A Measurement of the Proton Elastic Form Factors for 1 ≤ Q² ≤ 3 (GeV/c)² Citation Walker, Robert Christian David (1989) A Measurement of the Proton Elastic Form Factors for 1 ≤ Q² ≤ 3 (GeV/c)². Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1j3p-xf48. https://resolver.caltech.edu/CaltechTHESIS:10242013-084941401 Abstract We report measurements of the proton form factors, Gᵖ E and Gᵖ M , extracted from elastic electron scattering in the range 1 ≤ Q² ≤ 3 (GeV/c)² with uncertainties of < 15% in Gᵖ E and < 3% in Gᵖ M . The results for Gᵖ E are somewhat larger than indicated by most theoretical parameterizations. The ratio of Pauli and Dirac form factors, Q²(Fᵖ₂/Fᵖ₁), is lower in value and demonstrates less Q² dependence than these parameterizations have indicated. Comparisons are made to theoretical models, including those based on perturbative QCD, vector-meson dominance, QCD sum rules, and diquark constituents to the proton. A global extraction of the form factors, including previous elastic scattering measurements, is also presented. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Filippone, Bradley W. Thesis Committee: Filippone, Bradley W. (chair) Barish, Barry C. Frautschi, Steven C. Prince, Thomas A. Defense Date: 20 February 1989 Record Number: CaltechTHESIS:10242013-084941401 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10242013-084941401 DOI: 10.7907/1j3p-xf48 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 8007 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 24 Oct 2013 16:47 Last Modified: 07 Jan 2022 20:08 Thesis Files Preview PDF - Final Version See Usage Policy. 40MB Repository Staff Only: item control page

A MEASUREMENT OF THE PROTON ELASTIC FORM FACTORS FOR 1 ~ Q 2 ~ 3 (GeV / c) 2 Thesis by Robert Christian David Walker In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1989 (Submitted February 20 , 1989) 11 Dedicated to my family: My father, William, who lead me; My mother, Eleanor, who supported me; Afy brother, Thomas, who guided me; My sister, Diana, who lives through me. Ill Acknowledgments This work would not have been possible without the help of a great many people. Special thanks goes to my thesis advisor, Dr. Brad Filippone, who spent hundreds of hours helping me through the experiment, data analysis, and writing of this thesis. His patience, dedication, and wisdom have shown me the best qualities a physicist can achieve. I would also like to thank the scientists and staff of the Kellogg Radiation Laboratory and SLAC Experiment E140 who made this experiment possible. In particular, the spokespersons for E140, Dr. Arie Bodek and Dr. Steve Rock, deserve credit for their many hours of help. Thanks also goes to Dr. Sridhara Dasu for his many contributions to the data analysis presented here. Dr. Dave Potterveld provided much of the software that assisted in writing this thesis, along with many helpful answers to my numerous questions over the years. Thanks to you all. I would also like to thank my many friends, office mates, and room mates that I have had while at Caltech. Special thanks goes to Mike Hoenk, Tom Fric, and Doug Collins, who have provided many of the fond memories I take with me from Caltech. My love and gratitude also goes to Genesis Starr, who has greatly enriched my life here. My stay at Caltech could not have been as fulfilling without her friendship. My appreciation also goes to my "Pasadena parents," Shirley and John Fields, who gave me tremendous support and helped keep me sane. Finally, I would like to thank my parents and family; for without their love and support none of this would have been possible. IV Abstract We report measurements of the proton form factors, G~ and G~, extracted from elastic electron scattering in the range 1 ~ Q 2 ~ 3 (GeV fc? with uncertainties of <15% in G~ and <3% in G~. The results for G~ are somewhat larger than indicated by most theoretical parameterizations. The ratio of Pauli and Dirac form factors, Q2 (Ff / Ff), is lower in value and demonstrates less Q2 dependence than these parameterizations have indicated. Comparisons are made to theoretical models, including those based on perturbative QCD , vector-meson dominance, QCD sum rules, and diquark constituents to the proton. A global extraction of the form factors, including previous elastic scattering measurements, is also presented. v Table of Contents Acknowlegments Abstract . . . List of Figures List of Tables . Chapter 1: Introduction . lll . lV V lll X 1 1.1: Overview of Elastic Scattering 1 1.2: Derivation of Elastic Cross Section 1.3: Experimental Goals . . . . . . . 4 Chapter 2: Experimental Apparatus and Data Acquisition 2.1: Upstream Beam System 2.2: Targets 2.3: Spectrometer 2.4: Detectors 2.4.1: Cerenkov Counter 2.4.2: Wire Chambers . 2.4.3: Total Absorption Shower Counter 2.4.4: Plastic Scintillators 6 9 9 14 16 19 20 21 23 25 2.5: Electronics 2.5.1: Fast Electronics 25 25 2.5.2: Trigger 2.5.3: Data Acquisition System 2.6: Run Plan . . . . . . . . 27 29 2.7: On-Line Data Taking Procedure 31 Chapter 3: Data Analysis-! 3.1: Event Analysis . . 3.1.1: Tracking . . . 3.1.2: Shower Counter 3.1.3: Cerenkov 3.1.4: Electron Identification and Event Histogramming 3.2: Non-event Analysis . . . . . . . . . . . . . . . 30 33 33 33 37 41 42 43 vi 3.2.1: Acceptance . . . . . . 3.2.2: Efficiency and Dead-Time 3.2.3: Target Density 3.2.4: Incident Charge Chapter 4: Data Analysis-11 4.1: Histogram Analysis 43 52 56 59 62 62 4.1.1: !:1</J Sum . . . 62 4.1.2: !:iP/Po Integral 4.1.3: !:1() Average 63 64 65 4.2: Calculation of Raw Cross Sections 4.3: Subtraction of Aluminum Background 4.4: Radiative Corrections . . . . . . . 4.5: Combining Runs of Similar Kinematics 4.6: Dependence of Cross Sections on Kinematic Cuts 66 66 67 4.7: Elastic Form Factor Extraction 68 72 4.8: Systematic Uncertainties 4.8 .1: Point-to-Point Uncertainties 74 74 4.8 .2: Absolute Uncertainties 4.9: Final Values of the Form Factors 76 77 Chapter 5: Comparisons to Previous Data and Theories 5.1 : Previous Measurements . . . . . . . . . . . . . . 5.1.1: Overview of Previous Experiments . . . . . . . 19 5.1.2: Comparison to Previous Form Factor Measurements 5.1.3: World Fit to Elastic Form Factors . . . . . . . . . . 5.2: Theories and Parameterizations 5.2.1: Dipole Approximation and Form Factor Scaling 86 89 94 95 96 5.2.2: Vector Meson Dominance Models 5.2.3: Dimensional Scaling 5.2.4: Perturbative QCD 5.2.5: QCD Sum Rules 5.2.6: Diquark Models 5.2.7: Presentation of the Predictions 79 80 100 101 103 105 108 Chapter 6: Conclusions 112 Appendix A: Radiative Corrections 115 A.1: Overview 115 Vll A.2: Internal Corrections 116 A.3: External Corrections 119 A.4: Improvements to Internal Corrections 125 Appendix B: Kinematic Calibration B.1: Spectrometer Survey B.2: Wire Orbit B.3: Ionization Losses . . B.4: Beam Steering . . . B.5: Elastic Peak Calibrations B.5.1: Procedure for De-radiating Spectra B.5.2: Fits to De-radiated Spectra B.5.3: Absolute Energy Calibration B.5.4: Incident Energy Fluctuations B.5.5: Resolution . . . . . . . . 127 Appendix C: Subtraction of Aluminum Background C.1: Reversed Liquid Hydrogen Flow . . . . C.2: Determination of Aluminum Background C.3: Effect on Radiative Corrections C.4: Subtraction and Uncertainty Analysis 141 141 142 Appendix D: Data Tables D.1: Data From This Experiment .... D.2: Data From Other Experiments (World Fit) 155 References 166 127 128 129 130 131 132 135 136 138 138 146 152 155 159 Vlll List of Figures 1.1 2.1 2.2 Feynman diagram of elastic scattering A-line transport system Toroid calibration and readout circuit 2 10 13 Target assembly 8- GeV spectrometer 15 17 18 2.3 2.4 2.5 Spectrometer focal properties 2.6 2.7 2.8 Particle detectors Wire chambers Total absorption counter 2.9 3.1 Simplified electronics diagram Shower counter spectrum (inelastic data) 3.2 3.3 3.4 3.5 3.6 3.7 Shower counter spectrum (elastic data) Cerenkov pulse height spectrum Acceptance function (no cuts) Acceptance function (cuts) Acceptance target length effect vs. t:..PI Po and f:..() Acceptance target length effect vs. () 3.8 3.9 3.10 3.11 Acceptance target length effect vs. tl.</Jcut Wire orbit: dfl vs. E' Wire chamber efficiency vs. x hut Electronics dead-time 3.12 Average hydrogen density 3.13 Toroid readout comparison 4.1 Q2 and e distribution of data 4.2 Cross section vs. t:..<P 4.3 Cross section vs. ~)..() 4.4 Cross section vs. t:..PI Po-cut 4.5 Rosenbluth separation plots 4.6 4.7 4.8 x2 contour plot: Gi; vs. GtExtracted values of Gr;;IGD vs. Q2 Extracted values of G"M I GDI f.Lp vs. Q2 19 22 24 26 39 41 42 45 46 49 50 51 52 54 56 57 60 68 69 70 71 72 73 78 79 lX 5.1 5.2 Comparison of G~ I G Df.lp results to previous experiments 5.3 5.4 Sample c separation plots for world fit 5.5 5.6 Comparison of Gj;IGn results to previous experiments . . . . . . . Extracted values of G~IGnlf.lp vs. Extracted values of G~IGn vs. Q2 from world fit 1-p coupling in the VMD model . . . . . . . . . . Q2 from world fit 5.7 5.8 Feynman diagrams for factorization (in a,) of e-p scattering 5.9 Schematic of diquark model G~ prediction from QCD sum rules vs. data 5.10 Dipole form factor vs. Q 2 5.11 Theoretical models for G~ vs. Q 2 5.12 Theoretical models for G~ vs. Q 2 5.13 Theoretical models for Q2 (Ff IFf) vs. Q2 A.1 Higher order Feynman diagrams for radiative corrections A.2 A.3 Radiative effects on elastic peak spectrum Landau ionization spectrum A.4 B.1 B.2 B.3 B.4 B.5 B.6 LH2 target schematic C.1 Schematic of the LH2 target under proper running conditions C.2 Aluminum normalization factor vs. time C.3 C.4 C.5 Linearity of radiative corrections Absolute momentum calibration from wire orbit Measured missing mass squared histogram De-radiated missing mass squared histogram Peak position offsets and absolute energy calibration Peak position offsets and beam energy fluctuations Resolution measurements . . . . . . . . . . . . Ztgt spectrum . . . . . . . Extraction of ~z of the aluminum distribution 87 88 90 91 93 97 104 106 108 109 110 111 112 116 117 122 123 129 133 135 137 139 140 142 146 148 150 152 X List of Tables 2.1 Discriminator settings 2.2 Elastic data run plan 3.1 Reverse transport coefficients 3.2 Forward spectrometer transport coefficients from TRANSPORT 4.1 Form factors (statistical errors) 4.2 Systematic uncertainties . . . . . . . 4.3 Form factors (statistical and systematic errors) 5.1 Relative normalizations of previous experiments from world fit 5.2 Form factors from world fit . . . . . . A.1 Details of target geometry and materials D.1 Radiative corrections . . . . . . . . D.2 Cross section values D .3 Aluminum normalization factor (reversed flow) D.4 Aluminum normalization factor (proper flow) D .5 Janssens data (Q 2 :5 0.5 (GeV /c) 2 ) D.6 Janssens data (Q 2 ~ 0.5 (GeV /c) 2 ) D.7 Bartel, 1966, data D.8 Albrecht data D.9 Litt data . . D.10 Goitein data D .11 Berger data . D.12 Price data D .13 Bartel, 1973, data D.14 Kirk data D.15 Sill data D.16 Katramatou data 27 31 38 48 74 76 78 89 92 124 155 156 157 158 159 160 161 161 161 162 163 164 164 164 165 165 -1- Chapter 1 Introduction One of the fundamental problems addressed by nuclear and particle physics over the past 30 years is the underlying structure of the proton. It has been studied using both lepton and hadron probes of increasingly higher energy, leading to the quark/parton picture in force today. Elastic electron-proton scattering probes these constituents in a process which leaves them bound after the collision. The cross section for this process is described in terms of two functions, called the electric and magnetic form factors, G~ and G~. At low momentum transfers, G~ is related to the Fourier transform of the proton charge distribution, while G~ contains information about its magnetic moment distribution. At large momentum transfers, the form factors give important information about the quark structure within nucleons, and therefore about the nature of the strong force at moderate inter-quark separation. It is this quality of the strong force which will be studied here. 1.1 Overview of Elastic Scattering The Feynman diagram for elastic electromagnetic scattering of an electron from a spin-t particle (proton) is shown to lowest order in aem in Figure 1.1. The electron and proton, with initial four-momenta p and P, respectively, exchange a single photon of momentum q. The final state consists of the electron and proton with momenta p' and P', respectively. The requirements of momentum conservation at each vertex - 2- and initial and final states that are on mass-shell for both the electron and proton give the constraints: (1.1) (1.2) (1.3) e e p p Figure 1.1 - Feynman diagram of elastic scattering of an electron from a proton in the single photon exchange approximation. In the laboratory frame the electron has initial energy Eo, final energy E' , and scattering angle 0; and the proton is initially at rest. H the mass of the electron is -3- ignored (Eo ~me, E' sin 0 ~me), the four momenta can be written: P' = '"' where v Mp + v 0 ( -E' sine Eo- E' cosO ) (1.4) = Eo - E' and the initial electron direction has been chosen along the z-axis with the final electron momentum in the y-z plane. The four momentum transfer squared, defined by Q2 =-q =q'"'q'"', is: 2 (1.5) and is completely determined by the electron kinematics. The constraint that the final hadronic state contains only a single proton, W2 = P~P'~' = M'£, leads to the equation: (1.6) This equation is again only a function of the electron kinematics and provides a relationship between the initial and final electron energy and scattering angle that is necessary in order to guarantee that a single proton, and nothing else, exists in the final hadronic state. -4- 1.2 Derivation of Elastic Cross Section The electron current is given by [1 ,2] : /' = -e~(p')!I1.,P(p) (1. 7) where .,P(p) is the wave function of an electron with four momentum p . The proton current can be written as: J 11 = -e~(P') [Ff(Q 2 )111 + it 2 Ff(Q 2 )io-11 11 q 11 ] w(P) (1.8) p where the proton anomalous magnetic moment is Kp = J.Lp - 1 ::::::: 1.79. Terms in the proton current that contain 1 5 are ruled out due to parity conservation, and a linear term in q11 is ruled out by current conservation. The only invariant involved in the problem is Q 2 , so Ff and Ff have been introduced as arbitrary functions of Q 2 only. These are called the Pauli and Dirac form factors, respectively, and they contain the information about the internal structure of the proton. The helicity-conserving scattering amplitude is described by Ff, while Ff describes the helicity-fl.ip amplitude. They have been normalized at Q2 = 0 to: Ff{O) = Ff(O) = 1 (1.9) The differential cross section can be written [2] as: do- = IMI 2 1 1 4((p. P)2- m~Mi) "1 (27r ) 4 8 4 (p~ + P' 11 - p - P) d3 p' d3 P' (27r)32E' (27r) 32(Mp + v) x~~~~---~------- (1.10) -5- where the scattering amplitude M is related to the electron and proton currents [2] by: TJi = -i j j 11 ( - : 2 ) J 11 d 4 x = -i(21r) 4 84 (p' + P'- p- P)M (1.11) The differential cross section can be written in terms of the form factors, after integrating out the 8-functions that impose momentum conservation, as: (1.12) where the cross section has been averaged over initial spins and summed over final spins; i.e. an unpolarized beam and target have been assumed. (Note: the terms F1 and Ff, etc., will be used interchangeably in order to avoid cumbersome notation, such as Ff 2 ). The 8-function imposes the constraint that x = 1. Thus at fixed Eo and () the elastic cross section is a 8-function in E' and occurs at a value E' = E~ 1 = The Sach's form factors, Gk and G~, are defined in terms of Ff and Ff by: Gk(Q 2 ) = Ff(Q 2 ) - KprFf(Q 2) G~(Q2) = Ff(Q2) + KpFf(Q2) Q2 r= 4Mi (1.13) (1.14) (1.15) Note the normalizations of G~(O)=J.Lv and Gk(O)=l. These are referred to as the electric and magnetic form factors, respectively, because in the non-relativistic limit -6- they can be related [2] to the Fourier transforms of the charge and magnetic moment distribution of the proton. If equation 1.12 is integrated over E', the elastic cross section is: dJ _ (d(j) E' dD.- dD. mottEo (G11++rGL + 2rGM2 tan2 (0/2) ) T _ (d(j) E'_1_ dD. mott Eo 1 + T ( dt7) 2 (c E2 + ~G c M ) _ a cos (0/2) 2 (1.16) ( 1.1 7) 2 dD. mott- 4E5 sin4 (0/2) (1.18) -1 where c = (1+2(1+-r) tan2 (0/2)) is the longitudinal polarization of the virtual phot on 1.3 Experimental Goals There are two properties about the previous equation that should be noted. The first is the existence of the kinematic variable r that multiplies GL relative to G1. When r ~ 1 (small Q2 ), the cross section is dominated by G~, but for r ~ 1, G~ dominates. This Q 2 dependence is related to the fact that the magnetic force ("" 1/r3 ) dominates at short distances relative to the electric force ("" 1/r2 ). Since it is the large Q2 regime that is of interest here, G~ is relatively more difficult to extract from the cross sections than G~. The second thing of interest is the existence of the tan 2 ( () /2) term that multiplies GL but does not multiply G1. If several measurements of the elastic cross section are performed at fixed Q 2 but different (), it is possible to extract both G~ ( Q 2 ) and G~ ( Q2 ); this is called a Rosenbluth separation. Since G~ is only a small contribution to the cross section at large Q2 , uncertainties in the measured cross sections are magnified into larger fractional uncertainties in G1. In order to limit the uncertainties in G~, the following goals are the most important to achieve: -7- • Maximizing the range in 0 of the cross section measurements at each value of fixed Q2 . • Limiting the statistical uncertainties in dcr I dD.. • Limiting random systematic fluctuations in dcr I dD. . • Limiting the systematic uncertainties correlated with 0 at fixed Q2 . This means limiting any systematic uncertainties correlated with Eo, E', 0, beam current , cross section magnitudes, time, etc., since all of these quantities change when 0 is changed at a fixed value of Q 2 • It was this last point that was the most important in this experiment. Previous measurements [3,4] of G~ in this Q2 regime were frequently dominated by systematic uncertainties. These were usually due to uncertainties in normalization between data taken at different laboratories, or normalizations between different small and large angle spectrometers. Improvements made to the Stanford Linear Accelator Center (SLAC) beamline and detectors in End Station A, in order to perform a Rosenbluth separation of deep inelastic scattering cross sections [5-7], made it possible to substantially reduce the systematic uncertainties in the elastic cross sections measurements. These improvements included precise measurements of the incident beam energy and angle and an understanding of the spectrometer acceptance over a wide range of E'. A single spectrometer was rotated around the target pivot to measure cross sections at a wide variety of angles, including intermediate angles as a check on systematic effects. The deep inelastic scattering experiment [5] which was scheduled to be performed in End Station A required that elastic scattering data be taken to calibrate the incident beam energy (Eo = E' + Q 2 I2Mp at the elastic peak) as an additional technique to reduce systematic uncertainties. This is the data presented here. With a relatively small investment of time (the data presented here represents -8- ::::::: 5 days of beam time), substantial improvements over previous elastic scattering measurements were possible, along with precise calibrations of the experimental apparatus. -9 - Chapter 2 Ex peri menta I Apparatus and Data Acquisition 2.1 Upstream Beam System Electron pulses from both the Main Injector and the Nuclear Physics Injector (NPI) at SLAC [8] were used in this experiment. The Main Injector is located at the beginning of the beamline approximately 2 miles from the experimental hall, utilizing all 30 "sectors" of the linac to deliver the maximum electron energy. Energies of between 1-21 GeV can be achieved with peak currents :5 40 rnA. At the lower energies (< 6 GeV), the peak current is reduced due to the effects of beam breakup along the accelerator line. The NPI [9] was installed to provide high current beams at these lower energies and is located 6 sectors from the linac exit. It can provide beams of ~ 40 rnA peak current with energies between 0.65 and 4.5 GeV, and was therefore used at beam energies :5 4.25 GeV. The Main Injector was used at the higher energies. Beam pulses were typically 1.6 p.s in width and were operated at between 60-90 pulses per second (PPS). The beam was directed into the "A-line" (Figure 2.1) for delivery to the experimental hall, End Station A (ESA), by the beam operators at SLAC's Main Control Center (MCC). The energy of the beam was defined in the "A-bend" ; a set of eight identical dipole magnets that bent the beam in a horizontal plane (B10-Bl7). A set of slits (SLlO & SLll) then allowed only those electrons within the proper energy range to pass. By changing the width between the slits, the energy spread of the beam, which ranged from 0.1%-0.3% full width in this experiment, could be altered. -10- An additional identical dipole magnet, in series with the others, was maintained at MCC separate from the beamline. A rotating flip-coil was located at the nominal beam position inside this magnet and continuously monitored the field strength. The beam energy at the center of the A-bend slits was calculated and checked every few seconds, and its value was recorded on magnetic tape every few minutes. Additional quadrupole magnets were located in the A-bend and were used to minimize the beam spot size and divergence at the target position in ESA. C1 810 - 813 CO PM1 - PMS C Collimator PM PIAied m&CJ* a 8 Quadrupole Bendlno ma11181 Sl. Sit A Steering magnet RS Roler scraen L.....J.__J o 25 50 Meters Figure 2.1- Diagram of the A-line system used to transport the beam to ESA. The scale on the right is only approximate. It was the responsibility of the experimenters to perform the final steering of the beam to the target. This was accomplished by sending the beam through two sets of vertical and horizontal bending dipole steering magnets (A10-A13) after it left -11- the A-bend. The first set of magnets was located ~ 99 m upstream of the t arget; the second set was located ~ 53.5 m upstream. A set of two resonant microwave cavities were located immediately following the second set of magnets to measure the horizontal and vertical beam position. Two secondary emission wire arrays were located in the beam path ,....., 1 m upstream of the target. One wire array had its wires oriented in the vertical direction, the other in the horizontal. The wires were made of pure aluminum 0.13 mm in diameter and were spaced 0.4 mm apart. An LSI minicomputer continuously monitored the beam position at the cavity monitors and wire arrays throughout the experiment. This computer also controlled the current to a set of smaller auxiliary coils (trim coils) around the steering magnets and maintained the beam along the nominal beam axis continuously throughout the data taking. The beam steering system was calibrated after every energy change so the computer could properly adjust the current in the trim coils. Two zinc-sulfide (ZnS) roller screens, separated by,....., 10m, were located upstream of the target in ESA and could be rotated into the beamline at low beam pulse rates between data runs. The beam position could be observed on these roller screens by the experimenters through remote TV cameras. A ZnS target could also be inserted at the target position to allow t he beam position to be observed. Thus the experimenters could confirm that the beam transport system was operating properly. The total amount of incident charge in the beam was measured with a set of two identical ferromagnetic toroidal charge monitors placed around the beamline ,....., 10 m upstream of the target. When a pulse of electrons passed through the toroids, a time varying magnetic field was produced inside the ferromagnetic core. A wire was wrapped several times around the core and connected to an RC-circuit. A current was induced in this wire by the magnetic flux in the toroid and a damped resonant signal was created in this effective RLC-circuit with an amplitude proportional to the -12- total charge in the beam pulse. The RLC time constant was large compared to the width of a beam pulse. The output of the RC circuit was connected to a preamp located in a shielded box along the beamline ,....., 0.5 m from the toroid to minimize noise in the system. The output signal from the preamp was carried from ESA to Counting House A (CHA) where the electronics, experimenters, and computers for this experiment were located. There the signal passed through another set of amplifiers and the amplitude was measured by two different electronic circuits (see Figure 2.2). To allow for the wide range of beam currents measured, the gain of the amplifiers was altered over a range from 1-100. The first and original circuit measured the value of the signal after a time delay from the start of the beam pulse. This time delay was chosen to minimize sensitivity to the width of the beam pulse. The second and more recently installed circuit integrated the signal over the second half period. Before the experiment began, the starting and stopping times of this integration were set to minimize the sensitivity to the width of the beam pulse and drifts in the timing circuit. The toroids were calibrated by sending a pulse of charge through a wire that passed through the toroids (Figure 2.2) . A capacitor, whose capacitance was measured before the experiment to an accuracy of 0.1 %, was charged to a nominal voltage with a digital-to-analog converter (DAC) and was then discharged through the wire. An additional attenuator circuit was located near the toroids and could be remotely set to values of +1, +10, or +100, to allow for either large or small beam currents to be simulated. The resulting signal of the toroids was measured and the relationship between the incident charge and signal pulse height could be determined. A separate calibration system was used for each toroid, but occasionally the calibrators were temporarily switched so a cross calibration of the toroids could be performed. This -13- Beam pulse Proqr......J>l e q&J.n camac Proqr......J>l e delay and vidt~ End Sllllon A Interface Counting Ho._.. A Figure 2.2 - The toroid circuit used to measure the incident charge. Both calibration and readout circuits are shown. Two independent, identical such systems were used in this experiment. system was used to monitor any changes in the toroid system caused by temperature fluctuations, drifts in the amplifier gains, and shifts in the timing. Calibrations were done between data runs, which was typically every few hours. The calibrations were performed at gain and attenuator settings that roughly corresponded to the beam currents being used at that time of the experiment. Additional calibration runs were taken periodically at a variety of gains, DAC voltages, attenuator values, and output gain amplifier settings. These measured the linearity of the output circuit and checked for any changes in the calibration capacitors, DACs, or attenuators. Beam quality was monitored with two plastic scintillators mounted with phototubes in ESA. One scintillator was mounted along the beam pipe upstream from the -14- target and was used to measure the "bad spill" caused by the beam halo. The second scintillator was placed ,...,_ 10 m from the target at a scattering angle of ,...,_ 70° and measured the time structure of the beam ("good spill"). An oscilloscope displayed the anode signals which were monitored by MCC as diagnostic tools in tuning the beam. Analog-to-Digial Converters were also used to measure the spill monitors and were recorded on magnetic tape. Ideally the "bad spill" was kept near zero while the "good spill" had a square wave structure 1.6 J.LS in width to minimize dead-time effects and energy fluctuations. 