hut electronics rack (left) and patch panels (right).are sent to the Hall C floor patch panel via the hut patch, with the exception of theaerogel, which is sent directly from the detector to the floor patch. All the signalsare then sent to the Counting Room patch panel to be processed by the electronics(see Fig. 3.35). 88igure 3.35: HMS patch diagram from detectors to Counting Room.
Similar to the HMS drift chambers, the SHMS drift chambers are also read out byTDCs in a VXS Crate in the SHMS electronics hut (see Fig. 3.36).Figure 3.36: SHMS hut patch panel (left) and electronics racks (right).89he 224 shower counter signals are directly connected to 250 MHz flash ADCs[143], hereafter referred to as fADCs, in a separate VXS Crate (
ROC 04 ). Thepreshower signals (x14/side) pass through a 50:50 splitter where a part of the signalis fed to an fADC and the other part is partially summed in the hut and sent viathe hut patch panel to the Counting Room patch. The rest of the SHMS detectorsignals (HGC/NGC, hodoscopes, aerogel) are sent to the Counting Room via thehut patch panel to be processed by the electronics (see Fig. 3.37).Figure 3.37: SHMS patch diagram from detectors to Counting Room.
Once the detector signals arrive at the Counting Room patch (see Fig. 3.38(left)),they are processed by the NIM/CAMAC electronics (see Fig. 3.38(right)) to formthe single-arm or coincidence triggers for each spectrometer. The signals are alsosent to fADCs/TDCs to determine energy and timing information for individualdetectors. NIM or
Nuclear Instrumentation Modules and CAMAC or
Computer Automated Mea-surement and Control define a set of standard modular-crate electronics commonly usedin experimental nuclear/particle physics.
The XY scintillator arrays (hodoscope planes) form part of the standard HMS triggerconfiguration. Additional particle detectors may also be incorporated into the HMStrigger as required by different experiments. The gas ˇCerenkov and calorimetertriggers are used for e/π discrimination, whereas the aerogel ˇCerenkov trigger isused for π/K/p discrimination.
Hodoscopes Pre-Trigger
Each hodoscope plane consists of an array of scintillator paddles coupled to a PMTat each end (see Fig. 3.22), so each paddle reads out two signals. In Fig. 3.39, forexample, hodoscope plane h1X consists of 32 signals (16 paddles) read out in theCounting House (CH) patch. Each side of the plane (x16 signals/side) is fed into a64-channel input passive splitter (16 Ch./set). One-third of the signal amplitude is91igure 3.39: HMS hodoscopes electronics diagram.sent via a 16-channel ribbon cable to a 64-channel input Ribbon-to-BNC converter(16 Ch./set) fed into a 16-channel NIM input fADC. The remaining two-thirds ofthe signal amplitude is sent to a 16-channel input CAMAC discriminator unit. TheHMS discriminator thresholds and gate widths were set to -44.5 mV and 60 ns,respectively, but may be subject to change.The discriminated signals are sent via two ribbon cable outputs to C1190 TDCsand scalers (daisy-chained) and to a LeCroy 4564 CAMAC logic unit to form theplane pre-triggers. The logic unit takes four sets of 16-channel input ribbon cablesand forms a 16-fold OR for each set by default. Further boolean operations aredone through the module backplane by connecting a twisted pair cable to the pin92orresponding to the desired boolean operation. For the HMS hodoscope planepre-triggers, the boolean operations are as follows: h1X = h1X+ (16-fold OR) AND h1X- (16-fold OR) h1Y = h1Y+ (10-fold OR) AND h1Y- (10-fold OR) h2X = h2X+ (16-fold OR) AND h2X- (16-fold OR) h2Y = h2Y+ (10-fold OR) AND h2Y- (10-fold OR)Once a pre-trigger has been made for each plane, they are sent to a NIM/ECLconverter (Level Translator - Phillips Scientific (or P/S) Model 7126) via twistedpair cables. The NIM output is then sent to individual sets of a P/S Model 752NIM logic unit to adjust the widths of each of the plane pre-triggers as necessarybefore making coincidence. An XY hodoscope plane coincidence ( h1 = h1X OR h1Y , h2 = h2X OR h2Y ) is then made by feeding each hodoscope XY plane pairinto a P/S Model 755 NIM logic unit . A copy of each of the four individual planepre-triggers is also sent to another set of P/S Model 755 to make a 3/4 or 4/4plane coincidence (via a front-panel knob), which defines the standard hodoscopepre-trigger ( hHODO 3/4 ). An additional pre-trigger ( hSTOF = h1 AND h2 ) isformed by requiring the coincidence between any two of the front ( h1 ) and back( h2 ) scintillator plane pairs to measure the time-of-flight (TOF) between any of thetwo front and back planes. A copy of all the pre-triggers discussed above are sentto TDCs and scalers via a NIM/ECL converter for timing and counting information(see Fig. 3.39). The output widths of the P/S Model 755 logic units were set to ∼
50 ns for the HMS.See HCLOG entry https://logbooks.jlab.org/entry/3501357 . alorimeter Pre-Trigger Figure 3.40: HMS calorimeter electronics diagram.The HMS calorimeter consists of four layers of lead blocks. Layers A and B readout 26 PMT signals per layer (13 signals/side) while layers C and D read out 13signals/layer on one side. The first layer forms the preshower counter while all fourlayers (A, B, C and D) form the shower counter. Each layer is read out in theCounting Room patch and fed into 50:50 splitters. One output of the splitter isconnected to an fADC via a Ribbon-to-BNC converter (same as hodoscopes) whilethe other output is sent to P/S Model 740 NIM Linear FI/FO summing modules.Each side of a layer is summed first (hA+, hA-, hB+, hB-, hC and hD sums). Thesums are connected into a LeCroy Model 428F summing module where layers hA+/-and hB+/- are summed to form hA and hB sums. A copy of each layer sum is thensent to an fADC. The preshower sum ( preSh SUM ) is made from layer A, whilethe shower sum (
Shower SUM ) is made by summing all four layers. A copy of94he preshower and shower sums is also sent to an fADC channel. The preshowerand shower sums are also sent to a P/S Model 715 NIM discriminator unit to formthe preshower Low/High ( hPreSH LO , hPreSH HI ) and shower Low ( hShowerLO ) pre-triggers with thresholds -40 mV , -60 mV and -45 mV , respectively, withall gate widths set to 30 ns. A copy of the pre-triggers is sent to TDC and scalermodules for trigger timing and counting information. Gas ˇCerenkov Pre-Trigger
Figure 3.41: HMS gas ˇCerenkov electronics diagram. Same electronics diagramapplies for SHMS gas ˇCerenkovs.The HMS gas ˇCerenkov detector consists of a 1.5 m long cylindrical tank betweenthe first and second set of hodoscope planes (see Fig. 3.22). The tank is filled witha gas and has two spherical mirrors that focus the ˇCerenkov photons towards two 5-inch PMTs. The signals are read out in the Counting Room patch and pass througha 50:50 splitter. One output is fed into an fADC module via a Ribbon-to-BNCconverter. The other output is sent to a LeCroy Model 428F summing module, and95 copy of the sum is fed to an fADC. The sum is also sent to a P/S Model 715 NIMdiscriminator to form the ˇCerenkov pre-trigger ( hCER TRG ) with a thresholdand gate width set to -50 mV and 30 ns. A copy of the discriminated signal is alsosent to TDCs and scalers via a NIM/ECL converter for trigger and counting rateinformation.
Aerogel ˇCerenkov Pre-Trigger
The HMS aerogel ˇCerenkov detector signals are sent directly to the Hall C floorpatch panel and then sent to the Counting Room patch and connected to a 50:50splitter. One output leads to an fADC module via a Ribbon-to-BNC converter.The other output is sent to a summing module, and a copy of the sum is sent to anfADC. The sum is also sent to a NIM discriminator to form the aerogel pre-trigger( hAERO TRG ). A copy of the discriminated signal is registered by TDCs andscalers via a NIM/ECL converter for trigger and counting information purposes.The electronics diagram is the same as in Fig. 3.41.
HMS Single Arm Pre-Trigger
The HMS single arm pre-trigger is formed from the standard pre-trigger (hodoscopes)and a combination of other detector pre-triggers as required by the experiment. Thestandard and other experiment-specific pre-triggers are sent to a P/S Model 755NIM logic unit to form a final single-arm pre-trigger ( hHODO 3/4 , hEL REAL , hEL CLEAN ). A copy of every pre-trigger (shown in red in Fig. 3.42) is sent toscalers/TDCs (not shown). The final pre-triggers are sent to the front-end of theTI module in ROC 01 , which can receive up to 6 individual pre-triggers. A copyof the accepted trigger (Level 1 or L1 Accept) is sent via fiber optics cables to allthe ROCs associated with the HMS for data readout by all fADC/TDC modules.96igure 3.42: HMS single arm pre-trigger electronics diagram.A copy of certain final pre-triggers are also OR’ed and are ultimately distributedto all ROCs with fADC/TDC modules to function as a reference time associatedwith the L1 Accept. The reference time is subtracted from every channel in everyfADC/TDC module on an event-by-event basis to reduce intrinsic jitter and achievethe design resolution of the module. To guarantee that every event has an associatedreference time, the HMS standard pre-trigger ( hHODO 3/4 ) is OR’ed with the hEL-REAL pre-trigger to guarantee a reference time in the rare case where the hHODO 3/4 fails due to trigger inefficiency which is very small ( (cid:46)
The three planes (X1, Y1, X2) of scintillator arrays and the quartz plane (Y2) formpart of the standard SHMS trigger configuration (see Fig. 3.23). Additional particle97etectors may also be incorporated into the SHMS trigger as required by differentexperiments. The NGC and calorimeter triggers are used for e/π discrimination,whereas the HGC and aerogel ˇCerenkov triggers are used for e/π/p and π/K/p discrimination, respectively, depending on the gas pressure and aerogel materialused.
Hodoscopes Pre-Trigger
Figure 3.43: SHMS hodoscopes electronics diagram. It is important to note thatonly 18 of the 21 quartz bars are currently usable.Each hodoscope plane consists of an array of scintillator paddles (or quartz bars)coupled to a PMT at each end (see Fig. 3.26), so each bar reads out two signals.As shown in Fig. 3.43, for example, hodoscope plane S1X consists of 26 signals (1698addles) read out in the Counting House (CH) patch. Each side of the plane (x13signals/side) is connected to a 64-channel input passive splitter (16 Ch./set). One-third of the signal amplitude is sent via a 16-channel ribbon cable to a 64-channelinput Ribbon-to-BNC converter (16 Ch./set) and subsequently into a 16-channelNIM input fADC. The remaining two-thirds of the signal amplitude is sent to a 16-channel input CAMAC discriminator unit. The SHMS scintillator discriminatorsthresholds and gate widths were set to -30 mV and 60 ns, respectively, whereas thequartz plane discriminators thresholds and gate widths were set to -60 mV and 60ns, respectively.The discriminated signals are sent via two ribbon-cable outputs to C1190 TDCsand scalers (daisy-chained) and to a LeCroy 4564 CAMAC logic unit to form theplane pre-triggers. The logic unit takes four sets of 16-Ch. input ribbon cableand forms a 16-fold OR for each set by default. Further boolean operations aredone through the module backplane by connecting a twisted pair cable to the pincorresponding to the desired boolean operation. For the SHMS hodoscope planepre-triggers, the boolean operations are as follows:
S1X = S1XL (13-fold OR) AND S1XR (13-fold OR)
S1Y = S1YT (13-fold OR) AND S1YB (13-fold OR)
S2X = S2XL (14-fold OR) AND S2XR (14-fold OR)
S2Y = (cid:8) S2Y[1-16]T OR S2Y[17-21]T (cid:9)
AND (cid:8)
S2Y[1-16]B OR S2Y[17-21]B (cid:9)
Once a pre-trigger has been made for each plane, they are sent to a NIM/ECLconverter (Level Translator P/S Model 7126) via twisted pair cables to convert theECL signal (twisted pair) to a NIM signal. The NIM output is then sent to individualsets of a P/S Model 752 NIM logic unit to adjust the widths of each of the planepre-triggers as necessary before making a coincidence. An XY hodoscope planecoincidence ( S1 = S1X OR S1Y, S2 = S2X OR S2Y) is then made by connecting99ach hodoscope XY plane pair into a P/S Model 755 NIM logic unit . A copyof each of the four individual plane pre-triggers is also sent to another set of P/SModel 755 to make a 3/4 or 4/4 plane coincidence (configured via a front-panel knob)which defines the standard hodoscope pre-trigger ( pHODO 3/4 ). An additionalpre-trigger ( pSTOF = S1 AND S2 ) is formed by requiring the coincidence betweenany two of the front ( S1 ) and back ( S2 ) scintillator (or quartz) plane pair to measurethe TOF between any of the two front and back planes. A copy of the hodoscopepre-triggers are also sent to TDCs and scalers via a NIM/ECL converter for timingand counting information. PreShower and Shower Calorimeter Pre-Trigger
The SHMS preshower consists of two sets of fourteen PMT-coupled lead blocksoriented perpendicular to the shower counter blocks. The initial sum was done inthe SHMS electronics hut. The PMT signals in groups of four blocks were summedto form: preSh SUM [1-4]: [1-4]L + [1-4]RpreSh SUM [5-8]: [5-8]L + [5-8]RpreSh SUM [9-12]: [9-12]L + [9-12]RpreSh SUM [13-14]: [13-14]L + [13-14]RThe partial preshower signal sum was sent to the Counting Room patch where afinal sum was made. Two copies of the final sum were sent to a discriminator to formtwo preshower pre-triggers ( pPreSH HI , pPreSH LO ) with a lower and higherthreshold, respectively. The output widths of the P/S Model 755 logic units were set to ∼
100 ns for the SHMS.See HCLOG entry https://logbooks.jlab.org/entry/3501354 . ROC 04 fADCs in the SHMS detector hut.
Heavy and Noble Gas ˇCerenkov Pre-Trigger
The SHMS HGC detector consists of a 1 meter-long, 1.6 meters in diameter cylin-drical tank located between the front and back sets of hodoscope planes (see Fig.3.23). The tank is filled with a gas and has four thin spherical mirrors that focusthe ˇCerenkov light towards four 5-inch PMTs.The SHMS NGC detector consists of a 2 meter-long active length of argon/neongas tank located before the first drift chamber (see Fig. 3.23). The tank is filledwith a gas and has four overlapping mirrors that focus the ˇCerenkov photons to-wards four 5-inch PMTs. The electronics trigger setup for the SHMS ˇCerenkovs isshown in Fig. 3.45 and is very similar to the HMS ˇCerenkovs. Refer to Fig. 3.41and read the corresponding section for a full description of the corresponding triggersetup. 101igure 3.45: SHMS gas ˇCerenkovs electronics diagram.
Aerogel ˇCerenkov Pre-Trigger
The SHMS aerogel ˇCerenkov detector consists of a 110 × × . rectangularaerogel tray coupled to a diffusion box. The diffusion box has seven 5-inch PMTson each side which detect ˇCerenkov light produced by interactions with the aerogelmaterial. The detector is located between HGC and second set of hodoscope planes(see Fig. 3.23). The electronics diagram is the same as in Fig. 3.45. SHMS Single Arm Pre-Trigger
The SHMS single arm trigger is formed exactly as the HMS single arm trigger, withthe exception of the detectors pre-triggers involved which depend on the experiment.Refer to Fig. 3.42 and read the corresponding section for a detailed description ofthe electronic diagrams. 102igure 3.46: SHMS single arm trigger electronics diagram.
Coincidence Trigger Set-Up
In coincidence mode (see Fig. 3.47), the HMS and SHMS pre-triggers are sent to aNIM logic module where the first spectrometer pre-trigger that arrives will open acoincidence time window during which the second spectrometer pre-trigger may ormay not arrive in that time. This will determine whether two spectrometers pre-triggers are correlated with the event originated at the target. If the coincidencepre-trigger is formed, a copy is sent to scalers/TDCs while another copy is sentto the fron-end of the TI module in
ROC 02 which acts as the Trigger Master(TM) in coincidence mode. Once the TM accepts the coincidence trigger, multiplecopies of the L1 Accept are distributed to all HMS and SHMS ROCs (except
ROC ) via fiber obtics cables in all crates for data readout. An additional copy ofthe L1 Accept is also sent to the front-end of the TDCs in ROC 02 . Multiplecopies of the HMS/SHMS pre-triggers (reference times) are also distributed to theirrespective spectrometer ROCs with fADC/TDC modules to function as a referencetime associated with the coincidence trigger.Figure 3.47: Coincidence trigger electronics diagram.
The EDTM system is a new method used in Hall C to measure the total dead timeof the data acquisition (DAQ) system. It consists of introducing a controlled (fixedfrequency) pulse as near as possible to the detectors that form part of the trigger.Ideally, one would send the EDTM pulses at the detector level in the hut such that104oth the real physics and EDTM signals pass through the same electronics. Sincethis is not easy or practical to do, the EDMT logic pulses are injected at the triggerlogic level in the Counting Room.Figure 3.48: EDTM electronics diagram.By design, the EDTM is a real trigger as measured by the electronics and read-out systems. Since the EDTM is invasive to the trigger electronics, its frequencyshould be small enough to minimize the probability of blocking actual physics trig-gers, but sufficiently large to gather the necessary statistics for a precise dead time measurement during the course of a run.Figure 3.48 shows a simplified diagram of the EDTM signal distribution throughthe trigger electronics. The EDTM logic signals (purple) are injected into the triggerlogic where they mix with the physics pre-triggers (magenta). A separate copy ofthe EDTM is also sent to scalers/TDCs to be used in the dead time calculation. If105he EDTM makes it to the front-end of the Trigger Interface (TI) module and getsaccepted (L1 Accept), it has esentially measured both the electronics and computerdead time.Figure 3.49: Cartoon representation of EDTM (purple) and physics (magenta) pre-triggers at the TI module front-end.Figure 3.49 shows the random physics (magenta) and clocked (purple) EDTMpulses at an input channel of the TI front-end where the EDTM has been set toa sufficiently large frequency (1 /T clk ) to ensure that enough EDTM signals get ac-cepted in order to make a statistically significant and reliable dead time calculation.In this example, an EDTM signal has been accepted by the TI which triggered a BUSY signal for a time τ during which all other incoming pre-triggers are blockedcontributing to the DAQ computer dead time. The accepted pre-triggers are dis-tributed to all ROCs for data readout.Over the course of a run, the total dead time ( T TDT ), or alternatively, the totallive time ( T TLT ) in terms of the EDTM is defined as T TDT ≡ − T TLT = 1 − N edtm , acc N edtm , scl , (3.13)where N edtm , acc is the number of accepted EDTM counts obtained by requiring anon-zero hit on the EDTM TDC spectrum, and N edtm , scl is the number of EDTM106caler counts regardless of whether or not the EDTM was accepted. In reality,frequent beam trips occur during the course of a run which makes this calculationbiased since one can measure live times of ∼ ∼ T CLT = N phy , acc N phy , scl , (3.14)where N phy , acc is the number of accepted physics triggers obtained by requiring azero hit on the EDTM TDC spectrum (EDTM rejected by TI) and N phy , scl is thenumber of physics trigger scaler counts after having subtracted the EDTM scalercounts. The electronic live time can then be obtained from the following formula: T TLT = T CLT · T ELT , (3.15)where T ELT is the electroninc live time expressed as a fraction (not percent).In the E12-10-003 experiment, the data from the main analysis was read outby an unprescaled coincidence trigger which simplified the live time calculationsdiscussed above, since there was no need to divide by a prescale factor or accountfor simultaneous multiple input triggers. For a more detailed discussion on the livetime calculations and its correction factors see Refs. [144, 145].107HAPTER 4
GENERAL HALL C ANALYSIS OVERVIEW
The general Hall C analysis procedure for experiments in the 12 GeV era is discussed.The procedure outlines the first necessary steps in the data analysis regardless of thenature of the experiment. These include, but are not limited to, setting referencetime cuts, detector time window cuts, and performing detector calibrations. Opticschecks and optimization analysis of the SHMS reconstruction matrix using H( e, e (cid:48) ) p elastic data are discussed for this experiment. Finally, the data-to-simulation com-parisons of the spectrometer acceptance as well as the event selection criteria for H( e, e (cid:48) ) p elastics and the H( e, e (cid:48) p ) n reaction are shown. The first step in Hall C data analysis is to make sure the reference time cuts areset properly, as one needs to make sure the reference times correlated with thetrigger are selected. The reference time signal is defined as a copy of either oneor multiple OR’ed pre-trigger logic signals described in the electronics diagram ofFigs. 3.42, 3.46 and 3.47. The reference time is distributed to all fADCs andC1190 TDC modules of all Read-Out Controllers (ROCs). The fADC and TDCmodules register either analog (fADC) or discriminated logic (TDC) signals fromevery detector output as well as the corresponding reference time signal generatedby the trigger electronics. The main objective of the reference time signal is 2-fold: As with any hardware electronics signal, the reference time signal is arbitrary and is onlyuseful when compared relative to any other arbitrary signal such as to measure a relativetime. Before being sent to an fADC, the reference time logic signal must be converted into ananalog signal. This is done with a passive circuit. the reference time signal serves as a common stop (initiates a look-back win-dow) for all detector input channels in each fADC/TDC module • the reference time signal is used to determine time intervals from the rawdetector signals sent to the TDC moduleFigure 4.1: Cartoon illustrating the synchronization of the a detector signal withthe internal clocks of a C1190 TDC Module.Before modern TDCs such as the C1190 TDC [141], in the original fastbus TDCmodules [146] a L1 Accept (accepted trigger) was sent to the front-end of the moduleand acted as a common start time to all channels of the module as well as initi-ated data readout. The common start time was measured relative to the stop signalwhich was provided by the individual input channels on the TDC module. This timedifference was converted to a number and histogrammed to form a TDC spectrumof counts vs. channel number. 109n modern TDCs (see Fig. 4.1), the L1 accept, which has an intrinsic 4 nsjitter , is sent to the front-end to initiate data readout. The detector signal (green)sent to the front-end of the module serves as the TDC Start and is synchronizedwith the internal 40 MHz clock of the TDC (slow clock). The detector signal (TDCStart) latches onto the leading edge of the next 40 MHz clock cycle, which meansthe signal could have landed anywhere in a 25 ns range between the previous andnext clock cycle. As a result, a 25 ns jitter arises intrinsically when the raw TDCsignal is measured relative to the L1 accept, which has an additional 4 ns jitter.This means that the raw TDC detector signals cannot be determined better than29 ns resolution, which is well above the module specifications of 0.1 ns resolution.To improve the timing resolution, the TDC uses a second internal clock (fast clock)at 10 GHz or 0.1 ns periodicity. The reference time (copy of pre-trigger which iseffectively a delayed L1 accept) is sent to the TDC module at a time delay relativeto all the detector signals and effectively serves as the
TDC Stop , which latchesonto the leading edge of the next clock cycle of the 10 GHz high resolution clock,and initiates a look-back time window (usually a few µ s) corresponding to the fullTDC spectrum (e.g., see Fig. 4.2). This internal reference time, which is known toapproximately 0.1 ns, is subtracted from the raw detector signal TDC time, thereby,improving the timing resolution of the detector signals to ∼ hcana during the analysis replay. When using the reference time, hcana chooses thefirst hit in the time window if multiple hits are present per event. In this scenario, In electronics terminology, jitter refers to a small, irregular variation or unsteadiness inan otherwise periodic signal. good hit and the wrong reference time wouldbe chosen resulting in the wrong time being subtracted in the fADC/TDC spectra.By placing a reference time cut, the analyzer then considers the first hit after thecut, which is likely to be a good hit .As an example, consider the H( e, e (cid:48) ) p elastic coincidence run 3377, which hadthe highest SHMS rate of elastics taken during the E12-10-003 experiment.Figure 4.2: SHMS reference time spectrum for coincidence run 3377 of the E12-10-003 experiment. Background hits are shown in blue and good hits in red. Inset:Multiplicity histogram corresponding to the reference time spectrum.Figure 4.2 shows an uncorrected reference time spectrum in the SHMS drift cham-bers crate (ROC 06) where the red spectrum represents the prompt peak correspond-ing to the reference time signal events and the blue spectrum represents backgroundevents corresponding to signals other than the reference time. The inset plot showsthe multiplicity histogram corresponding to the TDC spectrum. The multiplicityhistogram shows the total number of counts or L1 accept ( y -axis) versus the number111f TDC hits (or multiplicity) corresponding to each event where the number of TDChits are defined as the total hits in the TDC readout window for any given event.For example, the inset of Fig. 4.2 shows that most of the events had a multiplictyof three, which means that for every event, the TDC readout window registered 3hits. In this case, the 3 hits correspond to three reference time signals, which wereOR’ed and used as an effective reference time. The other multiplicities ( > T at a givenphysics rate R is given by P ( λ ; k ) = e − λ λ k k ! , (4.1)where λ = R ∆ T and k is the number of TDC hits. From the multiplicity in Fig. 4.2,most reference time events had 3 good hits (HODO 3/4, EL-REAL, EL-CLEAN).For simplicity of the calculation, we redefine 3 good hits as a single good hit. Then,from Eq. 4.1 and Fig. 4.2, the probability of finding 2 hits within the drift chambertime window is P ( λ ; k ) = e − R ∆ T ( R ∆ T )
2! = 0 . P data = 13 346600602000 = 0 . ≤ ∼
19% of the events would have the incorrectreference time and a lower tracking efficiency by ∼ : T logicreftime , init ≡ p(h)HODO 3/4 OR p(h)STOF OR p(h)EL-REAL OR p(h)EL-CLEAN (4.4)where the logic pre-trigger signals described in Eq. 4.4 have been delayed in timerelative to each other and the p(h) refers to prefix used in the software to denotethe SHMS (HMS). Figure 4.3 shows a visual representation of Eq. 4.4 of how typicalreference time logic signals might appear on an oscilloscope. On January 2018, theSTOF trigger was removed from this definition . Finally, on August 2018, EL-CLEAN was removed from the reference time definition as well. It was determinedthat any pre-trigger that required the HODO 3/4 was unnecessary and redundantto have in the reference time definition so they were removed. See December 2017 HC-Log Entry https://logbooks.jlab.org/entry/3501198 . See HC-Log Entry https://logbooks.jlab.org/entry/3519686 . See HC-Log Entry https://logbooks.jlab.org/entry/3585301 . T logicreftime ≡ p(h)HODO 3/4 OR p(h)EL-REAL (4.5)as the EL-REAL did not require a HODO 3/4, and in the rare instances the latter ismissing, the former can be used as a reference time. The STOF was also completelyremoved from the trigger definition and the reference time was redefined as HODO3/4. Since STOF was removed, the EL-REAL now requires a HODO 3/4 and itwas determined that this reference time (EL-REAL) was no longer needed so it wasremoved from the reference time definition as well. Therefore, as of the current runperiod (Spring 2020), for the A1n/d2n experiment, the reference time is defined tobe: T logicreftime ≡ p(h)HODO 3/4 , which is the lowest level pre-trigger required toform all other pre-triggers.Figure 4.4 shows the reference time histograms with the set reference time cutsfor the 80 MeV/c setting of this experiment. The same reference time cuts usedfor the higher missing momentum setting (580/750 MeV/c) as there should not be114 significant shift in the reference time signals provided that there is no change inthe hardware (signal cable lengths, threhsolds, etc.) or the DAQ ROCs readoutwindow, which can only be modified by the experts and should not change duringthe course of an experiment. See Ref. [147] for a detailed list of the reference timesas well as instructions on how to set the reference time cuts in the analysis.Figure 4.4: SHMS (top panel) and HMS (bottom panel) reference time cuts forcoincidence run 3289 of the E12-10-003 experiment. The conversion from TDCchannel to time is ∼ The next step in the analysis procedure is setting up the detector time window cuts.These are necessary to reduce sources of background that slip into the detector timewindows when detecting the physics signals of interest. The time window cut ismade on a time difference between the fADC and TDC times on a PMT basis for115ll the detectors except the drift chamber, which cut on the raw drift times for eachplane. The time difference is defined in hcana asAdcTdcDiffTime = TdcTime[ipmt][jhit] - AdcPulseTime[ipmt][jhit]AdcTdcDiffTime = HodoStartTime - AdcPulseTime[ipmt][jhit]where the HodoStartTime is the hodoscope time projected at the focal plane , andthe (TdcTime[ipmt][jhit], AdcPulseTime[ipmt][jhit]) are the TDC and fADC pulsetime, respectively, for any given i th PMT and the j th hit within the correspondingi th PMT fADC/TDC look-back time window. The pulse times are timing signalscorresponding to a detector output that have been measured relative to the ref-erence time and are therefore considered corrected pulse times as opposed to theraw detector arbitrary pulse times that are sent to the front-end of the module. Ifthe event is truly a physics event originating from the target, then in principle, thetime difference should be a δ -function, however, due to the finite detector/moduletiming resolution, it has a finite width and gaussian shape. Events that are far awayfrom the main peak are clearly out-of-time indicating that the fADC pulse time andTDC time are NOT correlated with the same event, and a time window cut mustbe made. With respect to the drift chambers, a cut on the raw drift time spectrumis made to reduce the background from multiple TDC hits.Figure 4.5 shows typical examples of the detector time window cuts on an SHMShodoscope plane, a drift chamber plane and calorimeter block. The plots show dis-tributions with (red) and without (blue) a 3-hit multiplicity cut. A narrow peakis clearly distinguishable in all plots with the dashed lines representing the timewindow cut region. The detector time window cuts were determined for every PMT The focal plane is an imaginary mid-plane in-between the first and second drift chambers.Its nominal origin coincides with the focus of the spectrometer for momentum acceptance δ = 0%, Y tar = 0 cm. (see Fig. 3.24). T . After selecting the right reference times and setting proper detector time windowcuts, detector calibrations can be started. Ideally, one would use specific runs forcalibrations in which most of the focal plane (refer to footnote 7) is illuminated.Sometimes, a magnet de-focused run is used, however, one has to be careful as somecalibrations actually depend on reconstructed quantities at the target, and hence,knowledge of the reconstruction optics elements. In this case, it is recommended touse single-arm runs over coincidence runs, as the former will be less constrained andoccupy a larger region of the focal plane.117 .3.1 Hodoscopes
When a particle traverses a hodoscope plane (see Fig. 4.6), depending on the tra-jectory, any paddle (or quartz bar) could in principle be hit.Figure 4.6: Cartoon of individual scintillator paddles to illustrate the various timingcorrections applied. Note: Timewalk Effect illustration reprinted from Ref. [148].At this stage, the raw TDC signal has multiple unwanted timing offsets that mustbe subtracted to obtain the true arrival time of the particle at the hodoscope plane.The corrected TDC time is then used to determine the correct particle velocity, β = v/c . The general expression for the corrected TDC time for a hodoscope PMTcan be expressed as: t Corr = t RAW − t TW − t Cable − t prop . − t λ , (4.6)118here the corrected TDC time represents the particle arrival time at the scintillatorpaddle (or quartz bar). The corrections are summarized as follows: • Time-Walk Corrections , t TW : For analog signals arriving at the LeadingEdge Discriminators , the logic signal is produced when the signal crosses thediscriminator threshold and therefore depends on the signal amplitude (seeFig. 4.6). The fADCs do not have this disadvantage since they correct forFigure 4.7: Fit correlation between TDC pulse time and fADC pulse amplitude(left). Time-walk corrected pulse time versus fADC pulse amplitude shows no cor-relation (right).time-walk internally, and as a result, the fADC pulse time is not correlatedwith the signal amplitude. The algorithm used by the fADCs to effectivelyremove time-walk effects is similar to that of a constant fraction discriminator(CFD) timing algorithm. In the CFD algorithm, the logic signal is generatedat a constant fraction of the signal peak height which makes the discriminationof the signal independent of the pulse amplitude as illustrated in Fig. 4.7 ofRef. [148] or Fig. 7.4 of Ref. [149]. To correct for the TDC time walk, thefADC pulse time is used as a reference by taking the TDC-ADC pulse timedifference plotted against the fADC amplitude. A model function is fit to this119orrelation, and the parameters extracted are used to correct the TDC time(see Fig. 4.7). • Cable Time Corrections , t Cable : This correction takes into account the factthat the analog signal has to propagate across signal cables from the PMT allthe way into the Counting House electronics rack into the TDC. To determinethis correction, a correlation between time-walk corrected time and hodoscopepaddle track position is fit to extract the velocity of propagation across thepaddle, and the cable time offset. The propagation velocity is determined fromFigure 4.8: Fit correlation between track position along paddle and time-walk cor-rected (TDC-ADC) time difference used to determined the propagation velocityacross the paddle.the distance and time of the hit from the center of a paddle. The time isdetermined by taking half of the time-walk corrected TDC time differencebetween the two ends of a paddle. The half is to ensure that if the particlehits the edge of the paddle, half of the total propagation time across the entirepaddle is taken to obtain the time from the edge to the center. The hit distanceis determined by extrapolating the distance determined by the drift chambers120rom tracking. The correlation between time and distance is fit to extractthe propagation velocity and the cable time difference between the two ends(see Fig. 4.8). The cable time offset parameter is determined for all paddlesand the parameter is read by hcana, and added as a correction factor to thetime-walk corrected TDC time. • Hodoscope Plane Time Difference Corrections , t λ : This correction ac-counts for any additional time difference (other than the particle propagationtime to travel across the two paddles) between any two distinct scintillatorpaddles in different hodoscope planes.Figure 4.9: Illustration of all possible time difference combinations that are consid-ered in this correction.Six possible combinations between the four hodoscope planes are consideredwhen correcting for the time difference between any two of their paddles (seeFig. 4.9). The combinations of all six possible time differences were expressedas a system of 6 linear equations that were solved using the method of Single-Value Decomposition (SVD) to determine the calibration coefficients for eachindividual PMT. 121igure 4.10 shows the representative plots of the calibration results for the 580MeV/c setting of the E12-10-003 experiment. The β distribution shown uses thetracking information from the drift chambers, whereas the dashed lines use theformula: β = P c / (cid:112) m + P , where P c and m represent the spectrometer centralmomentum and particle mass, respectively.Figure 4.10: HMS/SHMS hodoscope calibration results for the 580 MeV/c settingof E12-10-003. The histograms are plotted using the drift chamber tracking infor-mation to determine β .The HMS hodoscope β distribution shows three distinctive peaks formed fromthe coincidence with the electrons in the SHMS. The β peak corresponding to theprotons are from quasi-elastic scattering off the liquid deuterium target, while thedeuteron and triton are produced from knockout reactions, most likely, quasi-elasticelectron scattering off the aluminum walls. The dashed lines were determined basedthe assumption that the particle mass is either that of a proton ( H), deuteron( H) or triton ( H) and momentum acceptance, δ = 0%. In the SHMS, the peak atexactly β = 1 clearly represents the electron as its mass is negligible compared to theenergy and momentum. See Ref. [150] for a detailed explanation of the hodoscopecalibration procedure. 122 .3.2 Drift Chambers When a charged particle traverses the drift chambers, it passes through 12 wireplanes, each surrounded by two cathode planes. The wire planes consist of alter-nating field and sense wires. The field wires and the cathode planes are kept at anegative voltage while the sense wires are kept grounded (0 potential). The potentialgradient creates an electric field oriented outwards from the sense wires.Figure 4.11: Illustration of a single drift cell (top view) in a drift chamber. Thedashed lines represent equipotential surfaces where the electric field is perpendicularto the contour. Figure adaptation from G. Niculescu.As the charged particle passes through a single drift cell, it ionizes the gas atomsin the chamber gas mixture, which causes the free electrons from the ionized gasto drift towards the sense wire producing a measurable current signal. The sensewire signals are pre-amplified and read out by 16-channel input discriminators whichproduce logic signals that are sent to the TDC via 16-channel ribbon cables. TheTDC registers the time when the signal is registered. This time contains the cabledelay it would have taken the signal to propagate across the sense wire, through the123ibbon cable and into the TDC if the particle would have passed through the sensewire itself. The drift time is the time it takes the free electrons to drift towards thesense wire and can be expressed as, t D = ( t meas − t REF ) − [( t wire + t cable ) (cid:124) (cid:123)(cid:122) (cid:125) t (cid:48) − t REF ] (4.7)where t meas is the measured time by the TDC, and t (cid:48) is the time it takes the signalto propagate across the sense wire, through the cable and into the TDC if the trackwere to pass directly through the sense wire. These times are measured relative toa reference time, t REF , used by the TDC as a common stop.A coarse reconstruction of the track can be carried out with only the knowledgeof the wires that fire from a physics event. Knowing the associated drift times of thewires that were hit, however, allows for a more precise track reconstruction, as thedrift time can be converted to a drift distance from the determination of the time-to-distance maps resulting from the drift chamber calibration as described below.For a collection of events illuminating all cells in any given wire plane, one ob-tains a drift time distribution for each sense wire that can be averaged over an entiregroup (up to 16 wires in a discriminator card) or over the entire plane to form adrift time distribution per plane (see Fig. 4.12).Associated with each drift-time spectrum is a quantity called “ t ”. The t corre-sponds to the location in the histogram where the ionized particle comes in contactwith the wire. If its value is anything other than zero nanoseconds (0 ns), it isinterpreted as the value by which the drift time must be shifted in order to assurethat t = 0 ns. All subsequent times in each drift time spectra are measured relativeto this time. 124igure 4.12: HMS drift time spectrum for plane 1x1. Inset: Fit of the leading edge ina drift time spectrum corresponding to a group of wires from a specific discriminatorcard of plane 1x1.Figure 4.13: HMS drift times versus wire number for plane 1x1 before t correction.Inset: Same as in Fig. 4.12. 125igure 4.14: Fit of the leading edge in an HMS drift time spectrum for the wirecard t .Figure 4.15: HMS drift times versus wire number for plane 1x1 after t correction.Inset: Drift time for wire group t correction. 126he t for each plane is determined by calculating the t for individual wires ineach plane and taking a weighted average. The t for individual sense wires (or wiregroup) is determined by a linear fit of the drift time spectra at around 20% of thepeak ± ∆ t for each sense wire (or wire group), where ∆ t is the fit range. The linearfit is then extrapolated to the horizontal axis (drift time), and this extrapolatedvalue is defined as t (see Fig. 4.14).Depending on the calibration method used, the t correction is applied on a sensewire basis from the individual wire fits, or on a wire group basis where the same t correction is applied to all wires of a group. The latter procedure is usually bettersince the edge wires have very low statistics that cause the fit to fail, whereas agroup of wires will have sufficient statistics for a successful fit.To determine the drift distances from the drift time spectra, it is assumed thatthe drift distances are uniformly distributed across the cell. This assumption isbased on the fact that a cell is uniformly illuminated with particles, and the ionshave an approximately uniform drift velocity, which implies there should be nopreferred drift distance for any ionized charge. Mathematically, the drift distance iscalculated as d drift ( τ = T ) = ∆2 (cid:82) T ≤ t max t F ( τ ) dτ (cid:82) t max t F ( τ ) dτ , (4.8)where ∆ is the cell width and F ( τ ) is the drift time distribution integrated from t =0 ns to some arbitrary time T ≤ t max where t max is the maximum drift timewithin a cell. In the limiting case of Eq. 4.8, the drift distance becomes d drift = , τ = 0 ns0 . , τ = t max , (4.9)which is the expected drift distance at the sense wire ( τ = 0 ns) and at the edgesof the cell ( τ = t max ). 127ue to the finite resolution of the TDC and other factors involved, the drifttimes are not determined to infinite precision and the integral in Eq. 4.8 becomes asum over a finite bin width, (cid:90) τ F ( τ ) dτ → (cid:88) bin( τ ) F ( τ ) (cid:124) (cid:123)(cid:122) (cid:125) bin content · ∆ τ (cid:124)(cid:123)(cid:122)(cid:125) bin width . (4.10)Re-writing Eq. 4.8 in terms of the finite sums in Eq. 4.10, one obtains bin( t + T ) (cid:80) bin( t ) F ( τ )∆ τ bin( t + t max ) (cid:80) bin( t ) F ( τ )∆ τ → N tot bin( t + T ) (cid:88) bin( t ) F ( τ ) (4.11)The ratio in Eq. 4.11 is the lookup value used to convert drift time to distance foran arbitrary drift time bin, T . The numerator represents the sum of all bin contentsup to a drift time T , and the denominator represents the sum over the bin contentsof all drift times up to t max , in a given plane. The bin width, ∆ τ , is a constantduring the sum, therefore it is cancelled, which simplifies the equation as a ratio ofthe sum of bin contents (up to some drift time) and the sum over all bin contents(up to a maximum, t max ), N tot .The results of this calibration are per-plane look-up tables that map any givendrift time to a drift distance in that plane. The drift distance for the X-plane ofHMS drift chamber 1 is shown in Fig. 4.16. As expected, the drift distances for allplanes are uniformly distributed across the cell width.The best way to determine the drift chamber performance is by measuringthe spatial resolution, or how well it can measure the position of particle tracks.This measurement is done through the determination of per-plane residuals. Fora particle traversing at least 4 planes of the chamber, a collection of space-points(X,Y) is measured based on the wires that fired. The space points are fit with astraight line such as to minimize the chi-square, and obtain a best fit. The line fit128igure 4.16: HMS drift distance for plane 1x1 before (red) and after (blue) calibra-tion of the drift maps.Figure 4.17: HMS fit drift residuals for plane 1x1 before (red) and after (blue) cali-bration of the drift maps. The standard deviation ( σ ) from the fit is representativeof the spatial resolution. 129s then compared to the measured space-point from the plane wires that fired, andthe difference is called the residual for that plane. The residuals are calculated onan event by event basis, and should be centered around zero (see Fig. 4.17).For the E12-10-003 experiment, typical residuals per plane were found to be onaverage ∼ µ m for the SHMS and ∼ µ m for the HMS drift chambers. SeeRef. [151] for details on the drift chambers calibration procedure. The calorimeter in each spectrometer is used primarily for particle identificationbased on the incident particle track momentum and the subsequent energy showersdetected by the PMTs coupled at the ends of the lead glass blocks. The signalsare sent to fADCs where the signal amplitude is proportional to the fADC channel,which is subsequently converted to the corresponding energy deposited at the PMT.In order to make the calorimeter trigger efficiency uniform across the calorimeterplane, the output signals were matched by adjusting the PMTs High Voltage (HV)to make the signal amplitudes as similar as possible [140]. This resulted in thePMTs gain being different across the vertical or dispersive direction since particleswith higher momentum (lower bend angles) impact the lower calorimeter blocksand deposit more energy (larger signal amplitude), whereas less energetic particlesimpact higher blocks and deposit less energy in the calorimeter. This results ina gain variation across the calorimeter plane that is approximately equal to thespectrometer momentum acceptance.The purpose of the calibration is thus to correct for the gain variations on aPMT basis across all the calorimeter blocks. The formula used for the deposited130nergy in the i th PMT is estimated to be [140]: (cid:15) i = c i · ( A i − A ped ,i ) · f ( y ) , (4.12)where c i is a calibration constant, A i is the raw fADC signal, A ped ,i is the correspond-ing fADC pedestal , and f ( y ) is a correction factor for the light attenuation acrossthe horizontal hut coordinate, y . The standard calibration algorithm minimizes thevariance between the total energy deposited in all channels ( E DEP = Σ e i ) relativeto the measured momentum of an incident electron at the face of the calorimeter.The algorithm was developed by Ts. Amatuni in the early 1990s.Figure 4.18 shows the representative calibration plots for the SHMS calorimeterusing the combined runs corresponding to the 580 MeV/c setting of E12-10-003. Theupper (A) and lower (B) left plots show the total energy deposition divided by thecentral spectrometer momenta ( E dep /P ) before and after calibration, respectively.Since the electron mass is negligible compared to its momentum, the total energy de-posited by the electron is approximately equal to its momentum before entering thecalorimeter and therefore its ratio is unity. The upper right plot (C) shows the cor-relation between the deposited energy by the electron in the PreShower and Showernormalized by the momentum of the best reconstructed track. The correlation onthis plot shows how the PreShower can be used to augment the electron detectioncapabilities of the SHMS calorimeter. Finally, the lower right plot (D) shows a cor-relation between the SHMS momentum acceptance ( δ ) and the normalized energydeposited in the calorimeter, which demonstrates that the energy deposited by theelectron is uniform across the entire calorimeter dispersive direction. See Ref. [152]for instructions on how to perform the calorimeter calibration. The pedestal is an electronic offset at the input to the digitization stage.