2.2 Targets A cylindrical liquid hydrogen target (LH2) 20 em in length [10] and 5.08 em in diameter (Figure 2.3) with side walls, entrance, and exit windows made of 0.076 mm aluminum was used to scatter electrons for this experiment. An identical, empty dummy cell (DUM) with an additional 1.16 mm of aluminum radiator added to both the entrance and exit windows was used to measure endcap contributions to the scattering. Additional aluminum was added to make the radiation length of the empty cell roughly equal to that of the hydrogen cell, and to increase the scattering rate from the empty cell. A detailed list of the target materials and dimensions is presented in Appendix A. Liquid hydrogen at 21° K and a pressure of 2 atm continuously flowed through the LH2 target. Heat deposited by the beam was removed by circulating the hydrogen through a heat exchanger with a liquid hydrogen bath. Contamination levels within the hydrogen were measured by mass spectroscopy to be~ 0.16% deuterium,< 0.37% H 2 0, and $ 0.1% other contaminants (per molecule). A liquid nitrogen cold trap filtered out any water from the gaseous hydrogen before it was liquefied and circulated into the target circulation system. A 4 em diameter aluminum tube 0.025 mm thick -15- LH2LH2 IN OUT Beam ~ ~7 ~-{ ~ Liquid Hydrogen Cell Figure 2.3- The liquid hydrogen and dummy targets. Note the flow of the liquid hydrogen. Additional radiators were placed over the entrance and exit endcaps of the dummy cell. was contained within the cell and was used as a flow guide. The hydrogen flowed into the target inside this flow guide and exited between the flow guide and the outer target wall. Circulation of the hydrogen was maintained by fan-like pumps at a flow rate~ 1 m/s. During part of the experiment the flow direction of hydrogen through the cell was accidentally reversed. The effects of this reversed flow are discussed in detail in Appendix C. Vapor pressure bulbs and platinum resistors were located at the entrance and exit of the flow guides to measure the hydrogen temperature and pressure. The ingoing and outgoing hydrogen density was calculated from these measurements, and was monitored every 10 sec. -16- These targets, along with other deuterium and solid targets [11], were mounted on a remotely controlled carousel that could be moved vertically and rotated in a horizontal plane to place any desired target into the beam line. This assembly was contained under vacuum within a scattering chamber that was an aluminum cylinder with 2.54 em thick walls, with entrance and exit windows as described below. An additional stack of lead shielding ,...., 20 em thick surrounded the chamber in most places to minimize the amount of radiation escaping the target area into the ESA. The beam entered the scattering chamber through a 5-in circular aperture made of 0.025 mm aluminum that isolated the chamber vacuum from the beamline vacuum. An extended snout attached to the scattering chamber allowed for electrons scattering at angles ~ 50° to exit the chamber through a thin 0.31 mm exit window. It also allowed the beam to exit the scattering chamber far enough downstream so that no electrons could scatter from the exit window into the spectrometer acceptance. The entire scattering chamber was mounted on a,...., 8ft diameter pivot area in ESA around which the spectrometer rotated. The targets were located directly above the center of the pivot along the beam line. 2.3 Spectrometer After the electrons scattered from the target, they were detected in the 8 GeV spectrometer [12,13] in the ESA (Figure 2.4). Electrons were focused and momentum selected by a series of three quadrupole and two vertical-bend dipole magnets. Immediately after the last quadrupole magnet was a lead-shielded hut in which the particle detectors were located. The spectrometer could be rotated around the target pivot on a circular track to allow only those electrons which had scattered at the desired angle to reach the detectors. -17- Shower counter o 1 2 3 Meters Figure 2.4- The 8-GeV spectrometer. The magnets of the spectrometer were tuned to focus particles with momentum E' and final angle 0 to vertical and horizontal positions, respectively, in the detector hut (Figure 2.5). Measurements of the position of the particle track at the two focal planes, one plane for E' (also referred to as P for momentum) and one for 0, could be transformed, to first order, to a measurement of E' and 0. The slope of the track through these focal planes also yielded a measurement of the out-of-plane angle, ¢, and the horizontal position along the spectrometer entrance window, Xsps, of the particle trajectory at the target. The spectrometer had an acceptance of roughly ±4% in tlP/Po, ±8 mr in tlO, ±30 mr in tl¢, and ±12 em in Xspg. Central values of 1 ~ E' ~ 8 GeV and 11.5° ~ 0 ~ 48° were used in this experiment. -18 - .. (a) . . ·.. '''··.,'·. ~ ·········· ii.P/P•-2t .................::.,:··· < :· .. ..·:·.:. - - - - .·..·· - - ii.P/P•O ~-f----....:;~ · · ·· · ..... . . . ,,-:-:" ..: : .. 13 . 7° .....................-"'· ~ .· ...·· · · · · · .:o.P/P•+2t P Focal Plane rnrnaarn 5 ...... e •+8 m.rad (b) --e-o •••• e --a m.rad ~ 9 Focal Plane z (111) Figure 2.5 - The (a) E' and (b) 9 focal properties of the 8-GeV spectrometer. Optical properties of the spectrometer were previously measured [14,15] in 1967 using a dark current electron beam from the accelerator with the spectrometer set at an angle of 0° with respect to the beam axis. Any changes in the optical properties of the spectrometer over the range of momenta used needed to be well underst ood to eliminate systematic errors correlated with E'. Wire orbit measurements [16,17] were done from 0.5 ~ E' ~ 9 GeV to calibrate the central ray of the spectrometer and to check for an E'-dependence to the transport coefficients. Results of the wire orbit study are presented in the Appendix B (E' calibration) and in the section on Acceptance in Chapter 3 (transport studies). -19- 2.4 Detectors The detector package was designed to detect electrons with > 99% efficiency and reject pions to one part in 105 . It was also required to measure both the position and angle of the particle tracks to ±2 mm and ±1 mr, respectively, at theE' and() focal planes. These constraints were achieved with three essential elements: a hydrogenfilled threshold Cerenkov counter, a set of ten multiwire proportional chambers, and a Ph-glass total absorption array (Figure 2.6) . Three sets of plastic scintillators were also included to add to the spatial segmentation of the detectors, to be used as fast trigger elements, and to assist in pion identification and rejection. L. 8c1At1llatora JP-rocal l'lue '1'-rocal / /l'lue ••t •• '::;. :··:; •: ':,: .. ··: h; . . h::: . : !:: :;: : 13.-f Cerenkov Wire Chambers Total Absorption Counter Figure 2.6- The particle detector elements located in the 8-GeV spectrometer. - 20- 2.4.1 Cerenkov Counter The Cerenkov counter entrance window was located at the end of the last quadrupole magnet of the spectrometer and was made as thin as possible to minimize the probability of pions causing "knock-on's" - electrons above the Cerenkov energy threshold that pions elastically scatter out of the aluminum window. The entrance and exit windows were made from aluminum and were 0.41 mm thick. The counter was 3.30 m long and was filled with hydrogen at atmospheric pressure with an index of refraction of 1.000140 (18] . Threshold energies for electrons and pions were 0.031 GeV and 8.4 GeV, respectively. Hydrogen gas was chosen to minimize the probability of pions producing knock-on's within the gas itself or creating scintillation light that could mimic an electron Cerenkov signal. It also contributed very little to the multiple scattering of the electrons and thus improved the resolution of the momentum and angle measurements, especially at low E'. Some of the Cerenkov radiation emitted by electrons appeared in the ultraviolet spectrum. Emphasis was therefore placed on eliminating oxygen in the counter since oxygen absorbs ultraviolet light. The Cerenkov counter was purged weekly by evacuating to < 5 mm Hg pressure, filling with nitrogen, evacuating again, and refilling with hydrogen. Leaking of oxygen through the edges of the phototube face and the rubber 0-ring against which it rested was limited by over-pressurizing the Cerenkov hood to 1.5 atm with nitrogen. A curved mirror 53 by 90 em m area was located 315 em from the entrance window and was used to focus the Cerenkov radiation onto the face of a RCA 8854 phototube located at the top of the counter. The mirror was aluminum with a 0.64 em backing of lucite and was resurfaced with a layer of MgF2 to eliminate oxidation on the surface. It was aligned within the counter with a laser to insure that the -21- Cerenkov light was properly focused onto the phototube face. A wavelength shifter was applied to the face of the phototube to increase its sensitivity to the ultraviolet . The output pulse of the phototube was shortened to 20 ns by placing a 5 ns delay clip line on the anode output that was terminated by a 6 db attenuator. 2 .4.2 Wire Chambers Following the Cerenkov counter were ten planes of multiwire proportional chambers [19](Figure 2.7). The wires were made of 20 p.m gold-plated tungsten and were spaced at 2 mm intervals. A "magic-gas" mixture of 65.75% argon, 30.00% isobutane ((CH3)2CHCH3), 4.00% dimethyl acetal formaldehyde (CH2(0CH 3)2), and 0.25% bromotrifluoromethane (CBrF3) continuously flowed through the chambers. The gas was contained within the chambers by windows made of 0.0762 mm of mylar coated with 0.0762 mm of aluminum. The typical operating voltage of the two cathode planes was 3.6 kV, with gaps between the anode wires and the planes of 4 mm in both directions. The catode planes were made of 0.051 mm of mylar coated with 0.0075 mm of aluminum. The chambers had an active region 35 em in height and 93 em in width, and spanned 1.81 m in the direction of the particle trajectory. Chambers were numbered from one to ten sequentially along the direction of the scattered electrons. Even numbered chambers had wires oriented along the horizontal direction to measure the vertical track position. These were called "P-chambers" since they were used to measure the particle momenta. There were a total of 176 wires in each of these chambers. A vertical teflon support wire near the center of the chambers suspended the wires to keep them from sagging. This wire caused a "' 2 em wide dead region near the center of each P-chamber. Staggering of the support wires guaranteed that no particle track could cross more than two of these dead areas. Chambers 1, 5 and 9 had the wires oriented at -30° from the vertical; chambers 3 and 7 were -22- oriented at +30° (viewed along the particle trajectory). These chambers measured the horizontal track position in addition to including redundancy, so that multiple tracks could be identified and separated, and were called "T-chambers" (for theta). They contained 480 wires with 2 mm spacing; however, the wires were electrically tied together in pairs so effectively 240 wires with a 4 mm spacing were used. The spectrometer E' and () focal planes were contained within the chamber area, as shown previously in Figure 2.6. ~' X 1 2 3 4 5 6 7 8 9 Figure 2. 7 - The orientation of the wires in the wire chambers. The support wires in the P-chambers are shown. 10 - 23- 2.4.3 Total Absorption Shower Counter The Ph-glass total absorption counter is shown in Figure 2.8. It was segmented both in the horizontal direction and along the particle trajectory. Electrons incident on the Ph-glass have a high probability of emitting bremsstrahlung photons which then convert into e+ e- pairs. These pairs will create more hard photons that create more pairs, resulting in an electromagnetic shower. Electrons and positrons will also create Cerenkov radiation which is detected by the phototubes mounted on the Phglass. Pions that enter the Ph-glass do not have a high probability of creating an electromagnetic shower because of their high mass. Instead, the hadronic showers they create will deposit only a portion of the pion's energy in the counter, and it will be deposited further towards the back of the counter, allowing for discrimination between pions and electrons. The first row of six F2-type Ph-glass blocks (20] were made of 45% Si02, 45% PbO, 6% K20, and 3% Na20, and were used as a pre-radiator (PR) to start the electromagnetic shower before the electrons entered the rest of the counter. These blocks had a radiation length of 3.22 em, a refractive index of 1.62, and were 32 em tall, 15.8 em wide and 10.4 em thick ( =3.2 r.l.). Aluminum blocks supported them 4 em off the mounting table so they were vertically centered with the other taller Ph-glass blocks used in the other rows. The maximum particle trajectory angle in the spectrometer was ±2.5° from the central axis, so the PR row was rotated by 5° around the vertical to eliminate the possibility of particles passing through the cracks between the blocks. XP 2041 phototubes were placed at the top of the blocks to detect Cerenkov radiation from the electromagnetic showers. The next 4 rows of SF5-type Ph-glass (21] were 40 em high, 14.6 em wide, and 14.6 em thick (=6.8 r.l.). The first three rows had 7 blocks, the last row had 6. - 24- ,. ee GG G ee e ee G e ee ee e ee e e ee e e ee ' '" rz X ..J Figure 2.8- The Ph-glass total absorption counter. Two of the rows of scintillators are also shown. Each row was staggered relative to the next so that the cracks between the blocks did not overlap. Phototubes were placed on the top of each block. Since the shower maximum occurred near the first row of blocks, an additional phototube was placed on the bottom of each of these blocks to maximize the shower detection efficiency and resolution. The rows were labeled TA, TB, TC and TD, from front to back; the top and bottom phototubes in the first row were called TAU and TAD, respectively. Individual phototubes were numbered 1-7 (or 1-6) from left to right when viewed along the particle trajectory. Clip lines were also placed on these phototubes to limit the signal pulses to 20 ns. The total thickness of the shower counter was 30.4 r.l. The FWHM resolution of the Ph-glass array was found to be 18%/ vfEi. -25- 2.4.4 Plastic Scintillators Plastic scintillators were used to detect all rrummum Iomzmg particles. One row of 6 vertical scintillators were located between wire chambers 7 and 8. Three horizontal scintillators were located between the shower rows PR and TA. These provided additional vertical seg"'Tientation to the detectors. A particle that created a shower in the PR would also give a larger pulse height in these scintillators than a single minimum ionizing particle would. This increased the detector's electron efficiency and pion rejection. A final set of three horizontal scintillators were located behind TD. These scintillators would detect those pions or muons that transversed the entire shower counter, while electrons would be totally absorbed in the counter. The three sets of scintillators were labeled SF, SM and SR, respectively. 2.5 Electronics 2.5.1 Fast Electronics Raw detector signals from the phototubes on the detectors were carried to the electronics in the CHA "' 100 m away by fast heliax cables (for trigger components) or regular coaxial cables (for other components). Pulses from each of the wires in the wire chambers were amplified, discriminated, and fed into a dual 450 ns delay flipflop circuit. This delayed signal could be latched into a bit register by a coincident "fast-latch" signal "' 75 ns from the event trigger to record the pattern of "hits" in the wire chambers. Commercially available CAMAC and NIM modules were used for the electronics in CHA. Attenuators were used to reduce by 50% the signals from the shower counter rows PR, TAU, TAD and TB at E' > 4 GeV to keep the signals from saturating the electronics at large momenta but still have reasonable resolution at small momenta. - 26- A simplified schematic of the electronics is shown in Figure 2.9. The electronic signals were divided using linear fan-out components. One output of the fan-outs went to a set of Analog-to-Digital Converters (ADC's) to record pulse height infermation. The other output was fed through discriminators to the trigger logic and other electronic elements; the discriminator threshold settings are given in Tab le 2.1. The outputs of the discriminators were 0. 7 V pulses with widths of 20 ns. These pulses were sent to scalers, fast latches, and as stop gates to Time-to-Digital Converters (TDC's ). In addition, the raw signals of individual components in each "row" of counters (SFl-6, PRl-6, TADl-7, etc.) were linearly added together to form single pulses for each row (SF, PR, TAD, etc.). These pulses went to ADC's and discriminators; then to scalers, latches, and TDC's. The trigger components (C, SF, SM, PR, TAD) were also sent to a fast trigger circuit to generate a trigger pulse, c 1 2 3 Pl. 4 5 6 ru 1 Dhcd•O iutor 2 rmmJ me OR. 5 Gate UL 6 7 OR. D Tilllll Pre- {7:\ Averaqelllllll.J Scaler 3 UD4 Out<] ~~c D Geller. IGGI .Ill. Ill IlL DR. DL Figure 2.9 - A simplified diagram of the electronics used. \.J - 27 - Component c SF PR SM TAU TAD TB TC TD SR Discriminator (mV) 40 40 60 40 60 60 30 40 40 40 Table 2.1 -The discriminator threshold settings for the various detector components. 2.5.2 Trigger The trigger was designed to fulfill several different purposes. It was necessary that it be more than 99.9% efficient for electrons over the entire range of momenta measured, 1 :::; E' :::; 8 GeV. Deep inelastic data [11] that was taken in parallel with this experiment was run with 1r / e backgrounds of up to 100:1; thus the trigger needed to have a pion rejection of > 99% to keep the trigger rate from being dominated by background pion events. Pion backgrounds were not significant for the elastic data presented here, however. It was also of interest to have a limited measurement of the detector response to pions so the detector performance could be better understood. There were three basic components to the trigger. Electron High (EL-H) was composed of a 3-out-of-4 (3/4) coincidence between C, PR, SM, and TAD. Each of these components had a high efficiency for electrons at high momenta. Since none of the components was absolutely required, this combination had a very high efficiency - 28- for electrons. C, PR and TAD also had good rejection of pions, so pion backgrounds did not significantly effect the trigger rate. However, at low momenta the electron shower in the shower counter was sometimes contained within the PR row of lead glass, and the EL-H trigger was effectively reduced to a 3/3 coincidence of C, PR and SM, with a corresponding reduction in efficiency. Electron Low (EL-L), which consisted of 2/3 of PR, SF and SM in coincidence with C, was instituted to give increased efficiency at low momenta. The inclusion of SF instead of TAD increased the electron efficiency to acceptable levels, and the absolute requirement of C made the trigger insensitive to pions. A logical OR of these two triggers (EL-H and EL-L) was formed to create the electron trigger (EL-20). During the time of the trigger pulse (20 ns), the electronics were not able to accept additional events. In order to measure the effect of this dead-time on the trigger rate, additional pulses (EL-40, EL-60, and EL-80) were formed that were identical to EL-20 except for a longer pulse width (40, 60, and 80 ns, respectively). The ideal trigger rate for a pulse width of 0 ns could be deduced from an extrapolation of the scaler rates of these pulses. A backup measurement of the electronics deadtime was included by creating signals that were a 3/3 coincidence of PR, TAD , and C (PTC-20, PTC-40, PTC-60, PTC-80). Another component of the trigger was added to systematically measure the pion background without a significant increase in the dead-time. A coincidence of SF and SM was formed, referred to as PION, that was efficient for any minimum ionizing particle passing through the detector package. This signal was then pre-scaled by a factor of 28 (PION-PRE) and included in the trigger. Pre-scaling kept this component from contributing significantly to the dead-time. These events were only used for studying the detector response to pions. - 29- A random pulse generator signal (RANDOM) fired approximately every 10 sec and was also included to monitor the pedestals of the electronics components. The coincidence of any of the three trigger components (EL-20, PION-PRE, RANDOM) with a beam gate generated a pre-trigger (PRE-TRIG). Limitations in the speed at which the computer could log data on magnetic tapes made it necessary to limit the trigger rate to once per beam pulse. PRE-TRIG therefore went through a circuit which would allow the trigger (TRIG) to fire only once per beam pulse. The trigger provided the gates for the ADC 's, generated start pulses for the TDC's, reset t he latches, and interrupted the VAX computer to perform the event data logging . It also generated a gate signal for reading out the wire chambers. 2.5.3 Data Acquisition System The main computer for the data acquisition system was a VAX 11-780 computer. This computer accumulated all the necessary hardware data and stored it on magnetic tape. Information that needed to be monitored on a periodic basis, such as t he spectrometer magnets, high voltage power supplies, accumulated scaler and toroid values, and the target positions were controlled through CAMAC interfaces. The VAX could also correct any drifts in the spectrometer magnets or high voltage power supplies. However, because of the high pulse rate (~ 180 pps), other dedicated computers were necessary for accumulating pulse-by-pulse information. Information from the beam steering system and the new toroid accumulat ors were read by an LSI-11 minicomputer. In addition, it corrected the beam steering, cleared the toroid electronics, and controlled the toroid calibration system. The VAX 11-780 computer would periodically (""' 5-10 min) receive the accumulated information from the LSI-11 and record it on magnetic tape. -30- Whenever the electronics was triggered, a computer interrupt to a PDP-11 was generated. This PDP-11 would read out all the ADC, TDC and latch informat ion from the electronics along with the wire chambers hits and record it in a buffer. The PDP-11 was able to handle this information at a rate of 180 Hz, and the buffer was large enough to store many event blocks at the same time. The VAX 11-780 could then read out the information from this event buffer and store it on magnetic tape. 2.6 Run Plan Extraction of the elastic form factors from the measured elastic cross sections required that data be taken at different scattering angles with Q2 held fixed. Physical blocking of the spectrometer by the downstream beam pipe limited the minimum spectrometer angle to ~ 11.5°. Scattering angles > 48° were ruled out because the spectrometer acceptance for events scattered from the ends of the target was reduced, resulting in unacceptably large systematic corrections. In addition, the exit snout attached to the scattering chamber only extended to ~ 50°. Final electron energies were limited by the spectrometer magnets to be 1 ~ E' ~ 8 GeV /c. Changes in the incident beam energy Eo would typically take 4-12 hours, while changes in the spectrometer momentum and angle could typically occur in 5-20 minutes. It was therefore desirable to minimize the number of changes in Eo . The incident and final energy and scattering angle were constrained by the kinematics of elastic scattering to have x = Q2 j2Mpv = 1. From these constraints, a run plan was developed (Table 2.2) . The small values of the cross sections for Q2 > 3 (GeV /c) 2 made it impossible to gather enough statistics without expending what was considered to be an unacceptably large amount of beam time. A data point at Q 2 = 1 (GeV /c) 2 was included to provide a calibration point with previous experiments and as a check of the systematic uncertainties. Additional -31- elastic data with reduced statistics was taken at each incident energy used for the inelastic data as a calibration of the incident energy. This data was taken at Q 2 of 1, 2, 2.5, or 3 (GeV /c) 2 whenever possible and was included in this analysis. Q2 (GeV /c) 2 Eo GeV E' () GeV 1.000 1.000 1.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.500 2.500 2.500 2.500 2.500 2.500 3.000 3.000 3.000 3.000 3.000 1.600 2.400 3.250 2.400 2.800 3.250 4.000 5.500 6.250 7.000 7.500 2.800 3.250 3.750 4.250 7.000 8.250 3.250 4.000 6.250 7.000 8.250 1.067 1.867 2.717 1.334 1.734 2.184 2.934 4.434 5.184 5.934 6.434 1.467 1.917 2.417 2.917 5.667 6.917 1.651 2.401 4.651 5.401 6.651 e- /pls Deg Peak I rnA Time hrs 45.000 27.326 19.374 46.557 37.437 30.783 23.824 16.465 14.272 12.598 11.685 45.912 36.927 30.446 25.949 14.421 12.014 43.910 32.456 18.487 16.194 13.428 4.35 1.30 0.58 40.00 31.88 19.36 10.23 4.21 3.02 2.26 1.91 40.00 40.00 40.00 25.00 7.26 4.75 40.00 33.75 23.75 19.97 12.77 0.10 0.10 0.10 0.07 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.03 0.05 0.09 0.08 0.10 0.10 0.02 0.03 0.09 0.10 0.10 0.62 0.62 0.62 0.86 0.62 0.62 0.62 0.62 0.62 0.62 0.62 2.06 1.16 0.70 0.75 0.62 0.62 3.99 2.11 0.72 0.62 0.62 Table 2.2 - Elastic data run plan. 2. 7 On-Line Data Taking Procedure The beam current was adjusted during each run to limit the number of triggers per pulse t o be no more than 0.1, limiting dead-time corrections to < 10%. Frequently the rate was much less than this due to limitations in the maximum peak -32- current available. Gain amplifiers on the charge monitors were then set so the monitors were operating in the linear region and toroid calibration data would be taken. The spectrometer was rotated in a clockwise direction to the desired angle and the momentum was set so that the center of the spectrometer corresponded to x = 1. Hysteresis effects were avoided by always setting the spectrometer magnets from lower to higher momenta, and degaussing whenever it was necessary to go back down to a lower momentum. The beam was observed on the roller screens to confirm that it was steered properly onto the LH2 target, which was then inserted into the beam line, and data taking began. Typically 20,000 electron counts were logged at each of the main kinematic points; 2000-10,000 electrons were recorded during the additional energy calibration runs. An additional 200-500 counts were measured with the 20 em DUM target in the beam to measure the background from scattering off the endcaps. - 33- Chapter 3 Data Analysis-! Transformation of the online data recorded on the magnetic tapes into the elastic form factors follows a series of distinct tasks. The first task was to make an event-byevent analysis to determine if the trigger was caused by a scattered electron passing through the detector. This was determined principally from the Cerenkov and shower counter information. Scattering kinematics (l:l.P /Po, 6.0, 6.</>) of the event were extracted from the wire chambers. This event data was then combined with other non-event information, such as the target density, integrated beam charge, detector efficiency, and dead-time, in order to determine the cross sections. Finally, the elastic form factors were extracted from the measured cross sections. Details of the event and non-event analysis are presented in this chapter. The cross section measurement and form factor extraction are presented in the following chapter. 3.1 Event Analysis 3.1.1 Tracking Track Fitting The pattern of wire hits in the wire chambers were recorded as a pattern of 1 's and O's, a 1 indicating a hit, a 0 indicating none. This data was then analyzed to determine the possible electron tracks that passed through the wire chambers. Due to the large number of wires and the three different orientations of the chambers, finding all possible tracks in the chambers was difficult . The task was simplified, however, -34- due to the low multiplicity of events in the chambers ("' 1), and the low probability of finding spurious tracks. First, all possible tracks between hit wires in different pairs of P-chambers were calculated. Tracks that were clearly spurious, i.e. those that were far outside of the spectrometer acceptance, were ignored. The other P-chambers were checked for any wire hits within ±4 wires of each of the tracks. This yielded the vertical coordinates of all possible tracks. Using this information, a similar process was followed with the T -chambers to find all possible tracks in the horizontal direction. In an initial pass of the data, searches were made for only those tracks that had associated hits in at least 7 of the 10 chambers, including 3 P-chambers and 3 T-chambers. This constraint limited the number of spurious tracks that could be found. If no tracks were found in this pass, a second search was made for tracks that had associated hits in at least 6 chambers, including 2 P-chambers and 2 T -chambers. This second search was included to reduce the inefficiency caused by the P-chamber support wires and was rarely needed. However, it was a much more CPU intensive search because of the greater number of spurious tracks that were found. H after these searches only one track was found, it was recorded as the particle track. Special cases of zero or multiple tracks are discussed below. 0-track events Occasionally a clear electron trigger had occurred, i.e. a large pulse height existed in both the shower counter and the Cerenkov, but no track was found in the wire chambers. A measurement of the individual efficiencies of each chamber, including the effects of the support wires in the P-chambers, indicated that < 0.1% of electrons that pass through the chambers should fail to leave a clean track. Yet 12% of all clean electron triggers yielded no tracks. A detailed, event-by-event, analysis -35- of a few hundred of these events was done by hand, and the results indicated that the wires near the top of the P-chambers were fired in a large percentage of these events. Since the active area of the wire chambers (35 em x 93 em) was smaller than the active trigger area (53 em X 90 em for the Cerenkov, 40 em x 102 em for the TA), it was concluded that these events were caused by electrons that were in the active trigger area but scraped the top of the wire chambers and were outside the fiducial region (see below). Thus they were not "good" events to be included in the analysis. Monte-Carlo calculations of the spectrometer acceptance indicated that events such as these were expected to occur. These events were excluded from the analysis, and no efficiency correction was included for them. Multi-track events In the case that multiple tracks were found in the wire chambers, it was desir- able to purge any tracks that were spurious or caused by pions and to try to find the "real" electron track associated with the trigger. Multiple tracks were found in approximately 10% of all events. Purging was done by forcing the multiple tracks to pass through a series of cuts until the "best" track was determined. Once the procedure had eliminated all but one track, the purging was stopped. Hit. was not possible to eliminate all but one track, one of the remaining tracks was chosen at random. In the case when two (or more) electrons did actually pass through the detectors within the same 20 ns period, it was also desirable to include only one of the tracks in the analysis since the correction for multiple events is already included in the correction for the electronics dead-time (see dead-time section). The hierarchy of multiple track purging was: • Tracks whose value of IXspsl was greater than ~50 em were eliminated since they did not point back to the spectrometer entrance. -36- • Tracks that did not intersect an area of the shower counter with a large shower pulse were presumed to be spurious or pions. • Tracks whose values oi D..Pf Po, D..O, or D..¢ were outside the spectrometer acceptance were purged. • Tracks that did not intersect a PR block with a large pulse height or an SF scintillator with a pulse corresponding to a minimum ionizing particle were presumed to be spurious or pions. • Pairs of tracks that had horizontal and vertical positions within 6 mm and 16 mm, respectively, of each other were assumed to be caused by the same particle. The track with the greatest number of hit chambers, or the best x2 for the track fitting if the number of hit chambers was the same, was kept; the other track was purged. Only 1% of real double track events, which were themselves only a small percentage of the total number of events, were expected to be purged from this cut. The number of events with multiple tracks after purging was usually < 0.1 %. Track Reconstruction Coordinates in the spectrometer hut were defined by: z-along the nominal particle trajectory, y-the direction of the vertical bend of the magnet, and X- perpendicular to the y-z plane in a right-handed coordinate system (Note: the target quantities Xtgt, Ytgt, and Ztgt were defined in a left-handed coordinate system). The particle track was parameterized in terms of its horizontal and vertical position along a plane z = constant (x and y) located between wire chambers 5 and 6, and its projected slope in the x-z and y-z planes (dx and dy). These parameters were then transformed into the particle characteristics at the target, which were the track location at the spectrometer entrance (Xsps and Ysp 8 , defined in a left-handed system like the target -37- coordinates), the track angles (flO and tl<jJ), and the particle's fractional momentum (tlP/ Po). Only four quantities were measured by the wire chambers, so it was necessary to eliminate one of the target quantities from the analysis in order to get a one-to-one mapping between the spectrometer and target coordinates. It was therefore assumed that the particle track originated along the beam axis in the vertical direction (Ysps=O). The beam spot size was .:S 3 mm in the vertical direction, so this was a good assumption. The target quantities reconstructed were thus tlP/ Po, tl(}, tl</J, and Xsps. To first order, the quantities are related by: tlP/Po ex y tl<P <X dy fl(} <X X Xsps ex x- dx (3.1) The units of the positions and angles are em and mr, respectively; the units of tlP/ Po are in %. The full set of second order reverse transport coefficients, including modifications after a more careful analysis (22] of the 1967 dark current optics data (14 ,15] was performed, are given in Table 3.1. 3.1.2 Shower Counter Electron events could be discriminated from pion events based on the amount of energy that was deposited in the shower counter. Raw signals from each phototube that were recorded by the ADC's were converted into a measurement of the shower energy of each event. The wire chamber information was used for each event to determine the path the particle followed through the shower array. The ADC pulse height signals from each phototube were proportional to the total energy deposited in each block; however, the proportionality constants were different because of the different gains of each phototube. It was necessary to determine these constants so the signal heights for each block along the particle's path could be linearly summed -38- X dx y dy X · X x · dx x·y X · dy dx· dx dx · y dx · dy y·y y. dy dy. dy 1 I::!.P/ Po f::!.() !::!.¢> X spa -0.00205 0.00245 -0.34275 0.00074 -0.00013 0.00012 0.00059 0.00005 0.00000 -0.00059 -0.00003 0.00020 0.00136 0.00004 0.00044 0.19387 0.02408 0.00050 -0.00419 0.00051 -0.00103 0.01485 -0.00098 0.00051 -0.01421 0.00082 -0.00012 0.00003 0.00001 0.00169 -0.03694 0.03954 -0.02689 -0.92820 0.01063 -0.01993 0.00034 0.00056 0.00930 -0.00037 -0.00052 -0.00525 -0.00083 -0.00009 0.00171 4.55362 -4.29185 -0.06007 -0.00142 0.01756 -0.03237 -0.00492 0.00133 0.01543 0.00850 -0.00106 -0.00411 -0.00019 -0.00005 0.16211 Table 3.1 - Second order reverse transport coefficients for the 8GeV spectrometer which give the target coordinates of an event in terms of the particle's track coordinates in the wire chambers. Units are in%, mr, or em, where appropriate. as a measurement of the total energy deposited. Corrections were also made for the effects of light attenuation within the blocks. The ADC pulse height signals from each phototube along the particle's path, plus the phototubes from the nearest adjacent blocks, were included in the sum to determine the energy deposited by each event. Adjacent blocks were included in order to capture any leakage due to the transverse spread of the shower in the Ph-glass ar ••.y. Enough rows of blocks were included to contain at least 98% of the longitudinal penetration of the shower in the lead glass [20]. The calibration coefficients for each phototube were determined using an iterative method of minimizing the shower -39 - 125 400 100 300 ~ ~ 0 tJ 75 0 tJ 200 50 100 25 1 0.5 SHTRK (a) 1.5 0.5 1 1.5 SHTRK (b) Figure 3.1 - (a) A high pion background (inelastic data) spectrum in the shower counter requiring a Cerenkov pulse above an ADC channel of 50. (b) The same spectrum requiring a Cerenkov pulse below an ADC channel of 50. spectrum width while requiring that the spectrum peak be located at a value of 1.0 (when normalized by E'). Corrections were made for the effects of light attenuation in t he vertical direction within the lead glass blocks. The attenuation was assumed to be linear in the distance between the track and the face of the phototube. The attenuation coefficients were determined by measuring the shift in each of the ADC spectrum peaks as a function of the distance of the tracks from each phototube and determining a first-order linear correction. The ADC pulse heights were scaled by the particle's momentum E' so the location of the shower spectrum peak was independent of E', in addition to the correction for the 50% attenuators that were inserted -40- in the signal cables for data taken at values of E' > 4 GeV. The normalized energy (SHTRK) was thus : Nrow itu,±l SHTRK = " ' " ' C IJ· ·[1- a IJ· ·(y~rkyt':'be)] x ADCij ~ ~ I IJ E' . (3.2) •=1 i=ii,ll: where Nrow is the number of rows of shower counter phototubes included, jt1k is the block in each row the track intersects (with the + or - of ± chosen depending on which adjacent block is closest to the particle track) , Cij is the phototube calibration constant, aij is the light attenuation coefficient, y:rk is the y position of the track in row i, Ytjbe is the vertical position of each phototube, and ADCij are the measured ADC values of each block. Nrow was 4 (PR, TAD, TAU, and TB) for E' < 5 GeV, and was 5 (PR, TAD, TAU, TB and TC) for higher momenta. A SHTRK spectrum, taken with deep inelastic data at E' = 1.08 GeV and a background rate of 1rje ~ 30, is shown in Figure 3.1(a). This spectrum only includes those events that fired the Cerenkov counter. The large electron peak at SHTRK~ 1 is clearly seen, with the low energy pion tail. Figure 3.1(b) is the same spectrum, but only includes those events that did not fire the Cerenkov. This shows the response of the shower counter to pion events. Figure 3.2 shows a typical spectrum cut on the Cerenkov under elastic running conditions (E' = 1.734 GeV) where the pion backgrounds are small. The line is a gaussian fit to the electron spectrum. The resolution of the shower counter was found to be ~ 18%/Vffi. This E' dependence is expected from the statistics of the electromagnetic shower ( # of photo-electrons oc # of particles in shower oc E'). -41- .-..... 6 0 X Ul E-< 4 ~ 0 u 2 SHTRK Figure 3.2- An elastic data spectrum in the shower counter, cut on a large Cerenkov pulse height. The solid line is a gaussian fit. 3.1.3 Cerenkov In Figure 3.3 a typical Cerenkov spectrum under the elastic scattering conditions is shown. The curve is a Poisson fit to the electron peak and indicates that 7.72±0.12 photoelectrons were typically produced in the phototube. This was consistent with the estimate of 7-9, which was predicted from the number of photons emitted by an electron [18) passing through 3.30 m of hydrogen gas ("' 46) and assuming a photo-conversion efficiency of 15-20%. - 42- 0 100 200 ADC CHANNEL II 300 400 Figure 3.3 - The Cerenkov pulse height spectrum for an elastic data sample. The curve is a Poisson fit to the spectrum indicating ~ 8 photoelectrons. 3.1.4 Electron Identification and Event Histogramming The definition of an electron event incorporated all of the above components and is stated here for completeness: • Electron Trigger. Only events that fired the EL-20 component of the trigger were included. • One-Track. It was required that one particle track be found in the wire chamhers. • Good Fiducial. A fiducial region was defined within the detector area and only those particles that passed through this region were included in the analysis. This - 43- cut was included in the definition of the acceptance function; it was therefore not necessary to include it in the efficiency correction. The fiducial region was defined by an area on the face of the PR from -43.67 to 43.7 em in x and -11.07 to 13.07 em in y, relative to the nominal axis in the hut. This corresponded to a 3.5 em inset from the edges of the shower array. • High Cerenkov. The Cerenkov pulse height was required to register above an ADC channel of 50. • High Shower Energy. The energy deposited in the shower counter was required to have SHTRK> 0.70. Electron events were then accumulated in a 3-dimensional histogram with coordinates (tlP/ Po,tlO,tl<jJ). The tlP/ Po dimension had 40 bins, ranging from -4% to 4%, with bin widths of 0.25%. The tlO dimension had 24 bins, ranging from -8 mr to 8 mr. The central 8 bins were 1 mr in width; the edge bins were 0.5 mr in width. The tl¢ dimension consisted of 14 bins, ranging from -40 mr to 40 mr. The central 4 bins were 10 mr in width; the edge bins were 4 mr in width. Edge bins were made smaller than the central bins to reduce errors caused by the acceptance function correction, which changed rapidly at the edges. 3.2 Non-event Analysis 3.2.1 Acceptance Physical apertures within the spectrometer blocked the passage of electrons that were not within the spectrometer acceptance. In a simple model, electrons that are emitted at a specific value of (tlP/ Po, tl(), tl¢, Xtgt, Yigt) will have either a 100% probability of reaching the detector hut or a 100% probability of being blocked by the - 44- spectrometer apertures and magnets. In this case the acceptance function is a fi vedimensional function that takes on a value of 1 in the area with perfect acceptance, and 0 in the area of perfect rejection. In practice, however, multiple scattering, the finite resolution of the wire chambers, and the assumption that Y'tgt=O smears out the edges of the acceptance function. In order to include a large amount of statistics in the data analysis it was necessary to correct the edges of the acceptance for the efficiency of electrons within these regions to reach the detector area. It was also necessary to correct the absolute value of the acceptance for any dependence on E', (), or the target length (i.e. Ztgt). Determination of Acceptance Function Corrections for the spectrometer acceptance were made with a histogram binned in ~PIPo, ~(),and~¢>, just as the event kinematics were stored. Each bin in the acceptance function, ACC3(~PIPo,~B,~¢>), contained the efficiency for detecting electrons with measured target characteristics ~PI Po , ~(), and~¢>. The acceptance function was generated from the deep inelastic data that was taken in parallel with this experiment and analyzed by S. Dasu [11]. Data from an Fe target (effectively a zero length target) at a variety of kinematics were measured and binned in histograms of ~pI Po, ~(), and ~¢>. A model of the distribution of events across the acceptance was generated for each run from the kinematic dependence of the cross section based on a modified fit to deep inelastic deuterium data [23], and included corrections for the Fermi motion of the nucleons, the non-zero value of R = <7LI<7T [6] the EMC effect [7] radiative corrections [11], and charge symmetric backgrounds [24]. A histogram of the "expected" number of counts in each bin (HIST3exp) was generated from this model and was normalized to the measured histogram (HIST3mea,) in the central region of the acceptance where the efficiency was - 45- expected to be effectively equal to 1. Both of these histograms were then summed over all runs. Runs in which the cross section had a strong kinematic dependence across the acceptance or large background contributions were excluded. A total of ,...... 10 6 deep inelastic scattering events were measured and included in this analysis. By comparing the two summed histograms, HIST3e:z:p and HIST3meas, the efficiency of each bin could be determined. The acceptance function was thus defined as: _ HIST3meas ( !:::..P /Po ,!:::..0 ,!:::..¢) ACC 3(!:::..Pj Po,!:::..O,!:::..¢) = HIST3exp(!:::..Pj Po,!:::..O ,!:::..¢) (3.3) The value of the acceptance function versus !:::..P/ Po, !:::..0, ?'1d !:::..¢ (summed over the other two variables in each case) is shown in Figure 3.4. i ! I ~ i 200 I !!. !!. t! t! ~ I I ~ 100 0 ....................~u..................._u..u " 8 -10 - D 0 M (mr) 5 10 -40 -20 0 20 t.rp (mr) Figure 3.4 - The acceptance function measured with deep inelastic data vs. t:J..PJP 0 , t:J..8, and t:J..<f>. Each histogram is summed over the entire range of the other two variables. 40 -46- From the acceptance function it was determined what region of the acceptance would be included in the analysis. Events from the far edges of the acceptance were excluded because of the large efficiency corrections that were necessary. Data was included only from the region -3 <f::lP/ Po< 3%, -6 <f::l0< 5 mr, and -24 <f::l</>< 24 mr. The one-dimensional projection of the acceptance function versus each of these three variables, with these cuts applied, is shown in Figure 3.5. It should be noted that most of the elastic scattering events occur at the elastic peak (f::lP/ Po~ 0), with the elastic tail extending below (f::lP /Po< 0). 1.0 0.11 0.11 0.11 i~ 0.4 i~ T ;I I I I I I I 0.8 - 0 .11 - 0.4 - II Tll: I I I I I I I - 1:; ~ 0 ~ 0.4 0.2 0.0 ...........J..J....L.......................................,_._, -10 -D 0 D 10 6.8 (mr) I I I o.a ,__ I I I I ,L ,IL 0.0 -40 -20 - ,I I I I I I I I I , II J 0 20 llt/J {mr) Figure 3.5 - The acceptance function with the cuts used in this analysis: -3 <AP/ Po< 3%, -6 <A8< 5 mr, -24 <AifJ< 24 mr. The dashed lines show the values of the cuts used for each of the variables. 40 -47- The acceptance function extracted from the deep inelastic data included dat a taken from a range of spectrometer momentum (1 < E' < 8 GeV /c) and scattering angles (11.5° < () < 48°). It was also taken exclusively from a near 0-length target. Systematic effects caused by the extended LH2 target or the changes in the spectrometer momentum and angle settings must also be included . Corrections to Solid Angle A Monte-Carlo simulation program of the 8-GeV spectrometer transport properties was written by D. H. Potterveld [25] and was modified for this analysis. In this simulation a uniform distribution of events in b..P /Po, /:)..(), and b..</> was produced at the target pivot position. Events were generated over a large enough area to completely cover the spectrometer acceptance. They were also generated uniformly along the target length, and were assumed to be distributed in a two-dimensional gaussian in the lateral beam direction, with O'z = O'y = 1 mm. These events were then transported through the spectrometer using forward transport coefficients generated with the SLAC program TRANSPORT [26]. The forward transport coefficients [9] from the target to the spectrometer hut are given in Table 3.2. Each event was checked to see if it collided with any of the apertures within the spectrometer. If it did intersect one of the apertures, it was presumed to be lost. If the event reached the spectrometer hut, it was linearly transported through the detectors, including mult iple scattering effects of the Cerenkov detector (windows, gas and mirror). The track trajectory within the wire chambers was then recorded. A random number generator was used to simulate the expected wire chamber resolution (based on the wire spacing) of O'z = 2.3 mm, O'y = 1 mm, O'dx = 2.8 mr, and O'dy = 1.25 mr. The measured trajectory characteristics were then transformed to the target characteristics using the reverse transport coefficients that were used in the event analysis. These events -48- were recorded in a three-dimensional histogram in !:iP/ Po, !:1(), and !:1¢>. This histogram was normalized to the total number of events initially generated at the target in each bin. b.P/ Po !:1() b.¢> Xsps Ysps X dx y dy 0.000 4.575 0.000 0.028 0.000 0.000 4.893 0.000 -0.189 0.000 -2.907 0.000 -0.014 0.000 -0.928 0.203 0.000 -1.090 0.000 0.000 Table 3.2 - Forward transport coefficients generated with the program TRANSPORT. These coefficients transform the target quantities to the position and angle of the traclc at the focal .planes (to first order). An acceptance function for a zero-length target at E' = 3 GeV was generated with the Monte-Carlo. These conditions roughly corresponded to the conditions under which the acceptance function was extracted from the data. This served as the normalization function for measuring systematic effects. All of the following acceptance functions are shown relative to this normalization function with the acceptance cuts, as discussed previously, applied. Surveys of the 8-GeV spectrometer indicated that the magnets and spectrometer did not move or rotate significantly when the spectrometer was rotated in 0, so no () dependence of the acceptance was anticipated for a zero-length target. However, such an effect might occur for an extended target . The Monte-Carlo simulation was used to generate the acceptance function for a 20 em target with the spectrometer set - 49 - 1.10 'i' u 1.05 0 <r " ~ 1.00 rz:l 0 0.95 (;' <I) .:!. "" ~ 0.90 -2 -4 0 &P/P 0 (%) 1.10 2u 1.05 ~ 1.00 e "" ...... <:;' rz:l 0 <I) • <r " ~ 0.95 0.90 -5 0 5 68 (mr) Figure 3.6 - The Monte-Carlo acceptance function for a 20 em target with a spectrometer angle of 45° relative to the nominal acceptance shown versus t:..P/Po and t:..8. at 45°. Events that are initiated from the target ends have a reduced efficiency for reaching the target hut. This effect can be seen in Figure 3.6 versus tlPI Po and f:l(). A slight reduction in efficiency was measured, but was independent of tlPI Po and f:l() within the acceptance cuts used. Deviations at the extreme values of f:l() can be seen, however. At this setting of the spectrometer angle the reduction in the overall acceptance was 0.32%. A similar analysis was done with other settings of the spectrometer angle. The reduction due to the effective target lengt h (measured along Xtgt, as seen from the spectrometer) was largest at the largest scattering angle. This is shown for different values of the spectrometer angle in Figure 3.7. A linear fit of this data was made -501.010 1.005 ::II u eu u 1.000 ~ Q;" u u < 0 .995 0.990 0 50 100 150 200 250 {Lsin{8)) 3 Figure 3. 7 - The reduction of the spectrometer acceptance due to target length effects, as determined from the Monte-Carlo, shown versus the 8 dependent variable [Lsin(8)]~, with L 19.972 em. The solid line is the linear fit described in the text, with a value of z = 1. 7 x 10- 11 em-~. = using the functional form Acccx: (1 - x[L sin(B)]2), where L = 19.972 em and x was a free parameter. Other functional forms for the (} dependence were tried, but this form yielded the best description of the data, and was the most reasonable form. The fit yielded a value of x = (1.7 ± 0.2) x 10-5 cm- 2 , with x2 /dof= 0.3. An analysis of deep inelastic data taken in parallel with this experiment with a 20 em deuterium cell (nearly identical to the LH2 target) measured [27] a value of x = (3.4 ± 7.5) x 10-5 cm- 2 , which is consistent with the Monte-Carlo value. Although the target length effect was found to be nearly independent of the t::.Pf Po and !::.