The calibration procedures for the threshold gas and aerogel ˇCerenkovs are verysimilar. The calibration is based on identifying the location of the single photo-electron (SPE) peaks relative to the pedestal of the corresponding PMTs. Afterpedestal subtraction, each SPE peak is fit with a gaussian. The mean of the gaus-sian represents the corrected fADC channel corresponding to the SPE peak, fromwhich a conversion factor between the SPE peak and fADC channel can be obtainedper PMT channel. Each detector calibration procedure employs slightly differentmethods to identify the SPE peaks as well as different particle identification require-ments. See Ref. [153] for more details on the gas ˇCerenkov detetcor calibrations. Forthe aerogel detector calibration, refer to the official Kaon-LT experiment page [154].132 .4 Hall C Coordinate System
Before discussing the spectrometer optics checks and optimization in the next section(see Section 4.5), the coordinate systems used to reconstruct the events at the targetreaction vertex must be introduced.Figure 4.19: Top view of the spectrometer and hall coordinate systems in Hall C.The hall (or vertex) coordinate system is denoted by the subscript “h” in Fig.4.19, where + z h is parallel to the incident beam direction, + x h is oriented beam-leftand denotes the horizontal beam position, and + y h points towards the ceiling anddenotes the vertical beam position. These coordinates are used to describe the beamposition at the reaction vertex. 133he spectrometer coordinate system is denoted by the subscript “s” where + z s is aligned with the spectrometer central ray rotated by the central spectrometerangle θ c , + x s points in the dispersive direction (towards the hall floor), and + y s is oriented beam-left in the non-dispersive direction. The detector hut coordinatesystem is defined by a simple spectrometer coordinate rotation about the y s -axis toalign + z s with the dipole bend.From Fig. 4.19 consider the following case in which the incident electron isaligned with the beam axis (shown in blue). The electron interacts with the targetfoil (gray slab) at the reaction vertex z v and scatters at angle θ c parallel to thespectrometer central-ray. The reconstructed event is projected along the dashed(black) line and is denoted as Y tar . In reality, the electron can also scatter rela-tive to the spectrometer central ray determined by the tangents, tan( φ ) = dy s /dz s (in-plane) and tan( θ ) = dx s /dz s (out-of-plane). Since the spectrometer apertureangles are usually very small (see Table 3.2), the tangents can be approximatedby tan( φ ) ≈ φ and tan( θ ) ≈ θ , which are commonly referred to as Y (cid:48) tar and X (cid:48) tar ,respectively. These derivatives are interpreted as angular distributions of the scat-tered particles relative to the spectrometer central-ray. Similar quantities can alsobe derived in the hut coordinate system using the focal plane variables, ( X fp , Y fp ),to obtain dX fp /dZ fp ≈ X (cid:48) fp and dY fp /dZ fp ≈ Y (cid:48) fp .In a more general case, the electron (shown in red) incident on the target isoffset by an amount x beam and scatters parallel to the central ray. As a result, thereconstructed Y tar is offset by an amount x beam cos( θ c ) which is geometrically equiv-alent to a spectrometer mispointing along y s . Furthermore, if the electron scattersat an arbitrary angle φ , the reconstructed Y tar is further offset by an amount L · φ .Combining all these offsets and adding an arbitrary y -mispointing offset ( y mispoint ),134 tar can be expressed in its most general form as Y tar + y mispoint9 = z v sin( θ c ) + x beam cos( θ c ) + L · φ = z v sin( θ c ) + x beam cos( θ c ) + [ z v cos( θ c ) − x beam sin( θ c )] Y (cid:48) tar = z v [sin( θ c ) + Y (cid:48) tar cos( θ c )] + x beam [cos( θ c ) − Y (cid:48) tar sin( θ c )] (4.13)Alternatively, it is also useful to express Eq. 4.13 in terms of the reaction vertex, z v = Y tar + y mispoint − x beam [cos( θ c ) − Y (cid:48) tar sin( θ c )]sin( θ c ) + Y (cid:48) tar cos( θ c ) . (4.14)Equation 4.14 is the most general form of the z -reaction vertex in terms of measur-able quantities. The difference between the z -reaction vertex in both spectrometerswas used as an event selection criteria for this experiment (see Section 4.6). The commissioning of the HMS/SHMS optics took place during the 2017-18 runperiod and underwent multiple revisions of the reconstruction matrix elements forboth spectrometers during that period [124, 125]. This section presents the opticsoptimization checks and procedures done on the HMS and SHMS for this experi-ment (E12-10-003) on April 2018. At the time, E12-10-003 also served as part of thegeneral optics commissioning as during data-taking, it was found that the SHMSQ3 magnet had an unnecessary correction in the matrix elements. As a result, thedata for this experiment is divided into two sections. Only the section after thecorrection in the SHMS optics was used in the optimization procedure. The spectrometer mispointing is defined as a parallel displacement of the spectrometercentral ray either horizontally ( y -mispointing) or vertically ( x -mispointing). The mis-pointing was determined by survey at various spectrometer angles and a function was fitto the x - and y -mispointing data [124, 125], separately. H( e, e (cid:48) ) p elastic runs were taken at different configurations such as to cover theentire HMS momentum range corresponding to the H( e, e (cid:48) p ) n reaction kinematics.The original and corrected H( e, e (cid:48) ) p kinematics are summarized below.Run HMSAngle [deg] HMSMomentum [GeV/c] SHMSAngle [deg] SHMSMomentum [GeV/c]3288 37.338 2.938 12.194 8.73371 33.545 3.48 13.93 8.73374 42.9 2.31 9.928 8.73377 47.605 1.8899 8.495 8.7Table 4.1: Original H( e, e (cid:48) ) p elastic kinematics in E12-10-003.Run HMSAngle [deg] HMSMomentum [GeV/c] SHMSAngle [deg] SHMSMomentum [GeV/c]3288 37.338 2.9355 12.194 8.53423371 33.545 3.4758 13.93 8.53423374 42.9 2.3103 9.928 8.53423377 47.605 1.8912 8.495 8.5342Table 4.2: Corrected H( e, e (cid:48) ) p elastic kinematics in E12-10-003.Spec δθ [rad] δφ [rad] X (cid:48) tar -offset[rad] Y (cid:48) tar -offset[rad]HMS 0.0 1 . × − . × − . × − SHMS 0.0 0.0 0.0 0.0Table 4.3: Spectrometer offsets determined from H( e, e (cid:48) ) p elastic run 3288 in E12-10-003. See Section 4.5.3 of this dissertation for more information.Since this is a coincidence experiment, the spectrometers are highly correlated,which makes the optics optimization more complicated as changes in one spectrom-eter can affect the other spectrometer. Based on the kinematics, it was determinedto focus on the HMS first, as the momentum is well below the dipole saturation ( ∼ .5.1 HMS Optics Check The procedure to check the HMS Optics involves determining whether a centralmomentum correction is needed and check that the HMS momentum fraction δ isindependent of the HMS focal plane variables for constant momenta. That is to say,that there should not exist a correlation between the momentum fraction and thefocal plane variables. HMS Central Momentum Correction
Since the H( e, e (cid:48) ) p reaction is used and the HMS is set to detect protons, one cancalculate the proton momentum as follows: P calc = 2 M p E b ( E b + M p ) cos( θ p ) M + 2 M p E b + E sin ( θ p ) , (4.15)where E b is the initial beam energy and θ p is the reconstructed proton angle. Themeasured proton momentum, P meas , depends on the δ from the following definition: δ
100 = P meas − P P → P meas = P (1 + δ
100 ) , (4.16)where P is the central momentum of the spectrometer and δ is the fractional devi-ation of the particle momentum from the central momentum in %.From the measured and calculated momentum, the momentum difference is de-fined as ∆ P = P calc − P meas . (4.17)In Eq. 4.17, it is assumed that the beam energy and HMS angle are well known,which may not entirely be true, but serves as the best available approximation. Themomentum difference, ∆ P , is determined for data and SIMC independently on anevent-by-event basis in terms of ( θ p , δ ). It is expected that ∆ P be near zero inSIMC, as the δ -reconstruction is well described by the TOSCA models, however137n data, this may not be the case, as the NMR probe location in the HMS haschanged since the 6 GeV era and the magnetic field for the central momentum maybe different from what is expected.Figure 4.20: Comparison of HMS momentum difference (∆ P ) between data andSIMC. The inset shows the calculated and measured HMS momentum distributionfor data.From the mean of the fit in Fig. 4.20, ∆ P data is ∼ P SIMC ,or equivalently, P measdata > P calcdata . The data momentum correction factor can be deter-mined as follows: f HMScorr = 1 − ∆ P SIMC − ∆ P data P , (4.18)and the corrected HMS momentum can then be expressed as P HMScorr = P HMSuncorr · f HMScorr . (4.19)Figure 4.21 shows the difference in the mean of the fit for data and SIMC beforeand after the HMS momentum corrections. After correction, the difference betweendata and SIMC is within ∼ P diff = ∆ P SIMC − ∆ P data , before andafter applying the momentum correction to data.Figure 4.22: HMS momentum difference for data, before (blue) and after (green)applying the momentum correction (2 nd iteration) to data.139ote that the corrections applied are after a second iteration, once the spectrometeroffsets were determined.Figure 4.22 shows the difference between the HMS calculated and measureddata momentum before and after applying a momentum correction during the 2nditeration. The corrected data momentum difference has clearly shifted towards zero,which indicates a successful momentum correction. HMS δ Check
To check the HMS delta ( δ ) reconstruction, the HMS fractional momentum isdefined as ∆ P frac = P calc − P meas P meas (4.20)and is plotted as a function of the HMS focal plane variables.Figure 4.23: HMS fractional momentum difference vs. focal plane variables for H( e, e (cid:48) ) p elastic run 3288. The HMS (or SHMS) δ is defined as δ ≡ P − P P , where P and P are the reconstructedparticle momentum ( P ) and central spectrometer momentum ( P ), respectively. H( e, e (cid:48) ) p elastic run 3371.Figure 4.25: HMS fractional momentum difference vs. focal plane variables for H( e, e (cid:48) ) p elastic run 3374. 141igure 4.26: HMS fractional momentum difference vs. focal plane variables for H( e, e (cid:48) ) p elastic runs 3377.Figures 4.23, 4.24, 4.25, and 4.26 show that ∆ P frac is uncorrelated across eachof the HMS focal plane variables, which demonstrates that the δ -reconstruction isalready optimized for the HMS. Now that the HMS optics are well understood, onecan move on to the SHMS optics checks. Similar to the HMS, the procedure to check the SHMS optics involves determiningthe central momentum correction and checking that the reconstructed δ is uncorre-lated with the SHMS focal plane variables. Additional checks on the Y tar , Y (cid:48) tar and X (cid:48) tar reconstruction variables may also be needed as the SHMS optics optimizationis still incomplete. 142 HMS Central Momentum Correction
Figure 4.27: Missing energy spectrum for H( e, e (cid:48) ) p elastic run 3288 before centralmomentum correction.To determine the SHMS central momentum correction, one starts with the missingenergy definition for elastic scattering on hydrogen, H( e, e (cid:48) ) pE m = ( E b − E (cid:48) ) + M p − E p , (4.21)where E (cid:48) and E p are the electron and proton final energies, respectively. Since itwas assumed that the beam energy, and the HMS momentum are well known, anydeviation from the expected missing energy is attributed to the electron momentumin the SHMS. The expected location of E m ideally would be at zero since H( e, e (cid:48) ) p is a completely determined system. However, due to the energy loss and radiativeeffects, the peak has a small offset from zero ( (cid:46)
10 MeV), which can be simulated.The measured and simulated E m are then compared to determine if the SHMS cen-tral momentum needs to be corrected. 143he SHMS central momentum was kept fixed during the entire experiment, whichwould suggest that the missing energy offset would be the same for the four elasticruns after the HMS momentum correction. This was found to be the case due tothe fact that the spectrometer offsets have not been determined at this stage. Alter-natively, it was decided to only focus on finding the central momentum correctionfor run 3288, as it was the closest kinematic setting to the H( e, e (cid:48) p ) n
80 MeV/csetting. This correction would then be applied to the remaining elastic runs.Assuming any variation in missing energy between data and SIMC was due tothe electron momentum, E (cid:48) , δE m δE (cid:48) = − → δE m = − δE (cid:48) , (4.22)where δE m = E SIMCm − E datam from the missing energy peak fit. The electron momen-tum correction is then E (cid:48) corr = E (cid:48) uncorr + δE (cid:48) = E (cid:48) uncorr (1 − δE m E (cid:48) uncorr ) , (4.23)where the correction factor is defined as f SHMScorr ≡ − δE m E (cid:48) uncorr . (4.24)After correcting the SHMS central momentum, the SHMS δ -reconstruction alsoneeds to be checked as a function of the SHMS focal plane variables, as it may needto be optimized. SHMS δ Optimization
To check the SHMS δ -reconstruction, similar to the HMS, one needs to determineand correct for any correlation that might exist between the reconstructed δ and each144f the focal plane variables. In general, from each of the measured trajectories atthe focal plane ( X fp , X (cid:48) fp , Y fp , Y (cid:48) fp ), one has to reconstruct the measured trajectoriesat the target, which are characterized by five quantities ( Y tar , X (cid:48) tar , Y (cid:48) tar , X tar , δ ),leading to an underdetermined system. To overcome this problem, one of the targetvariables has to be fixed (usually X tar ). Once the horizontal target position X tar has been surveyed, one can express each of the reconstructed target variables as apolynomial expansion of the focal plane variables. For our case, the δ componentcan be expressed as δ = (cid:88) ijklm D ijklm x i fp x (cid:48) j fp y k fp y (cid:48) l fp x m tar , (4.25)where i, j, k, l, m are the powers of the focal plane quantities and D ijklm are the matrix coefficients for a particular combination of powers where the target positionis typically set to x m tar = 0.To optimize the δ -component, we define the calculated δ as δ calc ≡ P calc − P P , (4.26)where the calculated electron momentum is determined from momentum conserva-tion to be P calc = E b + M p − E p . (4.27)From Eq. 4.27, the proton energy is E p = (cid:112) P + M and the electron momentumcan be approximated by P calc ∼ E calc . The measured proton momentum, P meas , isthe corrected HMS momentum determined in the previous section. Taking thedifference between the calculated and measured momenta, χ ≡ ( δ calc ( E b , P meas ) − δ meas ( x fp , x (cid:48) fp , y fp , y (cid:48) fp )) . (4.28)From Eq. 4.28, the SHMS δ -optimization is now a χ -minimization problem, wherethe goal is to find a set of matrix coefficients, D ijklm , that minimizes the differ-ence between the calculated and measured δ . For further details, see Ref. [155].145he χ -minimization procedure was done simultaneously on the four hydrogenelastic H( e, e (cid:48) ) p runs taken during this experiment. Each of these runs covered adifferent (also overlapping) region of the SHMS reconstructed δ with a coverage of − < δ < D ijklm matrix coefficients as opposed to having done the fits separately for each run,which resulted in matrix correction factors that would have to be applied to eachrun.Additionally, only the ( x fp , x (cid:48) fp ) focal plane terms were used in the fit since withelastic events there is a kinematic correlation between the momentum and scatter-ing angle that translates into a correlation between x fp and ( y (cid:48) fp , y fp ). So if one fits( y (cid:48) fp , y fp ) then one can be fitting this kinematic correlation and not an optics corre-lation [156].The δ terms that were used in the fit can be expanded from Eq. 4.25 to obtain, δ meas = D · x fp + D · x (cid:48) fp + D · x fp · x (cid:48) fp + D · x + D · x (cid:48) . (4.29)The coefficents were optimized for the first and second order ( x fp , x (cid:48) fp ) terms as wellas for the cross terms since the correlations observed were not completely linear asshown in Fig. 4.28. After fitting the correlations and determining the optimumcoefficients, these were updated in the SHMS optics parameter file, and the datawere re-analyzed.From Fig. 4.29, there is a noticeable improvement in the ( x fp , x (cid:48) fp ), as the corre-lations have been corrected, whereas in the ( y fp , y (cid:48) fp ), the effect is less noticeable asthese were not involved in the fit.Figure 4.30 shows that after the SHMS central momentum correction and opti-mization, there is a clear improvement in the missing energy spectrum.146igure 4.28: SHMS ( δ calc − δ meas ) vs. focal plane variables for H( e, e (cid:48) ) p elastic run3288 before δ -optimization.Figure 4.29: SHMS ( δ calc − δ meas ) vs. focal plane variables for H( e, e (cid:48) ) p elastic run3288 after δ -optimization. 147igure 4.30: Missing energy spectrum for H( e, e (cid:48) ) p elastic run 3288 after centralmomentum correction and δ -optimization.From Fig. 4.30, the improvements observed are: • Alignment of data missing energy to SIMC from the central momentum cor-rection • Narrower width in data missing energy from δ -optimization of the matrixcoefficients.The first bullet point is easy to understand, as the alignment is simply due to achange in the SHMS central momentum. The second bullet point can be understoodfrom the fact that since the SHMS δ meas matrix coefficients have been optimized,an event in the missing energy spectrum that would otherwise be reconstructed faraway from the main peak, is now reconstructed underneath the main peak resultingin an improvement in the resolution as well as in the recovered events. The SHMS δ matrix coefficients are directly associated with the SHMS measured mo-mentum on an event-by-event basis, so if these coefficients are optimized, the measuredSHMS momentum is optimized, which directly affects where the Missing Energy eventwill be reconstructed. HMS ( Y tar , Y (cid:48) tar , X (cid:48) tar ) Optimization
During the E12-10-003 experiment, an optics run with the centered sieve insertedwas taken after the optics in SHMS Q3 was fixed. These data were used to opti-mize the ( Y tar , Y (cid:48) tar , X (cid:48) tar ) components of the reconstruction matrix. The target usedconsists of three carbon foils positioned at (-10, 0, 10) cm to mimic the Hall Cextended target edges and center. The 3 foils provide events with known and fixed Y tar positions that are used to optimize the Y tar reconstruction whereas the sieveslit provides events with known and fixed sieve holes to optimize the X (cid:48) tar and Y (cid:48) tar reconstruction. The optimization code used can be found in Ref. [157].To check the optics, the SHMS δ vs. Y tar was plotted to verify how well the Y tar has been reconstructed across δ for each of the three foils. Below are the plotsshowing before and after the optimization.Figure 4.31: SHMS δ vs. Y tar for carbon sieve run 3286 before Y tar -optimization.149igure 4.32: SHMS δ vs. Y tar for carbon sieve run 3286 after Y tar -optimization.After the optimization, it is clear from Fig. 4.32 that there is almost no correla-tion as compared to before optimization. From the optimized variables ( Y tar , Y (cid:48) tar , X (cid:48) tar ),the last two are related to the in-plane and out-of-plane angles of the reconstructedparticle trajectory relative to the spectrometer central ray as discussed in Section4.4. Removing the correlation in Y tar improves the determination of the SHMS elec-tron scattering angle, which in turn corrects the location of the invariant mass ( W )peak as it depends on the electron angle. The optics optimization was originally done assuming there were no spectrometeroffsets. This is not true, however, as there were still some small mis-alignmentsobserved in the missing energy spectrum. An extensive study of the spectrometer150ffsets in Hall C has not been performed yet. We have estimated these offsets basedon observations in H( e, e (cid:48) ) p elastic run 3288, as it is closest in kinematics to thedeuteron 80 MeV/c setting. Central Angle Offsets
The central angle offsets refer to the angular offsets of the spectrometer central rayand can be classified as follows: • In-plane central angle offset ( θ c + δθ offc ), [h(p) thetacentral offset] • Out-of-plane central angle offset ( φ c + δφ offc ), [h(p) oopcentral offset]where the bracketed parameters represent the nomenclature in the analysis software. In-plane is parallel to the hall floor, whereas the out-of-plane is perpendicular tothe hall floor. The central angle offsets can be determined from the missing mo-mentum components of H( e, e (cid:48) ) p elastic events as these should ideally be centeredaround zero. In the hall coordinate system, the in-plane central angle offsets canbe determined by taking the fractional difference between the measured (data) andexpected (SIMC) X-component of the missing momentum as follows: δθ offc = P SIMCmx − P datamx P . (4.30)The out-of-plane central angle offset can be determined by taking the fractionaldifference between the measured (data) and expected (SIMC) Y-component of themissing momentum as follows: δφ offc = P SIMCmy − P datamy P . (4.31) The spectrometer central angle offset parameters can be found athallc replay/PARAM/(S)HMS/GEN/(s)hmsflags.param out-of-plane offset, the Y-component of the missing momen-tum agrees with simulation as shown in Fig. 4.34. With respect to the X-componentof the missing momentum, it was decided not to apply an in-plane angle offset as thiswould directly impact the location of the invariant mass peak. Alternatively, it wasdecided to apply a relative in-plane angle offset that would align the X-component.The relative angle offsets are discussed in the next section.
Relative Angle Offsets
The relative angle offsets refer to the angle offset relative to the spectrometer centralray and can be classified as follows: 152
In-plane relative angle ( Y (cid:48) tar + δθ off ) offset, [h(p)theta offset] • Out-of-plane relative angle ( X (cid:48) tar + δφ off ) offset, [h(p)phi offset]The Y (cid:48) tar offset is directly related to the spectrometer angle, and therefore has adirect impact on the electron/hadron kinematics, depending on which spectrometeris associated with the particle type. In E12-10-003, this offset was determined forthe HMS in order to align the X-component of the missing momentum as well asto improved the HMS central momentum correction. Recall that in Section 4.5.1 itwas assumed that the proton (HMS) angle was well known, which is not completelytrue.Figure 4.35 shows the relative out-of-plane angle distributions for all eventswithin the spectrometer acceptance. The zero value in the distribution representsevents whose trajectory was parallel to the central ray, whereas the events awayfrom the zero value represent those events that are at an out-of-plane angle relativeto the central ray. The X (cid:48) tar offset was determined by “eye”, using the mean of thedistribution.Figure 4.35: HMS X (cid:48) tar for run 3288 before (left) and after (right) applying the offsetcorrection. 153imilarly to the relative out-of-plane angles, the relative in-plane angles in the Y (cid:48) tar distribution (not shown) represent angles relative to the central ray, with the zero value representing particles parallel to the central ray. The Y (cid:48) tar offset was de-termined based on how well the X-component of the missing momentum betweendata and simulation were matched, as well as how well were the HMS momentumfrom data and simulation matched (see Fig. 4.21).After determining the spectrometer offsets, a second iteration of the HMS andSHMS Optics check procedure was performed to obtain improved results. Finally,the four H( e, e (cid:48) ) p elastic runs were used to determine the HMS momentum correc-tions for the H( e, e (cid:48) p ) n data, to be discussed in the next section. During the E12-10-003 experiment, the four H( e, e (cid:48) ) p elastic runs analyzed cov-ered the HMS momentum range such that the H( e, e (cid:48) p ) n measured momentum waswithin the range covered by the elastic data. From this knowledge, one can de-termine the H( e, e (cid:48) p ) n data momentum correction from a simple linear fit of the H( e, e (cid:48) ) p data.From Fig. 4.36, the momentum correction factor is plotted against the originalHMS central momentum and the four data points are fit with a straight line. Usingthe line fit, the H( e, e (cid:48) p ) n momentum correction for the three missing momentumsettings are determined from the H( e, e (cid:48) p ) n original HMS momentum setting.Tables 4.4 and 4.5 summarize the H( e, e (cid:48) p ) n kinematics before and after theSHMS (see Section 4.5.2) and HMS central momentum corrections. Since the SHMSmomentum was fixed during the experiment, the single correction factor determinedfrom the H( e, e (cid:48) ) p data analysis applies for all runs.154 it Results / ndof = 21 . . ± . ⇥ slope = . ± . ⇥
Table 4.4: Original H( e, e (cid:48) p ) n kinematics for E12-10-003. P m Setting [MeV/c] HMSAngle [deg] HMSMomentum [GeV/c] SHMSAngle [deg] SHMSMomentum [GeV/c]80 38.896 2.840 12.194 8.5342580 (set1) 54.992 2.1925 12.194 8.5342580 (set2) 55.000 2.1925 12.194 8.5342750 (set1) 58.391 2.0915 12.194 8.5342750 (set2) 58.394 2.0915 12.194 8.5342750 (set3) 58.391 2.0915 12.210 8.5342
Table 4.5: Corrected H( e, e (cid:48) p ) n kinematics for E12-10-003.155 .5.5 Spectrometer Acceptance Post-Optimization After optics checks and optimization for each spectrometer, the data and simulated(SIMC) reconstructed variables at the target were compared using H( e, e (cid:48) ) p elasticsrun 3288. The ratio of the data-to-simulation was also taken and is plotted below.The data were normalized by the total accumulated charge and corrected for exper-imental inefficiencies that will be discussed in Chapter 5. In addition, several eventselection cuts were applied for both data and simulation (see Section 4.6).Figure 4.37: SHMS target reconstruction after optics optimization of H( e, e (cid:48) ) p elas-tic run 3288 for the E12-10-003.Figure 4.37 shows a generally good agreement betweem data and simulation overa wide range of the spectrometer acceptance. The main issues seem to be at the156dges of the acceptance where differences beyond 20% (blue dashed line) can beobserved in the ratios. For the most part of the acceptance, the ratios indicatea discrepancy of ∼
10% between data and simulation yields with the exceptionof X (cid:48) tar , which seems to have a resolution issue as the simulation appears slighlynarrower than data. The overall integrated yield over the entire range, however,shows only a discrepancy of ∼ − H( e, e (cid:48) ) p events. It is important to keepin mind that the simulation program does not take into account the uncertaintiesdue to the elastic form factors that are used to simulate the hydrogen elastic events.Figure 4.38: HMS target reconstruction after optics checks of H( e, e (cid:48) ) p elastic run3288 for the E12-10-003. 157imilarly to the SHMS, the HMS reconstructed variables (see Fig. 4.38) showa generally good agreement between data and simulation with discrepancies below ∼
10% for most of the acceptance range, with discrepancies beyond ∼
20% at theedges.Given that these studies were done using the coincidence elastic H( e, e (cid:48) ) p data,determining systematic effects due to our knowledge of the spectrometer acceptancesis rather complicated given the correlations that exist between both spectrometerarms. Ideally, one would have to look at either single-arm elastic or deep-inelastic(DIS) data to carry out a complete spectrometer acceptance systematics study.Figure 4.39: SHMS target reconstruction of H( e, e (cid:48) p ) n run 3289 (80 MeV/c setting)for the E12-10-003. 158he spectrometer acceptance for the deuteron 80 MeV/c setting are shown inFigs. 4.39 and 4.40 since the kinematics were very close to that of hydrogen elastics,and were used to check the spectrometer acceptance before looking at the highermomentum settings. The data have been normalized by the total charge and cor-rected for inefficiencies and in addition, have also been integrated over the full rangeof neutron recoil angles ( θ nq ) for better statistical precision.Similar to the hydrogen, there is an overall good agreement between data andsimulation with up to ∼
20% difference (blue dashed line) in the yield over most ofthe acceptance range on both spectrometers. The Laget FSI model [60] was used inthe simulation.Figure 4.40: HMS target reconstruction of H( e, e (cid:48) p ) n run 3289 (80 MeV/c setting)for the E12-10-003. 159 .6 Event Selection A variety of cuts have been applied during the analysis of E12-10-003 to select true H( e, e (cid:48) ) p elastics and H( e, e (cid:48) p ) n events at the reaction vertex. Due to the similarityin kinematics between the hydrogen elastic (run 3288) and the deuteron 80 MeV/csetting (run 3289), similar cuts were placed on both hydrogen and deuteron data toselect good events. The same cuts were also placed on the simulation for a directcomparison, and ultimately, for the determination of the spectrometer phase spacefrom SIMC. An additional kinematic cut was placed on the deuteron data to selectevents at the highest momentum transfers ( Q ) allowed by the kinematics to furthersuppress MEC and IC contributions as stated in Section 2.5. The same cuts placedon the 80 MeV/c setting were also placed on the 580 and 750 MeV/c settings (notshown below) since the cut ranges were not affected by the change in kinematicsfrom lower to higher missing momenta. Figure 4.41: Missing energy cut on the 80 MeV/c setting of E12-10-003.160he primary cut used to select true H( e, e (cid:48) p ) n is a missing energy cut aroundthe deuteron binding energy ( ∼ . Figure 4.42: Momentum acceptance cuts on the 80 MeV/c setting of E12-10-003.To ensure that events are reconstructed in a momentum acceptance region wherethe optics reconstruction matrix is reliable, a cut is placed on the HMS momentumacceptance in the range − < δ HMS < − (cid:46) δ SHMS (cid:46) .6.3 HMS Collimator Cut
Figure 4.43: Data HMS collimator cut on the 80 MeV/c setting of E12-10-003.Inset: SHMS collimator geometry and reconstructed events.Figure 4.44: Simulated HMS collimator cut on the 80 MeV/c setting of E12-10-003.Inset: SHMS collimator geometry and reconstructed events.162o make sure that all events that enter the spectrometer pass through the collimator,and not re-scatter at the edges, a cut is placed on the HMS collimator. The insetplots represent the SHMS collimator geometry whose events are constrained bythe HMS collimator entrance. The events shown are projected at the collimatorentrance and are functions of the reconstructed variables ( Y tar , X (cid:48) tar , Y (cid:48) tar , δ ) and thesurveyed collimator position measured from the target center. It was preferred toput a geometrical cut rather than a cut on the reconstructed variables as the latterare subject to change when the spectrometer moves whereas the former is a fixedcut that defines the particles that enter the spectrometer. z v -Vertex Difference Cut Figure 4.45: Reaction z v -vertex difference cut on the 80 MeV/c setting of E12-10-003. 163o ensure real coincidence events are selected, a cut was made on the differencebetween HMS and SHMS z -reaction vertex (using Eq. 4.14) at ± z v -vertex can be significantly different between the two spectrometers, whichcontributes to the tails of the distribution (see Fig. 4.45). Additionally, the tailscan also arise from a bad Y tar reconstruction, as the z v -vertex is calculated from thisvariable. Figure 4.46: SHMS calorimeter cut on the 80 MeV/c setting of E12-10-003.164he SHMS calorimeter was used to separate electrons from background (pions),however, as it is shown in Fig. 4.46, the deposited energy in the calorimeter nor-malized by the incident particle track shows a very clean distribution with a peakat one indicating the detected particles were electrons. The clean electron samplecan be attributed to the low accidental trigger rates and low pion background in theSHMS to form these coincidences with the protons in the HMS.
Figure 4.47: Coincidence time cut on the 80 MeV/c setting of E12-10-003.To further clean the electron-proton coincidence sample of events, a coincidence cutwas made in the range 10 . < t coin < . Q ) Cut Figure 4.48: Four-momentum transfer ( Q )cut on the 80 MeV/c setting of E12-10-003.A kinematical cut on the 4-momentum transfer is made at Q = 4 . ± . to select events only at the highest possible momentum transfers. The previousdeuteron experiment in Hall A [55] measured the cross sections up to p r =550MeV/c and Q = 3 . ± .
25 (GeV/c) , whereas this experiment seeks to probe thedeuteron momentum distributions at extreme kinematics by moving beyond 500MeV/c recoil momenta at the highest Q , where MEC and IC are suppresed. The166ross sections for this experiment were also extracted at a lower kinematic bin of Q = 3 . ± . Q setting.167HAPTER 5 DATA CROSS SECTION EXTRACTION
In this chapter I discuss how the experimental cross section was determined forthis experiment as well as the various corrections applied to extract the yield. Inaddition, I will also describe the studies to determine the systematic uncertaintiesfrom various sources on the cross section. H ( e, e (cid:48) p ) n Cross Section
The deuteron cross section was introduced in Section 2.2 from a theoretical ap-proach, however, to make a direct comparison between theory and experiment, onemust also consider how the cross section is determined experimentally. Figure 5.1shows a simple cartoon of a typical coincidence experiment that will be used toderive the experimental cross section.Figure 5.1: Cartoon representation of a typical coincidence experiment.168onsider an electron beam with cross-sectional area A b [cm ] incident on a targetof length (thickness) ∆ L [cm] and density n t ([g]/[cm ]). The incident beam flux onthe target is then defined as J inc ≡ n b A b v b = dN inc dt , (5.1)where n b is the beam density in N inc / [cm] , N inc is the total number of incidentparticles, and v b is the electron velocity in [cm] / [s]. The beam flux is also interpretedas the instantaneous rate of the number of beam particles incident on an target area A b (beam current). The beam current is ususally measured in milli-Coulomb persecond (mC/s), which can also be expressed as the total number of incident electronsper second using the conversion factor: 1 mC/s ≡ e − /s. As the electronbeam passes through the target, each electron can interact with any target atomwithin the area A b . The total number of atoms that can potentially be scattered bythe beam within this area are determined to be N t ≡ n t A b ∆ L. (5.2)In reality, only a certain fraction of the total target atoms that can be scattered willinteract with the beam, which is characterized by the probability, p = N t σA b = n t ∆ Lσ, (5.3)where σ is defined as the cross section and describes the effective area of interactionfor a particular reaction out of a total area A b .From the number of interactions between the incident electrons and target atoms,the reaction rate J sct of the scattered particles is determined from the probabilitythat the beam flux interacts with the target atoms, which is defined as J sct ≡ p J inc = J inc n t ∆ Lσ. (5.4)169rom 5.4 a useful quantity to define is the experimental luminosity,
L ≡ J inc n t ∆ L (5.5)in units of [cm − ][s − ], which describes the total number of interactions that canbe produced from beam particles illuminating a specific target area. From theluminosity, the experimental cross section can be expressed in its simplest form as σ = J sct L , (5.6)where the cross section is a constant of proportionality between the luminosity andreaction rate.To determine the differential cross section in a typical coincidence experiment,consider a general A ( e, e (cid:48) p ) reaction where the incident electron knocks out a pro-ton and the ( A −
1) recoiling system is undetected. The scattered electron andknocked-out proton are detected in coincidence between each spectrometer withina finite angular acceptance region limited by either the spectrometer apertures orcollimator (if inserted). In reality, the acceptance region (phase space) coveredby the electrons in coincidence with protons is determined by the reaction kine-matics, which may be smaller than the full acceptance of the spectrometers. Forthe specific case of the deuteron break-up reaction, the phase space was deter-mined via a Monte Carlo simulation (SIMC) by randomly generating a set of values,( X (cid:48) e, tar , Y (cid:48) e, tar , X (cid:48) p, tar , Y (cid:48) p, tar , E (cid:48) ) , which along with a known beam energy, completelydetermines the final proton momentum. The volume formed by this hypercube wasdefined as V PS = N acc N gen ∆Ω e ∆Ω p ∆ E (cid:48) , (5.7) The primed variables represent relative spectrometer angles introduced in Section 4.4and E (cid:48) is the scattered electron energy (or approximately, its momentum). N acc is the number of accepted coincidence events that lie in the hypercubeand N gen is the total number of events generated by the random sampling process.The ∆Ω ( e,p ) = ∆ X (cid:48) ( e,p )tar ∆ Y (cid:48) ( e,p )tar defines the angular range of the electrons in coin-cidence with protons and ∆ E (cid:48) defines the range of the scattered electron momenta.The differential cross section can then be determined by dividing the total numberof detected electron-proton coincidences in the experiment by the Monte Carlo gen-erated phase space at the same reaction kinematics. By substituting Eqs. 5.5 into5.6 and dividing by Eq. 5.7 one obtains d σd Ω e d Ω p dE (cid:48) = J sct J inc n t ∆ L J corr (∆Ω e,p → d Ω e,p ) N acc N gen ∆Ω e ∆Ω p ∆ E (cid:48) , (5.8)where J corr (∆Ω e,p → d Ω e,p ) is a Jacobian matrix that is used to convert the solidangles from the spectrometer coordinates to spherical coordinates. The incident( J inc ) and reaction ( J sct ) rates can be integrated over the entire experimental runtime to obtain (cid:90) J inc dt ≡ (cid:90) dN inc dt dt = N inc , (5.9) (cid:90) J coin dt ≡ (cid:90) dN coin dt dt = N coin , (5.10)where N inc is the total number of incident electrons, which is normalized to 1 mCin SIMC and N coin is the true number of detected H( e, e (cid:48) p ) n coincidences, providedthat the event selection cuts defined in Section 4.6 have been applied. It is impor-tant to note that these coincidences, hereafter referred to as Y uncorr , have not beencorrected for detector inefficiencies. Therefore, to obtain the final reaction crosssection, the data yield has been corrected as follows: Y corr ≡ Y uncorr · f rad Q exptot · (cid:15) tLT · (cid:15) htrk · (cid:15) etrk · (cid:15) tgt . Boil · (cid:15) pTr , (5.11)where Q exptot is the total experimental accumulated charge from the electron beam atthe target, f rad is the correction due to radiative effects, and the (cid:15) i ’s are corrections171rom various experimental inefficiencies described in the following sections. Thenormalization of the data by the accumulated charge (yield/mC) is necessary for adirect comparison to the SIMC yield. Substituting Eq. 5.11 into Eq. 5.8, the finalaveraged experimental cross section for the H( e, e (cid:48) p ) n can be expressed as (cid:16) d ¯ σd Ω e d Ω p dE (cid:48) (cid:17) k = (cid:16) Y corr n t ∆ L J corr (∆Ω e,p → d Ω e,p ) N acc N gen ∆Ω e ∆Ω p ∆ E (cid:48) (cid:17) k , (5.12)where the “average” refers to the fact that the cross section has been calculated atthe center of the k th kinematic bin in question, where k = p r , Q , x Bj , θ nq , etc. Inreality, the true cross section must be determined at the averaged kinematics forthat bin. See Section 5.9 for a detailed discussion of the bin-centering corrections.Similar to the averaged data cross sections, the Laget model cross sections im-plemented in SIMC were determined using Eq. 5.12, where the simulated yield hadthe same cuts as the data yield, but the efficiencies were all (cid:15) i = 1 as SIMC doesnot simulate detector inefficiencies. (cid:15) htrk , (cid:15) etrk ) To account for the experimental yield loss due to a bad track reconstruction orthe selection of the wrong track by the Hall C tracking algorithm, the trackingefficiencies have been determined. The tracking efficiency per experimental run isgenerally defined as (cid:15) (htrk , etrk) ≡ N did N should , (5.13a) δ(cid:15) (htrk , etrk) ≡ √ N should − N did N should , (5.13b)where N did are the number of events for which there was at least one track formedby the drift chambers tracking algorithm given a specific criteria and N should are172he number of events where at least one track was expected but was not necessarilyreconstructed by the algorithm using the same criteria. For simplicity, we define thelogical operator AND as “ ∧ ” and the EQUALITY operator as “==” to be used inthe tracking criteria definition. For the electron arm (SHMS), the criteria for theformation of a track was defined as N should2 ≡ ( N goodScinHit == 1) ∧ ( β notrk > . ∧ ( β notrk < . ∧ (5.14)( E tot . norm > . ∧ ( N NGC , npeSum > . ,N did ≡ ( N should ) ∧ ( N DCtrk > . (5.15)For the hadron arm (HMS), a similar set of variables were used and defined asfollows: N should ≡ ( N goodScinHit == 1) ∧ ( β notrk > . ∧ ( β notrk < . ∧ (5.16)( E tot . norm < . ∧ ( N CER , npeSum < . ,N did ≡ ( N should ) ∧ ( N DCtrk > . (5.17)The variables for both spectrometers are defined as • N goodScinHit : Number of good scintillator hits, which can either be 1 or 0. Therequirement for one hit is that the candidate track passes through a fiducialregion in each XY hodoscope plane. The fiducial region is defined by requiringa TDC hit on a certain number of scintillator paddles that are adjacent to thepaddle the candidate track passed through. • β notrk : The hodoscope beta ( β = v/c ) is calculated without using trackinginformation , as this would make the drift chamber efficiency calculation biased. As an example the tracking criteria defined in Eq.5.14 can be read out as: “the totalnumber of events that should have passed the tracking criteria require that the totalnumber of good scintillator hits must be exactly 1 AND β notrk be between 0.5 and 1.5AND the total normalized calorimeter energy be below 0.6 AND the total number ofˇCerenkov photoelectrons be below 0.5” E tot . norm : The total energy deposited in the calorimeter normalized by thespectrometer central momentum. In the SHMS, a cut is made > . < . • N NGCER , CER , npeSum : The total number of photoelectrons from the NGC detec-tor in SHMS or HGC detector on the HMS. A cut is made > . < . • N DCtrk : Total number of tracks formed by the tracking algorithm. We requireat least one track to increment N did .In the tracking algorithm, it is possible that there may be multiple, but not neces-sarily real physics tracks that passed the criteria set above. In this case, Hall C usesthree distinct methods to select the best track: • scintillator hit method : Selects the best track as the track closest to the paddlehit in the last scintillator plane (S2Y). In addition, it also rejects tracks if theyfail certain criteria imposed on the hodoscope calculated β and normalizedcalorimeter energy. • best χ method : Selects the best track as the track fit with the lowest χ . • pruning method : Selects among multiple possible tracks, the track with thelowest χ after pruning (“cutting” or “trimming”) tracks that do not meetcertain criteria imposed on the reconstructed variables at the target, hodoscope β , focal plane time and number of PMT hits among others. For example, anevent with a bad X (cid:48) tar reconstruction might have the track with the lowest χ best χ method . The pruning method , however, would have pruned thispotential “good track” candidate, thereby reducing the probability of whatwould have otherwise been considered a “good track” by the best χ method .A judgement needs to be made regarding which are reasonable variables toprune on.For the E12-10-003, the best χ method was used to determine the tracking effi-ciencies. In general, the rates for the SHMS varied between ∼ ∼ .
8% for theHMS and 96 .
4% for the SHMS (see Fig. 5.2). . . . . . . . T r a c k i n g E ffi c i e n c y HMS TrkEff Average: 0.988SHMS TrkEff Average: 0.964
Tracking Efficiency vs. Run Number
80 (set1)580 (set1)580 (set2) 750 (set1)750 (set2)750 (set3)
Figure 5.2: Tracking efficiency of HMS (open) and SHMS (full) for the E12-10-003experiment. 175
300 3320 3340 3360 3380 3400120130140150160170 S H M S R a t e [ k H z ] Trigger Rates vs. Run Number
80 (set1)580 (set1)580 (set2)750 (set1)750 (set2)750 (set3)3300 3320 3340 3360 3380 34000 . . . H M S R a t e [ k H z ] P m = 80 MeV/c(scaled x 1 / . . . . . C o i n . R a t e [ H z ] P m = 80 MeV/c(scaled x 1 / . Figure 5.3: Trigger rates for the SHMS (top), HMS (middle) and coincidence trigger(bottom) during the E12-10-003 experiment. (cid:15) tLT ) Another source of inefficiency in the experimental yield arises from the total deadtime of the data acquisition (DAQ) system which is separated into an electronic andcomputer deadtime. The electronic deadtime arises from signal pile-up at the front-end of the electronic modules due to high rates, whereas the computer deadtimerefers to the amount of time the DAQ is unable to accept a pre-trigger.176n E12-10-003, the live time calculation was determined using the EDTM systemdescribed in Section 3.7.6. This method determines the electronic and computerlive time simultaneously by feeding a clocked EDTM logic signal into the triggerelectronics (mixed with the physics signals) and counting how many of them wereaccepted by the trigger interface. Using the formula from Eq. 3.13, the total EDTMlive time was determined to be on average ∼ .
7% (see Fig. 5.4). . . . . . . T o t a l L i v e T i m e Average Live Time: 0.927
Total EDTM Live Time vs. Run Number
80 (set1)580 (set1)580 (set2)750 (set1)750 (set2)750 (set3)
Figure 5.4: Total EDTM live time determined for the E12-10-003 experiment.The uncertainty on the EDTM system is not a straightforward calculation andneeds to be given more thought since 1) the numerator and denominator involved inthe calculation (see Eq. 3.13) are correlated and 2) the accepted number of EDTMlogic pulses (numerator) are governed by a random process that can be understoodfrom the fact that even though these are clocked pulses, they are mixed with thephysics pre-triggers, which are random and follow a poissonian distribution. Aneducated guess of ∼
3% relative uncertainty on the total live time was made basedon a crude estimate of the dominant source of electronic deadtime in the system.Given that the SHMS S1X hodoscope plane is the dominant source of electronic177ead time as it had the highest trigger rates (typically R ∼ τ ∼
60 ns, the electronicdeadtime can be approximated to be Rτ ∼ ∼ ∼
6% (or ± ∼ − (cid:15) tgt . Boil ) The exposure of a cryogenic target to the beam can cause density reductions in thebeam path or even localized boiling on the target due to the large amounts of heatdeposited by the beam. To minimize boiling, the beam can be rastered (“smearedout”) by up to 5x5 mm in area to re-distribute the heat deposited over a largerarea on the target (see Section 3.4.2). Even though the rastered beam minimizes thelocal boiling, there may still be a small reduction in the target density that resultsin the reduction of the data yield by a few percent.To correct the experimental data yield for target density, a series of dedicatedruns (see Table 5.1) were taken independently in each spectrometer (single-arm) atvarious beam currents. In this section I present the target density study resultsusing the HMS only, however, a similar study using the SHMS should be carriedout as a cross-check.The runs in Table 5.1 were taken at a spectrometer central angle and momentumsettings of 25 ◦ and -4.4 GeV/c (negative polarity), respectively, at a beam energy of E b = 10 . and LD targets, respectively, at various beam currents. To select electrons inthe HMS, the trigger was set to HMS EL-REAL (see Fig. 3.42). HMS Run Target Beam Current [ µ A]2093 Carbon-12 602094 Carbon-12 502095 Carbon-12 352075 LH Table 5.1: Target density reduction studies run list taken on April 02, 2018.The target density reduction analysis consists of determining the charge normal-ized yield as a function of the beam current to determine the yield loss per unitbeam current. To correctly determine the yield corresponding to a specific beamcurrent, a cut was made to select events that correspond to a stable beam currentperiod, and not to the beam ramp up or down periods.179o precisely select events corresponding to a certain beam current threshold,consider the following example illustrated in Fig. 5.5. In this example, the scalerreads (vertical black lines) are registered by the DAQ and dumped into the data-stream at either every 2 seconds or 1000 events.Figure 5.5: Illustration of scaler and event reads during a typical experimental run.As the run progresses, the average beam charge and time interval in betweenscaler reads can be determined from which the average beam current is calculatedfor each interval. A beam current low and high cut can then be set to select onlythose scaler read intervals for which the average beam current is within the cutlimits. In the data event loop, each event is then compared to the next scaler readand is discarded if it lies within a scaler read interval that did not pass the cut.After selecting only those events that passed the nominal beam current for eachrun, the charge normalized yields can be determined and plotted as a function ofthe average beam current. In this analysis, three separate charge normalized yieldswere determined, designed to test the computer live time and tracking efficiencycorrections. 180he yields were defined as Y scl = N scl Q tot , (5.18) Y no . trk = N acc Q tot · (cid:15) cpuLT , (5.19) Y trk = N acc Q tot · (cid:15) cpuLT · (cid:15) htrk , (5.20)where Eq. 5.18 is the yield calculated from the total number of HMS pre-triggerscaler counts normalized by the total charge, Eq. 5.19 defines the charge normalizedyield (using accepted HMS triggers) corrected for computer live time but does notuse tracking information in the event selection criteria, and Eq. 5.20 defines a chargenormalized yield that uses tracking information in the event selection criteria and istherefore also corrected for the tracking efficiency. The associated histograms usedto determine the counts for each of these yield calculations are shown below.Figure 5.6: Example of a BCM scaler current cut used to determine the yield.181igure 5.7: Example of the histograms used to determine the non-tracking (top)and tracking (bottom) yields. Top: x -axis shows the total deposited energy in thecalorimeter normalized by the central spectrometer momentum, E DEP /P c . Bottom: x -axis shows the HMS momentum acceptance, δ , in percent.Figure 5.8 shows the charge normalized yields using BCM4A (red) and BCM4B(black) beam current cuts in the event selection process. This study was done tocheck the behavior of both BCMs at very high currents. As can be seen from thenormalized yields using the LH and LD targets, above ∼ µ A, the BCM4Ayield is significantly lower than the BCM4B yield indicating that BCM4A begins tosaturate above this current (see Ref. [159]).182igure 5.8: Normalized tracking yields using BCM4A (red) and BCM4B (black)beam current cuts on carbon-12 (top), LH (middle) and LD (bottom) targets.183ue to the saturation of BCM4A at beam currents > µ A, it was decided to useBCM4B for the remaining target density studies.Figure 5.9: Linear fit of the charge normalized yields for carbon-12.The normalized yields for each of the targets were determined from Eqs. 5.18,5.19 and 5.20 and fit as a function of the BCM4B average beam current using thefit function Y norm = m · I beam + Y , (5.21)184here Y norm is the charge normalized yield ( y -axis), I beam is the average beam current( x -axis), and ( m , Y ) are the slope and y -intercept parameters, respectively. The fitresults are shown in Figs. 5.9, 5.10 and 5.11.Figure 5.10: Linear fit of the charge normalized yields for LH .A gradual improvement is observed in the χ fit from the top to bottom panelsfor each of the targets, indicating that the yield calculated using Eq. 5.20 gives thebest fit results. To apply the target density corrections, one needs to determine thefractional yield loss per µ A. This is done by normalizing the yield to the y -intercept,185 , which corresponds to the yield at 0 µ A to obtain a relative yield. From therelative yield and the slope, the fractional yield loss at any beam current withinthe fit range can be calculated. This correction is then applied to the experimentalyield. Figure 5.11: Linear fit of the charge normalized yields for LD .Figure 5.12 shows the fit results after normalizing the fit function by Y . Fromthe final fit results of the HMS target density studies, carbon-12 has a reductionin the yield of ∼ . /µ A, which shows almost non-existent density changes as186s expected from a solid target. Liquid hydrogen and deuterium, however, show asignificant reduction in the yield corresponding to ∼ . /µ A and ∼ . /µ A,respectively.Figure 5.12: Linear fit function normalized to the y -intercept, Y for each of thethree targets. Target Slope, m ≡ m/Y Slope Error, δ ( m )Carbon-12 -1.87x10 − − LH -6.34x10 − − LD -8.00x10 − − Table 5.2: Target boiling (or density reduction) studies fit results normalized to the y -intercept. 187he target density correction was applied on a run by run basis using the formula, (cid:15) tgt . Boil = 1 − m · I avg , (5.22a) δ(cid:15) tgt . Boil = (cid:113) I δ m + m δ I avg , (5.22b)where m is the slope of the corresponding target and I avg is the averaged BCM4Abeam current over the entire run, provided the beam current cuts have been ap-plied. It is important to note that the beam current cut on the data analysis wasless strict ( > µ A) than in the target density studies to avoid cutting out possiblecoincidences during the ramping up periods of the beam. . . . . . . . . T a r g e t B o ili n g F a c t o r Average Boiling Factor: 0.958 LD Boiling Factor vs. Run Number
80 (set1)580 (set1)580 (set2)750 (set1)750 (set2)750 (set3)
Figure 5.13: Target density correction of deuterium determined for the E12-10-003experiment.Figure 5.13 shows the target density correction factor for all data sets of theE12-10-003 experiment. The correction factor seems stable over all runs, which is aresult of the beam currents during this experiment were stable and within a smallrange of ∼ − µ A as shown in Fig. 5.14.188
300 3320 3340 3360 3380 3400Run Number4045505560657075 A v e r ag e B e a m C u rr e n t [ µ A ] Average Beam Current vs. Run Number
80 (set1)580 (set1)580 (set2)750 (set1)750 (set2)750 (set3)
Figure 5.14: Averaged beam currents (BCM4A) determined for the E12-10-003experiment. (cid:15) pTr ) In general A ( e, e (cid:48) p ) coincidence experiments there is a small ( ∼ few percent) proba-bility that the knocked-out proton never makes it to the required detectors to forma coincidence trigger with the electron. This is due to the fact that as the protonleaves the target, traverses various windows (e.g., the target cell window, the spec-trometer entrance/exit windows) and enters the detectors, it can lose energy and/oroutscatter and so fails to form a trigger or pass cuts.The proton absorption coefficient can be determined in theory from the knowl-edge of the material thickness and interaction length, which depends on the crosssection of the interaction process. In the E12-10-003 experiment, the proton ab-sorption coefficient was determined by taking several dedicated hydrogen elastic189uns (see Table 3.1) using either only a single-arm or a coincidence trigger at thesame kinematics. The general steps taken were: • Use the H( e, e (cid:48) ) p coincidence runs to determine the spectrometer acceptanceregion for both HMS/SHMS corresponding to ep elastics events (see Fig. 5.15). • Use the H( e, e (cid:48) ) p SHMS single-arm runs to determine the number of elec-trons that lie within SHMS the acceptance region corresponding to SHMSacceptance for ep elastics determined in the first step.Figure 5.15: Cartoon to illustrate how the proton absorption coefficient is deter-mined experimentally by selecting the ep elastics acceptance region.By selecting the SHMS acceptance region (single-arm) corresponding to the ep elas-tics, one is selecting the total number of electrons that should have been in coinci-dence (during the coincidence runs) with the proton in the HMS, but because theproton was absorbed by some material, never made it to form a trigger. The numberof electrons that did pass the requirements to be in coincidence with the protons190ere determined by imposing additional restrictions (“cuts”) on the HMS variables.It should be noted that even though only the SHMS singles triggers were used fordata readout, since these runs were taken with the DAQ in coincidence mode, theHMS detectors are also readout, which enables one to put cuts on the HMS relatedvariables. Coincidence H ( e, e (cid:48) ) p Acceptance Selection
To determine the acceptance region in the SHMS corresponding to elastic events,the momentum acceptance correlation between the two spectrometers is plotted inFig. 5.16 and shows that the SHMS momentum acceptance corresponding to elec-trons from ep elastics is completely determined by the HMS momentum acceptancefor protons.Figure 5.16: Momentum acceptance correlation between SHMS and HMS for coin-cidence H( e, e (cid:48) ) p data run 3248. Inset: Missing energy spectrum cut below 30 MeVto select true elastic events. 191igure 5.17: SHMS electron angular acceptance for coincidence H( e, e (cid:48) ) p data run3248.Figure 5.18: HMS proton angular acceptance for coincidence H( e, e (cid:48) ) p data run3248.Figure 5.17 shows the SHMS angular acceptance where the δ momentum ac-ceptance and missing energy cuts determined from Fig. 5.16 have been applied.192imilarly, for the HMS angular acceptance shown in Fig. 5.18, the same cuts onmissing energy and momentum acceptance have been applied. The dashed colorlines in both Figs. 5.17 and 5.18 define the angular acceptance cuts for ep elasticsevents. The SHMS acceptance cuts determined in this section will be applied to theSHMS singles ep elastics data in the next section. SHMS Singles H ( e, e (cid:48) ) p Acceptance Selection
To help suppress quasi-elastic electrons which scattered from the target cell walls,an aluminum dummy run was taken at the same kinematics as the hydrogen elas-tics data. Figure 5.19 shows the reconstructed events along the z -vertex with thetwo peaks representing the target cell end caps. A z -vertex cut of ± . z -vertex cut determination from aluminum dummy run 3254taken with SHMS singles trigger. 193igure 5.20: SHMS angular acceptance from electron singles run 3259. The redsquare is the elastic acceptance region determined from coincidence elastics data atthe same kinematics.Figure 5.21: Proton invariant mass ( W ) determined from SHMS electron singles run3259. 194fter doing the dummy target analysis, we now focus on the analysis of thesingle-arm H( e, e (cid:48) ) p elastic runs. Figure 5.20 shows the SHMS angular acceptancefrom electron singles. The boxed (red) region corresponds to the angular acceptancefrom ep elastics determined from the coincidence data. A cut was also placed onthe SHMS momentum acceptance determined from the elastic coincidence data aswell as the z -target cut mentioned above.Finally, a cut on the proton invariant mass W (reconstructed from H( e, e (cid:48) ) p electron singles) has been placed between 0.9 and 1.0 GeV/c to ensure electronsthat correspond to true ep elastics are selected (see Fig. 5.21).After determining the cuts to select electrons singles that truly originated fromhydrogen elastic scattering, we define the proton transmission coefficient as (cid:15) pTr ≡ e − did /(cid:15) htrk e − should , (5.23a) δ(cid:15) pTr ≡ (cid:112) e − should − ( e − did /(cid:15) htrk ) e − should , (5.23b)where e − should is the number of electrons that passed the above-mentioned cuts forwhich the correlated proton should have been detected in the HMS and e − did isa superset of (or contains) e − should with the additional requirement that the HMSreconstructed events are within the well defined HMS momentum acceptance ( | δ | < > e − did has also been corrected for HMS tracking efficiency (cid:15) htrk since HMS tracking-relatedvariables have been used in the determination of e − did . Using these definitions, thenumber of electrons singles that should have and did pass the cuts can be expressedas e − should ≡ ( δ SHMS ) ∧ (∆Ω SHMS ) ∧ (∆ Z tar ) ∧ (∆W) , (5.24) e − did ≡ ( e − should ) ∧ ( δ HMS ) ∧ (hTRIG 3/4) , (5.25)195here ∧ represent the logical AND operator, and the variables in parentheses rep-resent the applied cuts (∆Ω SHMS → angular acceptance cuts). Using these cuts,the ratio of e − did /e − should using Eqs. 5.24 and 5.25 was taken for the X (cid:48) tar and Y (cid:48) tar todetermine the variations in the proton transmission factor across the acceptance ofthe SHMS. The plots are shown in Figs. 5.22 and 5.23, respectively.Figure 5.22: Ratio of X (cid:48) tar from SHMS electron singles run 3259. Inset: X (cid:48) tar his-tograms before taking the ratio, where did is in red and should is in blue.Figure 5.23: Ratio of Y (cid:48) tar from SHMS electron singles run 3259. Inset: Y (cid:48) tar his-tograms before taking the ratio, where did is in red and should is in blue.196rom these ratios, the proton transmission factor is ∼
90% for the X (cid:48) tar ratio and ∼
95% for the Y (cid:48) tar near the center (Note: these early plots were not corrected for thetracking efficiency). At the edges, however, the ratio drops rapidly which indicatesthat the angular acceptance cuts need to be tightened. This is presumably due tothe HMS momentum acceptance rolling off at large X (cid:48) tar . The dashed blue linesindicate the region where the new SHMS angular acceptance cuts will be placed.In addition to the tighter acceptance cuts, a cut on the total energy deposited inthe SHMS calorimeter normalized by the central momentum was made to eliminateany possible pion background. The e − did also needed to be corrected for the HMStracking efficiency as this correction factor does not cancel in the ratio of Eq. 5.23a.The tracking efficiency for the analyzed run was determined to be 99.07% ( < W determined from SHMS elastic electron singles run3259.The final proton transmission factor was determined by taking the ratio of theintegrated invariant mass histograms reconstructed from SHMS electron singles (see197ig. 5.24). The W did and W should have their respective cuts, e − did and e − should , asdescribed above where the tighter acceptacne cuts, calorimeter cut and trackingefficiency corrections have been included.From Eqs. 5.23a, 5.23b and the number of electrons from Fig. 5.24, the finalproton transmission coefficient and its uncertainty were determined to be: (cid:15) pTr =0 . ± . Q tot ) To make a direct comparison between data and SIMC and determine the experi-mental cross section, the data must be normalized by the total experimental charge.The charge normalization for E12-10-003 was done on a data-set basis. That is,for each of the missing momentum sets, the runs were combined and scaled by thecorresponding accumulated charge. The precise determination of the uncertainty inthe charge is dependent on the BCM calibration, which is a work in progress. Forthis experiment, a conservative estimate of the relative uncertainty on the BCM4Acharge was determined to be dQ/Q = 0 .