(}cuts applied, it had a strong dependence on the t::.<P cut. This is shown -51- ~ ~ 6 ~ 6 ....... N I 8 0 "'I0 ....X 4 - ~ ~ 2 0 1 0 10 ~ ~ 20 30 40 50 li"'cu\ (mr) Figure 3.8- The acceptance parameter :z: (defined in the text) as a function of the t:&.l/Jc111 , measured by the Monte-Carlo. in Figure 3.8 where the parameter x, as defined above, is plotted versus the absolute value of the !:1</> cut applied ( -!:1</>cu.t <.5,!:1</><.5,!:1</>cu.t)· A sharp decrease between 24 mr and 36 mr can be seen. This was a motivating factor in the determination of the !:1</> cuts of ±24 mr. The acceptance could depend on the value of E' through two different effects. Multiple scattering at low E' could have potentially altered the acceptance function. The Monte-Carlo was used to simulate the acceptance function at E' = 1 GeV / c and E' = 8 GeV j c, and no significant changes were discovered. The transport coefficients could also have changed with E' as the field strengths in the spectrometer magnets changed, causing saturation or changes in the fringe fields. This was measured with -521.010 1.005 1.000 a c 8 '0 0.995 ......... c '0 0.990 0.985 0.980 0 2 4 E' (GeV/c) 6 a Figure 3.9- The solid angle, dO, vs. E' from the wire order data relative to the value assumed in the event-by-event analysis. the wire orbit technique [16,17]. The value of the solid angle d!l , defined as proportional to the product of the transport coefficients (tl.Otgt I Xwc)(tl.<l>tgt I dywc) , is shown in Figure 3.9 as a function of E' relative to the value of the transport coefficients used in the event analysis. It was found that the absolute normalization of the solid angle changed by "' 0. 7%, and varied with E' by "' 0.5%. 3.2.2 Efficiency and Dead-Time The efficiency for detecting electrons in each of the detector elements must be measured. Corrections were also necessary for the t ime the electronics and computer were "dead" and not capable of recording electron events. These corrections are discussed here. -53- Efficiency A Poisson fit to the Cerenkov spectrum of electrons (see Cerenkov section) indicated the number of photoelectrons was"" 8 with a peak position at ADC channel160. Below the ADC cut of channel 50, the area of the Poisson curve was 0.25%, indicating an efficiency of 99.75%. The Cerenkov efficiency was monitored on a run-by-run basis by analyzing events that deposited a large amount of energy in the shower counter and did not require the Cerenkov to fire the trigger. These runs indicated an efficiency of ~ 99. 7%. An analysis of pion events (low SHTRK) in a deep inelastic run indicated the pion rej~ction for this cut was "" 10 4 : 1. The wire chamber efficiency was determined by measuring the efficiency of each individual chamber and then computing a total efficiency factor. Individual _chamber efficiencies were calculated versus X hut to include the localized effects of the P-chamber support wires. On an event-by-event basis, it was determined if a given chamber was necessary for finding the track. H the chamber hit was necessary, the event was not included in the efficiency calculation. For those events that did not require that chamber, the percentage of events that had an associated chamber hit (±4 wires from the track intersection) was calculated. These efficiencies were stored in a histogram binned in Xhut with a bin width of 1 em. Individual efficiencies for a P-chamber and T-chamber are shown in Figure 3.10. The average efficiency of each chamber was "" 90-95%. The total wire chamber efficiency computed was 99.9% (Figure 3.10( c)), including the effect of the P-chamber support wires of "" 0.06%. The efficiency of the wire chambers was also measured on a run-by-run basis for electrons that pass near the center of the detectors (i.e. those events that had large pulse heights in the Cerenkov and the middle counters of SF, SM, PR, and TA). This eliminated events near the edges of the detectors. The ratio of 0-track/1-track events -54- ~ ~~Li!·: :~· ·:-~+: -: ~ ·~a -40 - 20 0 20 40 "~out (em) (a) s ] ~! -40 : : : ! ::· -20 :: 0 ; ~ :i~t ] : ' 20 40 "~uL (em) (b) ~H ~ >: :::: 0 S -20 : : 0 =: : : : : } :~ 0; 0 : ] 20 """'(em) (c) Figure 3 .10- The measured efficiencies of (a) a P-chamber, (b) aT-chamber, and (c) the entire wire chamber tracking system, shown vs. Zbt· under these conditions was a measure of the wire chamber efficiency and indicated an efficiency of 99.9%, in agreement with the previous analysis. The SHTRK spectrum, shown in the event section, was cut at a normalized value of 0. 7 with the electron peak at a value of 1.0. Assuming a gaussian distribution of events with a FWHM resolution of 18%/v'Jfi (as measured), the efficiency was 2: 99.94% forE' 2: 1 GeV. A run-by-run estimate of the SHTRK efficiency was also performed by cutting events on 1-track, Cerenkov, and a requirement that the track be in the center of the detector area (I~P/Pol< 3.5%, 1~01 < 6 mr, and 1~¢>1< 28 mr). It was assumed that all of the counts with 0.5 < SHTRK < 0.7 were electrons - 55- that failed the SHTRK cut. This gave a lower bound on the efficiency of 99.8%, in agreement with the gaussian analysis. The trigger efficiency was calculated from the efficiencies of each of the individual components, and was ~ 99.9988%. Thus any inefficiency due to the trigger was negligible. Dead-Time Dead-time due to the electronics was determined by comparing the scaler rates of a coincidence of PR, TAD, and C (PTC) with 20, 40, 60 and 80 ns gate widths. Scalers EL-20, 40, 60 and 80 were not used due to problems in the electronics that caused EL-20 and EL-40 to double pulse. A linear extrapolation of these scaler rates to 0 ns yielded the ideal scaler rate with no dead-time, as shown in Figure 3.11. The electronics dead-time ranged from between 0.00 to 0. 75%, and was consistent with expectations based on the measured electron rate and a 20 ns pulse width. The fact that the computer was limited to recording only one event per pulse also created a dead-time effect. After each computer trigger occurred, a veto pulse was created which lasted for the rest of the beam pulse. The rates of PTC-20 and PTC-40 were counted along with the rates of PTC-20 and PTC-40 in coincidence with no veto pulse (PTC-20V and PTC-40V). The ratio of the rates PTC-20/PTC-20V and PTC-40/PTC-40V measured the computer dead-time. These two measurements always agreed to 0.2% and were consistent with the measured electron rate per beam pulse. They were averaged to compute the total computer dead-time. -56- 35850 ';::? - 35800 u E-< ~ 35750 20 40 T0 (ns) 60 80 100 Figure 3.11 - A typical plot of PTC with 20, 40, 60 and 80 ns dead-times shown vs. the dead-time. 3.2.3 Target Density As the beam passed through the hydrogen in the LH2 target it deposited energy, resulting in heating that reduced the target density. The average density was determined by measuring the temperature and pressure of the circulating hydrogen as it entered and exited the target. This measurement is shown without any beam heating effects (measured while the dummy target was in the beamline) in Figure 3.12(a). At one point during the experiment, the pressure of the recirculating hydrogen was increased, which increased the density by 1.2%. The average density measured while the LH2 target was in the beam is shown in Figure 3.12(b). The shift of the density -57 I en I I I 10.0 I- - - - 7.5 rz. ~ rz. ~ ~ ~ ~ 0 r&l ~ I I I ILl 10.0 r- - 7.5 I- - 5.0 - - 2.5 - - 0 5.0 I- - 2.5 I- - 0.0 I 1.38 rl 1.3& P (g/cm') (a) I ,[ I 1.4 1.41 I ~ 0.0 l 1.38 ~ 1.39 I I 1.4 1.41 p (g/cm1 ) (b) Figure 3.12 - The average hydrogen density. The hydrogen flow pressure was altered during the experiment, shifting the average density from the lower peak to the higher peak. (a) The measured hydrogen density when data was taken with the dummy target. (b) The measure hydrogen density when the LH2 target was in the beamline. to lower values is shown, although the average density was never altered by more than 0.7%. The beam could also cause local density fluctuations by depositing enough energy along the beam axis to cause the liquid hydrogen to boil, creating small bubbles. As these bubbles expand during the 1.6 J.I.S pulse, the hydrogen density for electrons that arrive later in the beam pulse is lowered. Such local effects were not well measured by the average hydrogen density. Comparisons of the measured hydrogen cross sections at different beam currents and repetition rates, however, were sensitive to these effects. There was only one kinematic setting at which cross sections were measured with -58- significantly different (and large) beam currents. This data sample was limited due to the difficulty encountered in achieving high beam currents during most of the experiment. No significant limit on local beam heating effects could be determined from this data. A much better, and more realistic, limit could be determined by including the entire hydrogen elastic data set in the analysis of beam heating effects. Due to the low beam currents that were used during most of the experimental run, the average peak current , (I), and € were not well correlated. Local beam heating effects could be determined by dividing the measured cross sections by a model of the kinematic dependence and performing a linear fit to the peak current of the form a - b(I), where a was a normalization parameter and b indicated any beam heating effects. By using a variety of models the sensitivity of the beam heating parameter b to the choice of the elastic form factors used could be determined. Conversely, this method also demonstrated the sensitivity of the extracted form factors to local beam heating effects. Three different models for the form factors were used in this method. The first used the values of the form factors extracted from this experiment, assuming no beam heating effects, which indicated that Gt;;/Gn > 1. Values of a= 1.000 ± 0.003 and b = (0.7 ± 2.2) x 10- 4 (mA)- 1 were extracted, with x2 /dof-0.9. By fixing a = 1 a value of b = (0.5 ± 1.4) x 10-4 (rnA)- 1 was extracted with x 2 /dof=0.9. Models using the fit to the form factors of Hohler (28] (Gl£/Gn < 1) and the dipole approximation (Gt;;/Gn = 1) were also used (see Theory section). With these fits the value of the normalization parameter a differed from 1 by up to a few percent, as expected. But the measured values of the boiling parameter were b = (0 .6 ± 2 .2) x 10- 4 (mA)- 1 (for Hohler) and b = (0.7 ± 2.2) x 10- 4 (mA)- 1 (for the dipole); the x2 /do£ were 1.3 -59- and 1.1, respectively. The small value of band the increased value of x2 fdof for the different fits indicated that (I} and £ were not well correlated. and the limits on the boiling effect were not sensitive to the values of the form factors used in this method. The maximum boiling effect was limited to 0. 7% at 37 rnA, with a maximum effect of 0.3% at a normal operating current of 15 rnA. Similar results were obtained for an analysis of the beam heating effects correlated with the average current rather than the peak current. 3.2.4 Incident Charge The total charge incident on the target during each run was measured usmg the two independent toroidal charge monitors. Six different readouts of the toroids (three for each monitor) were accumulated on a pulse-by-pulse basis and periodically stored on magnetic tape. This gave a rough measurement of the total number of pet aelectrons (PE= 10 15 e-) that were incident on the target. Corrections for drifts in the gains and the calibrators were necessary to achieve a more accurate measurement. Analysis of the toroid calibration runs was necessary to determine these corrections. Data from the toroid calibration runs were periodically accumulated. These runs measured time-dependent drifts in the gain of the toroids, as well as the gain relative to a nominal setting (= 1). They also measured the non-linearity of the toroid readout circuit with respect to the pulse charge passing through the toroids, and provided redundancy for self-calibration of the calibrator elements themselves (the DAC and attenuator). Cross calibrations were periodically performed by switching the calibrators between their two respective toroids. A systematic study of the calibrations was performed m the offline analysis. Errors in the software for the two newer readout systems made this data unreliable; it was therefore decided to only use the older readouts in the data analysis. These - 60- readouts have a long history of reliability. The analysis indicated that the toroic ~m amplifiers were generally stable over time at the 1% level. However, a few occasional, sudden shifts in the gains of the toroids did occur. The largest such shift, - 15%, occurred near the middle of the experiment when the beam was momentarily badly mis-steered. It was discovered that an electronic chip in both readout systems had burned out. This chip allowed for the polarity of the signal pulse to be switched and was redundant with other elements of the system. It was decided to simply replace this chip with a set of copper wires, causing an increase in the gain of the toroids. Intensive calibration data were taken at this period of time, and the shift in the gain was well measured. 125 100 75 ....0 ... 50 25 0 ~~-L~--~~~~--~~-L~~~u_~~--~~~~__J -2 -1 0 1 TOR2-TOR1 (,;) Figure 3 .13- Comparison of the two independent toroid readouts (TOR2 and TOR1). The average of the two readouts was used in the analysis. The readouts agreed to -0.08 ± 0.21% on average. 2 -51- Corrections for time dependent drifts were made by linearly interpolating the calibration corrections taken immediately before and after each run. The time between these calibration runs was "" 8 hours, and the gains were typically stable over such time periods to "" 0.2%. Non-linearities in the gain amplifiers were determined by periodically measuring the gains with many different calibration charges, and then interpolating for the average beam pulse charge used during the run. This correction was "" 0.1 %. The calibration attenuators were measured to deviate from their nominal values by "" 0.1% and were linear (:S: 0.1% level). The DAC's were calibrated before and after the experiment with a DVM and were shown to be highly linear. The average of the corrected values of the two independent toroids was used as the measure of the total incident charge, and the readouts agreed to typically -0.08 ± 0.21% (see Figure 3.13). -62- Chapter 4 Data Ana lysis- II In this section the process of converting the event information into the the cross sections will be described. Corrections were made for the acceptance, target density, detector efficiencies, and dead-time effects. Contributions to the scattering from the aluminum within the target were subtracted. Effects of higher order (in Oem) radiative processes were also included. The form factors were then extracted from the cross sections using a Rosenbluth separation technique. Systematic uncertainties will be discussed at the end of the chapter. 4.1 Histogram Analysis 4.1.1 ~f/J Sum The histogram generated by the Pass-1 analysis was binned in ~PI Po, ~() , and ~f/J. Due to the slight kinematic dependence of the cross section on ~f/J, this was the first variable that was integrated out. ~f/J for ~¢Jlo S.~f/JS.~¢Jhi, with ~¢Jhi The histogram was summed over = -~¢Jlo = 24 mr. The (~PI Po,~O,~f/J) event histogram was thus converted into a (~PI Po,~O) histogram. The acceptance function was similarly converted from 3-dimensions into 2-dimensions. The physical scattering angle, O$catt, was a combination of the in-plane angle, Oo+~O, and the out-of-plane, angle ~f/J. They were related by: cos(Oscatt) = cos(Oo + ~()) cos(~f/J) ( 4.1) -63- To lowest order () scatt depends only on 6.¢J 2 . Thus for t:l.tjJ ~ 1 know ledge of the average value of 6.¢J 2 is sufficient to correct for this dependence. An array in 6.() of the average value of 6.¢J2 was stored so this correction could be performed later in the analysis. This histogram was defined as: (4.2) The typical value of PHI2 was I"V (14 mr) 2 , which was in agreement with a MonteCarlo simulation of the spectrometer acceptance. The correction to the cross section due to PHI2 was usually small (I"V 0.5%), and was performed in this manner because it was desirable to place the acceptance cuts along constant values of 6.() and t:l.tjJ, rather than along lines of constant ()scatt· 4 .1. 2 t:l.P I Po Integral The ( t:l.P I Po ,flO) histogram was summed over tlP I Po to obtain the elastic cross section as a function of 6.(). It was necessary to integrate over the elastic tail caused by higher order radiative effects. There were, however, background processes at low values of t:l.P I Po which must be excluded. It was also desirable to limit the tlP I Po integration to the region where the acceptance corrections were small, and eliminate those regions where events were contributed only from scattering from the aluminum end caps. The lower bound of the integration, Plo, was set to a minimum tlP I Po value of -3% to avoid the low efficiency acceptance edges. The threshold for 1r-production occurred at a value of missing mass squared W 2 = (Mp + M1r) 2 ~ 1.152 (GeVIc) 2 • Thus Plo was also limited to be above a value which corresponded to W 2 = 1.12 ( GeVIc? . This allowed for a clear separation of the elastic scattering events from inelastic events, including resolution effects. The high flPI Po cut, Phi, was chosen to -64- correspond to a constant vV 2 far enough above the elastic peak, W 2 = M~ ::::::: 0.88 ( GeVIc ) 2, to include the entire elastic peak width caused by resolution effects. The values of Plo and Phi were always placed at the edges of the histogram bins to avoid making approximations that are necessary when placing cuts within bins, and they were calculated for each value of fl.(). Corrections were then made for the acceptance efficiency of each bin. The number of events, and the associated uncertainty, used for calculating the elastic cross section for each fl.() bin was defined by: Ner ( Phi(t:..B) fl.()) = """' L...J HIST2( !:l.PI Poi ,!:l.O) t:..Pf Po;=Plo(t:..B) ACC2(!:l.PI Poi,!:l.O) Phi ( 6.0) [fl.Ner(!:l..0)]2 = L HIST2(!:l.PIPoi,fl.()) 2 t:..P/ Po;=Pio(t:..B) [ACC2(!:l..PI Poi,!:l.())] ( 4.3) 4.1.3 fl.() Average Corrections were made to the elastic scattering cross section in each fl.() bin for the kinematic dependence of the cross section relative to the center of the acceptance (!:l.O=O). This made the expectation value of the cross section independent of fl.(), so the values could then be averaged across the acceptance based on their statistical weight resulting in the cross section value at !:l.O=O. The raw cross section (including an overall constant, CRo, that includes dead-time, target thickness, etc.) was: dO' - dO. raw = CRoO'o ""' tr L...Ji t:.,;2 O'Q = ---:-'-' 'Li t:..~~• (4.4) where: O'j= 1 (!:l.¢>hi - !:l.(ho) Ner(!:l..Oi) O';jodel(Oo) 8()j (j;rdel (oi) 1 fl.N el ( fl.()i) O''Jodel (Oo) !:l.O'i = ~-------:- _ _,;,_~ --li:l~....,..:...~ ( fl.t/>hi - !:l.tf>lo) 8()i O''Jodel (()i) ()i = cos- 1 [cos(Oo + !:l.Oi)cos( .jPHI2(!:l..Oi))1 (4.5) - 65- where (j';iodel is the e-p elastic scattering cross section using a dipole approximation for the form factors including the effect of the Plo cut of the radiative tail, and 8()i is the width of the f:l.() bin in mr. Corrections for the value of(¢}) were included in the definition of Bi. In order to make the analysis of the dummy target. runs consistent with the way the AI endcap contributions were handled in the LH2 target runs, the dummy runs were averaged using the same model cross section as the LH2 runs, and the statistical weighting factor, t:l.Nel• was estimated using the dipole cross section for hydrogen and the acceptance function evaluated at x = 1. 4.2 Calculation of Raw Cross Sections The target thickness, incident charge, dead-time corrections, and efficiencies, were contained within the constant CRo in the previous section, defined by: CRo = 1 DT 0.60225 (Beam· EJJ · tLH2 · Acccor) (4.6) where DT is the electronics and computer dead-time, Beam is the total incident charge (in units of PE= 10 15 electrons), E f f is the efficiency of the electron cuts and trigger, iLH2 is the thickness of the hydrogen in the LH2 target (g/cm2 ), and Acccor includes the corrections for the reduction of the acceptance at large spectrometer angles due to the target length and the momentum dependence of the spectrometer optics described in previous sections. The constant in the denominator was calculated from Avogadro's number and the relevent units such that the final cross sections were measured in nb/sr. In order to treat the measurement of the dummy target runs consistent with the treatment of the aluminum endcap backgrounds of the hydrogen runs, iLH2 was set equal to 19.972 ern x0.0.0707 gfcrn3 for the dummy target runs. -66- 4.3 Subtraction of Aluminum Background After the computation of the elastic cross section for each run, the contribution to the scattering rate from the AI endcaps and flow guides was subtracted. It was necessary to normalize the subtraction to the scattering rate above the elastic peak region of the LH2 runs in order to determine the amount of material the deformed flow guides contributed to the target thickness. After the determination of this normalization, the measured contribution of the aluminum scattering from the d ummy target in the elastic region was subtracted from the elastic cross section of the LH2 target. The effect of this material was also included in the calculation of the radiative corrections to an accuracy of ~ 0.2%. Typical corrections to the cross sections for the aluminum background (under the conditions of the reversed hydrogen flow) was 3.0 ± 0.3% ; the typical effect of the deformed flow guides on the radiative corrections was 2.0 ± 0.2%, and was negatively correlated with the background subtraction, resuiting in a smaller combined uncertainty in the total correction for the aluminum background. Details of this subtraction are presented in Appendix C . 4.4 Radiative Corrections Corrections for higher order processes in aem, which affect the scattering amplitude beyond the single photon exchange that was assumed in the introduction, were also included. Bremsstrahlung, vacuum polarization, vertex, ionization, and two photon exchange effects were included as corrections to the principle scattering vertex itself, as were radiative processes within the rest of the target material. The procedure for calculating the radiative corrections is outlined in Appendix A. A single number, RCOR, related the ideal one-photon exchange cross section to the higher order cross section that was measured: dO' -= dO 1 dO' xRCOR dOraw (4.7) -67- This correction depended on the scattering kinematics, target geometry, and the f}.P j Po-cut applied to the elastic tail. The kinematics corresponded to the values (Eo, Bo) at the center of the spectrometer acceptance. A list of the radiative corrections is shown in Appendix D. 4.5 Combining Runs of Similar Kinematics In order to extract the form factors from the cross section measurements using the Rosenbluth separation technique, it was necessary to have cross section measurements at the same value of Q2 but different values of e. Actual data, however, is frequently taken at similar, but slightly different, values of Q2 or e due to slight inaccuracies in the setting of the beam energy or spectrometer angle during the data taking. The distribution of the Q2 and e values is shown in Figure 4.1, represented by the crosses. Small corrections were applied to each cross section to account for the difference between the actual kinematics, ( Qr, ei), and the "average" kinematics ( Q5, eo) of each of the kinematic data subsets, represented by the circles in the preceding figure. These corrections were made using a model cross section assuming a dipole form of the form factors: (4.8) The size of these corrections was usually < 1%. Different runs at each (Qfi , eo) were then averaged together, weighted by their statistical uncertainty, to arrive at a single measured cross section at each kinematic point . The average x 2 jdof of these averages was 1.03. Values of the cross sections at each kinematic setting are given in Appendix D. -683.5 Cll a 3.0 Q) 2.5 s EB ~ !) ~ + e 2.0 !) d) !JEBI!IB 1.5 1.0 0.5 0.6 0.7 0.8 0.9 1 £ Figure 4.1- The values of Q 2 and£ for each of the data runs are represented by the crosses. The circles represent the nominal values that were used in the form factor extraction. 4.6 Dependence of Cross Sections on Kinematic Cuts If all the corrections have been properly applied, the measured cross sections should be independent of the D.P/ Po, 6.(} and 6.¢> cuts used in defining the acceptance region. Any dependence of the cross sections on the kinematic cuts is therefore a good indicator of any errors in the acceptance function and target length corrections, the radiative corrections, or the aluminum background subtraction. The cross section as a function of 6.¢>, averaged over all runs and normalized to the central region (-10 :::; 6.¢> :::; 10 mr), is shown in Figure 4.2. The dashed line is the Monte-Carlo prediction of the 6.¢> dependence due to target length effects. Variations in the cross sections are well accounted for by the Monte-Carlo in the -69- region of the acceptance that was used in the cross section analysis (- 24 ::=; 6.¢> ::=; 24 mr). A similar analysis separated in small, medium, and large() runs indicated that the Monte-Carlo simulation described the 6.¢> cross section variation in the region of interest at all spectrometer angle settings. I I I ,. m- - ..m- - -Ill- - - -m - - -ar- - -~ 1.00 \ / \ I \ I ~ 0.75 0 \ I ' I I $ <l 0 ~ ........ $ 0.50 <l 0 ~ 0.25 o.o~4~0~~-L~--2~0~~~~~0L-L-L-L-L-2~0~~~~~40 6~ Cmrl Figure 4.2 - The measured cross section as a function of L!!l..l/>, averaged over all runs and normalized for small L!!l..l/> values. The dashed line is the prediction for the L!!l..l/> dependence of the cross section from a Monte-Carlo calculation of the acceptance including target length effects. The cross section, as a function of 6.(), is shown in Figure 4.3. Figure 4.3(a) is averaged over all runs with small scattering angles (11.5° ::=; () ::=; 20°), while Figure 4.3(b) is large scattering angle data (40° ::=; () ::=; 50°). Both were normalized in the region -6 ::=;6.0 ::=; 5 mr. Dips can be seen in the large angle runs near 6.0=-8 -70- and 6.0=6.5 mr. The Monte-Carlo simulation predicted such an effect in these regions, again due to the target length effects at large angles, although it significantly underestimated their magnitude. Data taken at medium scattering angles also displayed such structures, although smaller in magnitude. These dips occurred in the far edges of the acceptance region where the acceptance efficiency was ;S; 50%, and the acceptance corrections were large and rapidly changing. In the region of the final acceptance cuts ( -6 '5,.6.0 '5:. 5 mr) the cross sections appear to have no systematic 6.0 dependence, as predicted by the Monte-Carlo. This indicated a good reliabil- ity in both the acceptance function and the cross section model used to correct the kinematic 6.0 dependence of the cross section. 1.2 1.0 0 b ~ 0.8 .!!.. b 0.8 -5 0 5 M (mr) (a) 1.2 m m 1.0 Ill Ill 0 ~ 0.8 m ';" m 0.8 -15 0 t.8 (m.r) (b) Figure 4.3- The cross sections as a function of A.fJ. (a) At small scattering angles. (b) At large scattering angles. The cross sections are independent of A.(J within the final acceptance cuts. - 711.010 1.005 I I - 0 0.. ........ -e; 1.000 b 0.995 1-- .... . .. .. .. b ........ 0.. ........ 0 .990 I ~ e; I .. ll' ! ~ ( . . .. .. .... . . .. . p' ... ... ' - I I I I -4 -3 -2 -1 0 ~P/Pout Figure 4.4 - The ~pI P 0 -cut dependence for the measurement of a typical cross section, as discussed in the text. The data is normalized to the cross section value corresponding to a ~pI Pocut of -3.0%. The error bars include only the statistical error relative to this normalization point. The dotted lines are an estimate of systematic uncertainties from other sources. A typical cross section as a function of the !:::..P/ Po-cut, Plo, is shown in Figure 4.4, normalized to the cross section value measured with Plo = -3.0%. Only statistical error bars are shown relative to the normalization point; correlations exist between the values of the other points. The cross section is independent of the elastic tail cutoff within statistical fluctuations. Changes in the raw cross section (before radiative corrections) were "' 20% between the lowest and highest cuts shown. The average x2 /dof between the cross sections evaluated with the lowest !:::..P/ Po-cut relative to those with the highest !:::..P/ Po-cut was 26.4/22 for all the data runs. If systematic uncertainties due to the kinematic calibration of the elastic peak are included (dotted - 72- line) the average x2 /dof was reduced to 18.3/22. This indicated a high reliability to the D.P /Po- dependence of the radiative corrections and the acceptance function , as well as proper subtraction of the aluminum background. 2 2 Q = 1.00 Q =2.00 0 .029 0.100 0 .028 0 .095 ~ 0.027 ; 0.090 0.026 b b 0 .085 0 .025 0.080 0 .6 0 .7 0.8 0 .024 0.6 0.9 0.7 E 0.8 0 .9 E Q2 =3.01 Q2 =2. 50 0.0180 0 .0115 0.0175 0.0110 0.0 170 I b 0.0165 b 0.0160 I 0 .0105 0.0100 0.0155 0.0150 0.6 0 .7 0.6 0.0095 0.6 0.9 0.7 0.8 0.9 E E Figure 4.5 - The linear Rosenbluth fits used for extracting the form factors. 4.1 Elastic Form Factor Extraction The relationship between the cross sections and form factors is: d(j E' ( G~ = (jmottdO Eo 1 - + rG~ + 2rGM 2 2 ) tan (0/2) +r ( 4.9) This can be rewritten as: (jnorm =c:(1 Eo 1 d(j + r) E' -O'mott -dO =rG~ +c:G~ ( 4.10) (4.11) -731 where E was defined previously as E = (1+2(1+T) tan2 (0/2))- and is the longitudinal polarization of the virtual photon, and <7norm is a function of Q2 and c; only. A linear fit of <7norm to E at fixed Q2 has TGi£ as the intercept and G~ as the slope. Graphs of these fits to the data are shown in Figure 4.5. The average x2 /dof was 0.62. 1 1.25 1.5 1.75 2 2.25 (Gl/Gn)2 Figure 4.6 - The x 2 contour plot of G], vs. GJ for Q 2 = 3.0. The point is located where x 2 is a minimum, the ellipse is the contour for x 2 = x!u. + 1, and the assigned uncertainties in G], and GJ are shown by the error bars. In the determination of the uncertainties .6-Gi£ and .6-G~, the correlations between Gl£ and G~ involved in the fit have been included. This is shown in the x2 contour plot in Figure 4.6, for the Q2 = 3 fit. The error bars are also shown and clearly include the effect of correlations. In table 4.1, the values of TGi£ and G~ and their statistical uncertainties are given along with the x2 / dof. -74- Q2 (GeV/c) 2 G2 E rG2M x 2 /dof 1.000 3.094E-02 ± 4.292E-03 6.655E-02 ± 3.522E-03 0.50 2.003 6.677E-03 ± 8.427E-04 2.105E-02 ± 6.905E-04 0.67 2.497 3.002E-03 ± 5.398E-04 1.389E-02 ± 4.357E-04 1.01 3.007 2.094E-03 ± 4.598E-04 8.942E-03 ± 3.596E-04 0.19 Table 4.1 -The extracted values of the form factors with statistical errors. 4.8 Systematic Uncertainties Systematic uncertainties in the scattering kinematics and the operation and understanding of various components of the experimental equipment must be included with the statistical uncertainties of the form factors. The systematic uncertainties can be separated into point-to-point uncertainties, which can change independently from run-to-run or at different kinematics (such as fluctuations in Eo), and absolute uncertainties, which are the same for each data run (such as the absolute value of the acceptance) . A presentation of the estimates of the systematic uncertainties and their effects on the form factors is given here. 4.8.1 Point-to-Point Uncertainties Effects caused by the various point-to-point uncertainties on the values of the form factors were calculated using a Monte-Carlo program. In this program, the parameter in question was adjusted randomly on a run-by-run basis according to a gaussian distribution with a mean of zero and a standard deviation given by the uncertainty in that particular parameter. Changes in the cross section values were then calculated and new values of the form factors were extracted. This process was repeated "' 1000 times, and the mean difference (6ft) and standard deviation ( o") of - 75- the form factor distribution were determined (in all cases I:::..J.l ~ o"). The value of~ was used as the estimate of the systematic uncertainty of the form factors. Individual contributions to the total systematic uncertainty were summed in quadrature to estimate the total uncertainty. Fluctuations in Eo were typically ±0.05%, and were measured with the elastic peak analysis to an accuracy of ±0.03%. The scattering angle was determined to an accuracy of 0.003°, based on the accuracy of the wire array and beam cavity monitors in the beam steering system, and the survey of the spectrometer and wire chambers. The effect of the uncertainties in the incident energy and scattering angle were estimated using the dipole approximation to the form factors. Comparison between the two identical toroid monitors indicated an uncertainty in the incident charge measurement of 0.2%. Uncertainty of the target density was dominated at large beam currents by local beam heating effects, and was estimated at 0.5% at the maximum current of 40 rnA. An average uncertainty of 0.2% was assigned to each cross section to account for target density effects. A systematic uncertainty was applied to each cross section equal to half the size of the angle dependent correction for target length effects ( ±0.35 sin2 ( 0) %) as a conservative estimate. An additional uncertainty was included for the E' dependence of the acceptance that was measured by the wire orbit of 0.1 %. No e dependent uncertainty of the radiative corrections was included due to difficulty in obtaining a proper estimate of its size. The dominant uncertainties in the radiative corrections were due to ignoring the two-photon exchange terms [29], which have been limited in absolute magnitude (30,31] to be less than < 1%, and the approximations of higher order effects by exponentiating bint [32], which is again estimated to be < 1% in absolute magnitude. The table of radiative corrections are provided in Appendix D to allow for improved analysis as the theory of radiative corrections is improved. -76- Quantity Uncertainty Incident Energy (rel) Scattering Angle Incident Charge Target Density Endcap Subtraction Acceptance (fJ) Acceptance ( E') ±0.03% 0.003° 0.20% 0.20% 0.20% <0.20% 0.10% Pnt-to-Pnt Systematic Statistical Incident Energy Scattering Angle Rad. Cor. Acceptance Absolute Systematic 0.07% 0.003° 1.00% 1.00% ~a/a ~Gj;/GD ~Gj;/G D (typical) Q2=1 Q2=3 ±0.34% 0.10% 0.20% 0.20% 0.20% 0.10% 0.10% ±2.2% 0.7% 1.8% 1.8% 1.8% 0.9% 0.9% ±4.4% 1.3% 2.6% 2.6% 2.6% 1.3% 1.3% 0.50% 0.80% 4.1% 7.1% 6.7% 13.8% 0.67% 0.10% 1.00% 1.00% 0.3% 0.5% 0.5% 0.5% 0.8% 1.2% 0.6% 0.6% 1.57% 0.9% 1.7% Table 4.2 - The individual systematic uncertainties and their average effect on cross sections ( tr) and ~at Q, = 1 and Q, = 3. 4.8.2 Absolute Uncertainties Absolute systematic uncertainties have a small impact on the uncertainties of Gj;, Ff, and any of the ratios of form factors since they have very little c dependence and thus have relatively small effects compared to statistical or point-to-point systematic fluctuations. However, the absolute values of cross sections and the determination of G~and Ff are sensitive to the absolute calibrations since these have relatively small uncertainties from other effects. These absolute uncertainties are discussed here, and are surrunarized in Table 4.2 . A systematic uncertainty equal to the size of the average incident energy correction based on the elastic peak analysis, 0.07%, was included for the absolute value of E 0 . Absolute determination of the scattering angle was limited to ±0.005° based -77- on the survey of the spectrometer and beamline, the precision of the beam steering system, and the measurement of the spectrometer central ray by the wire orbit. An absolute uncertainty of 0.5% was assigned to the measurement of the integrated incident beam current based on the uncertainty of the charge on the calibration capacitor and previous cross calibrations [33) between the toroids and a Faraday cup that indicated an agreement of better than 0.5%. The acceptance was measured [16 ,17) to ±1.0% by the wire orbit and the deep inelastic data [ll). Absolute radiative correction uncertainties were estimated [32) at 1%. Errors caused by using an elastic scattering cross section model based on the dipole approximation in various steps of the analysis were estimated by redoing the entire analysis, but using a form factor model based on the Hohler fit [28). The form factor values changed by~ 0.16% in G~ and "' 0.03% in G~ when this was done, indicating that the extracted values were very insensitive to the form of this model. 4.9 Final Values of the Form Factors The values of G~/Gn and G~/Gn/f.Lp are plotted in Figure 4.7 and 4.8, respectively, with f.Lp = 2.79 and Gn = 1/(1 + Q2 /0.71) 2 • Statistical uncertainties are represented by the inner error bar; the outer error bar is statistical and systematic added in quadrature. Values of the form factors are presented, including both statistical and systematic errors, in Table 4.3. Other combinations of the form factors that are commonly used, such as Ff, Ff, (p,pG~/G~) 2 , Rei = ~(G~/G~) 2 , and Q2 (Ff / Fi) are also given. -78- 2.0 1.5 1.0 0 .5 0.0 1 0 3 4 Figure 4. 7 - The extracted values of ~/Go vs. Q 2 • Q2 (GeV/c) 2 1.000 2.003 2.497 3.007 G~JGD 1.020E+O ± 7.1E-2 ± 4.0E-2 ± 9.1E-3 1.193E+O ± 7.5E-2 ± 4.0E-2 ± l.SE-2 1.118E+O ± l.OE-1 ± S.SE-2 ± 1.7E-2 1.254E+O ± 1.4£-1 ± 6.7E-2 ± 1.7E-2 G~JGD/J'p 1.006E+O ± 2.7E-2 ± l.SE-2 ± 7.3E-3 1.007E+O ± 1.7E-2 ± 8.8E-3 ± 7.4E-3 1.024E+O ± 1.6E-2 ± 9.3E-3 ± 7.5E-3 l.OOSE+O ± 2.0E-2 ± 9.8E-3 ± 7.4E-3 pP 2.441E-1 ± 6.7E-3 ± 3.8E-3 ± 1.9E-3 1.341E-1 ± l.lE-2 ± 6.0E-3 ± 1.2E-3 5.494E-1 ± 5.9E-2 ± 3.3E-2 ± 3.5E-3 4.650E-l ± 8.9E-2 ± S.OE-2 ± 5.3E-3 1.028E+O ± 2.0E-l ± l.IE-1 ± 1.2E-2 1.218E-1 ± 2.2E-3 ± 1.2E-3 ± l.OE-3 3.942E-2 ± 3.0E-3 ± 1.6E-3 ± 4.6E-4 6.479E-I ± 6 .0E-2 ± 3.2E-2 ± 8.4E-3 3.172E-l ± S.OE-2 ± 2.7E-2 ± 7.1E-3 1.404E+O ± 2.2E-1 ± 1.2E-1 ± 3.2E-2 9.013E-2 ± 2.0E-3 ± 1.2E-3 ± S.OE-4 2. 785E-2 ± 2.3E-3 ± 1.4E-3 ± 3.3E-4 7.715E-1 ± S. IE-2 ± 4.7E-2 ± l.OE-2 2.160E-1 ± 4.6E-2 ± 2.6E-2 ± S.SE-3 1.192E+O ± 2.5E-1 ± l.SE-1 ± 3.2E-2 7.182E-2 ± l.SE-3 ± 9.0E-4 ± 6.0£-4 1.705E-2 ± 2.1E-3 ± l.lE-3 ± 2.1E-4 7.138E-1 ± l.lE-1 ± 5.2E-2 ± 9.8E-3 2.342E-l ± 6.1E-2 ± 3.0E-2 ± 5.7E-3 1.556E+O ± 4.0E-l ± 2.0E-1 ± 3.8E-2 1 pP 2 Q2(FVFP) (GeV/c)~ Rei (JJpG~/G~) 2 Table 4.3 -Values of the form factors, including statistical, pointto-point systematic, and absolute systematic uncertainties. - 79 1.15 1.10 1.05 ... 1.00 ~ Q t.!l ......... ...t.!l=- 0.95 0 .90 0.85 0 1 2 Q2 (GeV/c) 2 3 4 Figure 4.8 - The extracted values of G"t I Go I'"'' vs. Q 2 • Chapter 5 Comparisons to Previous Data and Theories 5.1 Previous Measurements In this section the data from selected previous measurements of e-p elastic scattering will be presented. The experiments are limited to those that provide a suitable range of c values to allow for the form factor extraction at Q2 > 1 ( GeV / c )2, although a significant body of lower Q 2 data will also be included. First a brief overview of each of the experiments will be presented, including a few of the important experimental details, the prescription used for calculating radiative corrections, the kinematic range covered, and the relative uncertainty in cross sections. See Appendix A for the details -80- of the radiative corrections procedure used for this experiment. Comparisons will be made between the form factors extracted from this work and some of the previous experiments. All the previous data will then be combined with the data from t his experiment and a global fit to the form factors will be performed. The extracted form factors will be presented for a wide range of Q2 values, along with the relative normalizations for the different experiments. The cross sections used from these previous experiments are presented in Appendix D. 5 .1.1 Overview of Previous Experiments Eleven different experiments were chosen as representing the essential elastic e-p data taken in the last 25 years for Q 2 > 1 ( GeV / c )2. These experiments are presented here in chronological order under the name of the principal author. Kinematics and cross sections for each of the experiments are listed in Appendix D. Janssens, et al.,1966 [34] Data was taken at the Stanford Mark III linear accelerator with a 0.953 em liquid hydrogen target. Scattered electrons were detected in a 72 inch double-focusing magnetic spectrometer with a momentum acceptance of 3.5%. Cross sections were measured at 25 different Q2 , of which 20 were suitable for performing an e separation, covering a range of Q 2 of 0.15 ::=; Q 2 ::=; 0.86 (GeV /c) 2 . The incident energy ranged from 0.25 ::=; Eo ::=; 1.0 GeV. Typically 3-5 e points were taken at each Q 2 , covering an angular range of 45° ::=; () ::::; 145°. Single e points were taken at () ~ 145° and Q 2 values of 1.01, 1.09, and 1.17 (GeV /c) 2 . Radiative corrections were calculated using the method of Tsai [29] for the internal contributions and of Schwinger [35,36] for the external contributions. Uncertainties in cross section values were typically 5.0%, although they were slightly larger at the higher Q 2 points. An overall normalization uncertainty of 1.6% was also reported . -81 - Bartel, et al., 1966 [37] This experiment was the first of two performed by Bartel and collaborators at DESY that will be presented here. An electron beam provided by the 6-GeV synchrotron was focused onto a 5 em liquid hydrogen target. Scattered electrons were detected by a magnetic spectrometer that could be rotated around the target pivot. The spectrometer had an angular acceptance of ±12 mr in the horizontal plane, ±9 mr in the vertical plane, and a momentum acceptance of ±2%. A total of 8 different Q 2 measurements were done in the range 0.39 ~ Q 2 ~ 4.1 (GeV /c) 2 . The incident energy ranged from 3.6 ~ Eo ~ 5.9 GeV. The measurements were all done at relatively small scattering angles, 10° ~ () ~ 25°. Usually only one c point was taken at each Q 2 , and the maximum spread in angle at a fixed Q 2 was only 4°, so an independent extraction of the form factors from this data alone was not possible. Radiative corrections were made using the method of Meister and Yennie [38]. T ypical uncertainties in cross sections were 3.0%, with an additional absolute normalization uncertainty of 3.5%. Albrecht, et . al., 1967 [4] This data was also taken at DESY, with an electron beam incident on a cylindrical liquid hydrogen target [39]. A doubly focusing spectrometer was used to detect the final electrons. Only three measurements of the elastic cross sections were done; at Q2 = 1.95, 2.92, and 9.56 (GeV /c) 2 , all performed at()~ 76°. The incident energy ranged from 1.7 <Eo< 6.2 GeV. Again, no independent extraction of the form factors from this experiment was possible. However, the first two data points could be combined with the data of Bartel 1966 to perform an c separation. This extraction was dominated by the normalization uncertainties . The highest Q2 data point was used to extract G~ under the assumption G~=G~ / J.Lp· Radiative corrections were -82 - carried out using the prescription of Meister and Yennie [38]. The two lower Q2 cross sections had uncertainties of 4.3%; the highest Q2 point uncertainty was 14.1 %. An overall normalization uncertainty of 4% was also included. Litt, et al. 1970 [40] This experiment was performed at SLAC with electrons of incident energies 3.0 ::; Eo ::; 10 GeV scattered from a 23 em liquid hydrogen target. The same 8-GeV spectrometer in End Station A that was used in the present experiment was used by Litt, eta/. Cross sections were measured at Q2 = 1.00, 1.50, 2.00, 2.50, and 3.75 with scattering angles of 12° ::; (}::; 42° . Corrections were made for straggling in the target using the prescription of Eyges [41] and internal radiative processes using the prescription of Tsai [29], which was found to be consistent with Meister and Yennie [38], within the experimental error bars. Typically 3-5 c points were measured at each Q2 . Cross section uncertainties were ,..__ 1.5-2.0%, with a normalization uncertainty of 4.0% . Goitein, et al., 1970 [42] This experiment, performed by Harvard University at the Cambridge Electron Accelerator, used a liquid hydrogen target. Both the scattered electron and the struck proton were detected. The electron spectrometer had a momentum acceptance of 14% with a solid angle of 0.83 msr. Scattering angles ranged from 19° ::; (} ::; 34°, with the proton recoil angle ranging from 24 ° ::; (}P ::; 65°. Good agreement was found between the cross sections measured with the electron arm only and those measured with the additional requirement of a coincident proton. The incident beam energy ranged from 1.6 ::; Eo ::; 6.0 GeV. A total of six Q 2 -values were measured, from 0.27 ::; Q 2 ::; 5.9 (GeV/c) 2 • Once again, a sufficient c range was not available to extract the form factors from this experiment alone. Internal corrections were done -83- with the method of Meister and Yennie [38], and bremsstrahlung corrections with the method of Bjorken [43]. Cross section uncertainties were ""' 4%, with an overall normalization uncertainty of 2.2%. Berger, et al., 1971 [44] This experiment was performed at the Physikalisches Institut of the University of Bonn. A liquid hydrogen target 5 em in diameter [45] scattered electrons into a magnetic spectrometer with solid angle 0.507 msr that could be rotated around t he target pivot. The incident energy ranged from 0.66 :::; Eo :::; 1.91 GeV. Data were taken at approximately 14 different Q2 , from 0.1 :::; Q2 :::; 1.95 (GeV/c) 2 . Some of the very low Q 2 data was taken exclusively at small angles, (} ~ 30°, to provide a normalization to other experiments. Anywhere from 3 to 14 different c: points were taken at the medium and high Q2 values, covering a maximum angular separation of 25° :::; (} :::; 111°. At the highest Q 2 points, only relatively large (} measurements were performed, with the intention of combining this data with other experiments performed at smaller scattering angles and extracting the form factors. Radiative corrections were done with the prescription of Meister and Yennie [38] for the internal, including the energy dependence of the elastic cross section in the internal bremsstrahlung terms [46], and Heitler [4 7) for the bremsstrahlung contributions. The cross section uncertainties ranged from 2-6%, with a normalization uncertainty of 4%. Price , et al., 1971 [48) This experiment was also performed at the Cambridge Electron Accelerator with an experimental group similar to the one of Goitein, et a/. Data was usually taken at(}~ 90° at 6 different Q 2 values ranging from 0.25 to 1.75 (GeV/c) 2 • The incident energy ranged from 0.45 :::; Eo :::; 1.6 GeV. The radiative corrections were calculated -84- using the equivalent radiator method of Mo and Tsai [32], with additional corrections applied from Meister and Yennie (38] and Yennie, Frautschi, and Suura (49] . Typical cross section uncertainties were "'4% with a normalization uncertainty of 1.9%. Kirk, et al., 1973 (13] This experiment was performed at SLAC in the End Station A using the 8 GeV spectrometer. The principal purpose of this experiment was to extract G~\1 at large momentum transfers (1 :$ Q2 :$ 25 (GeV /c) 2 ), under the assumption that G~=G~ / Jlp, where the contribution of G~ to the cross section was expected to be small. Incident energies ranged from 4 :$ Eo :$ 17.3 GeV with scattering angles 12° :$ 0 :$ 35°. Five liquid hydrogen targets were used, and were vertical cylinders 8-32 ern in diameter. Radiative corrections were carried out with the .prescription of Tsai (29] for the internal part and Eyges (41] for the external part . Additional correction terms (such as the contribution to the internal corrections due to the energy dependence of the elastic cross section) are described by Kirk, et al. (13]. Typical cross section uncertainties were 2% at low Q2 , and the absolute normalization uncertainty was estimated at 4%. Although most of this data was far beyond the range of Q 2 where a reliable determination of G~ is possible for the purposes of this paper, data at low values of Q2 are used to provide a cross calibration of this experiment with other experiments. In addition, some of the higher Q2 data (Q 2 = 5.1 and 10 (GeV /c) 2 ), combined with large angle measurements from some of the other experiments at similar Q2 values, is used to determine an upper limit on the value of G~. This is the first data on G~ for Q2 values above 4 (GeV /c) 2 . Bartel, et al., 1973 [3] This second set of data . measured by Bartel, et al., was also done at DESY. Three spectrometers were used to measure the e-p elastic cross sections. A small -85- angle electron spectrometer was set at () ~ 12°. The solid angle was 0.44 rnsr, and the momentum acceptance was ±2.5%. A large angle electron spectrometer was set at () = 86°, with a solid angle of 21.0 rnsr and a momentum acceptance of ±5.5%. An additional recoil nucleon detector was rotated to an angle corresponding to an electron scattering angle of () = 86°. This was used as an independent measurement of the scattering cross section. Data was taken at Q 2 = 0.67, 1.00, 1.17, 1.50, 1.75, 2.00, 2.33, and 3.00 (GeV /c) 2 , corresponding to incident electron energies of 0.8 ~ Eo ~ 6.5 GeV. Radiative corrections were calculated from Meister and Yennie [38] for the internal and Mo and Tsai [32] for the external. Contributions from the energy dependence of the elastic cross section in the internal corrections [32] were also included. The major source of uncertainty in this experiment was the cross section normalization between the small and large angle spectrometers, which was estimated at 1.5%. Typical cross section uncertainties were 2-4%, with an absolute normalization uncertainty of 2.1 %. Sill, et al., 1986 (50,51] This experiment was performed at SLAC in the End Station A using the 8 GeV spectrometer. The principal purpose of this experiment was to extract G~ at large momentum transfers (2.8 ~ Q 2 ~ 31.3 (GeV/c) 2 ), similar to the experiment of Kirk, et al.. Electron beams with energies of 5.5 ~ Eo ~ 21 GeV were incident on one of two liquid hydrogen targets with lengths of 25 and 65 ern. Scattering angles ranged from 21° ~ 0 ~ 33°. Radiative corrections were carried out using the prescription of Mo and Tsai [32] for the internal and Tsai [52] for the external. Typical cross section uncertainties were 3-4%, with a normalization uncertainty of 3.6%. Again, most of this data was far beyond the range of Q2 where a reliable determination of G~ is possible for the purposes of this paper, but data at Q 2 = 2.9 and Q2 = 3.7 (GeV /c) 2 is used to provide a cross calibration of this experiment with other experiments; and -86- some of the higher Q 2 data (Q 2 = 5.0 and 9.7 (GeV/c) 2 ) IS used to determine an upper limit on the value of G~ at large Q 2 . Katramatou, et al., 1987 [53,54] This data was taken at SLAC m the End Station A as subset of e-d elastic data. A 180° back angle electron spectrometer and a 0° nucleon recoil spectrometer were built on the floor of the experimental area, with solid angles 22.5 and 5.9 msr, respectively, and momentum acceptances of ±4% and ±1.5%, respectively. At these kinematics the cross section is completely dominated by G~; there is no contribution from G~ . Thus the cross section measurement is an unambiguous measurement of the magnetic form factor and no Rosenbluth separation is necessary. The incident beam ranged in energy from 0.5 :5 Eo :5 1.3 GeV, which allowed measurements over a range of 0.5 :5 Q2 :5 1.8 (GeV /c) 2 . Various liquid hydrogen targets were used with lengths from 10 to 40 em. Radiative corrections were carried out using the equivalent radiator approximation of Tsai [52] for the internal contributions combined with the effect [52] of bremsstrahlung and Landau straggling in the external radiators of the target . The effect of bremsstrahlung energy loss before the scattering vertex and the energy dependence of the elastic cross section was included. The cross sections uncertainties ranged from 2-3%, with an absolute normalization uncertainty of 1.8%. 5.1.2 Comparison to Previous Form Factor Measurements In this section a comparison will be made between the form factors that were measured in this experiment and the form factors that were extracted from the previous data. Only those experiments that measured a sufficient c; range to extract the form factors will be included; no cross normalizations between small and large () -871.15 1.10 l 1.05 ... 1.00 :t .......... Q t.!l .......... l'o)l t.!l 0.95 •-ru. work ••Jan.