02 (or 2%) [160].
The H( e, e (cid:48) ) p is the ideal reaction to study spectrometer acceptances, as well as de-termining spectrometer/kinematical offsets and misalignments. This is possible dueto the wide variety of data that exists at different kinematics, which has enabled thedetermination of the electric ( G Ep ) and magnetic ( G Mp ) form factors very preciselyover a wide kinematic range [161]. 198igure 5.25: Fully corrected experimental data to SIMC yield ratio for four H( e, e (cid:48) ) p elastic data.In the E12-10-003, four H( e, e (cid:48) ) p elastic data points taken at kinematics thatcovered a significant part of the SHMS acceptance ( δ SHMS : −
10 to 12%) were mainlyused for optics optimization (see Section 4.5). In addition, the hydrogen data werealso used to check how well they would agree with the calculated cross sectionsusing the form factor parametrization from Ref. [158]. This was achieved by takingthe ratio of the fully corrected data yield to the SIMC yield, which is equivalentto the ratio of the cross sections (see Fig. 5.25). The yields were determined byintegrating the invariant mass W in the range [0.85, 1.05] GeV using similar eventselection cuts as described in Section 4.6. There is a significant drop in the ratiowith increasing SHMS trigger rates, which is currently not well understood, butlikely due to unresolved issues with the tracking efficiency and/or the electronicdeadtime at very high rates. However, with respect to the first two data points, theratio is very close to one. 199e have decided to focus on the second data point (run 3288), as it is theclosest in kinematics to the 80 MeV/c deuteron data. From this data point, the H( e, e (cid:48) ) p elastic cross section has been measured with a normalization systematicerror of about 3% at a Q region where the absolute cross section is known at the ∼
6% level (shaded gray error) determined from the uncertainties in the form factors(Supplemental Materials of Ref. [161]). We quote the precise ratio of this data pointto be: R ≡ Y data Y SIMC ± δ stats ± δ (norm)syst ± δ ( G Ep ,G Mp )syst = 0 . ± . ± . ± . ∼ .
036 (see Table 5.4), but it does not tell us the normalization systematicerror better than the 5 .
6% error from the form factors. For this reason, we havedecided not to normalize the deuteron data to the ep elastics run 3288. f rad ) Radiative effects contribute significantly to the determination of the experimentalcross sections. These effects refer to the process by which the electron interactswith the electric field of a nearby nucleus causing the electron to change its ve-locity and emit either real or virtual photons known as bremsstrahlung radiation.This process can be further divided into either external or internal bremsstrahlungradiation. In external bremsstrahlung , the electron interacts with the electric fieldof a nucleus other than the nucleus involved in the scattering process, whereas in internal bremsstrahlung , it interacts with the electric field of the same nucleus itscatters off. Furthermore, the electron can radiate before and (or) after the (hard)200cattering process. As a result, the reaction kinematics and the cross section canbe modified significantly and the events might not necessarily be reconstructed atthe vertex, but at some other kinematics. This process becomes evident in theso-called radiative tails present in some of the reconstructed histograms, for exam-ple, the reconstructed missing energy in Fig. 4.41 or the invariant mass in Fig. 5.21.Figure 5.26: Examples of internal bremsstrahlung photons (red) for a general N ( e, e (cid:48) ) N . In the top diagrams, the electron emits a real bremsstrahlung photon(a) before and (b) after scattering off a nucleus N . In the bottom diagrams, theelectron exchanges a virtual bremsstrahlung photon with (c) the nucleus N and (d)between its own initial and final state.Theoretical models do not account for these radiative effects in their calculationsof the cross sections, therefore, the experimental data must be corrected before a201omparison with theory can be made. The theoretical calculations for radiativeeffects were first carried out by Schwinger [162] and later improved by Mo andTsai [163]. The simulation program SIMC uses the radiative correction formulasfor coincidence ( e, e (cid:48) p ) reactions calculated using the Mo and Tsai formulation byR. Ent et al. in Ref. [164]. A brief report describing how radiative effects are sim-ulated in SIMC can be found in Ref. [165]. Figure 5.26 shows a Feynman diagramrepresentation of a typical radiative process involving internal bremsstrahlung.In the E12-10-003 experiment, the radiative corrections were applied by multi-plying by the ratio of non-radiative to radiative SIMC yields, f rad , k = Y norad Y rad , (5.27)where k is an arbitrary kinematic bin that is defined based on the choice of kinematicfor binning the cross sections. For convenience, the radiative and non-radiative yieldhistograms were binned in k=( p r , θ nq ) bins and a ratio was taken and multiplied bythe experimental yield per data set, also binned in k=( p r , θ nq ). As an illustrativeexample, Fig. 5.27 shows a 2D histogram of the ratio between the non-radiative andradiative SIMC yields. The inset shows a vertical projection along p r for the bin θ nq = 35 ± ◦ , where the vertical axis of the inset corresponds to radiative correctionfactor, f rad .Figure 5.28 shows the experimental data yield binned in missing momentumbefore and after applying the radiative correction factor of the inset plot of Fig.5.27. After radiative corrections, it is clear from Fig. 5.28 that there is a significantincrease in the number of coincidence counts per mC for each bin in p r . These“recovered” counts are interpreted as the number of coincidences that should havebeen detected if it were not for the radiative effects modifying the reaction kinematicsat the vertex, as shown in Fig. 5.26. 202igure 5.27: 2D histogram of radiative correction factor, f rad , binned in ( p r , θ nq ) forthe 80 MeV/c setting of E12-10-003. Inset: Y-projection of θ nq between 30 and 40degrees.Figure 5.28: Missing momentum yield for θ nq = 35 ± ◦ before and after radiativecorrections for the 80 MeV/c setting of E12-10-003.203 .9 Bin Centering Corrections ( f bc ) As mentioned in Section 5.1, the experimental cross sections are averaged over akinematic bin k, which was re-defined as k=( p r , θ nq ) in Section 5.8. In reality,( p r , θ nq ) has a sub-range of kinematics over which the theoretical cross section canvary rapidly so the question arises as to which kinematic value within this bin shouldbe associated with the experimental cross section.We define the bin-centering correction factor for the k=( p r , θ nq ) kinematic bin as f bc , k ≡ σ model (¯k)¯ σ model (k) , (5.28)where the numerator is the theoretical cross section calculated at the averaged kine-matics (calculation is external to SIMC) and the denominator is the theoretical crosssection determined from SIMC, averaged over the kinematic bin k=( p r , θ nq ). TheLaget FSI model was used for the theoretical cross section calculations. See Section5.10 for systematic studies using the PWIA and FSI Laget models.The bin-center corrected data cross sections at each kinematic bin were deter-mined by multiplying the average experimental cross section by the correction factorat each kinematic bin as follows: σ databc , ¯k ≡ ¯ σ data , k · f bc , k , (5.29)where σ databc , ¯k is the bin-centered corrected data cross section evaluated at the aver-aged kinematics (¯k) over bin k=( p r , θ nq ).Figure 5.29 shows an illustrative example of how the bin-centering correctionswere done for this experiment. The blue and green bins represent the bin contentof the average data and model cross sections, and the theoretical curve representsthe same model cross section evaluated at the averaged kinematics, denoted by abar. The orange bin represents the bin content of the data cross sections after ap-plying the bin-centering corrections. The advantage of performing the bin-centering204orrections this way is that one can avoid time-consuming Monte Carlo calculationswhen comparing the experimental data to different theoretical models.Figure 5.29: Cartoon illustrating bin-centering calculation for this experiment.The averaged kinematics on this experiment were determined from SIMC at thereaction vertex for every ( p r , θ nq ) bin since the vertex quantities are corrected forenergy loss at the target. The average for a kinematic quantity X for the kinematicbin k=( p r , θ nq ) was calculated as ¯ X k = ( (cid:80) i X i w i ) k ( (cid:80) i w i ) k , (5.30)where X i is the value of the kinematic quantity X for the event i and the modelcross section was used as weight ( w i ).Figures 5.30 and 5.31 show the bin centering correction versus p r ±
20 MeV/cbins at θ nq =35 and 45 ± ◦ bins using either the PWIA or FSI models from Ref. [60].205igure 5.30: Bin-centering correction factor at θ nq = 35 ± ◦ for each p r settingof E12-10-003. The dashed reference lines indicate ±
10% (black) or ±
20% (red)deviation from unity. The correction factor was calculated by taking the ratioof cross sections (see Eq. 5.28) either within the PWIA (full data points) or byincluding FSI (empty data points) for each dataset (see Section 3.3). The theoreticalcross section calculations were by J.M. Laget [60] using the Paris potential [41].Figure 5.31: Bin-centering correction factor at θ nq = 45 ± ◦ for each p r setting ofE12-10-003. See Fig. 5.30 for definition of lines and data points.206t the low missing momentum setting (80 MeV/c), the corrections are relativelylarge (ignoring the lowest missing momenta at ∼
20 MeV/c) with correction factorsbetween 10-20% for recoil momenta up to p r ∼
240 MeV/c whereas at higher recoilmomenta up to ∼ p r , whereas the smaller corrections athigher missing momenta are representative of a smaller (less steep) fall-off as willbe shown in the reduced cross sections presented in Chapter 6. A study of the sensitivity of the bin-center corrected experimental cross sectionsto variations in the event selection cuts described in Section 4.6 was carried out.Comparing two data subsets one needs to take into account the fact that part of thedata are correlated. To determine if the variation in each of the cuts contributes toa systematic effect and whether this contribution is significant enough to be consid-ered as a systematic error, we use the approach by R. Barlow described in Ref. [166].Consider a cross section measurement done two different ways (i.e., apply differ-ent cuts). Let the measurements and their statistical uncertainties be: ( σ expbc , ± δσ expbc , )and ( σ expbc , ± δσ expbc , ) where one of the measurements is a subset of the other. Thedifference and its associated uncertainty can be expressed as,∆ ≡ σ expbc , − σ expbc , , (5.31a) σ ≡ ( δσ expbc , ) − ( δσ expbc , ) , (5.31b)where the error of the difference between the two measurements is found by takingthe difference of their variance. As demonstrated in Ref. [166], this error accounts207or the possible correlation between the two measurements. By taking the ratio R Barlow ≡ ∆ σ ∆ , (5.32)a criterion imposed on R Barlow determines whether the difference is significant enoughto be considered as a systematic error or sufficiently small that it may be ignored.This criterion requires knowledge of the correlation between the subsets, but ingeneral, as suggested in Ref. [167]: if R Barlow < < σ ∆ ) the test passes andif R Barlow > > σ ∆ ), the test fails and the discrepancy must be added as asystematic error. For 2 < R Barlow <
4, a judgement must be made.In E12-10-003, for each of the event selection cuts, the difference between thedata cross section corresponding to the largest cut and each of the subset cuts wastaken and divided by the corresponding difference in their variances to the halfpower to determine the ratio for each ( p r , θ nq ) bin. As an example, the results for asingle θ nq bin on each of the event selection cuts are shown in Figs. 5.32-5.36. .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . p r [GeV] − − − R = ∆ / σ ∆ E m Cut Systematics, θ nq = 35 ± ◦
80 MeV ( E m : 30 MeV)580 MeV (set2)750 MeV (set1)80 MeV ( E m : 40 MeV)580 MeV (set2)750 MeV (set1)80 MeV ( E m : 50 MeV)580 MeV (set2) MeV750 MeV (set1) MeV80 MeV ( E m : 60 MeV)580 MeV (set2) MeV750 MeV (set1) MeV Figure 5.32: Systematic effects of the missing energy cut. The inner dashed andouter solid lines represent the ∆ = ± σ ∆ and ± σ ∆ boundaries, respectively.208 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . p r [GeV] − − − R = ∆ / σ ∆ Z tar Cut Systematics, θ nq = 35 ± ◦
80 MeV ( | Z tar | < | Z tar | < | Z tar | < | Z tar | < | Z tar | < Figure 5.33: Systematic effects of the Z tar difference cut. The lines are described inFig. 5.32. .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . p r [GeV] − − − R = ∆ / σ ∆ HMS Collimator Cut Systematics, θ nq = 35 ± ◦
80 MeV (Scale: 1.0)580 MeV (set2) MeV750 MeV (set1) MeV80 MeV (Scale 0.9)580 MeV (set2)750 MeV (set1)80 MeV (Scale 0.8)580 MeV (set2)750 MeV (set1)
Figure 5.34: Systematic effects of the HMS collimator cut. The lines are describedin Fig. 5.32. 209 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . p r [GeV] − − − R = ∆ / σ ∆ Coincidence Time Cut Systematics, θ nq = 45 ± ◦
80 MeV580 MeV (set1)580 MeV (set2)750 MeV (set1)750 MeV (set2)750 MeV (set3)
Figure 5.35: Systematic effects of the coincidence time cut. The lines are describedin Fig. 5.32. .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . p r [GeV] − − − R = ∆ / σ ∆ SHMS Calorimeter Cut Systematics, θ nq = 45 ± ◦
80 MeV580 MeV (set1)580 MeV (set2)750 MeV (set1)750 MeV (set2)750 MeV (set3)
Figure 5.36: Systematic effects of the SHMS calorimeter cut. The lines are describedin Fig. 5.32. 210he Barlow ratio shown in these plots are used mainly as a check that the deviationsin the cross section due to applied cuts are within a reasonable boundary ( < σ ∆ )such that the systematic effects can be neglected. Some important points to keepin mind on the interpretation of these systematic studies results are: • As can be seen from some of these plots, for example, in Fig. 5.35 or 5.36,the data points are scarce, which indicates that there was no difference in themeasured cross sections giving a ratio of zero. This is understood from thefact that this experiment was very clean of any background sources, and sotaking the difference in cross sections with and without these cuts does notaffect the final result at all. • For the missing energy cut, we do expect the ratio to change, as we are chang-ing how much of the radiative tail we include in the cut, so a different numberof counts is expected. The ratio was found to be more spread out in the 80MeV/c setting, however, it was mostly within the boundaries at the highermissing momentum settings. This could be attributed to the low numberof statistics in the larger settings and hence larger variances (and statisticalfluctuations). • Some of the ratios might be slightly negative. So one asks, how it can be thatwhen a subset cut cross section is subtracted from the cross section of thetotal set, we end up with negative? Which could only mean that the subsetcross section is larger. One possible explanation is that this is due to theradiative correction factor becoming smaller with larger cuts, which meansthat a very tight cut (subset) can have a larger cross section due to a largerradiative correction factor. Another possibility is that a smaller cut reducesthe acceptance of the spectrometer.211
If the variances of the two measurements are almost the same, this can givevery large values of the ratio, as the denominator is the difference in thevariances. . . . . . . . p r [GeV] − − − R = ∆ / σ ∆ Radiative Systematics, θ nq = 35 ± ◦
80 MeV580 MeV (set1)580 MeV (set2)750 MeV (set1)750 MeV (set2)750 MeV (set3)
Figure 5.37: Systematic effects of the model dependency of radiative corrections.The lines are described in Fig. 5.32. . . . . . . . p r [GeV] − − − R = ∆ / σ ∆ Bin-Centering Systematics, θ nq = 35 ± ◦
80 MeV580 MeV (set1)580 MeV (set2)750 MeV (set1)750 MeV (set2)750 MeV (set3)
Figure 5.38: Systematic effects of the model dependency of bin-centering corrections.The lines are described in Fig. 5.32. 212n additional study was carried out to determine the magnitude of the systematiceffects due to the model dependency of the radiative and bin-centering correctionsapplied to the experimental data. In this case, the Barlow ratio was calculated fromthe difference between the experimental cross sections using the Laget PWIA andFSI models for both radiative and bin-centering corrections. In other words, theexperimental cross sections were radiation and bin-center corrected using both theLaget PWIA and FSI models and the difference between these models was compared.Figures 5.37 and 5.38 are shown for a single bin in θ nq and demonstrate that theeffects of model dependency on the correction factors are negligible on the measuredcross sections. Starting from the fully corrected experimental cross section, σ exp = σ expuncorr · f · f ... · f n , (5.33)where σ expuncorr is the uncorrected data yield ( Y uncorr ) divided by the SIMC phase space,and f (cid:48) n s are the correction factors, which are re-defined for covenience as f = 1 (cid:15) tgt . Boil target density factor (5.34) f = 1 (cid:15) pTr proton transmission factor (5.35) f = 1 (cid:15) etrk electron tracking efficiency (5.36) f = 1 (cid:15) htrk hadron (proton) tracking efficiency (5.37) f = 1 (cid:15) tLT total live time (5.38) f = 1 Q exptot total accumulated charge (5.39)213 = f rad radiative correction factor (5.40) f = f bc bin-centering correction factor (5.41)The systematic uncertainty on the cross section due to the uncertainty in each ofthese correction factors are added in quadrature as( dσ exp ) = (cid:88) i =1 (cid:16) ∂σ exp ∂f i (cid:17) df i . (5.42)From Eq. 5.33, the derivative with respect to factor f i is ∂σ exp ∂f i = σ exp f i . (5.43)Substituting Eq. 5.43 into Eq. 5.42, one obtains( dσ exp ) = ( σ exp ) (cid:88) i =1 (cid:16) df i f i (cid:17) (5.44)From Eq. 5.44, one can see that the overall relative error on the cross section due tothe normalization uncertainties is simply the quadrature sum of the relative errorsof each normalization factor. For the radiative and bin-centering correction factors,the relative error is zero as determined from the systematics studies in the previoussection.The normalization uncertainties on the cross sections were determined per dataset by averaging each normalization factor and its associated uncertainty over allruns of said set. The average was taken as there were no significant fluctuationsin the factors over all runs for each data set. The results from each data set wereadded in quadrature for overlapping missing momentum bins once the reduced crosssections were combined for all sets. For the relative normalization uncertainties thatdid not vary over the entire experiment, that is, the uncertainties associated withthe proton transmission factor, the total live time and charge, these were added inquadrature as an overall factor to the final error (see Tables 5.3 and 5.4).214 r [MeV/c] (cid:15) htrk (cid:15) etrk (cid:15) tgt . Boil (cid:15) pTr (cid:15) tLT Q exptot [mC]80 0.989 0.965 0.958 0.953 0.908 142.140580(set1) 0.990 0.965 0.960 0.953 0.929 1686.83580(set2) 0.987 0.964 0.959 0.953 0.929 1931.77750(set1) 0.988 0.964 0.957 0.953 0.924 5329.49750(set2) 0.989 0.962 0.956 0.953 0.923 1894.01750(set3) 0.989 0.962 0.956 0.953 0.924 1083.70Table 5.3: Summary of the averaged normalization correction factors (or efficiencies)in fractional form and the total accumulated charge per data set. p r [MeV/c] δ(cid:15) htrk /(cid:15) htrk δ(cid:15) etrk /(cid:15) etrk δ(cid:15) tgt . Boil /(cid:15) tgt . Boil δ(cid:15) pTr /(cid:15) pTr δ(cid:15) tLT /(cid:15) tLT δdQ exptot /Q exptot
80 0.0344 0.0413 0.3948 0.4951 3.0 2.0580(set1) 0.3999 0.7586 0.3766 0.4951 3.0 2.0580(set2) 0.4786 0.6041 0.3842 0.4951 3.0 2.0750(set1) 0.5329 0.7155 0.4013 0.4951 3.0 2.0750(set2) 0.4719 0.7089 0.4196 0.4951 3.0 2.0750(set3) 0.5127 0.7584 0.4150 0.4951 3.0 2.0
Table 5.4: Summary of relative systematic error on the measured cross sections dueto the normalization factors (units are in percent).
The determination of the experimental cross sections depends on the spectrometerkinematics. In particular for this coincidence experiment, the determination of thebeam energy ( E b ), final electron angle and momentum ( θ e , k f ) and either the protonangle ( θ p ) or momentum ( p f ) completely determines the deuteron reaction kinemat-ics. Therefore, how well can we measure these quantities (kinematic uncertainties)determines how well can we measure the experimental cross sections.Ideally, the kinematic uncertainties can be determined by taking a series of dedi-cated H( e, e (cid:48) ) p elastic singles for each spectrometer arm at a wide range of kinemat-ics as well as beam energies. The data can then be simultaneously fit to determinethe kinematical offsets as well as the kinematic uncertainties. In reality, this is very215ifficult to do due to the availability of the beam as well as the required time forthese runs to take place in a very busy experimental schedule at Jefferson Lab.In E12-10-003, we had a limited and usable hydrogen elastic data set (runs 3288,3371 and 3374) taken in coincidence, which we used to simultaneously determine thekinematical uncertainties using the procedure described in Ref. [168]. This methodbasically consisted of a general χ -minimization procedure using a matrix approachthat enables one to extract the variance-covariance matrix that contain the uncor-related (diagonal) as well as the correlated (off-diagonal) kinematical uncertainties.Of the several models described, we used the results from MODEL 2 of Ref. [168],which simultaneously fit the elastic data to determine the kinematical as well asthe correlated uncertainties on ( E b , k f , θ e , θ p ). Tables 5.5 and 5.6 summarize thekinematical and correlated uncertainties on each of the variables.Kinematic Uncertainties δθ e δθ p δk f / k f × − δE b / E b × − Table 5.5: Kinematic uncertainties corresponding to the diagonal elements of thecorrelation matrix. Correlated Kinematic Uncertaintiescov( E b , k f ) 6.838 × − cov( E b , θ e ) -1.213 × − [rad]cov( E b , θ p ) -6.267 × − [rad]cov( k f , θ e ) -1.488 × − [rad]cov( k f , θ p ) -7.014 × − [rad]cov( θ e , θ p ) 8.432 × − [rad] Table 5.6: Kinematic uncertainties corresponding to the off-diagonal elements of thecorrelation matrix. 216s a side note, recall from the beam energy measurements made in Section3.4.1, that the relative error in the beam energy was determined to be δE b /E b =4 . × − as compared to δE b /E b = 7 . × − determined from Ref. [168].These two measurements are very close to each other and on the same order ofmagnitude at the ∼ − level. Given the small effect that variations in the beamenergy (by itself, and not correlated) has on the deuteron cross sections, we decidedto use the more conservative and slighly larger value determined from Ref. [168].To determine the systematic effects of the kinematical uncertainties on the mea-sured cross sections, the derivatives of the cross section with respect to each ofthe kinematic variables are needed. For this experiment, a table of the cross sectionderivatives has already been calculated using the Laget FSI model. These derivatveshave been calculated using the averaged kinematics determined per data set as input.Using the standard error propagation formula, the table of cross section derivativesand the kinematical uncertainties, the full kinematic uncertainty contribution to theexperimental cross section can be expressed as( δσ expkin ) = (cid:16) dσdθ e δθ e (cid:17) + (cid:16) dσdθ p δθ p (cid:17) + (cid:16) dσdk f δk f k f k f (cid:17) + (cid:16) dσdE b δE b E b E b (cid:17) + 2 dσdE b dσdk f cov( E b , k f ) + 2 dσdE b dσdθ e cov( E b , θ e ) + 2 dσdE b dσdθ p cov( E b , θ p )+ 2 dσdk f dσdθ e cov( k f , θ e ) + 2 dσk f dσdθ p cov( k f , θ p ) + 2 dσdθ e dσdθ p cov( θ e , θ p ) . (5.45)By including the covariance errors from Eq. 5.45, the overall kinematics uncer-tainty can actually be reduced in the case where any two variables are anti-correlated(“-” covariance sign) as it is the case from most of the variables in Table 5.6.As an example, Fig. 5.39 shows the model cross section derivatives wirh respectto each kinematic variable and Fig. 5.40 shows the systematic contribution to themeasured cross section due to the kinematics as well as the correlated uncertaintitesfor θ nq = 35 ± ◦ . 217 . . . . . . p r [GeV/c] − − − − − δ σ L ag e t , f s i [ % / m r a d o r % / M e V ] Cross Section Derivatives: 80 MeV (set1), θ nq = 35 ± ◦ dσdθ e dσdθ p dσdE f dσdE b Figure 5.39: Laget FSI model cross section derivatives at θ nq = 35 ± ◦ for the 80MeV/c setting.In Fig. 5.39, the electron scattering angle uncertainty ( δθ e ) clearly has the largesteffect on the cross sections with a ∼ − p r ∼
300 MeV/c,whereas the other kinematics have a (cid:46)
5% effect per mrad (or MeV). This largeeffect on the electron angle is attributed to the Mott cross section, σ Mott , whichhas a dependence of σ Mott ∝ ( θ e ) on the electron angle. Therefore, it is crucialthat we determine the SHMS (electron arm) angle to much better than 1 mrad ofuncertainty. The variations on the higher momentum settings are still dominatedby the electron angle but to a much lower extent of ∼ E b , θ e ) in cyan, cov( E b , k f ) in218avy blue, and cov( k f , θ e ) in green. Two out of these three major contributors, how-ever, are anti-correlated so when the errors are added in quadrature, the final resultis actually smaller, as can be seen by the total kinematic systematic error (color red). . . . . . . p r [GeV/c] − − − δ σ k i n [ % ] Kinematic Systematic Errors: 80 MeV (set1), θ nq = 35 ± ◦ dσ θ e dσ θ p dσ k f dσ E b [ dσ θ e , dσ θ p ][ dσ θ e , dk f ][ dσ θ e , dE b ][ dσ θ p , dk f ] [ dσ θ p , dE b ][ dk f , dE b ] dσ kintot Figure 5.40: Kinematic systematics relative error contributions to the experimentalcross sections at θ nq = 35 ± ◦ for the 80 MeV/c setting. The overlall relative erroron the cross section is shown in red.The kinematic systematic errors, such as that shown on Fig. 5.40, were deter-mined poin-to-point in ( p r , θ nq ) bins for each missing momentum setting, and addedin quadrature for overlapping p r bins. The overall systematic uncertainty in thefinal cross section was determined by the quadrature sum of the normalization andkinematic systematic uncertainties. This result was then added in quadrature tothe statistical uncertainty (20-30% on average) to obtain the final uncertainty in219he cross section. See Appendices A1 and A2 for a summary of the relative sta-tistical and systematic uncertainties for each ( p r , θ nq ) bin at Q = 3 . ± . Q = 4 . ± . , respectively. 220HAPTER 6 RESULTS AND DISCUSSION
In this chapter I will discuss how the experimental reduced cross sections wereextracted. Finally, I will present the final experimental reduced cross sections com-pared to various theoretical models as well as their cross section ratios and discussthe implications of these results. H ( e, e (cid:48) p ) n Reduced Cross Sections
From the theoretical cross sections introduced in Eq. 2.11 under the PWIA assump-tion, we define the experimental (or theoretical) reduced cross section evaluated ata kinematic bin k=( p r , θ nq ) as( σ exp , thred ) k ≡ ( σ exp , th ) k ( E f p f f rec σ cc1 ) k , (6.1)where σ exp , th is the fully corrected data (or theoretical) cross section, ( E f , p f ) are thefinal proton energy and momentum, respectivelty, f rec is a recoil factor introducedin Eq. 2.12, and σ cc1 = σ cc1 ( G E p , G M p ) is one of two variations of the deForest [73]off-shell cross section that describes the scattering between an electron and a looselybound (“free”) proton.By dividing by the kinematical and recoil factors as well as the deForest crosssection, which depends on the proton elastic form factors, most of the kinematicaldependencies found on the cross sections are cancelled leaving only a dependenceon the neutron recoil momentum, p r . Therefore, the reduced cross sections areclosely related to the genuine momentum distributions of the nucleons provided thekinematics have been chosen such that the PWIA is dominant. In analogy, this isalso observed in H( e, e (cid:48) ) p , where the elastic cross sections are divided by σ Mott to221xtract the proton elastic form factors which describe the internal structure of theproton.It is important to note that each of the kinematic variables in the denominator ofEq. 6.1, similar to the fully corrected cross sections ( σ exp , th ) determined in Section5.9, were also determined at the averaged kinematic setting for each kinematic bin in( p r , θ nq ). Finally, the reduced cross sections calculated for each missing momentumdata set were combined for overlapping bins in ( p r , θ nq ) to gain better statisticalprecision. H ( e, e (cid:48) p ) n Momentum Distributions
The experimental and theoretical reduced cross sections have been combined foroverlapping bins in p r for each data set and are shown in Figs. 6.1-6.10. Theratio of the experimental and theoretical reduced cross sections to the CD-BonnPWIA model has also been evaluated where the inset plot shows a more detailed(close-up) representation of the ratios taken. In addition to the results presentedat Q = 4 . ± . , reduced cross sections have also been determined at Q = 3 . ± . for comparison. For θ nq = 35 ◦ , ◦ and 75 ◦ we have plottedthe reduced cross sections from the previous deuteron break-up experiment per-formed in Hall A [55] at Q = 3 . ± .