••D.8 0•Utt 0 .90 o.aer,... "•Bartel 11173 C•Katramat.ou 0.85 0 1 2 Q2 {GeV/c) 2 3 4 Figure 5.1- The measurement of Gt/Go/J.', vs. Q~ from this work, along with the data of Janssens, Litt, Berger, Bartell973, and Katramatou. measurements of different experiments will be done. That will be performed in the following section. In Figure 5.1, the results for G~/Gn/ iJ.p are shown for this work along with the results of the c:-separations of Janssens, Litt, Berger, and Bartel-1973. The data of Katramatou is also included since at () = 180° the cross section is only dependent on G~ . No c:-separation was required for this data, which eliminates many of the systematic uncertainties involved in a Rosenbluth extraction of the form factors. Our data tends to fall somewhat below the results of Bartel-1973 and Berger, but is in good agreement with Katramatou and Litt. Janssens' data lies at too low a value of Q2 to make a meaningful comparison, except to point out that the data point -88- at Q2 =0.86 (GeV /c) 2 seems anomalously large. Results for G~ are sensitive to the absolute normalization of cross sections, which have not been included in the error bars. Results for G~/GD are shown in Figure 5.2 for the same experiments as the G~ plot, with the exception of Katramatou. Our data tends to lie above the data of Bartel 1973 with a discrepancy of about two sigma. This is strongly correlated with the discrepancy in the G~ results. Our data is in moderately good agreement with the results of Berger and Litt. 2.0 ••Thill work •-J~eM 0•LILL o-a.raar li•Barlel 1873 1.5 t 1.0 I 0.5 0.0 0 1 3 Figure 5.2 - The measurement of~/ Go vs. Q 3 from this work, along with the work of Janssens, Litt, Berger, and Bartel 1973. 4 -89- Experiment T/ b..ry Janssens Bartel, 1966 Albrecht Litt Goitein Berger Price Kirk Bartel, 1973 Sill Katramatou 0.983 ± 0.009 0.994 ± 0.011 0.976 ± 0.025 1.005 ± 0.004 0.973 ± 0.010 0.994 ± 0.007 0.966 ± 0.012 1.002 ± 0.009 0.993 ± 0.006 1.006 ± 0.018 1.017 ± 0.011 0.016 0.035 0.040 0.040 0.022 0.040 0.019 0.040 0.021 0.036 0.018 Table 5.1 - Relative normalizations between each of the previous experiments and this work. Absolute syste~atic uncertainties for each of the experiments are also given. 5.1.3 World Fit to Elastic Form Factors All of the data from the previous experiments (compiled in Appendix D) was combined with the data of this experiment to perform a global fit. From this fit , Gt;; and G~ were extracted at 28 different values of Q2 , along with the ten normalization constants between each of the experiments and this work. Cross sections were corrected for differences between the measured and nominal Q 2 values using the dipole approximation (typically "' 10%). After the form factors were extracted, the fit was re-iterated using these new values of the form factors. Values of the extracted form factors did not change significantly. The fit was done by minimizing x 2 , where x2 was defined by: bG).,(~p + ••;•GHQ]l] b.. a norm }2 -90- @ a "' 0.20 0.11 0.18 0.10 0.16 0.09 ; 0.14 0.08 b 0.07 0.12 0.10 0 0.2 0.4 0.6 0.06 0.6 0 0.2 0.4 0.6 0 .8 0 .6 0.8 £ (a) (b) 0.012 0 .07 ; 0.011 0.06 I b b 0.010 0.05 0 0 .2 0.4 0.6 0.009 0 .6 0 0 .2 0.4 E £ (c) (d) Figure 5.3 - Four typical £ separation plots for the form factor extraction at: (a) Ql 0.60; (b) Q:J 1.00; (c) Q'J 1.20; and (d) q:J = 3.00 (GeV/c)l. = = = (5.1) where the definitions are similar to those of equation 4.10. The values of 1Ji are the normalization constants, and l:11Ji is the absolute cross section uncertainty given for each experiment. The cross section error bars (~a-:/o~m) included both statistical and point-to-point systematic uncertainties. All the variables in the above equation (1Ji , G~(Q]), and G~(QJ)) were simultaneously fit to minimize x2 . The total x2 for the fit was 172 for 211 ·degrees of freedom, indicating a conservative estimate of the systematic uncertainties. A few typical £ separation plots are shown in Figure 5.3. Relative normalization constants for the experiments are given -91- in Table 5.1. Values of Gj)Gn and G~/Gn//lp are given in Table 5.2 at each Q 2 value, along with the x2 /dof that was contributed from each of the Q 2 points. This data is also plotted in Figures 5.4 and 5.5. 1.15 1.10 1.05 ... ::l. 1.00 ......... C!J"' ............,. 0.95 C!J 0.90 0.85 0 1 3 4 Figure 5.4 - The values of~ I Go I~-&, vs. Q 3 from the world fit. The data for Q 3 > 4 (GeVlc) 3 are not shown. The normalization constants are well within the systematic uncertainty assigned by each experiment; only Price shows more than a one sigma deviation from a value of 1Ji = 1.0. In addition, excellent agreement is found between the normalizations of experiments done at the same facilities : the SLAC 8 GeV spectrometer data (Litt, Kirk, Sill, and this work), DESY experiments (Bartel 1966, Albrecht, and Bartel 1973), and the CEA work (Goitein and Price). This result, coupled with the small values of x2 /dof for the fit, tends to indicate that a single normalization constant for - 92- Q 2 (GeV /c) 2 G~/GD G~/GD/ Jl-p x2 /do£ 0.155 0.180 0.195 0.230 0.270 0.290 0.310 0.350 0.400 0.430 0.470 0.505 0.545 0.600 0.660 0.700 0.740 0.780 0.860 1.000 1.200 1.500 1.750 2.000 2.400 3.000 3.750 5.100 9.600 1.017 ± 0.028 0.957 ± 0.026 0.935 ± 0.026 1.012 ± 0.034 0.953 ± 0.023 0.997 ± 0.045 0.964 ± 0.026 0.931 ± 0.067 0.976 ± 0.017 1.016 ± 0.064 0.973 ± 0.056 0.969 ± 0.026 0.980 ± 0.071 0.961 ± 0.016 0.969 ± 0.044 1.074 ± 0.086 1.051 ± 0.155 0.956 ± 0.030 1.212 ± 0.094 0.967 ± 0.023 1.012 ± 0.032 1.048 ± 0.031 0.017 ± 0.071 1.105 ± 0.04 7 1.063 ± 0.062 1.050 ± 0.099 1.358 ± 0.186 < 2.17 (95% CL) < 3.24 (95% CL) 0.919 ± 0.030 0.952 ± 0.018 1.003 ± 0.031 0.933 ± 0.028 0.948 ± 0.017 0.931 ± 0.024 0.948 ± 0.019 0.966 ± 0.026 0.963 ± 0.012 0.965 ± 0.026 0.968 ± 0.021 0.983 ± 0.010 0.979 ± 0.022 0.990 ± 0.007 0.988 ± 0.019 0.979 ± 0.022 1.012 ± 0.032 1.007 ± 0.010 1.015 ± 0.009 1.028 ± 0.007 1.025 ± 0.007 1.032 ± 0.007 1.043 ± 0.008 1.025 ± 0.010 1.032 ± 0.010 1.032 ± 0.012 0.996 ± 0.037 1.047 ± 0.083 0.875 ± 0.097 0.17/1 0.96/5 1.83/3 0.88/2 7.91/6 0.44/3 2.22/4 0.07/1 9.91/15 0.54/1 1.68/4 3.84/3 4.60/3 11.9/25 2.78/3 1.43/4 0.80/1 20.2/16 12.8/3 14.0/16 5.44/14 7.49/14 8.00/9 14.6/17 14.7/17 11.1/11 3.54/6 0.38/3 0.60/1 Table 5.2 - The extracted values of the form factors from the world fit at each Q 2 , relative to the dipole fit. The x 2 / dof for each of the Q 2 values is also shown. The error bars include statistical and point-to-point systematic uncertainties. An overall normalization uncertainty of 1-2% has not been included. -932.0 1.5 1.0 0.5 0.0 0 1 3 Figure 5.5 - The values of d'£/Go vs. Q 2 from the world fit. The upper limits at Q 2 > 4 (GeV /c) 2 are not shown. each experiment was a good correction for the systematic differences between each of the data sets. Values of G~ are very similar to previously published results. For Q2 < 1 ( Ge V / c) 2 it lies below the dipole, rising quickly to a point that is a few percent above the dipole. At larger values of Q 2 , it again falls below . the dipole, although much of that data has not been included here. The data on G~, which have substantially reduced error bars compared to previous results, is nearly constant relative to the dipole; in fact, it tends to .rise slightly above the dipole approximation. This is in sharp contrast to the interpretation generally given to previous experimental results, which have been used as an indication that Gl£/Gn tends to decrease with increasing Q2 above Q 2 = 1 (GeV/c) 2 . However, these results were usually"' 1-o- away from a -94- value of 1.0, and were dominated by systematic uncertainties involved in cross normalizations between different spectrometers or different experiments. These systematic uncertainties affected the data at the different Q2 values in a highly correlated way. Data from this experiment alone, which is dominated by statistical uncertainties, combined with this world fit, would seem to indicate that Gj;/Gn does not fall below 1.0 for Q 2 < 4 (GeV /c) 2 . At Q 2 = 0.86 (GeV /c) 2 , the value of Gj; seems somewhat anomalous. This is related to the anomalous G~ value extracted from Janssens, and is reflected in the value of x2 / dof for this point. 5.2 Theories and Parameterizations A number of different approaches have been developed to try to understand the nucleon elastic form factors. Low Q2 data have been interpreted in terms of the spatial distributions of the charge and magnetic moment distributions, such as the rms radius of the proton. Moderate Q 2 data have been viewed in the light of vector meson dominance which models the virtual photon as a sum of massive vector mesons. The Q2 dependence of the form factors is thus, in part at least, caused by the Q2 dependence of the meson propagator terms. Models involving higher twist effects, such as diquarks, also make specific predictions for the elastic form factors at moderate Q2 • Large Q 2 data have been compared to perturbative QCD predictions and dimensional scaling laws, and has been interpreted as confirmation of the quark/parton model of the nucleon and the success of quark counting rules. These various approaches will be discussed here, and comparisons to measurements will be made at the end of this section. - 95- 5.2.1 Dipole Approximation and Form Factor Scaling The elastic form factors of point particles, such as muons, have no Q2 dependence. But for particles with non-zero size, such as protons, the form factors decrease with increasing Q2 as the virtual photon begins to probe distance scales comparable to or smaller than the size of the particle. The dipole approximation is a lowest order attempt to incorporate the non-zero size of the proton into the form factors. It is assumed, in this model, that the proton has a simple exponential spatial charge distribution: p(r) = poe-r/ro (5.2) is the norwhere ro is the scale of the proton radius ( (r 2 ) = 12r5) and Po - __!_,.. Srr"' 0 malization constant. The form factors are related, in the non-relativistic limit, to the \ Fourier transform of the charge and magnetic moment distribution. The dipole form factor, Gn, can thus be defined by: Gn(q2) = J eiq·rp(r)d3r 1 (5.3) where q 2 is the three-momentum of the virtual photon squared. In the non-relativistic limit (v <t:: Mp), the approximation q 2 = Q2 + v 2 = Q2 (1 + zMp) ~ Q2 can also be made. If it is also assumed that the magnetic moment distribution has the same spatial dependence as the charge distribution (i.e. form factor scaling), we achieve the dipole approximation to the form factors: 2 1 Gn(Q ) = (1 + Q2r5)2 = G~(Qz) = G~ ( Qz)/f.Lp (5 .4) -96- Previous measurements of e-p elastic scattering have indicated a best fit value of r5 = (0.24 fm? = 1/0.71 (GeV/c)- 2 , indicating an rms radius of .J7li} ~ 0.81 fm. Measurements of G~ and G~ agree with the dipole approximation, as well as form factor scaling, to 10% for Q 2 < 1.5 (GeV jc) 2 . 5.2.2 Vector Meson Dominance Models Vector meson dominance models [55,56] approximate the e-p scattering vertex by assuming the primary mode of coupling between the virtual photon and the proton is through a vector meson. The form factor for the 1-p coupling through a vector meson can be written as the meson propagator term times a VNN coupling term: (5.5) where Mv is the vector meson mass, and FvNN is the meson-nucleon form factor. This coupling is shown in Figure 5.6. The VMD model contains both iso-vector and iso-scalar meson components to the photon, thus there are iso-vector and iso-scalar components to the form factors . These can be written in terms of the Pauli and Dirac form factors, F1 and F2, and are related to the nucleon form factors by: Ff = F{s + F[V Ff = pJS + pJV piV F1n -_ FIS 1 1 (5 .6) Different models incorporate different numbers of vector mesons in the calculations, and some include a bare photon coupling in addition. Two typical models will be discussed here. - 97 - p v p Figure 5.6- Vector meson dominance picture of the coupling of a photon to a proton. G. Hohler, et . al. [28] The VMD model developed by G. Hohler, et al., incorporates terms involving the exchange of p and w mesons, plus higher terms that are associated with the ¢, w' and p' mesons. Contributions from the p were determined from 1r-N scattering data and 2-7r final state data, as well as F1r( Q2 ) . Masses and coupling strengths of the higher mesons were free parameters that were fit to the previous proton form factor data. Expressions for the form factors were thus: (5.7) -98- FP = (~) 0.955 + 0.090(1 + Q 2 /0.355)-2 lp 2 1 + Q2 /0 .536 F.p = (~) 5.335 + 0.962(1 + Q2 /0.268)- 1 Zp 2 1 + Q2 /0.603 (5 .8) where V refers to the heavier meson states. The value used for the mass of thew was fixed at 0.7826 GeV. Many different fits were done to various form factor data sets, and the parameters presented here (from fit 5.3) were the best fit to the proton data set alone: a1(w) = 0.67 az(w) = 0.04 a1 (VI) = -0.39 az(Vl) = -1.88 a1(Vz) = -0.54 az(V2) = 0.24 M1(V1) = 0.96 GeV M2(Vi) = 1.14 GeV M1(V2) = 1.66 GeV M2(V2) = 3.19 GeV lachello, Jackson, and Lande {57} Another VMD model, developed by Iachello, Jackson, and Lande (IJL), incorporated terms from p, w, and </>exchange, in addition to a contribution from a bare photon coupling. An intrinsic proton form factor, g(Q 2 ), was also included. The equations used for F1 and F2 were: -99- where an attempt has been made to include the finite width of the p meson, r P• into the formalism using the techniques of Frazer and Fulco (58]. The widths of the other mesons are small and were ignored. The function a(Q 2 ) was defined: (5.10) Different forms of g(Q 2 ) were attempted in the fit, but only the one which gave the best fit to the form factor data (and the proper asymptotic behavior within the QCD formalism) will be considered here. This form was a dipole: (5.11) The parameters were fit to elastic proton and neutron form factor data. The fit had fixed values of Mp = 0.765 GeV, Mw = 0.784 GeV, and M¢> = 1.019 GeV. When the effect of the p-meson width was ignored (f p=O), the x 2 jdof was large (....., 2-3). However, with fp = 112 MeV, the x 2 was much smaller("-' 1). The following parameters were extracted: 1 = 0.25 (GeV /c)- 2 {3p = 0.672 f3w = 1.102 !3¢> = 0.112 0'¢> = -0.052 with x2 /dof=0.924. Note that the value of 1 for the intrinsic nucleon form factor was much smaller than the value of r5 extracted for the dipole form factor (r5 = 1.41 -100- 5.2.3 Dimensional Scaling Dimensional scaling predicts (59] that in the absence of an internal mass scale the exclusive cross section of a process AB ---+ CD can be expressed by: (5.12) where n is the total number of lepton, photon, or elementary quark fields carrying a finite fraction of the momentum in the particles A, B, C, and D; s is the center of mass energy squared, t = -Q 2 , and f is a function of the ratio tj s. For ep ---+ ep scattering, n = 8 (initial state (1e, 3q) + final state (1e, 3q) = 8 fields). Thus the prediction of dimensional scaling for ep elastic scattering is, in the limit of large t and fixed tj s: (5 .13) Fixed tj s corresponds to a fixed center-of-mass scattering angle. This can alternatively be expressed in the limit of high Q2 in the fixed target frame by Eo "' Q2 , The cross section du / dt is related to du /dO, and thus to Ff which dominates at large Q2 , by: (5.14) -101- therefore: (5.15) Thus dimensional scaling predicts that Ff"' 1/ Q 4 in the limit of high Q 2 . Since Ff is related to the helicity non-conserving part of the scattering amplitude, it is suppressed in the limit of high Q 2 by a factor of"' m~/ Q 2 relative to Ff where mq is the quark mass scale. Dimensional scaling thus predicts that Ff"' 1/Q 6 in t he limit of high Q 2 . This can be stated alternatively as Q4 Ff"' constant and Q2 ( Ff / Ff) "' constant. These predictions are similar to those of Perturbative QCD (PQCD) [60], neglecting factors of a_,( Q2 ). The question as to what Q 2 is a "high" Q 2 is a controversial one within the literature. More discussion of this question will appear in the following two subsections. Data from e-p elastic scattering [51] seem to indicate the onset of scaling in the range of 5- 10 (GeV /c) 2 . Data [61] on pion cross sections exhibit scaling behavior for Q2 > 1 (GeV/c?, while neutron data [62] exhibits the behavior above 3 (GeV/c? . Recent data [63] on the photodisintegration of the deuteron also indicate scaling above 1 (GeV /c) 2 • 5 .2.4 Perturbative QCD M. Gari and W. Kriimpelmann [60] have attempted to combine the low Q2 phenomenology of VMD with the high Q2 predictions of PQCD. In this model the intrinsic proton vertex form factors, Ff and Ff, follow a monopole type Q2 dependence at low Q 2 , where meson physics dominates. But at high Q2 they have a dependence of Ff"' ljQ 4 and Ff"' ljQ 6 (modified by log(Q 2 / A2 )), as predicted by PQCD . These form factors are also modified by contributions from vector meson propagator terms. For this model only the p and w mesons were included, and no at tempt was made to -102- include the p resonance width. This leads to a parameterization of the iso-scaler and iso-vector form factors of: (5.16) A~ pQCD _ I - A~ A~ + Q2 A~ + Q2 FQCD _ A~ pQCD 2 -A~+ Q2 I (5.17) ln(A~+Q2) A2 2 Q = Q AQCD 2 (5 .18) ln(~) QCD where Kv = Kp- Kn and "'~ = Kp + Kn, and Kp and Kn are the anomalous contributions to the proton and neutron magnetic moments, respectively. A1 is the scale of the proton wave function ("' 0.8 GeV) and A2 is the scale below which the meson dynamics dominate and above which the quark dynamics dominate. The parameter Q2 contains the logarithmic dependence of the strong coupling constant. This model was fit to proton and neutron form factors extracted from the measured electron scattering cross sections using the experimental values Mp = 0.776 GeV and Mw = 0.784 GeV. The following values of the parameters were extracted: A1 = 0.795 GeV A2 = 2.27 GeV gp = 0.377 Kp = 9w = 0.411 Kw fp fw AQcD 6.62 = 0.163 = 0.29 GeV -103- with a x2 /dof of 0.43. These parameters are in good agreement with other experiments (Kp = 6.1±0.6 [64,65] and Kw = 0.14±0.2 [66] from pion-nucleus scattering) and SU(3)F predictions (gpfgw =sin e;v'3, g~ = 0 fore near ideal mixing angle [60] and i\1 "' 0.8 GeV from a strong interaction coupling scheme [67,68]). Perturbative QCD effects begin to dominate the form factors at a value of Q 2 =A~= 5.15 (GeV/c) 2 . 5.2.5 QCD Sum Rules A model proposed by Radyushkin [69] attempts to reproduce the form factor scaling behavior that has been measured in various experiments by using only the soft components of the proton wave function and without invoking the large Q 2 assumptions of scale invariance and lowest order PQCD. In this model e-p scattering is decomposed into the components shown in Figure 5.7. The higher order diagrams are suppressed by a factor of o:$(Q 2 )/rr for each gluon exchange that occurs. Thus diagram (a) is naively expected to be the dominant contribution to the form factors. At large Q 2 , however, diagrams (a) and (b) are suppressed due to the momentum imbalance of the quarks and the corresponding small overlap between the intermediate quark wave function and the final proton wave function. Diagram (c), which allows the quarks to achieve similar momenta will then dominate, and is expected to fall like 1/Q4 due to the dimensional scaling laws [59] discussed before. There should exist an energy scale, Q5, where diagram (a) is dominant for Q2 < Q5, while (c) is dominant for Q2 > Q5. Radyushkin calculates the contribution from diagram (a) (to be discussed below), and finds that it behaves like 1/Q6 in the asymptotic limit. He therefore concludes 2 that the scale of Q5 might be of the order [1/(o:$(Mp)/rr) 2 ][0(1 (GeV /c) )] "' 100 ( GeV j c ) 2 . This is well above the range of presently available data, which have been used as an indication of the validity of the scaling model. -104- + +... Figure 5. 7 - The Feymnan diagrams for factorization (in a,) of e-p elastic scattering. In this model the coupling of the electromagnetic current to a current with the same quantum numbers as the proton is calculated (Figure 5.7(a)) . The hadronic spectral density, p(st, 82, Q 2 ), where s1 and s2 are the mass squared of the incoming and outgoing hadronic current, respectively, was determined. At large values of s1 and s2 where the hadronic resonances are broad the hadronic density should correspond to the perturbative free quark density. At small values of s1 and 82 nonperturbative effects will cause p to become a S function located at the mass of the ever, that these non-perturbative effects only cause the strength to be concentrated within a local range in s1 and 82 and do not effect the overall magnitude. Thus the -105- contribution of the proton, the lowest mass hadron with the proper quantum numbers, can be related to an integral of the perturbative contribution, pPert, over a range of 0 ~ s1 ~so and 0 ~ s2 ~so, where so is the duality constant. The value of so was fixed (70,71] at so= 2.3 (GeV/c) 2 • The form factors were computed to be: (5.19) where u = s1 + s2 + Q 2 , z = J u 2 - 4sls2, eu = 2/3, ed = -113. AN is related to s 0 by (27r) 4 )..~ = s~l12. The results of these calculations agree with the data [51] on G~ fairly well up to Q 2 "'20 (GeV lc) 2 (see Figure 5.8). It is not until Q 2 "' 30 ( GeVIc ) 2 that the full asymptotic 1I Q6 falloff begins to occur. In this limit the 4 3 magnetic form factor behaves like G~- ~;o. It is important to note that, although the normalization G~(O)=l is fixed in this theory (through the quark charges eu and ed), the normalization of G~(O)=JLp is not required, since JLp is not included in any of the equations. 5.2.6 Diquark Models The diquark model of the nucleon was motivated, in part, by recent polarized elastic p-p experiments (72] performed at Brookhaven that indicated an unexpectedly large number of helicity non-conserving events in elastic proton-proton scattering. Non-perturbative effects are one way to describe such helicity flips. The diquark model of Anselmino, Kroll, and Pire (AKP) [73] is an attempt to include such higher twist effects. -106- 0.5 0.4 0.3 a. ~ a. :I 0.2 C!l •a 0.1 0.0 0 10 30 40 Figure 5.8 - A comparison between data [51] on Gt and the model of Radyushkin [69] using quark-hadron duality and QCD sum rules. In this model it is presumed that of the three quarks within the nucleon, two of them form a tightly coupled diquark state (r "' 0.1-0.3 fm), while the third quark is more loosely bound (see Figure 5.9). These diquarks could exist in either a scaler or (axial-)vector state. The proton form factors can be derived in this model, treating the diquark as an elementary particle, and are given by: 2 2 GP (Q2)_81rCp{jd d ""*( )a.,(Q )Fs(Q )<P () M - 3Q2 x Y'f'SY (1-x)(1-y) sx 2 - Q 2 Fv(Q 2) jdx dy <Pv(Y) 8m . as(Q 2)(1- x)(1- y)Fv(Q 2) <Pv(x)} xy F~(Q2 ) =- ~ ~: Fv(Q 2) j dx dy <Pv(Y) 2 -107- . as( Q2)Fv( Q2) <Pv(x) xy (5.20) where Q2 = (1- x)(l- y)Q 2 and Q2 = xyQ 2, m is the mass of the vector diquark (assumed to be 580 MeV), and CF = 4/3 is the color factor. Fs and Fv are the diquark form factors and are given by: F ( Q2) = as( Qz)Q~ s Q5 + Q2 F (Q2) = as(Q2)Qi v Qi + Q2 (5.21) where Q5 and Qf are related to the size of the scaler and vector diquarks (similar to the dipole form factor for the proton). <Pv and <Ps are the longitudinal momentum wave functions of the diquarks. Under exact SU(6) symmetry they would be related by <Pv = <Ps. The asymptotic behavior of the form factors, G~"" 1/Q 4 and Ff"" 1/Q 6 , is also predicted by this model. AKP found good agreement with the data [51] on G~ for 10:::; Q2 :::; 30 (GeV/c) 2 with Q5 = Qf ~ 3.2 GeV2 and cP& = <Pv = o(x- 0.4) when the form factors were normalized to the data at Q2 = 15 (GeV /c) 2 • An identical calculation was performed for this paper, and a similar agreement with the data was found when a multiplicative normalization constant of~ 0.16 was applied. However, G~ was found to be approximately related to G~ by: (5.22) for 1:::; Q2 :::; 100 (GeV/c) 2 . The large value of G~ relative toG~ occurs because of the negative value of Ff. At very large Q 2 ("" 107 (GeV /c) 2 ), the asymptotic limit stated by AKP of G~~G~ is achieved. This is far beyond the valid range of this model, and it occurs because the contribution of Ff goes to zero due to the extra power of a&(Q 2 ) in the vector diquark contributions relative to the scaler diquarks, but this falloff occurs only logarithmically. -108- (&) (b) Figure 5.9 - (a) Schematic picture of a proton in the diquark model. (b) Feynman diagram for scattering from a diquark within the proton. 5.2.7 Presentation of the Predictions The dipole form factor, Gn(Q 2 ), is plotted in Figure 5.10 versus Q2 . This parameterization describes the dominant Q2 behavior of the form factors. Other predictions for the form factors will be presented relative to the dipole form factor. The sharp falloff with Q 2 is clearly seen, leading to an even sharper falloff in the cross section at large Q2 since the cross section depends on the square of the form factor. Various parameterizations for G~ are shown in Figure 5.11. The data from various experiments are plotted for comparison. The models of Gari (60], Hohler [28], and IJL [57] clearly fall below our data. However, since these theories had free parameters that were fit to previous data, which was dominated by Bartel's 1973 -109- 1.00 0.75 0.50 0.25 0 .00 0 1 3 4 Figure 5.10- The dipole form factor, Gv(Ql), versus Ql, data [3], slight adjustments could be made to these parameters to account for the new data. Radyushkin [69] shows good agreement with our data. The diquark theory of AKP [73] has not been plotted because it is off scale; in the range of 1 ::; Q2 ::; 3 ( Ge VIc) 2 , this model predicts Gj; I Gn ,..., 4-5, and the value stays above 4 for Q2 < 8 (GeVIc? (the world fit limits Gj;IGn < 3.2 for Q2 < 10 (GeVIc) 2 at the 95% C.L.). Predictions for G~IGnl Jf.p are shown in Figure 5.12. The curves of Gari, Hohler, and IJL go above the data of this experiment for reasons that are strongly correlated with the discrepancies in the G~ values. Radyushkin's theory disagrees for Q2 ::; 2.5, but is clearly getting results that are nonphysical in this region (i.e. the required normalization of G~(O) = J.lp is violated). However, the value of G~IGn/ J.lp reaches a minimum of~ 0.65 at Q2 ~ 0.16, and rises back up to~ 0.94 at Q2 = 0. -110- 2.0 ••'l'h.lal work ••Jan.Hn. 0•Litt o.aer1 ar •-Bartel 11173 1.5 ... -Carl - Hohler .. lJL - · - Ra4yuahldn 1.0 e, ........ ...... e, --0 .5 0.0 0 1 3 Figure 5.11 - The models for ~· The lines correspond to: solid=Gari, dash=Hohler, dot=IJL, dotdash=Radyushlcin. Radyushkin attempted to eliminate some of these low-Q 2 problems by assuming the phenomenological form factor scaling relationship ( G~=G~ I J.Lp) and extracting both form factors from the Ff calculation, which is expected to be more accurate at low Q2 . This does tend to reduce the disagreement in the range of Q2 I"V 2-3 ( GeVIc ) 2 (G~IGniJ.LP = 1.01 at Q2 = 2 (GeVIc) 2). The model of AKP is not plotted, but falls from a value of 1.15 at Q2 ~ 1 to 0.85 at Q2 ~ 3 (GeV lc) 2 , and is clearly ruled out by the data in the range of Q2 ~ 3 (GeVIc) 2 . In figure 5.13 the values of Q 2 (Ff IFf) are plotted. Our data clearly flattens out much more quickly than most of the theories; only Radyushkin shows a similar behavior. The best fit value to the slope of the data of this experiment for Q 2 2: 2 (GeV/c)2 is 0.08 ± 0.11 (GeV/c)- 2 , the value for Gari in this region is 0.19, while - 1111.15 1.10 --- - 1.05 ... ::t 1.00 ........ "' t.!l ........ D. )II .1' 0 .95 "' .1' t.!l .1' "' I' I' 0 .90 / / 0 .85 "' "' 0 1 Q 1 ••Tb18 work •·J~u <:l•Utt O•Beraer IC•Bartel 11173 O•l'&t.ramatou 2 (GeV/c) 1 3 Figure 5.12 - The models for ~. The legend is the same as Figure 5 .11. Radyushkin predicts 0.14. The AKP model is not plotted, but predicts Q2 (Ff / Ff)"' -0.4 in this Q 2 regime due to the fact that Ff is negative in this model. -112 1.50 ••Thll work ••Janaaeu 1.25 o.~~er,.,. --Ga:rl - - Hohlor - . - " .<lyuahldn X•ButeJ 1&73 1.00 0 .75 - N 0 ;<11 0 .50 ~ 0.00 0 1 Figure 5.13 - The models for Q 2 (F!/.Ff). same as the previous figures. 3 4 The legend is the Chapter 6 Conclusions We have extracted the proton form factors from e-p elastic scattering for 1 :::; Q 2 :::; 3 (GeV fc) 2 using a Rosenbluth separation technique covering an angular range of cross section measurements of 11 .5° :::; (} :::; 48°. Statistical uncertainties were :::; 1% in the cross sections, 7-11% in G~ and 1.5-2.5% in G~. The systematic uncertainties were smaller than the statistical uncertainties; most previous form factor extractions -113- have been dominated by systematic uncertainties. The absolute normalization uncertainty for the cross sections was 1.6%. The total uncertainty in G~ was reduced by "' a factor of two over previous measurements in this Q2 regime. Improvements could be made, however, with a better theoretical understanding of the radiative corrections used in the analysis; particularly the corrections arising from two-photon exchange and 0( a 4 ) contributions. A comparison between our data and previous measurements indicate marginal agreement with the previous DESY data and good agreement with previous SLAC data. However, our data for G~ rises above the dipole approximation, while the DESY data tended to fall 1-u below the dipole. Most parameterizations of the data have also fallen below the dipole, but this has been ruled out, both by our data and the previous SLAC data. Our measurements are also in marginally good agreement with the ansatz of form factor scaling (G~';:::jG~/J.I.p)· The measurements of Q2 (FfjFf) are lower in magnitude and demonstrate less Q2 dependence then the previous DESY and Bonn measurements and most parameterizations, but are in good agreement with the predictions of dimensional scaling. They are also in good agreement with the predictions by Radyushkin based on QCD sum rules. The results clearly rule out the diquark model in this range. A global fit to the data at Q2 > 1 (GeV /c) 2 shows a good consistency between the different data sets within the normalization errors. The extracted values of G~ have error bars reduced by a factor of "' 3 over previous measurements. This data once again rules out the possibility of any significant decrease of G~ / G D below 1 for Q2 < 4 (GeV jc) 2 and generally indicates an increase in G~ relative to the dipole with increasing Q 2 . The results are in good agreement with the QCD sum rule prediction of Radyushkin for G~. The global fit also limits G~/Gn < 3.2 for Q2 < 10 (GeV /c) 2 -114- at the 95% C.L. This clearly rules out the diquark predictions for this Q2 range and tends to support the ansatz of form factor scaling which has been frequently used by other experiments to extract G~ from single angle cross section measurements. The extraction of the proton elastic form factors at relatively large Q2 with substantially reduced systematic uncertainties has proven to be possible. Measurement of the form factors at larger Q 2 (> 5 (GeV /c) 2 ) would be an important test of the predictions of dimensional scaling and would enhance our knowledge of the proton substructure and the strong force. A dedicated experiment with greater amounts of beam time than was possible in this experiment should be able to perform such measurements. More precise data at larger values of f) would also be important in order to reduce the statistical and systematic uncertainties involved in the Rosenbluth separation. Such an experiment has been approved at the ESA at SLAC, and should be taking data shortly. Improved calculations of the radiative corrections are also necessary if better measurements of the form factors are to prove possible. Two photon exchange terms, which require an understanding of the contributions arising when the proton is off mass-shell in the intermediate state, need to be estimated. Explicit calculation of the 0( o: 4 ) terms, rather than the relatively simple estimate made by exponentiating the O(o: 3 ) contribution, would prove useful. And a better understanding of the energy dependent effects arising from internal bremsstrahlung, beyond the equivalent radiator approximation used in this work, could also be important, due to the strong c dependence of these corrections. Systematic uncertainties could be further reduced through better calibration of the incident energy. -115- Appendix A Radiative Corrections A.l Overview Elastic scattering does not consist of simply the one-photon exchange process outlined in the introduction. Higher order processes in O'em also affect the cross section. These processes can be split into two general categories. Internal effects are those that occur as a part of the primary e-p scattering vertex, such as vacuum polarization, vertex, or internal bremsstrahlung. External effects are those caused by secondary scattering from rest of the material in the target, such as bremsstrahlung or ionization losses. (Figure A.l). These processes cause the elastic cross section to be changed from the simple 8-function at E' = E~1 = Eo/[1 + 2 t sin 2 (0/2)] (modified by resolution effects and the energy spread of the beam) to an asymmetric peak with an extended elastic tail at lower energies (see Figure A.2). The radiative tail extends down to values of E' where other processes ( 1r-production, ~-resonance , etc.) also occur. Therefore, t he integration of the cross section must be cut off at a value of E' = E~ 1 - ~E, with the value of ~E chosen to be small enough to exclude these other background processes, but large enough so that the value of the integral is not sensitive to the exact shape of the energy resolution function of the detectors. Radiative corrections depend on the det a ils of the target materials and geometry. T hey also depend in a complicated way on the kinematics of the scattering. Therefore they are included as part of the experimental corrections, so that the measured cross -116- X (a) X X (b) (c) X (d) <•I Figure A.l - The higher Feynman diagrams used in the calculation of the radiative corrections. The internal corrections: (a) vacuum polarization, (b) vertex, (c) two-photon exchange, (d) internal bremsstrahlung. The external corrections: (e) electron bremsstrahlung. sections correspond to the simple one-photon exchange in the Born approximation presented in the introduction. Exact details of the calculation of the corrections are given below. A.2 Internal Corrections The radiative corrections procedure of Mo and Tsai [32] were used to make the internal corrections for this experiment. For a detailed presentation of the calculations involved, see ref [29]. The results are stated here for completeness. - 117 - /•p'" ' .... E' (b) E' (a) Figure A.2 - (a) Elastic peak in the simple, one-photon exchange approximation (including resolution effects) as a function of E' at fixed E 0 and 8. Other processes (1r production, etc. ) occur at E' values well below the elastic peak. (b) The elastic peak including higher order radiative effects. The elastic spectrum is integrated from E;1 to E;1 - L:J.E. The radiative corrections were parameterized by Dint, which related the cross section of 0( a~m) to the one-photon exchange cross section by: (A.l) Higher order corrections were approximated by exponentiating Dint : dal . -da -e 6'"'df!mea" df! 1--y (A.2) -118- Although this approximation for the higher order terms was strictly valid only for the infrared divergent terms, the error caused in t he nondivergent terms was estimated [32] to be small(:::=; 0.7% at Q 2 = 20 (GeV/c) 2, and smaller at our kinematics). Contributions from vacuum polarization (Figure A.l (a)) and electron vertex diagrams (Figure A.1(b)) were calculated explicitly. Only the infrared divergent contributions from proton vertex (Figure A.1(b)) and two-photon exchange (Figure A.1(c)) diagrams were calculated; the nondivergent components of these were estimated to be small < 1% [32,30]. Contributions from these diagrams (neglecting the infrared divergent terms that were not completely cancelled) were: (A.3) Contributions from the internal bremsstrahlung diagrams (Figure A.l(d)) were somewhat more difficult to calculate. It was assumed that ~E(l + 2Eo/ Mp) ~ E' to simplify the calculation. Corrections for this approximation will be discussed later in the section on improvements to the internal corrections. The value of bint , including the bremsstrahlung diagrams, was: -a bint = 1r (28 13 Q Q Eo ---In(-)+ (ln(-)- 1 + 2Z ln 77)(2ln(-)- 3ln 17)) 9 6 2 2 m2e m e2 ~E - ~( E' -Eo) - z21n( Ep) + z2ln( Mp )( ~ ln( 1 + .84) - 2) E' Mp 17~E .84 1 - .84 + Z 2 { l_ln( 1 + .84) ln( E4 + Mp) _ ~[-( E4- Mp )1( 1 + .84 )1]} .84 2 1- .84 2Mp E4 + Mp 1-.84 + Z[~( _ Mp- E') _ ~( Mp(Mp- E') ) + ~( 2E'(Mp- E') ) ~ 2E&-~~ 2E' E4 - MpEo 2E&-~~ Mp + lnl Eo(Mp- 2E') lln(2E')] _ Z[~( _ E4- E') _ ~( !vfp(E4- E') ) + ~( 2Eo(E4- E') ) E 2~&-~E 2~&-~E - 119- with the definitions of Eo = incident electron energy, E' = final electron energy, me = mass of the electron, TJ = Eo/ E', Mp = mass of the proton, E4 = final energy of the proton, /34 = final velocity of the proton, Z = charge of the target particle ( = 1), and t1E = the E' cutoff of the elastic tail. The infrared divergent terms completely cancelled in this calculation. The Spence functions, ~' is defined: ~(x) = {x -ln 11- Yldy lo Y (A.5) and was integrated numerically in order to calculate Sint· The typical value of Sint was "' -0.17. A.3 External Corrections Radiative effects that occur external to the principle scattering are caused by bremsstrahlung in the target material and the effects of the Landau tail of the ionization energy loss spectrum. Corrections were made based on the work of Tsai (52). - 120- Bremsstrahlung Spectrum An electron of energy E will emit bremsstrahlung photons in the field of a nearby nucleus. The probability of losing energy w (w ~ 0.8E, E ~ 0.1 GeV) after passing through a thickness of t radiation lengths is: PB(w,t)dw = f(1 ~ bt) [(~)bt] ~ [q)(~)]dw q)(~) = 1- (~) +~(~r (A.6) ( A.7) where q) is the shape of the bremsstrahlung spectrum and normalized to q)(O) = 1. The parameter b ~ 4/3 is given [74] more precisely by: b = ~{ 1 + ~ [~: ~] [ln(184.15Z~)] 17 = ln(1194Z~)/ ln(184. 15Z~) -I} (A.8) (A .9) where Z is the charge of the nucleus . The cross section for an electron of initial energy Eo to elastically scatter from a proton and have final energy E' = E~ 1 - l::l.E', when the electron is emitting bremsstrahlung photons with ti and t 1 radiation lengths of material before and after scattering, respectively, is: cPa( Eo, E') [t::..Eo [t::..E' . , , da(Eo - wo) wo , 1 dD.dE' = Jo dwo Jo dw Ps(wo, ti)Ps(w , t 1) df! 8(w + R- b..E ) (A.10) where the 8-function fixes the final energy at E' = E~ 1 - b..E', and the effect of the proton recoil is included through the parameter R which relates l::l.Eo = Rl::l.E' (R ~ (if, )2 ). Since Ps(w , t) is strongly peaked at w ~ 0, the double integral can be - 121- approximated by separating it into two single integrals; one near w 0 ~ 0 and one near w' ~ 0 (energy peaking approximation) . These integrals can be explicitly calculated, leading to the equation: d2 cr(E0 , E') (Rt::..E')b;t; (t::..E')b't' 1 1 dD.dE' = Eo E~ 1 f(1 + biti) f(1 + btt1) x [dcr(Eo) bttf 4>(t::..E') dcr(Eo- Rt::..E') biti "'(Rt::..E')J dD. t::..E' E'el + dD. t::..E' 'P Eo (A.ll ) Straggling and Ionization Loss Spectrum The energy loss spectrum due to ionization for an electron of energy E passing through material in the ultra-relativistic limit is shown in Figure A.3. This spectrum is described by a Landau distribution [75] which is defined: 1 1~T+ioo 4>L(A) = - . 21rz ~T-ioo eAu+ulnudu. (A.12) where cr > 0 is a real number. The parameter A [75,52] is defined: A= t::..E- t::..Emp (A.13) e e= 0.154(Z/A)txo MeV (A.14) where Z is the number of protons/nucleus in the material, A is the mass number, and tx 0 is the thickness in g/ cm2 . The most probable energy loss is t::..Emp and is defined in Appendix B. At large A the tail of 4>L(A) falls like"' 1/A2 • The correction for the Landau tail was small for this experiment. Under these conditions a correction to the cross section, C LN, can be calculated to a good approximation by: C LN = 1 - roo dw 4> L ( .,,~) - 1roo dw 4> L (: ) l:!..E' 1Rl:!..E' <,f (A .15) -1220 .20 0.15 -,., ~ 0.10 oeo 0.05 0.00 -5 5 0 10 15 A Figure A.3 - The Landau ionization energy loss spectrum. The parameter A is defined in the text. where C LN is the percentage of the ionization spectrum that scatters with final energy E' > E~ 1 - LlE' (again using the energy peaking approximation), and the subscripts i and f on erefer to initial and final material thickness. Recall that Eo and E' have already been corrected for the most probable energy loss (Appendix B). The value of CLN was typically 99.8% (6.E' /e,...., 1000). Calculation of External Corrections The external radiative corrections can be parameterized by a number bext, analogous to the internal correction parameter, which is defined by relating the integral of A.ll, along with the correction for the Landau tail, to the lowest order cross section: 12 ') Oeztdu(Eo) = CLN X [ E'el dE'a-u(Eo,E e dO. j E' -6.E dO.dE' e/ (A.l6) - 123- The integrand on the right hand side diverges like l/(l::iE') 1-bt as !::iE' -+ 0 due to the bremsstrahlung spectrum. Therefore when 8ext was calculated the bremsstrahlung contribution was integrated analytically in the region where !::iE' is small (E~ 1 -E' ::; 3 MeV) . In this region the cross section could be treated as a constant. The contribution at larger !::iE' was numerically integrated. The energy dependence of du /dO was approximated using the dipole approximation. T1 ·-t----- --------:::r~<§l--~·- L ~~======~·· .= . ...=~~-~-~-~~~® Figure A.4 - The schematic of the LH2 target used for calculating the radiation lengths before and after the scattering, t1 and t1, for scattering along the target length. The value of 8ext depended on where in the target the electron was assumed to scatter (through the variables ti, t,, ei, and e,). It was presumed that the electrons scattered from the center of the target in the transverse direction (Xtgt=Yigt=O), and had an equal probability of scattering anywhere in the longitudinal direction along -124- the target (0 :=:;Ztgt:=:; 20 em). Steps of 0.5 em were made along the target length, and the amount of material before and after the scattering, including all entrance and exit windows, etc., was calculated. The model used for the target is shown in Figure A.4, and the material properties and thicknesses are given in. Table A.l. The effect of the flow guides that were accidentally bent into the beam path is discussed in Appendix C. The value of e 5•zt averaged over the length of the target was used in the corrections. Typical values of bext were "' -0.18%. Item 1 2 3 * 4 5 6 7 8 9 10 11 12 13 Name Wire Arrays Hyman End cap Flow Guide Liquid H2 Flow Guide Cell Wall Insulation Epoxy Solder Window Air 8GeV Window Endcap Material Aluminum Al 6061 Al 5052 Al 5052 Hydrogen Al 5052 AI 5052 Mylar Epoxy Sn,Pb,Ag AI 6061 Nz,Oz Al 6061 AI 5052 (em) p (g/cm3 ) 0.00664 0.00254 0.00762 2.70 2.70 2.68 0.00254 0.00762 0.00635 0.01270 0.00457 0.03175 82.55 0.02718 0.00762 2.68 2.68 1.39 1.39 8.81 2.70 0.0012 2.70 2.68 T Xo (g/cm2 ) 24.01 23.39 23.63 23.63 63.05 23.63 23.63 39.95 39.95 7.744 23.39 36.98 23.39 23.63 L = 19.972 em, Le ~ 1.90 em, L~ : : : : 1.27 em, D = 5.08 em * see Appendix C Table A.l -The thicknesses and radiation lengths of the target materials. - 125- A.4 Improvements to Internal Corrections Because of the reduced systematic and statistical uncertainties of the present experiment, uncertainties in the radiative corrections at the level of ~ 1% caused by some of the approximations used in the previous sections were unacceptably large. In calculating the value Dint , Tsai assumed that [29) ~E(1 + 2E0 f Mp) ~ E' when the internal bremsstrahlung contributions were included. This approximation is not valid to the ~ 1% level necessary for this experiment, and an error is introduced in calculating the contribution to the elastic tail from the first diagram shown in Figure A.1 (d). In this case, the electron emits a bremsstrahlung photon before scattering from the proton, giving it a lower energy and thus an enhanced scattering cross section; by as much as 50% for the kinematics considered here. Therefore the size of the elastic tail has been underestimated, and this error increases with larger values of the radiative cutoff ~E. The effect of this approximation can be estimated, however. Internal bremsstrahlung corrections can be approximated [76) by treating the single proton as an external radiator, with equivalent radiator thickness: (A.17) External radiative corrections due to bremsstrahlung (equation A.ll) can be calculated with these equivalent radiator thicknesses, and the effect of the energy dependence of the cross section can be estimated by calculating the elastic tail with ~ approximated with the dipole form factors relative to the same calculation assuming a constant cross section (no energy dependence). This contribution to the internal correction, 8~nt• is defined by: 6~nt = e - J dE' cPa( Eo, E') dipole/ dO.dE' J dE' d2a(Eo , E') con.!t dO.dE' (A.18) - 126- This correction ranged from 0.2-3.0%, and had a strong .6.£'-cut and e dependence. \Nithout this correction term, the cross sections showed (77Ja strong("-' 1-2%) dependence on the .6.P/Po cut. Vacuum polarization contributions from.J.L+ J.l- and qq loops were also neglected. The muon loop was calculated from: 2a [ 5 4M 2 b~ac =-;- -g + 3 Q; + 2 ( 1 2M ) 3- 3 Q; ( Q2 ln MJ (1 + )] y~ 1 + -qf-) (A.19) where Mp. = 0.10566 GeV is the muon mass. This contribution is the same as the electron loop contribution in the limit of Q2 ~ M;. Quark loop corrections (78] were performed using a simple fit to the previous equation for all flavors of quarks (including the color factor) which was valid for 1 ::::; Q 2 ::::; 64 (GeV jc) 2 : (A.20) An additional correction by Schwinger [79] was included by Tsai [52] to correct for the noninfrared divergent part of the soft photon emission cross section. A sign error in Tsai's paper (52] has since been corrected [80], giving the term: 8sch 2 -a [71" 6- ~(cos 2 (0/2))] =--:;- (A.21) All these effects were then included in the definition of 8int: (A.22) Combining all of the above effects, the radiative correction, RCOR, is defined : (A.23) This value of RCOR was used to correct the measured cross sections for the higher order radiative effects. Different values of the radiative corrections used for this experiment are tabulated in Appendix D. - 127- Appendix B Kinematic Calibration It was important to have systematic uncertainties in Eo, E', and () well con- trolled since these are correlated with c. The A-bend and spectrometer energies were calibrated, and the incident beam position and angle and spectrometer angle were measured. The ionization energy loss of the electrons in the target was calculated, and the centroid of the incident beam energy distribution was determined. These considerations are discussed below. B.l Spectrometer Survey The spectrometer angle was measured with respect to the nominal beam axis before the experimental data run. The spectrometer was set at an angle of approximately 70°, and the angle was measured relative to a 70° scribe mark on the ESA wall. Due to a problem with the spectrometer angle encoder just before the experiment was about to begin, the survey needed to be done quickly and without access to the pivot. The encoder was set to only the approximate value of the angle. A more extensive survey of the spectrometer was performed after the experiment at eight different angles from 12° to 46°. These measurements indicated an offset of 0.049° in the encoder setting. The measurement of the spectrometer angle was accurate to A pointer was attaced to the spectrometer that pointed to a scribe line on the ESA floor when the spectrometer was set at 30.000°. A video camera mounted on the spectrometer allowed this pointer to be periodically observed, guaranteeing that -128- the angle encoder value did not change during the experiment. Previous extensive surveys of the spectrometer indicated that the angle encoder functioned properly over the entire range of spectrometer angle settings. In addition, these measurements indicated that the pivot position was stable to 0.27 ± 0.07 mm when the spectrometer was moved, and that the vertex position (where the spectrometer central ray intersects the beam path) was stable to 6.9 ± 0.1 mm. The position of the wire chambers in the detector hut was also measured relative to the nominal central ray. These measurements were accurate to ± 1 mm in the wire chamber position and ±0.5 mr in the relative wire chamber rotations. B.2 Wire Orbit The wire orbit [16,17], in addition to studying the transport characteristics of the spectrometer, was able to calibrate the l::iP/ Po and !::10 values of the nominal central ray of the spectrometer. The data indicated that the nominal ray in the detector hut had an offset in the horizontal angle at the target of -0.010°. This offset was possibly caused by a rotation in one or more of the quadrupole magnets relative to their nominal positions. The difference between the central value of the spectrometer momentum and the nominal momentum setting is shown in Figure B.l. The initial calibration of the central momentum was determined with dark current studies [14], conducted in 1967, with the spectrometer set at zero degrees (relative to the beamline). These studies were carried out at 3, 6, 8 and 9 GeV. From this data a fifth order polynomial for the current in the spectrometer magnets, as a function of momentum, was determined. This polynomial was used to include the effects of residual fields at zero current as well as saturation effects at high momentum settings. Data from the wire orbit indicated that this current equation gave the proper momentum setting at the points that were -129- fit (3, 6, 8, and 9 GeV), but residual discrepancies on the order of ±0.1% still exist. Two sets of wire orbit data were taken with different tensions placed on the wire. These data were in good agreement in the area of the data overlap (0.5-4.0 GeV), indicating that systematic errors were well controlled. Measurements were taken at each of the momenta several times, indicating a repeatability of the spectrometer momentum setting to ±0.025%. 0.15 0.10 -- 0.05 ~ 0 0.. 0.00 ......... 0 0.. I 0.. !!I -0.05 -0.10 -0.15 0 2 4 P 0 (CeV) 6 e Figure B.l - The absolute momentum calibration of the spectrometer central momentwn (P) relative to the nominal setting (Po) from the wire orbit. B .3 Ionization Losses Electrons in the beam lost energy as they passed through the target material due to the ionization of atomic electrons within the target. The nominal value of -130- the incident beam energy (and spectrometer momentum setting) was correct ed for the most probable energy loss an electron would experience as it entered (exited) the target by subtracting (adding) the energy loss calculated from the equation [81-83] valid in the ultra-relativistic limit: (B.l ) where NA is Avogadro's constant, t = the length of material in (g/cm2 ), Z = t he atomic number of the nucleus, MA is the atomic mass of the nucleus, p is the density of the material in (g/cm3 ), and t1Emp is given in MeV. The most probable energy loss was averaged over the target length assuming a uniform probability of scattering along the target. The typical size of this correction was 2.5 MeV. B.4 Beam Steering Offsets of the beam position and angle at the pivot can cause offsets in the measured momentum and angle of the scattered electron in the detector hut. The SLAC computer program TRANSPORT was used to estimate the effects of beam position and angle on the various kinematic quantities measured. The beam's horizontal and vertical angle were related to the measured values of 1:10 and 1:1¢>, respectively, by a simple 1:1 relationship. Offsets in the beam's vertical position, Yigt, caused offsets in the detected momentum fraction; this offset was related by d(t1P/Po)=0.3183 x ytgt, where !:1P /Po is in % and Yigtis in em. The target reconstruction algorithm assumed that the scattering takes place at Xtgt=O, so offsets in the beam's horizontal position caused offsets only in the measured target position along the beam, Ztgt· These were related by the simple geometric relationship Ztgt= tan(O)Xtgt· It was not important -131- for the kinematics of the scattering to know the precise target position so this correction was ignored. For certain data runs the beam was deliberately displaced by 3 mm in either the vertical or horizontal direction. The offsets in the observed elastic peak positions for these runs were consistent with the TRANSPORT predictions. Only data which was taken with the beam steering operating properly are included in this analysis , so the above corrections were typically extremely small. The beam position was known at both the wire array and microwave cavity monitor to better than ±1 mm. This lead to a typical uncertainty in the beam position and angle at the pivot of ±0.1 em and ±0.03 mr, respectively, causing an uncertainty in !:lP/ Po of 0.03% and in !:lO and !:l¢ of 0.03 mr. B.S Elastic Peak Calibrations The kinematic constraint that elastic scattering occurs at x = 1 allows a useful method to calibrate the incident and final electron energy. Measurements of the elastic peak positions at a variety of kinematics yields a measurement of the incident energy relative to the spectrometer momentum setting. Combining this information with the absolute calibration of the spectrometer provided by the wire orbit, an absolute calibration of the beam energy was possible. These measurements also indicated differences between the average beam energy in a particular run and the nominal central value measured by the flip-coil caused by a nonsymmetric distribution of the beam within the energy defining slits. A run-by-run correction of the absolute beam energy can thus be determined. -132- B.S.l Procedure for De-radiating Spectra The ( t::..P/Po, !::..0, !::..