25 (GeV/c) .The error on the cross sections have been determined by adding the statisticaland systematic errors in quadrature. See Appendices A1 and A2 for the tabulatedreduced cross sections and the associated statistical and systematic errors for each( p r , θ nq ) bin at Q = 3 . ± . Q = 4 . ± . , respectively.222 − − σ r e d [ f m ] Reduced Cross Section, θ nq = 5 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d .
00 0 .
05 0 .
10 0 . . . . . . Figure 6.1: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 5 ± ◦ . Top panel: Theblue lines represent the theoretical calculations by J.M. Laget [60] using the Parispotential [41] denoted by JML and the green/magenta lines are calculations fromM. Sargsian [59] using either the AV18 (green) [42] or CD-Bonn (magenta) [43] po-tentials denoted by MS. The dashed lines are calculations within the PWIA and thesolid lines are calculations including FSI. Bottom panel: The dashed reference (ma-genta) line refers to MS CD-Bonn PWIA calculation (or momentum distribution)by which the data and all models are divided. Inset (bottom panel): Close-up plotof the reduced cross section ratio shown in the bottom panel.223 − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 15 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . . . . . Figure 6.2: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 15 ± ◦ . The lines aredescribed in Fig. 6.1. 224 − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 25 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d . . . . . . . . . . Figure 6.3: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 25 ± ◦ . The lines aredescribed in Fig. 6.1. 225 − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 35 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) Q = 3 . ± .
25 GeV (Hall A) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d . . . . . . . . . . . . Figure 6.4: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 35 ± ◦ . The lines aredescribed in Fig. 6.1. 226 − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 45 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) Q = 3 . ± .
25 GeV (Hall A) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d . . . . . . . . . . . . . . . . Figure 6.5: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 45 ± ◦ . The lines aredescribed in Fig. 6.1. 227 − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 55 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d . . . . . Figure 6.6: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 55 ± ◦ . The lines aredescribed in Fig. 6.1. 228 − − − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 65 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d . . . . Figure 6.7: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 65 ± ◦ . The lines aredescribed in Fig. 6.1. 229 − − − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 75 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) Q = 3 . ± .
25 GeV (Hall A) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d .
00 0 .
25 0 .
50 0 .
75 1 . Figure 6.8: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 75 ± ◦ . The lines aredescribed in Fig. 6.1. 230 − − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 85 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d . . . . . . . . . . . . Figure 6.9: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 85 ± ◦ . The lines aredescribed in Fig. 6.1. 231 − − − σ r e d [ f m ] Reduced Cross Section, θ nq = 95 ± ◦ JML Paris PWIAJML Paris FSIMS AV18 PWIAMS AV18 FSIMS CD-Bonn PWIAMS CD-Bonn FSI Q = 4 . ± . (Hall C) Q = 3 . ± . (Hall C) . . . . . . . p r [GeV/c] R = σ r e d / σ C D − B o nn P W I A r e d . . . . . . . . Figure 6.10: H( e, e (cid:48) p ) n reduced cross sections at θ nq = 95 ± ◦ . The lines aredescribed in Fig. 6.1. 232 .3 Discussion of Results Even though the focus of this experiment was to extract the H( e, e (cid:48) p ) n reducedcross sections at 35 ◦ < θ nq < ◦ where the FSI are significantly reduced, it hasbeen possible to determine the reduced cross sections at a different range of neutronrecoil angles, 5 ◦ < θ nq < ◦ and 55 ◦ < θ nq < ◦ , for comparison. In general, thereis an overall good agreement between the Halls A and C data ( θ nq = 35 ◦ , ◦ and75 ◦ ) even though they were taken at different Q kinematics. This can be under-stood from the fact that the kinematical dependencies of the cross sections havebeen divided out in the reduced cross sections, as mentioned in Section 6.1. Thegood agreement between both experiments at lower p r gives us confidence in themeasurements made at the higher missing momentum settings for which no previ-ous data exist.For all recoil angles shown in Figs. 6.1-6.10, at recoil momenta p r ≤
250 MeV/c,the reduced cross sections are well reproduced by all models when FSI are included.The agreement at p r ≤
250 MeV/c can be understood from the fact that this re-gion corresponds to the long-range part of the
N N potential where the One PionExchange Potential (OPEP) is well known and common to all modern potentials.Beyond p r ∼
250 MeV/c at θ nq = 35 ◦ and 45 ◦ , the JML Paris and MS AV18models significantly differ from the MS CD-Bonn calculation. In this region, theJML Paris and MS AV18 cross sections are dominated by the PWIA and withingood agreement of each other up to p r ∼
700 MeV/c. The MS CD-Bonn basedcross sections, in contrast, are generally smaller than those calculated with the JMLParis or MS AV18 wave function in this region. In addition, for θ nq = 35 ◦ , the MSCD-Bonn cross sections are dominated by the PWIA up to p r ∼
800 MeV/c whilefor θ nq = 45 ◦ , FSI start to contribute already above p r ∼
600 MeV/c.233t θ nq = 75 ◦ and p r >
180 MeV/c, FSI become the dominant contribution to thecross sections for all models that exhibit a similar behavior (smaller fall-off) thateliminates any possibility of extracting the momentum distributions.The difference between the deuteron wave functions with CD-Bonn, Paris andAV18 potentials [169] is how the
N N potential is modeled based on the empirical
N N scattering data. The CD-Bonn model is based on the One-Boson-Exchangeapproach in which the nucleon-meson-meson couplings are constrained to describethe
N N scattering phase shifts extracted from the data. The interaction poten-tial represents the static limit of this potential. The Paris and AV18 models arephenomenological in which the Yukawa type interaction is introduced and parame-ters are fit to describe the same
N N scattering phase-shifts. The major differencebetween CD-Bonn and Paris/AV18 potentials is that the former predicts a muchsofter repulsive interaction at short distance that results in a smaller high momen-tum component in the deuteron wave function in momentum space. The effect ofthese local approximations on the
N N potential are shown in Fig. 2 of Ref. [43].To quantify the discrepancy observed between data and theory at higher miss-ing momenta for θ nq = 35 ◦ and 45 ◦ , the ratio of the experimental and theoreticalreduced cross sections to the deuteron momentum distribution calculated using theCD-Bonn potential is shown in the lower subplot of Figs. 6.1-6.10. For θ nq = 35 ◦ and 45 ◦ , the data are best described by the MS CD-Bonn FSI calculation for recoilmomenta up to p r ∼
700 MeV/c and ∼
600 MeV/c, respectively, with a ratio of R ∼ . − R ∼ − θ nq = 75 ◦ which indicates a significantreduction in FSI at forward θ nq angles. Furthermore, the agreement between theHalls A and C data supports the Hall A approach of selecting a kinematic regionwhere recoil angles are small and FSI are reduced.234t larger recoil momenta, where the ratio is R > θ nq = 35 ◦ ,FSI start to dominate for missing momenta typically above 800 MeV/c for the MSCD-Bonn calculation, while the other models predict still relatively small FSI below900 MeV/c. At θ nq = 45 ◦ , the FSI dominance starts earlier for all models above 800MeV/c and for the MS CD-Bonn based calculation, above 600 MeV/c.Overall, it is interesting to note that none of the calculations can reproduce themeasured p r dependence above 600 MeV/c in a region where FSI are still relativelysmall ( < θ nq = 75 ◦ , FSI are small below p r ∼
180 MeV/c, but do not exactly cancel thePWIA/FSI interference term in the scattering amplitude, which results in a smalldip in this region in agreement with the data. At p r >
300 MeV/c, the data werestatistically limited as our focus was on the smaller recoil angles. The Hall A data,however, show a reasonable agreement with the FSI from all models, which gives usconfidence in our understanding of FSI at smaller recoil angles.
This experiment extended the previous Hall A cross section measurements [55] onthe H( e, e (cid:48) p ) n reaction to very high neutron recoil momenta ( p r >
500 MeV/c) atkinematic settings where FSI were predicted to be small and the cross section wasdominated by the PWIA and sensitive to the short range part of the deuteron wavefunction. The experimental and theoretical reduced cross sections were extractedand found to be in good agreement with the Hall A data. Furthermore, the MSCD-Bonn model was found to be significantly different than the JML Paris or MSAV18 models and was able to partially describe the data over a larger range in p r . At235igher missing momenta, however, all models were unable to describe the missingmomentum dependence of the data. Additional measurements of the H( e, e (cid:48) p ) n would be required at a wider range in central missing momentum, as stated inthe original proposal [100], to reduce the statistical uncertainties in this very highmissing momentum region ( p r >
500 MeV/c) and to better understand the largedeviations observed between the different models and data.236IST OF REFERENCES[1] Harold C. Urey, F. G. Brickwedde, and G. M. Murphy. A Hydrogen Isotope ofMass 2.
Phys. Rev. , 39(1):164–165, Jan 1932. https://link.aps.org/doi/10.1103/PhysRev.39.164 . 1[2]
The existence of a neutron , volume 136. The Royal Society, June 1932. http://doi.org/10.1098/rspa.1932.0112 . 1[3] W. Heisenberg. ¨Uber den Bau der Atomkerne. I.
Zeitschrift f¨ur Physik ,77(1):1–11, Jan 1932. https://doi.org/10.1007/BF01342433 . 1[4] W. Heisenberg. ¨Uber den Bau der Atomkerne. II.
Zeitschrift f¨ur Physik ,78:156–164, March 1932. https://doi.org/10.1007/BF01337585 . 1[5] W. Heisenberg. ¨Uber den Bau der Atomkerne. III.
Zeitschrift f¨ur Physik ,80:587–596, September 1933. https://doi.org/10.1007/BF01335696 . 1[6] H. Bethe, Peierls R., and Hartree Douglas R. Quantum theory of the diplon.
Proceedings of the Royal Society , 148:146–156, 1935. https://doi.org/10.1098/rspa.1935.0010 . 1[7] M. Garcon and J. W. Van Orden. The deuteron: structure and form factors,2001. https://arxiv.org/abs/nucl-th/0102049 . 1[8] C. Lattes, G. Occhialini, and C. Powell. Observations on the Tracks of SlowMesons in Photographic Emulsions.
Nature , 160:486–492, 1947. https://doi.org/10.1038/160486a0 . 2[9] J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias. An ElectricalQuadrupole Moment of the Deuteron.
Phys. Rev. , 55(3):318–319, Feb 1939. https://link.aps.org/doi/10.1103/PhysRev.55.318 . 2[10] J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias. An ElectricalQuadrupole Moment of the Deuteron The Radiofrequency Spectra of HD andD Molecules in a Magnetic Field.
Phys. Rev. , 57:677–695, Apr 1940. https://link.aps.org/doi/10.1103/PhysRev.57.677 . 2[11] G. M. Murphy and Helen Johnston. The Nuclear Spin of Deuterium.
Phys.Rev. , 46:95–98, Jul 1934. https://link.aps.org/doi/10.1103/PhysRev.46.95 . 4 23712] V. I. Zhaba. Deuteron: properties and analytical forms of wave function incoordinate space, 2017. https://arxiv.org/abs/1706.08306 . 5[13] EUGENE GARDNER and C. M. G. LATTES. Production of Mesons bythe 184-Inch Berkeley Cyclotron.
Science , 107(2776):270–271, 1948. https://science.sciencemag.org/content/107/2776/270 . 5[14] Mituo Taketani, Seitaro Nakamura, and Muneo Sasaki. On the Method of theTheory of Nuclear Forces.
Progress of Theoretical Physics , 6(4):581–586, 081951. https://doi.org/10.1143/ptp/6.4.581 . 5[15] M. Naghdi. Nucleon-Nucleon Interaction: A Typical/Concise Review.
Physicsof Particles and Nuclei , 45(5):924971, Sep 2014. http://dx.doi.org/10.1134/S1063779614050050 . xiv, 6[16] Mitsuo Taketani, Shigeru Machida, and Shoroku O-numa. The Meson The-ory of Nuclear Forces, I*: The Deuteron Ground State and Low EnergyNeutron-Proton Scattering.
Progress of Theoretical Physics , 7(1):45–56, 011952. https://doi.org/10.1143/ptp/7.1.45 . 5[17] K. A. Brueckner and K. M. Watson. The Construction of Potentials in Quan-tum Field Theory.
Phys. Rev. , 90:699–708, May 1953. https://link.aps.org/doi/10.1103/PhysRev.90.699 . 5[18] K. A. Brueckner and K. M. Watson. Nuclear Forces in Pseudoscalar MesonTheory.
Phys. Rev. , 92:1023–1035, Nov 1953. https://link.aps.org/doi/10.1103/PhysRev.92.1023 . 5[19] Saburo Fujii, Junji Iwadare, Shoichiro Otsuki, Mitsuo Taketani, Smio Tani,and Wataro Watari. The Meson Theory of Nuclear Forces, II*: High EnergyNucleon-Nucleon Scattering.
Progress of Theoretical Physics , 11(1):11–19, 011954. https://doi.org/10.1143/PTP.11.11 . 5[20] Junji Iwadare, Shoichiro Otsuki, Ryozo Tamagaki, Shigeru Machida,Toshiyuki Toyoda, and Wataro Watari. Part I. Development of Pion The-ory of Nuclear Forces.
Progress of Theoretical Physics Supplement , 3:13–31,01 1956. https://doi.org/10.1143/PTPS.3.13 . 6[21] Mitsuo Taketani. Introduction. Nuclear Forces and Meson Theory.
Progressof Theoretical Physics Supplement , 3:1–12, 01 1956. https://doi.org/10.1143/PTPS.3.1 . 6 23822] Yoichiro Nambu. Possible Existence of a Heavy Neutral Meson.
Phys. Rev. ,106:1366–1367, Jun 1957. https://link.aps.org/doi/10.1103/PhysRev.106.1366 . 6[23] Alexander Abashian, Norman E. Booth, and Kenneth M. Crowe. PossibleAnomaly in Meson Production in p + d Collisions.
Phys. Rev. Lett. , 5:258–260, Sep 1960. https://link.aps.org/doi/10.1103/PhysRevLett.5.258 .6[24] Norio Hoshizaki, I Lin, and Shigeru Machida. Nonstatic One-Boson-ExchangePotentials: .
Progress of Theoretical Physics , 26(5):680–692, 11 1961. https://doi.org/10.1143/PTP.26.680 . 6[25] A. R. Erwin, R. March, W. D. Walker, and E. West. Evidence for a π − π Resonance in the I = 1, J = 1 State. Phys. Rev. Lett. , 6:628–630, Jun 1961. https://link.aps.org/doi/10.1103/PhysRevLett.6.628 . 6[26] B. C. Magli, L. W. Alvarez, A. H. Rosenfeld, and M. L. Stevenson. Evidencefor a T = 0 Three-Pion Resonance. Phys. Rev. Lett. , 7:178–182, Sep 1961. https://link.aps.org/doi/10.1103/PhysRevLett.7.178 . 6[27] Norio Hoshizaki, Shoichiro Otsuki, Wataro Watari, and Minoru Yonezawa.Nuclear Forces and Bosons: The Sakata Model and One-Boson-Exchange-Potentials.
Progress of Theoretical Physics , 27(6):1199–1220, 06 1962. https://doi.org/10.1143/PTP.27.1199 . 6[28] Shuzo Ogawa, Shoji Sawada, Tamotsu Ueda, Wataro Watari, and MinoruYonezawa. Chapter 3 One-Boson-Exchange Model.
Progress of TheoreticalPhysics Supplement , 39:140–189, 01 1967. https://doi.org/10.1143/PTPS.39.140 . 6[29] T. Hamada and I.D. Johnston. A potential model representation of two-nucleon data below 315 MeV.
Nuclear Physics , 34(2):382 – 403, 1962. . 7[30] Roderick V Reid. Local phenomenological nucleon-nucleon potentials.
An-nals of Physics , 50(3):411 – 448, 1968. . 7[31] Alexander Gersten, Richard H. Thompson, and A. E. S. Green. RelativisticCalculation of Nucleon-Nucleon Phase Parameters.
Phys. Rev. D , 3:2076–239083, May 1971. https://link.aps.org/doi/10.1103/PhysRevD.3.2076 .7[32] G. Schierholz. A relativistic one-boson-exchange model of nucleon-nucleoninteraction.
Nuclear Physics B , 40:335 – 348, 1972. . 7[33] K. Erkelenz. Current status of the relativistic two-nucleon one boson ex-change potential.
Physics Reports , 13(5):191 – 258, 1974. . 7[34] K. Holinde and R. Machleidt. Momentum-space OBEP, two-nucleon andnuclear matter data.
Nuclear Physics A , 247(3):495 – 520, 1975. . 7[35] R. Machleidt. Nucleon-nucleon potentials in comparison: Physics or polemics?
Physics Reports , 242(1):5 – 35, 1994. . 7[36] A.D. Jackson, D.O. Riska, and B. Verwest. Meson exchange model for thenucleon-nucleon interaction.
Nuclear Physics A , 249(3):397 – 444, 1975. . 7[37] W. N. Cottingham, M. Lacombe, B. Loiseau, J. M. Richard, and R. VinhMau. Nucleon-Nucleon Interaction from Pion-Nucleon Phase-Shift Analysis.
Phys. Rev. D , 8:800–819, Aug 1973. https://link.aps.org/doi/10.1103/PhysRevD.8.800 . 7[38] M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, P. Pires, andR. de Tourreil. New semiphenomenological soft-core and velocity-dependentnucleon-nucleon potential.
Phys. Rev. D , 12:1495–1498, Sep 1975. https://link.aps.org/doi/10.1103/PhysRevD.12.1495 . 7[39] R. B. Wiringa, R. A. Smith, and T. L. Ainsworth. Nucleon-nucleon potentialswith and without ∆(1232) degrees of freedom.
Phys. Rev. C , 29:1207–1221,Apr 1984. https://link.aps.org/doi/10.1103/PhysRevC.29.1207 . 7[40] R. Machleidt, K. Holinde, and Ch. Elster. The bonn meson-exchange modelfor the nucleonnucleon interaction.
Physics Reports , 149(1):1–89, 1987. . 724041] M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, J. Cˆot´e, P. Pir`es, andR. de Tourreil. Parametrization of the Paris N − N potential. Phys. Rev. C ,21:861–873, Mar 1980. https://link.aps.org/doi/10.1103/PhysRevC.21.861 . xxii, xxiii, 7, 26, 206, 223[42] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla. Accurate nucleon-nucleonpotential with charge-independence breaking.
Phys. Rev. C , 51:38–51, Jan1995. https://link.aps.org/doi/10.1103/PhysRevC.51.38 . xxiii, 7, 26,223[43] R. Machleidt. High-precision, charge-dependent Bonn nucleon-nucleon poten-tial.
Phys. Rev. C , 63:024001, Jan 2001. https://link.aps.org/doi/10.1103/PhysRevC.63.024001 . xxiii, 7, 26, 223, 234[44] W. U. Boeglin. Coincidence experiments with electrons.
Czech. J. Phys. ,45:295–335, 1995. https://doi.org/10.1007/BF01698011 . 8[45] M. Croissiaux. Electron-Proton Coincidences in Inelastic Electron-DeuteronScattering.
Phys. Rev. , 127(2):613–616, Jul 1962. https://link.aps.org/doi/10.1103/PhysRev.127.613 . 8[46] Paul Bounin and Michel Croissiaux. Expriences de concidences (e, p)sur le deutrium.
Nuclear Physics , 70(2):401–414, 1965. . 8[47] Werner Boeglin and Misak Sargsian. Modern studies of the Deuteron: Fromthe lab frame to the light front.
International Journal of Modern Physics E ,24(03):1530003, 2015. https://doi.org/10.1142/S0218301315300039 . xiv,8, 9, 11, 29[48] K.I. Blomqvist et al. Large recoil momenta in the D(e,ep)n reaction.
PhysicsLetters B , 424(1):33 – 38, 1998. . xiv, 9, 10, 24[49] Werner Fabian and Harrtmuth Arenhvel. Pion exchange currents and isobarconfigurations in deuteron electro-disintegration below pion threshold.
Nu-clear Physics A , 258(3):461–479, 1976. . xiv, 9, 10, 27[50] Y. Antufev, V. Agranovich, V. Kuzmenko, and P. Sorokin. Expriences deconcidences (e, p) sur le deutrium.
Yadernaya Fizika , 22(2):236–241, 1975. https://inis.iaea.org/search/search.aspx?orig_q=RN:10422786 . 924151] M. Bernheim, A. Bussire, J. Mougey, D. Royer, D. Tarnowski, S. Turck-Chieze,S. Frullani, G.P. Capitani, E. De Sanctis, and E. Jans. Momentum distributionof nucleons in the deuteron from the d(e, ep)n reaction.
Nuclear Physics A ,365(3):349 – 370, 1981. . 9, 10[52] S. Turck-Chieze et al. Exclusive deuteron electrodisintegration at high neutronrecoil momentum.
Physics Letters B , 142(3):145–148, 1984. . 9, 24[53] W.-J. Kasdorp et al. Deuteron Electrodisintegration at High Missing Mo-menta.
Few-Body Systems , 25(1):115–132, Dec 1998. https://doi.org/10.1007/s006010050098 . 10, 24[54] P. E. Ulmer et al. H( e, e (cid:48) p ) n Reaction at High Recoil Momenta.
Phys.Rev. Lett. , 89:062301, Jul 2002. https://link.aps.org/doi/10.1103/PhysRevLett.89.062301 . 10, 24[55] W. U. Boeglin et al. Probing the High Momentum Component of the Deuteronat High Q . Phys. Rev. Lett. , 107:262501, Dec 2011. https://link.aps.org/doi/10.1103/PhysRevLett.107.262501 . 10, 12, 13, 24, 28, 166, 222, 235[56] K. S. Egiyan et al. Experimental Study of Exclusive H( e, e (cid:48) p ) n ReactionMechanisms at High Q . Phys. Rev. Lett. , 98:262502, Jun 2007. https://link.aps.org/doi/10.1103/PhysRevLett.98.262502 . 10, 24, 28[57] Misak M. Sargsian. Selected Topics in High Energy Semi-Exclusive Electro-Nuclear Reactions.
International Journal of Modern Physics E , 10(06):405457,Dec 2001. http://dx.doi.org/10.1142/S0218301301000617 . xv, 11, 24, 27,28, 31[58] L. L. Frankfurt, M. M. Sargsian, and M. I. Strikman. Feynman graphs andgeneralized eikonal approach to high energy knock-out processes.
Phys. Rev. C ,56:1124–1137, Aug 1997. https://link.aps.org/doi/10.1103/PhysRevC.56.1124 . 11, 24, 28[59] Misak M. Sargsian. Large Q electrodisintegration of the deuteron in thevirtual nucleon approximation. Phys. Rev. C , 82:014612, Jul 2010. https://link.aps.org/doi/10.1103/PhysRevC.82.014612 . xiv, xxiii, 11, 27, 28,223 24260] J.M. Laget. The electro-disintegration of few body systems revisited.
PhysicsLetters B , 609(1):49 – 56, 2005. . xiv, xxii, xxiii, 11, 27, 29, 30, 159, 205,206, 223[61] Sabine Jeschonnek and J. W. Van Orden. New calculation for H( e, e (cid:48) p ) n atGeV energies. Phys. Rev. C , 78:014007, Jul 2008. https://link.aps.org/doi/10.1103/PhysRevC.78.014007 . xiv, 11, 27[62] Samuel S.M. Wong.
Introductory Nuclear Physics . Wiley-VCH, Hoboken, NJ,1998. 13, 27[63] K. Sh. Egiyan et al. Observation of nuclear scaling in the A ( e, e (cid:48) ) reaction at x B > Phys. Rev. C , 68:014313, Jul 2003. https://link.aps.org/doi/10.1103/PhysRevC.68.014313 . 13[64] K. S. Egiyan et al. Measurement of Two- and Three-Nucleon Short-RangeCorrelation Probabilities in Nuclei.
Phys. Rev. Lett. , 96:082501, Mar 2006. https://link.aps.org/doi/10.1103/PhysRevLett.96.082501 . 13[65] R. Shneor et al. Investigation of Proton-Proton Short-Range Correlations viathe C( e, e (cid:48) pp ) Reaction. Phys. Rev. Lett. , 99:072501, Aug 2007. https://link.aps.org/doi/10.1103/PhysRevLett.99.072501 . 13[66] Nadia Fomin, Douglas Higinbotham, Misak Sargsian, and Patricia Solvignon.New Results on Short-Range Correlations in Nuclei.
Annual Review of Nuclearand Particle Science , 67(1):129159, Oct 2017. http://dx.doi.org/10.1146/annurev-nucl-102115-044939 . 13[67] P.E. Ulmer et al. Short-Distance Structure of the Deuteron and ReactionDynamics in H(e,e’p)n. , 2001.
Jefferson Lab Proposal E01-020 . 13, 20[68] C. Bochna et al. Measurements of Deuteron Photodisintegration up to 4.0GeV.
Phys. Rev. Lett. , 81:4576–4579, Nov 1998. https://link.aps.org/doi/10.1103/PhysRevLett.81.4576 . 13[69] E. C. Schulte et al. Measurement of the High Energy Two-Body DeuteronPhotodisintegration Differential Cross Section.
Phys. Rev. Lett. , 87:102302,Aug 2001. https://link.aps.org/doi/10.1103/PhysRevLett.87.102302 .13 24370] E. C. Schulte et al. High energy angular distribution measurements of theexclusive deuteron photodisintegration reaction.
Phys. Rev. C , 66:042201,Oct 2002. https://link.aps.org/doi/10.1103/PhysRevC.66.042201 . 13[71] M. Mirazita et al. Complete angular distribution measurements of two-body deuteron photodisintegration between 0 . Phys. Rev. C ,70:014005, Jul 2004. https://link.aps.org/doi/10.1103/PhysRevC.70.014005 . 13[72] S. Boffi, C. Giusti, and F.D. Pacati. Nuclear response in electromagneticinteractions with complex nuclei.
Physics Reports , 226(1):1 – 101, 1993. . 20,21, 24[73] Taber De Forest. Off-shell electron-nucleon cross sections: The impulseapproximation.
Nuclear Physics A , 392(2):232–248, 1983. . 22, 221[74] Werner U. Boeglin. Private communication, October 2019. 22[75] Hassan F. Ibrahim.
The H ( e, e (cid:48) p ) n Reaction At High Four-Momentum Trans-fer . PhD thesis, Old Dominion University, 5115 Hampton Blvd, Norfolk, VA23529, 12 2006. https://digitalcommons.odu.edu/physics_etds/100 . xiv,25[76] E. E. Salpeter and H. A. Bethe. A Relativistic Equation for Bound-StateProblems.
Phys. Rev. , 84:1232–1242, Dec 1951. https://link.aps.org/doi/10.1103/PhysRev.84.1232 . 26[77] A. Bianconi, S. Jeschonnek, N.N. Nikolaev, and B.G. Zakharov. Final stateinteraction effects in scattering.
Physics Letters B , 343(1-4):1318, Jan 1995. http://dx.doi.org/10.1016/0370-2693(94)01436-G . 27[78] William P. Ford, Sabine Jeschonnek, and J. W. Van Orden. Momentumdistributions for H( e, e (cid:48) p ). Phys. Rev. C , 90:064006, Dec 2014. https://link.aps.org/doi/10.1103/PhysRevC.90.064006 . 27[79] J.M. Laget. Pion photoproduction on few body systems.
Physics Reports ,69(1):1 – 84, 1981. 29[80] S. Galster, H. Klein, J. Moritz, K.H. Schmidt, D. Wegener, and J. Bleck-wenn. Elastic electron-deuteron scattering and the electric neutron form fac-244or at four-momentum transfers 5 fm < q <
14 fm . Nuclear Physics B ,32(1):221 – 237, 1971. . 29[81] O. Gayou et al. Measurement of G E p /G M p in → e p → e → p to Q = 5 . . Phys. Rev. Lett. , 88:092301, Feb 2002. https://link.aps.org/doi/10.1103/PhysRevLett.88.092301 . 29[82] R. J. Glauber. Cross Sections in Deuterium at High Energies.
Phys. Rev. ,100:242–248, Oct 1955. https://link.aps.org/doi/10.1103/PhysRev.100.242 . 30[83] E. D. Bloom, D. H. Coward, H. DeStaebler, J. Drees, G. Miller, L. W. Mo,R. E. Taylor, M. Breidenbach, J. I. Friedman, G. C. Hartmann, and H. W.Kendall. High-Energy Inelastic e − p Scattering at 6 o and 10 o . Phys. Rev. Lett. ,23:930–934, Oct 1969. https://link.aps.org/doi/10.1103/PhysRevLett.23.930 . 32[84] M. Breidenbach, J. I. Friedman, H. W. Kendall, E. D. Bloom, D. H. Coward,H. DeStaebler, J. Drees, L. W. Mo, and R. E. Taylor. Observed Behaviorof Highly Inelastic Electron-Proton Scattering.
Phys. Rev. Lett. , 23:935–939,Oct 1969. https://link.aps.org/doi/10.1103/PhysRevLett.23.935 . 32[85] Franz Gross. Making the Case for Jefferson Lab.
Journal of Physics: Confer-ence Series , 299:012001, May 2011. https://iopscience.iop.org/article/10.1088/1742-6596/299/1/012001 . 32, 33[86] Charles E. Reece. Continuous wave superconducting radio frequency elec-tron linac for nuclear physics research.
Phys. Rev. Accel. Beams , 19:124801,Dec 2016. https://link.aps.org/doi/10.1103/PhysRevAccelBeams.19.124801 . 33[87] Christoph W. Leemann, David R. Douglas, and Geoffrey A. Krafft. THECONTINUOUS ELECTRON BEAM ACCELERATOR FACILITY: CEBAFat the Jefferson Laboratory.
Annual Review of Nuclear and Particle Sci-ence , 51(1):413–450, 2001. https://doi.org/10.1146/annurev.nucl.51.101701.132327 . 33[88] Michael Drury, Edward Daly, G.K. Davis, John Fischer, Christiana Grenoble,John Hogan, Lawrence King, Joseph Preble, Wang Haipeng, Anthony Reilly,John Mammosser, and Jeffrey Saunders. Summary Report for the C50 Cry-245module Project. 01 2011. . 33, 35[89] R. D. McKeown. The Jefferson Lab 12 GeV Upgrade.
Journal of Physics:Conference Series , 312(3):032014, September 2011. https://doi.org/10.1088 . 34, 36[90] Fulvia C. Pilat. The 12 GeV Energy Upgrade at Jefferson Laboratory.9 2012. http://accelconf.web.cern.ch/AccelConf/LINAC2012/papers/th3a02.pdf . xv, 34, 35, 36[91] R Rimmer, R Bundy, Gary Cheng, Gianluigi Ciovati, W Clemens, E Daly,J Henry, W Hicks, P. Kneisel, S Manning, R Manus, F Marhauser,J Preble, Callie Reece, Kotoga Smith, Mircea Stirbet, L Turlington, andK Wilson. JLab CW Cryomodules for 4th Generation Light Sources. 022020. . xv, 35[92] Volker D. Burkert. The JLab 12GeV Upgrade and the Initial Science Program,2012. https://arxiv.org/abs/1203.2373 . 36[93] Reza Kazimi et al. CEBAF injector achieved world’s best beam quality forthree simultaneous beam with a wide range of bunch charges. 35, 7 2004. http://accelconf.web.cern.ch/AccelConf/e04/PAPERS/TUPLT164.PDF . 37[94] R. Kazimi et al. Four Beam Generation for Simultaneous Four-Hall Oper-ation at CEBAF. In
Proc. of International Particle Accelerator Conference(IPAC’16), Busan, Korea, May 8-13, 2016 , number 7 in International Parti-cle Accelerator Conference, pages 4240–4242, Geneva, Switzerland, June 2016.JACoW. http://jacow.org/ipac2016/papers/thpoy060.pdf . 37[95] R. Kazimi et al. Operational Results of Simultaneous Four-Beam Deliveryat Jefferson Lab. In
Proc. 10th International Particle Accelerator Conference(IPAC’19), Melbourne, Australia, 19-24 May 2019 , number 10 in InternationalParticle Accelerator Conference, pages 2454–2457, Geneva, Switzerland, Jun.2019. JACoW Publishing. http://jacow.org/ipac2019/papers/wepmp053.pdf . 37[96] M. F. Spata. 12 GeV CEBAF Initial Operational Experience and Challenges.In
Proc. 9th International Particle Accelerator Conference (IPAC’18), Van-couver, BC, Canada, April 29-May 4, 2018 , number 9 in International Parti-246le Accelerator Conference, pages 1771–1775, Geneva, Switzerland, June 2018.JACoW Publishing. http://jacow.org/ipac2018/papers/weygbd1.pdf . 38[97] S.A. Wood. 2019 Version: Jefferson Lab Hall C Standard Equip-ment Manual, 2019. https://hallcweb.jlab.org/safety-docs/current/Standard-Equipment-Manual.pdf . xv, xvi, 38, 42, 51, 52, 53, 55, 57, 58, 59,66, 67, 68, 81, 82[98] H. Fenker. Hall C KPP Demonstration, March 2017. https://hallcweb.jlab.org/DocDB/0008/000847/001/Hall_C_KPP_Demonstration-Mar2017-v8-final.pdf . 39[99] HallC-Collaboration. Conceptual Design Report Hall C 12 GeV/c Upgrade,2002. . 39, 64[100] W. U. Boeglin et al. Deuteron Electro-Disintegration at Very High MissingMomenta. , 2014.
Jefferson Lab Proposal E12-10-003 . 39, 236[101] S.P. Malace et al. Precision measurements of the F structure function atlarge x in the resonance region and beyond. , 2009. Jefferson Lab Proposal E12-10-002 . 39[102] D. Dutta et al. The Search for Color Transparency at 12 GeV. , 2006.
Jefferson LabProposal E12-06-107 . 39[103] D. Gaskell et al. Detailed studies of the nuclear dependence of F in light nu-clei. ,2009. Jefferson Lab Proposal E12-10-008 . 39[104] D. Gaskell and J. Arrington. SIMC - Physics Monte Carlo for Hall Cand Hall A, 2009. https://hallaweb.jlab.org/data_reduc/AnaWork2009/simc_overview.pdf . 40[105] P.E Ulmer, I. Karabekov, A. Saha, P. Bertin, and P. Vernin. Absolute BeamEnergy Determination At CEBAF, 1993. https://cds.cern.ch/record/258842/files/P00021008.pdf . 43247106] C. Yan, R. Carlini, and D. Neuffer. Beam Energy Measurement Using theHall C Beam Line, 1993. https://accelconf.web.cern.ch/accelconf/p93/PDF/PAC1993_2136.PDF . 43[107] John R. Arrington.
Inclusive Electron Scattering From Nuclei at x > andHigh Q . PhD thesis, California Institute of Technology, 1200 E CaliforniaBlvd, Pasadena, CA 91125, June 1998. . 44[108] C. Yan, P. Adderley, D. Barker, J. Beaufait, K. Capek, R. Carlini, J. Dahlberg,E. Feldl, K. Jordan, B. Kross, W. Oren, R. Wojcik, and J. VanDyke. Super-harp A wire scanner with absolute position readout for beam energy mea-surement at CEBAF. Nuclear Instruments and Methods in Physics ResearchSection A: Accelerators, Spectrometers, Detectors and Associated Equipment ,365(2):261 – 267, 1995. . 45[109] Mark K. Jones. Private communication, January 2020. 45[110] M.K.Jones. HallC Wiki: Table of 12 GeV Beam Energy Measurements (HallC Wiki), 2019. https://hallcweb.jlab.org/wiki/index.php/Table_of_12_GeV_Beam_Energy_Measurements . 45, 46[111] Klaus Wille. Synchrotron Radiation, February 2013. https://indico.cern.ch/event/218284/contributions/1520454/attachments/352184/490697/JUAS2013_Synchrotron_Radiation_1.pdf . 46[112] Dave J. Mack. Private communication, July 2020. 50[113] C. Yan, N. Sinkine, and R. Wojcik. Linear beam raster for cryo-genic targets.
Nuclear Instruments and Methods in Physics Research Sec-tion A: Accelerators, Spectrometers, Detectors and Associated Equipment ,539(1):1 – 15, 2005. . 52, 266, 267, 268, 269[114] Argonne National Lab. Experimental Physics and Industrial Control System,2019. https://epics.anl.gov . 52[115] Dave J. Mack. Private communication, February 2020. 54248116] K. B. Unser. The parametric current transformer, a beam current moni-tor developed for LEP.
AIP Conference Proceedings , 252(1):266–275, 1992. https://aip.scitation.org/doi/abs/10.1063/1.42124 . 54[117] D. J. Mack. Beam current monitors for Hall C.
AIP Conference Proceedings ,269(1):527–530, 1992. https://aip.scitation.org/doi/abs/10.1063/1.42965 . 54[118] Gregory Smith. Cryotarget Training, 2016. https://userweb.jlab.org/~smithg/target/Qweak/HallACTgt_Training.pdf . xvi, 59, 61[119] Silviu Covrig. Private communication, March 2020. xvi, 61[120] John Arrington et al. The Science and Experimental Equipment for the 12GeV Upgrade of CEBAF. 1 2005. .64[121] T. Horn. SHMS Optics Update. https://slideplayer.com/slide/5985891/ . 66[122] Steven Lassiter. Private communication, March 2020. xii, 67[123] M.K.Jones. HallC Wiki: How to operate and monitor the Spectrome-ter Magnets, 2019. https://hallcweb.jlab.org/wiki/index.php/How_to_operate_and_monitor_the_Spectrometer_Magnets . 68[124] Holly Szumila-Vance. Commissioning the HMS optics in the 2017-18 run pe-riod, 2018. https://hallcweb.jlab.org/DocDB/0009/000998/002/hmsSat.pdf . 69, 135[125] Holly Szumila-Vance. Notes on the SHMS optics in the 2017-18 run period,2019. https://hallcweb.jlab.org/DocDB/0010/001007/001/shmsNote.pdf . 71, 135[126] B. Pandey. Status Update on the New HMS Wire Chamber. .73 249127] M.E. Christy, P. Monaghan, N. Kalantarians, D. Biswas, and M. Long.SHMS Drift Chambers. https://hallcweb.jlab.org/document/howtos/shms_drift_chambers.pdf . 73[128] c (cid:13) . 76[129] c (cid:13) Rexon Components & TLD Inc. RP-408. . 76[130] Corning Inc. High Purity Fused Silica (HPFS R (cid:13) ). . 76[131] G. Niculescu, I. Niculescu, M. Burton, D. Coquelin, K. Nisson, and T. Jarell.SHMS Hodoscope Scintillator Detectors. https://hallcweb.jlab.org/document/howtos/shms_scintillator_hodoscope.pdf . 79[132] S. Malace, G. Niculescu, I. Niculescu, A. Amidouch, S. Danagoulian, andD. Mack. SHMS Hodoscopes. . 79[133] Julian T. Wilson. Quartz Hodoscope: Assembly, Calibration, and Data Analy-sis. Master’s thesis, North Carolina A&T State University, Greensboro, NorthCarolina, 2014. 79[134] Wenliang Li. Heavy Gas Cherenkov Detector Construction for Hall C atThomas Jefferson National Accelerator Facility. Master’s thesis, Universityof Regina, October 2012. https://ourspace.uregina.ca/handle/10294/3818 . xvi, 81[135] C. Cothran, D. Day, and J. Mitchell. A threshold gas Cerenkov detector forCEBAF’s Hall C High Momentum Spectrometer. Bulletin of the AmericanPhysical Society , 28:1007f, 1995. https://inis.iaea.org/search/search.aspx?orig_q=RN:28002126 . 81[136] D. Day. Preliminary Design of the SHMS Noble Cerenkov Detector. https://hallcweb.jlab.org/DocDB/0009/000933/001/shms-cerv6.pdf . 82[137] R. Asaturyan, R. Ent, H. Fenker, D. Gaskell, G.M. Huber, M. Jones, D. Mack,H. Mkrtchyan, B. Metzger, N. Novikoff, V. Tadevosyan, W. Vulcan, and250. Wood. The aerogel threshold Cherenkov detector for the High MomentumSpectrometer in Hall C at Jefferson Lab.
Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers, Detectors and Asso-ciated Equipment , 548(3):364 – 374, 2005. . 84[138] T. Horn, H. Mkrtchyan, S. Ali, A. Asaturyan, M. Carmignotto, A. Dittmann,D. Dutta, R. Ent, N. Hlavin, Y. Illieva, A. Mkrtchyan, P. Nadel-Turonski,I. Pegg, A. Ramos, J. Reinhold, I. Sapkota, V. Tadevosyan, S. Zhamkochyan,and S.A. Wood. The Aerogel erenkov detector for the SHMS magnetic spec-trometer in Hall C at Jefferson Lab.
Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers, Detectors and As-sociated Equipment , 842:28 – 47, 2017. . 84[139] M. Tanabashi et al. Review of Particle Physics.
Phys. Rev. D , 98:030001, Aug2018. https://link.aps.org/doi/10.1103/PhysRevD.98.030001 . 84[140] H. Mkrtchyan et al. The lead-glass electromagnetic calorimeters for the mag-netic spectrometers in Hall C at Jefferson Lab.
Nuclear Instruments andMethods in Physics Research Section A: Accelerators, Spectrometers, Detec-tors and Associated Equipment , 719:85100, Aug 2013. http://dx.doi.org/10.1016/j.nima.2013.03.070 . 85, 87, 130, 131[141] CAEN. Technical Information Manual: V1190 A/B VX1190 A/B MULTIHITTDCs. Technical report, 08 2011. https://coda.jlab.org/drupal/system/files/pdfs/HardwareManual/TI/TI.pdf . 88, 109[142] J. William Gu. Description and Technical Information for the VME Trigger In-terface (TI) Module. Technical report, Jefferson Lab, 07 2019. https://coda.jlab.org/drupal/system/files/pdfs/HardwareManual/TI/TI.pdf . 88[143] Hai Dong. Description and Instructions for the Firmware of ProcessingFPGA of the ADC250 Boards Version 0x0C0D. Technical report, 02 2017. https://coda.jlab.org/drupal/system/files/pdfs/HardwareManual/fADC250/FADC250_Processing_FPGA_Firmware_ver_0x0C0D_Description_Instructions.pdf . 90[144] D. Mack. Live Time Calculations, 2019. https://hallcweb.jlab.org/DocDB/0010/001001/002/EDTMnonPoissonBiasCorrectionv2.pdf . 107251145] Eric Pooser. Live Time Calculations, 2019. https://hallcweb.jlab.org/DocDB/0010/001022/001/pooser_live_time.pdf . 107[146] Teledyne LeCroy. 1877 Multihit Time-to-Digital Converter, 1996. https://teledynelecroy.com/lrs/dsheets/1877.htm . 109[147] Carlos Yero. General Hall C Analysis Procedure in 12 GeV Era,2019. https://hallcweb.jlab.org/DocDB/0010/001032/001/analysis_notes.pdf . 115, 117[148] Eric J. Pooser.
The GlueX Start Counter & Beam Asymmetry Σ in Single π Photoproduction . PhD thesis, Florida International University, 11200 SW 8thSt, Miami, FL 33199, March 2016. https://digitalcommons.fiu.edu/etd/2450 . xviii, 118, 119[149] W.R. Leo.
Techniques for Nuclear and Particle Physics Experiments: A Howto Approach , pages 319–320. Springer-Verlag, 1 edition, 1987. 119[150] Carlos Yero. Brief Document On the Hall C Hodoscopes Calibration,2018. https://hallcweb.jlab.org/DocDB/0009/000970/001/hodo_calib.pdf . 122[151] Carlos Yero. HMS/SHMS Drift Chambers Calibration, 2017. https://hallcweb.jlab.org/DocDB/0008/000863/003/HallC-Software-Workshop_pdf.pdf . 130[152] Vardan Tadevosyan. SHMS Calorimeter Calibration, 2017. https://hallcweb.jlab.org/DocDB/0008/000865/002/pcal_calib.pdf . 131[153] Ryan Ambrose and Garth Huber. Calibration of the SHMS Cherenkovs (HGC& NGC) , 2017. https://hallcweb.jlab.org/DocDB/0008/000893/001/HGC_Calibration.pdf . 132[154] V. Berdnikov et al. Calibration of the SHMS Aerogel Cherenkov Detector,2020. https://redmine.jlab.org/issues/478 . 132[155] Jure Bericic. Notes on HMS Optics, 2017. https://hallcweb.jlab.org/DocDB/0008/000849/001/HMS_optics_notes.pdf . 145[156] Mark K. Jones. Private communication, July 2019. 146252157] Holly Szumila-Vance. SHMS optics, 2019. https://github.com/hszumila/SHMS_optics . 149[158] J. Arrington. Implications of the discrepancy between proton form factormeasurements.
Phys. Rev. C , 69:022201, Feb 2004. https://link.aps.org/doi/10.1103/PhysRevC.69.022201 . 157, 199[159] D. Biswas, D. Bhetuwal, and D. Mack. Fall 17-Spring 18 BCM Overview andParameter Recommendations, 2018. https://hallcweb.jlab.org/DocDB/0009/000968/014/BCMSummaryFall2017Spring2018.pdf . 182[160] Dave J. Mack. Private communication, November 2019. 198[161] Zhihong Ye, John Arrington, Richard J. Hill, and Gabriel Lee. Proton andneutron electromagnetic form factors and uncertainties.
Physics Letters B ,777:8 – 15, 2018. . 198, 200[162] Julian Schwinger. Quantum Electrodynamics. III. The Electromagnetic Prop-erties of the Electron—Radiative Corrections to Scattering.
Phys. Rev. ,76:790–817, Sep 1949. https://link.aps.org/doi/10.1103/PhysRev.76.790 . 202[163] L. W. MO and Y. S. TSAI. Radiative Corrections to Elastic and Inelastic epand up Scattering.
Rev. Mod. Phys. , 41:205–235, Jan 1969. https://link.aps.org/doi/10.1103/RevModPhys.41.205 . 202[164] R. Ent, B. W. Filippone, N. C. R. Makins, R. G. Milner, T. G. O’Neill, andD. A. Wasson. Radiative corrections for ( e, e (cid:48) p ) reactions at GeV energies. Phys. Rev. C , 64:054610, Oct 2001. https://link.aps.org/doi/10.1103/PhysRevC.64.054610 . 202[165] D. Dutta. Radiative Corrections - The SIMC Way, 1999. https://userweb.jlab.org/~johna/SIMC_documents/documents/radcor.ps . 202[166] Roger Barlow. Systematic Errors: facts and fictions, 2002. https://arxiv.org/abs/hep-ex/0207026 . 207[167] Roger Barlow. Systematic Errors in Particle Physics, 2017. https://indico.cern.ch/event/591374/contributions/2511753/attachments/1429002/2193943/01_PWA-Barlow.pdf . 208253168] Carlos Yero. Update on Spectrometer Offsets Determination Using H(e,e’p)Elastics, 2019. https://hallcweb.jlab.org/DocDB/0010/001036/002/HC_SoftwareMeeting_Oct03_2019_pdf.pdf . 216, 217[169] Misak Sargsian. Private communication, May 2020. 234[170] Mark K. Jones. Private communication, February 2020. 268254PPENDICES
Appendix A: Reduced Cross Section Data Table
The H( e, e (cid:48) p ) n reduced cross sections are tabulated for fixed Q and θ nq bins. The p r , bin represents the neutron recoil (missing) momentum central bin with a bin widthof ± .
02 GeV/c and the p r , avg represents the recoil momentum averaged over each p r , bin . The uncertainties in the reduced cross sections ( σ red ) are expressed as relativestatistical ( δσ stat ), normalization ( δσ norm ), kinematic ( δσ kin ) and systematic ( δσ syst )and total ( δσ tot ) where δσ = δσ + δσ and δσ = δσ + δσ . A1. Reduced Cross Sections at Q = 3 . ± . Table A1: θ nq = 5 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03027 6.37735E+00 33.23 3.66 3.40 5.00 33.610.060 0.05398 2.07666E+00 22.38 3.66 5.18 6.34 23.26Table A2: θ nq = 15 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.02958 5.72716E+00 18.91 3.66 3.61 5.14 19.590.060 0.05397 2.17760E+00 13.01 3.66 5.46 6.57 14.570.100 0.08841 8.09013E-01 24.49 3.66 6.83 7.75 25.69255able A3: θ nq = 25 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03069 5.29384E+00 14.29 3.66 3.41 5.00 15.140.060 0.05562 1.95390E+00 9.28 3.66 5.70 6.78 11.490.100 0.09083 5.07866E-01 14.93 3.66 7.36 8.22 17.040.420 0.42023 1.26426E-04 37.87 3.86 0.43 3.88 38.070.460 0.46019 8.69872E-05 43.33 3.98 0.55 4.02 43.51Table A4: θ nq = 35 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.02981 5.75153E+00 12.70 3.66 2.77 4.59 13.510.060 0.05656 1.97889E+00 7.55 3.66 5.60 6.69 10.090.100 0.09245 4.77584E-01 9.78 3.66 7.48 8.33 12.850.140 0.12870 1.16331E-01 25.84 3.66 7.18 8.06 27.060.340 0.34540 2.95606E-04 36.50 3.86 1.07 4.00 36.720.380 0.38192 2.09371E-04 20.20 3.86 0.93 3.97 20.580.420 0.42582 1.09994E-04 18.63 4.09 1.22 4.27 19.110.460 0.46156 9.26702E-05 15.50 4.09 1.16 4.25 16.070.500 0.50001 3.99584E-05 19.22 4.21 1.28 4.40 19.720.540 0.53895 4.28367E-05 16.35 4.21 1.28 4.40 16.930.580 0.57799 3.07679E-05 17.99 4.21 1.30 4.41 18.520.620 0.61847 2.82866E-05 19.84 4.01 1.18 4.18 20.270.660 0.65751 2.00139E-05 27.60 4.01 1.26 4.21 27.920.700 0.69694 2.55271E-05 36.10 3.90 1.08 4.05 36.330.740 0.73689 3.82137E-05 44.95 3.77 0.78 3.85 45.12256able A5: θ nq = 45 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03068 6.49330E+00 11.19 3.66 2.46 4.41 12.030.060 0.05804 1.80699E+00 6.22 3.66 5.23 6.39 8.910.100 0.09531 3.64016E-01 7.30 3.66 7.08 7.97 10.800.140 0.13365 8.33640E-02 13.32 3.66 6.86 7.77 15.430.180 0.16738 4.94696E-02 33.93 3.66 6.25 7.24 34.700.340 0.34768 4.20104E-04 36.82 3.86 1.72 4.22 37.060.380 0.38332 3.01566E-04 17.06 3.86 1.50 4.14 17.560.420 0.42526 1.61313E-04 13.62 3.98 1.61 4.29 14.280.460 0.46385 1.25089E-04 10.97 4.21 1.97 4.65 11.910.500 0.50159 9.86100E-05 9.51 4.21 1.96 4.64 10.590.540 0.54044 7.40245E-05 8.97 4.21 2.00 4.66 10.110.580 0.57974 6.26061E-05 8.39 4.21 2.10 4.70 9.620.620 0.61922 4.38124E-05 9.09 4.21 2.19 4.75 10.260.660 0.65889 3.68714E-05 9.49 4.21 2.33 4.81 10.640.700 0.69869 3.28269E-05 10.07 4.21 2.52 4.91 11.200.740 0.73850 3.02874E-05 10.81 4.21 2.68 4.99 11.910.780 0.77856 2.62080E-05 12.42 4.11 2.70 4.91 13.350.820 0.81836 1.64611E-05 16.49 4.21 3.21 5.30 17.320.860 0.85822 1.61986E-05 17.65 4.21 3.49 5.47 18.480.900 0.89831 1.67930E-05 18.69 4.12 3.58 5.46 19.470.940 0.93852 9.87289E-06 25.96 4.01 3.56 5.36 26.510.980 0.97853 7.98887E-06 30.33 4.11 4.41 6.03 30.921.020 1.01844 6.65554E-06 35.45 4.01 5.48 6.80 36.091.060 1.05818 6.25747E-06 38.86 4.01 25.32 25.64 46.56257able A6: θ nq = 55 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03135 5.68088E+00 10.16 3.66 1.76 4.06 10.940.060 0.05799 1.71593E+00 5.42 3.66 4.45 5.76 7.910.100 0.09654 3.21520E-01 5.85 3.66 5.97 7.01 9.130.140 0.13567 8.93298E-02 8.62 3.66 5.88 6.92 11.060.180 0.17490 2.23853E-02 18.36 3.66 5.14 6.31 19.410.380 0.38595 5.59781E-04 25.63 3.86 1.96 4.33 25.990.420 0.42326 2.84214E-04 16.63 3.86 1.77 4.24 17.160.460 0.46538 2.16829E-04 11.82 3.98 2.00 4.45 12.630.500 0.50462 1.91949E-04 9.00 4.21 2.51 4.90 10.250.540 0.54230 1.69112E-04 7.40 4.21 2.58 4.94 8.900.580 0.58117 1.25412E-04 7.05 4.21 2.73 5.02 8.660.620 0.62050 9.92010E-05 6.74 4.21 2.87 5.10 8.450.660 0.66011 9.09309E-05 6.21 4.21 3.09 5.22 8.110.700 0.69960 8.10597E-05 6.05 4.21 3.34 5.38 8.090.740 0.73933 6.87813E-05 6.23 4.21 3.56 5.51 8.320.780 0.77900 6.30654E-05 6.40 4.21 3.85 5.71 8.580.820 0.81856 5.16239E-05 7.20 4.21 4.18 5.93 9.330.860 0.85833 5.00020E-05 7.59 4.21 4.43 6.11 9.740.900 0.89803 3.77447E-05 9.31 4.21 4.81 6.39 11.290.940 0.93774 2.20978E-05 13.24 4.21 5.20 6.69 14.840.980 0.97741 2.37352E-05 14.04 4.21 5.65 7.04 15.711.020 1.01709 2.09706E-05 17.02 4.21 7.60 8.69 19.111.060 1.05682 1.30169E-05 25.95 4.01 25.11 25.43 36.33258able A7: θ nq = 65 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03201 5.46553E+00 9.55 3.66 1.27 3.87 10.300.060 0.05898 1.72017E+00 4.87 3.66 2.83 4.63 6.720.100 0.09658 3.37215E-01 4.86 3.66 4.24 5.60 7.420.140 0.13579 7.71106E-02 7.00 3.66 4.18 5.56 8.940.180 0.17593 1.60043E-02 13.98 3.66 3.31 4.94 14.830.220 0.21683 3.86961E-03 31.70 3.66 2.54 4.46 32.020.260 0.25715 3.21213E-03 35.77 3.66 1.97 4.16 36.010.420 0.42498 8.41352E-04 43.89 3.88 2.11 4.41 44.110.460 0.46269 4.15285E-04 30.13 3.88 1.95 4.34 30.440.500 0.50230 5.38070E-04 15.63 3.86 1.86 4.28 16.200.540 0.54122 2.97346E-04 14.39 3.86 1.89 4.30 15.010.580 0.58318 2.43462E-04 12.73 3.98 2.40 4.65 13.550.620 0.62058 2.56870E-04 11.14 3.98 2.50 4.70 12.090.660 0.66056 1.78182E-04 13.00 3.98 2.68 4.80 13.860.700 0.69892 1.46847E-04 14.18 4.09 3.30 5.26 15.130.740 0.73716 1.17390E-04 17.61 3.98 3.03 5.00 18.300.780 0.77603 7.64768E-05 26.01 3.98 3.26 5.14 26.510.820 0.81601 8.07370E-05 33.59 3.86 2.91 4.83 33.930.860 0.85494 7.90722E-05 46.40 3.86 3.11 4.95 46.66Table A8: θ nq = 75 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03171 5.06893E+00 9.13 3.66 0.76 3.74 9.860.060 0.05974 1.57799E+00 4.50 3.66 1.54 3.97 6.010.100 0.09702 3.07899E-01 4.23 3.66 1.77 4.07 5.870.140 0.13679 5.87380E-02 5.97 3.66 1.65 4.02 7.200.180 0.17715 1.75816E-02 10.67 3.66 1.10 3.82 11.340.220 0.21708 4.50003E-03 21.85 3.66 0.82 3.75 22.170.260 0.25832 1.96842E-03 30.79 3.66 0.55 3.70 31.010.300 0.29956 1.38562E-03 32.14 3.66 0.36 3.68 32.350.340 0.34019 1.22510E-03 32.70 3.66 0.27 3.67 32.900.460 0.45934 7.70540E-04 40.98 3.66 0.31 3.67 41.15259able A9: θ nq = 85 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03231 4.79539E+00 8.87 3.66 0.84 3.76 9.630.060 0.05981 1.57054E+00 4.30 3.66 0.53 3.70 5.670.100 0.09790 2.51174E-01 3.88 3.66 0.88 3.76 5.410.140 0.13759 5.62479E-02 4.99 3.66 1.15 3.84 6.290.180 0.17779 1.61925E-02 7.74 3.66 1.24 3.86 8.650.220 0.21862 5.88967E-03 12.69 3.66 1.28 3.88 13.270.260 0.25921 2.77418E-03 21.35 3.66 1.30 3.88 21.700.300 0.29994 1.88288E-03 19.32 3.66 1.24 3.86 19.710.340 0.33991 1.35843E-03 20.48 3.66 1.13 3.83 20.840.380 0.37952 9.67146E-04 24.63 3.66 0.99 3.79 24.920.420 0.41918 3.76107E-04 40.86 3.66 0.89 3.77 41.03Table A10: θ nq = 95 ± ◦ at Q = 3 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03213 5.03786E+00 8.99 3.66 1.11 3.83 9.770.060 0.06053 1.50083E+00 4.27 3.66 1.98 4.16 5.960.100 0.09861 2.56864E-01 3.63 3.66 2.57 4.48 5.770.140 0.13869 5.57178E-02 4.38 3.66 3.07 4.78 6.480.180 0.17838 1.64757E-02 6.26 3.66 3.08 4.79 7.880.220 0.21883 7.18822E-03 8.31 3.66 3.04 4.76 9.580.260 0.25901 2.93957E-03 12.22 3.66 2.77 4.59 13.060.300 0.29899 1.24017E-03 18.02 3.66 2.40 4.38 18.540.340 0.33891 9.18492E-04 21.36 3.66 2.04 4.19 21.770.380 0.37883 9.05995E-04 22.41 3.66 1.67 4.03 22.770.420 0.41818 3.81181E-04 37.84 3.66 1.45 3.94 38.04260
2. Reduced Cross Sections at Q = 4 . ± . Table A11: θ nq = 5 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03259 5.03296E+00 42.52 3.66 4.31 5.65 42.900.060 0.06183 1.29738E+00 14.47 3.66 5.46 6.58 15.890.100 0.09878 2.97336E-01 10.63 3.66 5.28 6.42 12.420.140 0.13646 6.46574E-02 16.60 3.66 5.02 6.21 17.730.180 0.17414 2.42630E-02 32.29 3.66 4.87 6.10 32.86Table A12: θ nq = 15 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03223 4.95610E+00 27.92 3.66 4.08 5.48 28.460.060 0.06248 1.37967E+00 8.42 3.66 5.29 6.43 10.600.100 0.09853 3.01980E-01 6.37 3.66 5.21 6.36 9.000.140 0.13669 7.71973E-02 8.79 3.66 5.18 6.35 10.840.180 0.17523 2.30324E-02 17.23 3.66 5.34 6.47 18.410.220 0.21438 1.16646E-02 33.63 3.66 5.59 6.68 34.29Table A13: θ nq = 25 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03215 7.38449E+00 24.01 3.66 3.15 4.83 24.490.060 0.06256 1.30063E+00 6.82 3.66 4.61 5.89 9.010.100 0.09921 3.00250E-01 5.06 3.66 4.88 6.10 7.930.140 0.13785 7.18342E-02 6.82 3.66 5.21 6.36 9.330.180 0.17685 2.79423E-02 10.91 3.66 5.59 6.68 12.790.220 0.21664 1.24979E-02 17.84 3.66 5.81 6.86 19.120.260 0.25638 2.73724E-03 44.80 3.66 5.63 6.71 45.300.540 0.54008 2.40551E-05 42.51 3.86 0.41 3.88 42.690.580 0.58068 1.87702E-05 39.24 3.98 0.54 4.02 39.450.620 0.62026 1.87233E-05 31.00 4.21 0.75 4.28 31.290.660 0.65915 1.05431E-05 40.99 3.98 0.62 4.03 41.190.740 0.73882 1.43833E-05 35.55 4.01 0.72 4.08 35.790.780 0.77924 8.40120E-06 42.55 4.01 0.74 4.08 42.74261able A14: θ nq = 35 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03181 4.00473E+00 20.91 3.66 2.52 4.45 21.380.060 0.06217 1.18093E+00 6.32 3.66 3.36 4.97 8.040.100 0.09914 2.88897E-01 4.60 3.66 4.09 5.49 7.170.140 0.13875 7.58109E-02 5.77 3.66 4.76 6.00 8.330.180 0.17780 2.00561E-02 10.12 3.66 5.22 6.38 11.970.220 0.21766 8.63454E-03 15.72 3.66 5.46 6.57 17.040.260 0.25754 2.78087E-03 28.92 3.66 5.25 6.40 29.620.460 0.46420 9.60399E-05 45.41 3.86 0.72 3.92 45.580.500 0.50193 5.99945E-05 28.20 3.86 0.71 3.92 28.470.540 0.54070 4.24077E-05 23.10 3.86 0.72 3.92 23.430.580 0.58204 1.86561E-05 26.66 4.09 1.09 4.23 27.000.620 0.62089 2.27245E-05 18.76 4.21 1.33 4.42 19.270.660 0.66007 1.23351E-05 21.67 4.09 1.26 4.28 22.090.700 0.69960 1.24011E-05 18.34 4.21 1.52 4.48 18.880.740 0.73908 9.99662E-06 19.13 4.21 1.63 4.52 19.660.780 0.77906 1.15967E-05 18.11 4.11 1.63 4.42 18.650.820 0.81867 6.11389E-06 23.62 4.21 1.91 4.62 24.070.860 0.85839 7.10916E-06 22.31 4.21 2.06 4.69 22.800.900 0.89856 4.18936E-06 31.34 4.01 1.97 4.47 31.660.940 0.93832 3.55695E-06 35.39 4.01 2.17 4.56 35.69262able A15: θ nq = 45 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03147 6.12525E+00 21.33 3.66 2.06 4.20 21.740.060 0.06241 1.33303E+00 6.20 3.66 1.87 4.11 7.440.100 0.09921 2.56214E-01 4.58 3.66 2.79 4.60 6.490.140 0.13844 6.81144E-02 5.64 3.66 3.74 5.23 7.690.180 0.17838 2.08912E-02 8.57 3.66 4.24 5.61 10.240.220 0.21825 5.24976E-03 16.75 3.66 4.36 5.69 17.690.260 0.25782 2.83066E-03 24.31 3.66 4.10 5.50 24.930.500 0.50340 1.01408E-04 39.86 3.86 1.22 4.04 40.070.540 0.54176 6.88359E-05 25.86 3.86 1.25 4.05 26.180.580 0.58257 4.36631E-05 22.56 3.98 1.59 4.28 22.960.620 0.62125 3.99820E-05 19.20 3.98 1.69 4.33 19.690.660 0.66104 2.35369E-05 19.55 4.21 2.37 4.83 20.140.700 0.70033 3.32595E-05 13.85 4.21 2.56 4.93 14.700.740 0.73983 1.78367E-05 16.45 4.21 2.74 5.03 17.200.780 0.77941 1.78128E-05 15.02 4.21 2.99 5.16 15.880.820 0.81913 1.49669E-05 15.46 4.21 3.25 5.32 16.350.860 0.85887 8.16579E-06 20.51 4.21 3.50 5.47 21.220.900 0.89857 1.12660E-05 17.45 4.21 3.85 5.71 18.360.940 0.93846 7.33177E-06 23.14 4.12 3.83 5.63 23.820.980 0.97812 1.13579E-05 19.69 4.09 4.14 5.82 20.531.020 1.01796 6.95598E-06 25.22 4.21 6.49 7.74 26.38Table A16: θ nq = 55 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03172 5.18815E+00 21.36 3.66 0.65 3.72 21.680.060 0.06218 1.39568E+00 6.48 3.66 0.77 3.74 7.480.100 0.09928 2.88239E-01 4.72 3.66 0.58 3.71 6.000.140 0.13833 7.06831E-02 5.86 3.66 1.67 4.03 7.110.180 0.17855 2.00666E-02 9.24 3.66 2.35 4.35 10.210.220 0.21843 7.04340E-03 14.24 3.66 2.59 4.49 14.930.260 0.25868 2.68690E-03 21.99 3.66 2.40 4.38 22.420.300 0.29881 1.20335E-03 33.39 3.66 2.02 4.18 33.660.660 0.66016 1.11707E-04 38.02 3.86 1.89 4.29 38.260.700 0.69951 5.88594E-05 45.24 3.86 2.01 4.35 45.450.740 0.73909 9.00726E-05 33.47 3.86 2.12 4.40 33.76263able A17: θ nq = 65 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03116 5.29860E+00 25.05 3.66 2.00 4.17 25.400.060 0.06157 1.38526E+00 7.47 3.66 2.78 4.60 8.770.100 0.09847 2.94809E-01 5.23 3.66 2.33 4.34 6.790.140 0.13752 7.09719E-02 6.39 3.66 1.25 3.87 7.470.180 0.17772 1.89440E-02 9.85 3.66 0.38 3.68 10.520.220 0.21785 4.84834E-03 17.33 3.66 0.46 3.69 17.720.260 0.25887 2.20160E-03 24.19 3.66 0.67 3.72 24.480.300 0.29937 1.11342E-03 32.38 3.66 0.66 3.72 32.600.340 0.33983 1.10078E-03 33.42 3.66 0.56 3.70 33.620.380 0.37986 7.81849E-04 40.87 3.66 0.45 3.69 41.04Table A18: θ nq = 75 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03205 4.86919E+00 29.30 3.66 4.72 5.98 29.900.060 0.06131 1.38732E+00 8.83 3.66 5.60 6.69 11.080.100 0.09788 2.68460E-01 6.34 3.66 4.58 5.87 8.640.140 0.13715 6.42418E-02 7.43 3.66 3.55 5.10 9.010.180 0.17692 1.72753E-02 10.83 3.66 2.58 4.48 11.720.220 0.21724 5.28407E-03 17.61 3.66 1.75 4.06 18.070.260 0.25809 3.33159E-03 22.42 3.66 1.23 3.86 22.750.300 0.29969 1.24744E-03 35.03 3.66 0.92 3.77 35.230.340 0.33965 1.09621E-03 38.01 3.66 0.72 3.73 38.200.380 0.37969 1.41521E-03 35.62 3.66 0.58 3.71 35.82Table A19: θ nq = 85 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03071 5.11718E+00 36.38 3.66 6.22 7.22 37.090.060 0.06108 1.16687E+00 11.57 3.66 7.25 8.12 14.130.100 0.09839 2.72231E-01 8.45 3.66 6.04 7.06 11.010.140 0.13708 6.28358E-02 9.87 3.66 5.20 6.36 11.740.180 0.17669 1.61772E-02 14.42 3.66 4.32 5.66 15.490.220 0.21679 5.34681E-03 22.55 3.66 3.65 5.17 23.140.260 0.25716 4.56447E-03 26.89 3.66 2.92 4.68 27.30264able A20: θ nq = 95 ± ◦ at Q = 4 . ± . p r , bin [GeV / c] p r , avg [GeV / c] σ red [fm ] δσ stats [%] δσ norm [%] δσ kin [%] δσ syst [%] δσ tot [%]0.020 0.03048 3.39458E+00 46.80 3.66 6.23 7.23 47.350.060 0.05984 1.63840E+00 16.85 3.66 8.27 9.05 19.120.100 0.09783 2.41467E-01 14.49 3.66 7.19 8.07 16.580.140 0.13606 8.15315E-02 17.95 3.66 6.17 7.18 19.330.180 0.17524 3.55247E-02 26.77 3.66 5.23 6.38 27.520.220 0.21358 3.30502E-02 46.09 3.66 4.55 5.84 46.46265 ppendix B: The Hall C Fast Raster Figure B.1: Cartoon showing how the fast-raster system works. The beam bunchesfeel a kick (time-varying magnetic force) along the ( x, y ) coordinates due to a time-varying magnetic field and form a rectangular pattern at the target.The Hall C fast raster (FR) system consists of a set of X and Y air-core magnets.A linear (triangular) waveform generator provides a ∼
25 kHz [113] time-varyingcurrent to the FR magnets coils which produces an oscillating magnetic field trans-verse to the beam axis. As the electron beam bunches pass by the FR-X and FR-Ymagnets, they are deflected by a magnetic force (“kick”) along the x and y axis, The first linear raster in Hall C has been operating since August 2002. The advantage ofthe linear (triangular) waveform over the original sinusoidal waveform is the higher linearvelocity and suppression of the turning time at the crests and troughs of the oscillatingmagnetic field. This suppression significantly reduces the dwelling time of the field at theedges of the raster and contributes to less beam energy being deposited at the boundaries.As a result, the overheating generated in the cryogenic target due to the edges of theraster is reduced. For a detailed description of the linear raster, see Ref. [113]. x ,∆ y ) at the target effectively smears outthe beam in a rectangular raster pattern (See Fig. B.1).The deflection angle of the electron beam bunch along ( x, y ) -axes at the targetcenter is given by θ ∆ x,y = ∆ x, y/L FR , (B.1)where the length from the center of the target to the average distance between FR-Xand FR-Y magnets is L FR = 13 .
56 m. To estimate the deflection of the beam at thetarget, substitute Eq. B.1 into the equation of motion for a charged particle undera magnetic field (Eq. 3.3) to obtain p e = C k (cid:82) B y,x ( t ) d(cid:96) ∆ x, y/L FR . (B.2)From the FR-magnet specifications in Table 1 of Ref. [113], the field integral is givenby (cid:90) B y,x d(cid:96) = 8 . × − [T][m][A] × I x,y . (B.3)Substituting Eq. B.3 in Eq. B.2 and solving for ∆ x and ∆ y ,∆ x, y = 0 . × I x,y p e , (B.4)where ∆ x, y is the electron beam bunch deflection in [mm] at the target center, I x,y is the FR-magnet current in [A] and p e is the electron momentum in [GeV/c]. Sinceeach FR-magnet has two coils and each coil gives equal deflection, both coils willgive double deflection, therefore I coil x,y = I x,y /
2. Using this expression, Eq. B.4 canbe expressed as ∆ x, y = 0 . × I coil x,y p e , (B.5) The deflection equations of the FR-X and FR-Y magnets have the same form and aredenoted by the ( x, y ) subscripts to differentiate between the magnets. x, y )-axes are defined as twice the deflectionangle, R x,y ≡ x, y = 1 . × I coil x,y p e . (B.6)From Table 2 of Ref. [113], the maximum operating current of each FR-magnet is I peak ≈
100 A, and assuming that running the power supply at 80% of the voltageis best for the long-term lifetime of the power supply, I peak ≈
80 A [170]. Themaximum operating current per coil is then given by I coil , max x,y = I peak x,y / R ( x,y ) , max = 52 . p e . (B.7)Therefore, for a 5-pass beam ( p e =10.6 GeV/c) the maximum possible raster size is R x, max × R y, max = 5 × . (B.8)From these results and Eq. B.6, the maximum deflection is ∆ x max , y max = R ( x,y ) max / . L tgt is given by θ max∆ x ∓ = ∆ x max L FR ∓ L tgt , (B.9) θ max∆ y ∓ = ∆ y max L FR ∓ L tgt . (B.10)Inserting the numerical values ∆ x, y = 2 . L FR = 13560 mm, and L tgt = 100mm (10-cm long target) in Eqs.B.9 and B.10, θ max∆ x ∓ , θ max∆ y ∓ = (1 . × − , . × − ) rad . (B.11)From Eq. B.11 the maximum deflection angle at both ends of the target along the( x, y ) axes is negligible for the maximum possible raster size, therefore, the raster is268pproximately uniform across the target length. This experiment used a raster sizeof 2 × across a 10-cm long target at 5-pass, therefore, the deflection anglesare expected to be even smaller than in Eq. B.11.To detect and process the raster signals, the fast-raster system is equipped with2 current probes and a field pickup probe. The current probes are used for precisemeasurements of the magnet current amplitude and direct current (DC) offset de-tection and the field pickup probe is used to detect the variations in the rampingmagnetic field. The signals are sent to a beam raster monitor in the hall that com-pares the setting and readback parameters to create a fast shutdown detection forthe machine safety operation [113]. The signal from the field pickup probe is sent toan ADC module in the Hall C Counting House and the raw ADC signals are thenfurther processed by the analysis software.269ITACARLOS YERO PEREZBorn, Santiago de Cuba, Cuba2010 A.A., BiologyMiami-Dade CollegeMiami, Florida2014 B.Sc., PhysicsFlorida International UniversityMiami, Florida2014–2015 Graduate Research AssistantFlorida International UniversityMiami, Florida2015–2016 Graduate Teaching AssistantFlorida International UniversityMiami, Florida2016–2018 Graduate Research AssistantFlorida International UniversityMiami, Florida2018 M.Sc. in PhysicsFlorida International UniversityMiami, Florida2018–2020 Ph.D Candidate in PhysicsFlorida International UniversityMiami, FloridaPUBLICATIONS AND PRESENTATIONSPooser, E., Barbosa, F., Boeglin, W., Hutton, C., Ito, M.M., Kamel, M., Khetarpal,P., Llodra, A., Sandoval, N., Taylor, S., Whitlatch, T., Worthington, S., Yero, C. andZihlmann, B. The GlueX Start Counter Detector. Nucl.Instrum.Meth. A927 :330-342, May 2019.Yero, C. et al. (2020).
Probing the Deuteron at Very Large Internal Momenta .Manuscript in preparation for Physical Review Letters (PRL).270ero, C. (2019, March).
Deuteron Electro-Disintegration Experiment at Hall C .Workshop on Quantitative Challenges in SRC and EMC Research, Boston, Mas-sachusetts.Yero, C. (2019, October).
First Cross Section Results of D ( e, e (cid:48) p ) n at Very HighRecoil Momentaat Very HighRecoil Momenta