¢>) histograms that were generated in the Pass-1 analysis were converted into one-dimensional missing mass squared (W 2 ) histogram. The values of Eo, E', and 0 that were used included all the corrections noted above. Corrections were made for acceptance effects and the !::..0 dependence of the cross section across the acceptance. A typical W 2 histogram measured with the LH2 target is shown in Figure B.2. Similar measurements were done with the aluminum D UM target, and these were then subtracted from the LH2 histograms to exclude the contributions from endcaps and flow guides. These spectrum have all been checked to guarantee that the distribution of events in the super-elastic regions was statistically consistent with a flat, zero contribution. Radiative effects, as described in the Appendix A, shift the cross section from lower to higher W 2 . When this effect is folded in with the resolution of the spectrometer and the spread in the incident beam energy, the W 2 value where the maximum of the elastic peak occurs is shifted to W 2 > M'i. This effect must be included before the peak position can be determined. The unfolding procedure was based on the de-radiating prescription of P.N. Kirk, et a/. [13]. Radiative tails of the lower W 2 bins were subtracted from the higher W 2 bins on a bin-by-bin basis. First a low W 2 bin which was not contaminated by radiative tails from smaller W 2 bins was chosen, such as the first bin above the super-elastic cutoff (see Appendix C). This bin was corrected for the total radiative losses expected from the radiative corrections procedure of Mo and Tsai [29,32,52] (see Appendix A), and the contributions of this radiative tail were subtracted from each of the higher W 2 bins. H we let o-1 = the measured cross section in the lowest ( radiatively uncontaminated) bin, O"i = the measured cross section in bin i, RCOR(W2 ) = the -133 - '- +.> 30 "' m: ........ ...c m: ,5 c: -o ........ 0 20 m -o m m 10 ID ID ID ~.7 0.8 0.9 1 W2 CGeV/cl 2 Figure B.2 - A typical missing mass squared histogram measured with the LH2 target. The elastic peak position, neglecting radiative effects, should occur at w:~ M} 0.88035 (GeV /c):J (dashed line). = = radiative correction for a missing mass squared cut of W 2 (with the assumption x = 1 corresponds to W 2 =Mi), and oW2 be the histogram bin width, the radiative contributions from this first bin can be unfolded by: (B.2) 1 dw' ai :=ai- a~ X Ri c5W2 Ri = [RCOR(ioW 2 - (B .3) w'+Mi)- RCOR((i -l)b'W2 - w' + M;)] (BA) where cr' indicates the components of a new, partially radiatively corrected W 2 histogram. - 134- The second bin above the super-elastic cutoff ( o-~) has thus been corrected for the contribution of the radiative tail from the lower W 2 bin (o-D, and can now be radiatively corrected with the same procedure. This was done to succesively higher W 2 bins, until the entire spectrum had been de-radiated. Error bars assigned to each bin must be handled carefully to include the effects of correlations between the vV2 bins after de-radiating. Confusion can be avoided by relating the components of the completely de-radiated spectrum (o-?) directly to those of the measured spectrum (O"i) with the matrix equation: 0"0 O"b O"n3 0"0 n al a2 - 0 0 0 a3 a1 a2 al an an-1 an-2 al = Rl 0 0 0 (B.5) where: 1 n an = -al L R;an-i+l (B.6) i=2 The uncertainty squared of each bin of the de-radiated spectrum (~o-?) 2 can then be computed by simply taking the sum of the squares of the uncertainties of the measured bins (~o-;) 2 weighted by the proper component of the de-radiating matrix (aLj+l). A typical de-radiated spectrum is shown in Figure B.3. Note that the error bars shown for each bin are the proper statistical error bar for that particular bin. Binto-bin correlations exist in the values of the de-radiated cross sections, however, and thus the uncertainties involved when comparing values between bins are somewhat smaller than a naive interpretation would indicate. These correlations are especially -135- ....'rn"' ......... ~ c 60 40 ~ ......... b ~ 20 Figure B.3 - A typical W 3 histogram. after the effects of the radiative tail have been unfolded. The solid line is a least-squares fit of a gaussian to the spectrum. The dashed line is the expected value of the peak position, W 3 = Mj = 0.88035 (GeV jc) 3 • strong at the higher W 2 bins where large subtractions of the elastic radiative tails have been made. B .5.2 Fits to De-radiated Spectra Each de-radiated spectrum was fit to a gaussian shape. Only the central 7 bins near the peak maximum were fit . From these fits the gaussian height Ao, peak position W6, and peak width 6w2 were extracted for each run, along with their uncertainties. The typical x 2 I dof for these fits was 0.35. The x 2 I dof to the entire spectrum, as compared to only the central bins included in the fit, was typically 0.65. These small values of x 2 I dof were caused by the large correlations between the values in - 136- different bins, and thus indicated a overly conservative estimate of the uncertainties of the extracted parameters. From these measured values of the peak positions, the calibration of the incident and final energies could be carried out. B.5.3 Absolute Energy Calibration An offset in the measured peak positions, W~, relative to the kinematic constraint, vV 2 = 1\1~, can be related to offsets in the kinematic variables. This offset , dW 2 = W~ - M~, is related to the offsets in the incident and final energies and scattering angle by: dW2 = E 0 aw2 dEo E' aw2 dE' aw2dO aEo Eo + aE' E' + ao 2 aw 2 aEo = 2Mp- 4E' sin (0/2) ~~: = -2Mp- 4Eo sin2 (0/2) aw2 = -4EoE' sin((}/2) cos((} /2) ---a8 (B.7) The contribution of the d(} term was very small due to the high precision of the scattering angle known from the spectrometer survey, wire orbit, beam steering, and event tracking and was neglected. An estimate of the necessary energy offsets needed to explain the peak offsets can be performed by setting one of the energy offsets to zero, and assuming that the offset was due entirely to the other energy. The results of this analysis are shown in figure B.5. A fit to a constant, average offset in either the incident energy or the final energy was performed with the above data. H the offsets were assumed to be entirely due to dE0 j Eo, the average offset (dashed line (a)) was found to be -0.146%, with a x2 fdof=2723/61. H the offsets were assumed to be entirely due to dE'/ E', the average offset (dashed line (b)) was found to be 0.111%, with a x2 /dof=1614/61. -137 - 0. 1 I 0 .0 - 0 1 1-0:2 II) II) :=_ - ~ - - $-- -0.3 1- tf I 1 ~ til - I <j) tll II) .....; -IB a; ; - - - - - - IB- - _,_ ~ - - • - ~ ' - Eo (GeV) (a) 0 .3 I I I 0.2 g i'-1 '~ 0.1 0 .0 -0.1 I I I 2 4 8 a E' (GeV) (b) Figure B.4 - One-parameter analysis of the peak positions offsets. (a) The offset in the incident energy, dEo/ Eo assuming no offset in the final energy. (b) The offset in the final energy, dE'/ E', assuming no offset in the incident energy. These large values of x2 were caused by fluctuations of the peak positions around these average offsets, and were consistent with a typical ±0.05% fluctuation in the incident beam energy. Since the full width spread in the incident beam energy was up to 0.3%, fluctuations of this order were not surprising. The different kinematic dependence that the offsets dEo/ Eo and dE'/ E' have on the peak offset dW2 can, in principle, allow for a two-dimensional separation of these absolute offsets. However, since the kinematic terms are highly correlated and the fluctuations in the average incident beam energy tend to be significant compared to the average offset, it was difficult to get separate constraints on both the incident and final energies. It was therefore decided to correct the final energy by 0.055%, and the - 138- incident energy by -0.07%. The 0.055% shift in the spectrometer momentum was consistent with the uncertainty in the absolute calibration of the wire orbit (""' 0.05%) and the accuracy of the beam steering system (""' 1 mm). Since the cross section was only a function of the incident energy and scattering angle ( x = 1), shifts in the final energy had no direct effect on the measured cross sections. A systematic uncertainty of ±0.07% was included in the absolute incident energy calibration to account for the incident energy correction, but this uncertainty had only a small effect on the final extracted form factors. B.5.4 Incident Energy Fluctuations After the absolute energy calibration was accomplished, small fluctuations in the peak positions were still observed (see Figure B.5). These fluctuations were, as discussed above, consistent with "' 0.05% run-to-run fluctuations in the incident energy. The incident energy for each run was corrected assuming these peak fluctuations were caused entirely by differences between the average energy of the beam and the central value defined by the A-bend slits. Run-to-run fluctuations in the beam steering system(- 1 mm) and in the spectrometer momentum calibration("'-' 0.025%), could also contribute to these peak position fluctuations. A 0.03% systematic uncertainty was assigned to the point-to-point relative energy calibration to account for these uncertainties. B.5.5 Resolution The widths of the elastic peaks were also extracted from the de-radiated peak spectrum. Contributions to the width of the elastic peaks were from the incident energy spread (tl.E / Eo=full width), multiple scattering of the electrons in the detector hut (principally from the Cerenkov mirror) , and the resolution of the wire chambers -1390 .3 0.2 0.1 + ¢ CD ~ ~ 0 .0 <I> <I> coco C> r&l ......... C> r&l <I -0. 1 *f -0.2 -0.3 2 4 B 6 E 0 (GeV) Figure B.5 - Run-to-run fluctuations in the relative incident beaiil energy extracted from elastic peak locations. ("' 2 mm) A rough model based on these effects gave an expected peak width (in t:l.P /Po) of: Cup= Jo];. + o~., + ~c 0 0 _ 0 - 0.14% E' Eo - 1 !:lEo Eo (8E') 2../3 Eo E' 8Eo I x=l m"'- Owe= 0.07% (B.8) where Oexp is an absolute number, and all energies are in GeV. A comparison between the measured peak widths and this model is shown in Figure B.6. In addition to these intrinsic effects, an additional contribution to the width of the peaks was caused by -140- 2.0 1.5 . ~ - -- -- - 1- 1.0 '0 t ........ :1 •a '0 0 .5 0 .0 0.05 0.1 0 . 15 oup 0.2 0.25 Figure B.6 - Comparison between the measured elastic peak widths in i!a.P /Po ( 6-•) and the expected peak widths ( 66:q) that was based on a rough model described in the text. The solid line is also described in the text. the 0.25% !:l.P /Po bin width of the three-dimensional histogram generated in the Pass-1 analysis that was converted to the W 2 histogram. This effect must also be included: (B.9) and is represented by the solid line in Figure 8 .6. The model agrees well with the data when this effect is included. Resolution effects caused by the intrinsic width of the peaks (8exp) were included in the corrections for the radiative tail. These corrections were usually very small("'-' 0.01%). -141- Appendix C Subtraction of Aluminum Background Contributions to the detected electron rate from the LH2 target involved scattering from both the liquid hydrogen and the Al endcaps of the target. In addition, an accidental reversal of the flow of the liquid H2 through the Al flow guides of the target caused the flow guides to be bent into the path of the beam during part of the experiment. It was important to know how much aluminum was present in the beam path and subtract this background from the measured cross section. This material also contributed to the external radiative corrections (see Appendix A). The effect this material had on the radiative processes of the electrons scattered from the hydrogen will also be discussed. C.l Reversed Liquid Hydrogen Flow During the experimental run, it was noticed that additional electrons were detected above the elastic peak (v < Q 2j2."-1p) that were not accounted for from the endcap contributions alone. An on-line analysis of this problem lead to the discovery that the hydrogen flow through the target was incorrect. In Figure C.l a schematic of the hydrogen target with the correct flow path is shown: the hydrogen flows in through the center flow guide (0.00254 em thick aluminum) and out between the flow guide and the target wall. The pressure inside the flow guide, PO, is larger than the pressure outside, Pl. However, when the hydrogen flows in the opposite direction, the pressure Pl > PO, and the flow guide is pushed inwards. The combination of the flexibility of the flow guide, the flow rate of the hydrogen (> 1 m/s), and the -142- difference in the cross sectional areas of the inner and outer flow paths (inner area ,...._ 12.6 cm2 , outer area,...._ 7.7 cm2 ) could easily cause the flow guide to be bent into the beam path. Correction of the flow direction before the end of the experiment substantially reduced the rate of scattering above the elastic peak in subsequent data runs, giving support to the theory that the flow guides were, in fact, the cause of this additional background. LH2LH2 IN CXJT L Flow Guide -------------P~l~~--~--._------~ Beam ) J Pot Figure C.l - Schematic of the LH2 target under proper running conditions. C.2 Determination of Aluminum Background A "super-elastic" cross section was defined as the cross section in the reg1on v < Q2 /2Mp. While electrons scattered from hydrogen were kinematically forbidden -143- in this region, scattering from the Al endcaps was allowed due to the Fermi momentum [84) of the nucleons within the aluminum nucleus. Thus the super-elastic cross section was proportional to the total amount of aluminum in the beam path. The statistics in the super-elastic region were typically rather low; usually """" 100 counts. Thus the bin-by-bin statistics were very low and did not easily lend themselves to a statistical analysis. For this reason a model of the quasi-elastic plus deep inelastic scattering cross section from aluminum was developed to assist analysis of the distribution of events within the acceptance. The quasi-elastic model was based on a y-scaling formalism [85,2S] fit to previous data. The inelastic contribution was taken from the fit [23] to deuterium data and was Fermi smeared. Higher order radiative processes were also included using the procedure of Mo and Tsai [32,52) In order to make the kinematic corrections for variations in the beam energy from run-to-run, it was desirable to define the super-elastic cross section at the center of the acceptance region (6..PI Po=0,6..0=0). The super elastic cross section was defined (including an overall constant, SEo , which incorporates units, dead-time, etc.): J2uSE NsE dD.dE' = SEo x ACCsE x RsE v'NSE J2 usE 6..(dD.dE') = SEo x ACCsE x RsE (C.l) with: SE NsE = I: HIST2( 6..PI Poi, 6..()i) (C.2) A.P/Poi,A.8j EAsw ACCsE = E Ew (C.3) sw RsE = Ew uModel(6..P/ Po=0,6..()=0) Esw (C.4) -144- where NsE is the total number of counts detected in the super-elastic region, ACCsE is the acceptance (weighted by the cross section) in (mr 2 · GeV), and RsE is the ratio of the model cross section at the center of the acceptance to the cross section averaged over the super-elastic region. The following definitions have been used: SE EAsw = 2: ACC2(~P/ Poi, ~Bj)uModel(~P/ Poi, ~Bj)WGTij (C.5) uModei(~P/ Poi, ~Bj )WGTij (C.6) WGTii (C.7) t::.P/Po;,t::.Bi SE :Esw = 2: t::.P/Po;,t::.Bi SE :Ew = 2: t::.P/Po;,t::.Bi and: (C.S) The notation is similar to that of Chapter 4. Kinematic cuts on the super-elastic region were defined as -6 ~~()~ 5 mr and Phi ~~p/Po~ 3%. The constant SEo was analogous to CRo in Chapter 4, and included corrections for dead-time, etc. It was defined by: SEo = 1 DT 0.6022 (Beam· Ef f ·tAl · Acccor) (C.9) where tAl is the thickness of only the aluminum endcaps in the target: 0.01524 g/cm2 for the LH2 target and 0.2469 g/cm2 for the DUM target. The model used for computing the acceptance and the centering correction was checked against previous measurements [85,25] of e-Al inclusive scattering in this Q2 regime and was found to agree, in absolute cross sections, to 10%. The super-elastic cross section values changed by < 1% when a uniform cross section in E' and () ( umodel instead of this more realistic model. = 1) was used -145- These cross sections were then used to extract the total amount of aluminum present in the LH2 target. If t~? is defined as the thickness of the endcaps alone and t!{?' as the total thickness of the aluminum in the beam path of the LH2 target (in g/ cm2 ), they can be related to the measured super-elastic cross sections of the DUM and LH2 targets by: (C.10) Remember the super-elastic cross sections were defined assuming only the Al endcap contributions. A normalization factor, DF, can be defined as the ratio of measured to nominal aluminum thickness in the LH2 target: (C.ll) A factor of one would indicate that the aluminum in the target was entirely accounted for from the endcap contributions. A plot of DF versus time is shown in Figure C.2. It is clearly seen that when the LH2 flow was restored to the proper direction ( t ~ 39 days), the dummy factor was reduced to a factor close to one. The average factor before the flow was corrected was 4.43 ± 0.12 (x 2 /dof=64/52); after the correction it was 1.28 ± 0.12 (x2 /dof=10/7). Deviations of the normalization from 1.0 after correction of the flow direction were accounted for by the contamination of the liquid hydrogen by deuterium (D/H~ 0.16%, measured in atoms, leads to D/Al~ 13%, measured in g/cm2 where Al represents only the endcap contributions of the LH2 target) and uncertainties in the measured endcap thickness ofthe LH2 target {"' 10%). The total LH2 target thickness under optimal conditions (DF= 1) was 2.41% r .1. (1.453 g/cm2 ); with DF= 4.43, the thickness was increased to 3.01% r.l. (1.593 g/cm2 ). For comparison, the DUM target thickness was 2.80% r .l. (0.662 g/cm2 ). - 146- 8 I l 6 rz. Q 4 2 f 0 0 20 t 40 Time (Days-11/1/85) 60 Figure C.2- The normalization factor (DF, defined in the text) for the dummy target subtraction as a function of time. C.3 Effect on Radiative Corrections The additional aluminum in the beam path not only affects the scattering rate by directly scattering electrons itself; it also affects it indirectly by external radiative effects (see Appendix A) as the electrons pass through the material before scattering from the hydrogen. However, while material inserted in the beamline at the upstream end of the target (defined here as Z = -10 em) can cause electrons to produce additional bremsstrahlung photons before scattering from the hydrogen, material at the downstream end (Z = +10 em) will have no effect. It is therefore important to understand not only how much additional material was placed in the beam path (DF), but also the distribution of the aluminum within the target. -147- The flow guide extended almost the entire length along the target and was supported at the upstream end. The downstream end was completely free, and was only supported by the pressure of the hydrogen flowing through the target. Since the upstream end could not be bent into the beam path while the downstream end could, it was expected that the aluminum distribution might plausibly be shifted in the +Ztgtdirection relative to the hydrogen distribution. Although DF was a direct measure of the total quantity of aluminum along the beam path, there was no way to know the precise details of the distribution across the cross sectional area of the beam or along the beam path. This is not a problem, however, if the radiative corrections are linear with respect to the amount of material the beam passes through. If it is assumed that a fraction of the beam, a, passed through an additional amount of material from the flow guides, D.~AI, and the remaining 1 - a fraction of the beam passed through no additional material (average amount of material= ~tAl = (DF -1)ti}f2), the linearity of the radiative corrections can be tested by calculating RC(a) = aRCOR~ +(1-a) RCORo. The subscript on a RCOR indicates the additional amount of material placed in the path of the incident beam ( "t;" in the notation of Appendix A). It can be easily shown that if the radiative corrections are linear: RCOR~ a ~tAl = RCORo(1 +a--) a (C.12) where a is a constant, then: ~tAl RC(a) = aRCORo(1 + a - ) + (1- a)RCORo a = RC0Ro(1 + a~tAl) (C.13) -148 1.020 1.015 1.010 -tS u 1.005 e:: 1.000 0.995 0.990 0 0 .2 0 .4 0.6 0.8 1 a Figure C.3 - A plot of RC(a) vs. a, as defined in the text, normalized at a = 1 for a typical set of kinematics and running conditions. and RC(a) is independent of a. A plot of RC(a) for a typical kinematic setting is shown in Figure C.3. The fact that RC(a) is nearly independent of a for a> 0.10 indicates that the radiative corrections do not depend much on the distribution of the aluminum across the cross sectional area of the beam; in fact, it indicates that the radiative corrections do not depend much on the details of the distribution along the beam direction; only the mean position of the distribution along the beam affects the radiative corrections. This can be shown by calculating the average radiative correction of a target of length L, cross sectional area A (weighted by the beam distribution), and volume V = A XL, -149- with a distribution of additional material p(x,y,z): !v ~ !v (RCOR) = ~ = dx dy dzRCOR(x, y, z) dx dy dzRCORo [1 +a = RCORo [1 = RCORo [1 !v +a~ !v +a~ dx dy dz 1z 1L dz' p(x, y, z 1)] dz'O(z- z')p(x, y, z')] 1 dx dy dz'(L- z )p(x,y, z')] (C.l4) which is only a function of the total amount of additional material and its mean position along the (average) beam path. The wire chamber tracking information not only measured the !::iP /Po, /::i() and !::i<j> values of a particular track, but also the horizontal (in plane) position of the partide at the spectrometer entrance, Xsps. This quantity is related to the position along the beamline that the track originated from by Ztgt=Xsps/sin(O). It was assumed that the track originated along the beamline, rtgt=O. The measurement of the mean Z value of the distribution, ( Z) sE, of the super-elastic events was a measurement of the mean position of the aluminum in the target. A typical spectrum of Ztgt is shown in Figure C.4. Systematic offsets in this quantity can be caused by an offset in the target position on the pivot, a rotation in the wire chambers (Xspscx: Xwc - dxwc), and errors in the transport coefficients. A similar measurement of the (Z) EL of the events in the elastic region was dominated by the scattering from the hydrogen, and was sensitive to identical systematic effects. The difference between these two quantities, !::iZ = {Z)sE- {Z)EL ~ (Z)Al- (Z)H2 measured the shift of the mean of the aluminum distribution relative to the hydrogen distribution. Systematic effects with -150- 2.0 1.5 -.... .., 0 X 1.0 C/l E-o ~ 0 u 0.5 0.0 -40 -20 0 20 40 Ztct (em) Figure C.4- A typical spectrum in Ztgt· regard to the exact placement of the target on the pivot, the orientation of the wire chambers, and the transport coefficients were cancelled. The super-elastic and elastic kinematic regions occupied different areas of the acceptance. Since the Ztgt measurement was related to the x - dx track coordinates in the wire chambers, it was possible that acceptance effects correlated with the scattering kinematics (the x and y positions in the wire chambers) could affect the value of 6.Z. The Monte-Carlo program that was used to study the spectrometer acceptance (see Acceptance section) was also used to study this effect. This model has been shown to be a reasonably good approximation to the physical spectrometer acceptance function. The value of {Z) was computed as a function of 6.P/Po and 6.(). This calculation indicated a typical difference between the (Z) in the super-elastic -151- region and the elastic region of ~ 1 em due to acceptance effects. Corrections were made to the measured {Z)sE and {Z)EL based on this calculation. A check of this correction was done using the dummy target, and will be discussed below. Figure C.5(a) shows the values of ~z measured with the LH2 target as a function of time (for t ~ 39 days, before the flow was corrected). The average offset was 1.8 ± 0.4 em, (x 2 /dof=l.1). Since the scattering from the DUM target in both t he elastic and super-elastic region was from the same material, no such offset should be observed with this target. An identical analysis of the DUM target gave an average offset of 0.8 ± 0.7 (x 2 /dof=0.6), which was consistent with zero, indicating that systematic effects were well controlled. Without the Monte-Carlo based corrections for the acceptance effects, the measured offset with the DUM target was 1.5 ± 0.7. This indicated a good reliability to the Monte-Carlo. The super-elastic events were scattered from the endcaps, which were centered on the target, and the flow guides, which caused the ~z displacement. The average value of the dummy factor was 4.43 while the hydrogen flow was reversed, indicating that on average 23% of the super-elastic scattering was from the endcaps, and 77% of the scattering was from the flow guides. The offset of the Al flow guide distribution was thus taken to be 1.29 x (1.8 ± 0.4) = 2.3 ± 0.5 em from the cent er of the target . A model of the aluminum distribution of the flow guides was developed that had both the correct amount of material and the proper mean position along the beamline. The aluminum was presumed to be uniformly distributed across the beam cross section. The density was scaled for each run to achieve the proper normalization of material to account for the measured dummy factor. It was also assumed that the aluminum was uniformly distributed along the beamline from Z = -5.4 em to Z = +10 em, where Z = 0 em was defined as the center of the target and therefore the -15210 I I 10 I ( - 5 - 5 j( ~( e ~ -; 1\ v I I 1\ lSI -5 .. ~ - 0 1\ '1 ~ v e - 0 f- lSI v I I 1\ lSI v - -5 -1 o LL..LJ....LJ_I.u<..ILJ...Jil...L...L..JL.L..I....L.I. I.L.L...I.I..J.....L.J....u 0 6 10 16 Tlme (d.ayw) 20 26 6 (a) 10 Time (dayw) (b) 16 20 Figure C.5 - Measured values of A.Z as a function of time. (a) With the LH2 target. (b) With the DUM target. center of the hydrogen distribution. This model had the proper amount of aluminum with the proper (Z) AI value and, due to the high linearity of the radiative corrections, properly incorporated the radiative effects of the flow guides to ±0.2%. C.4 Subtraction and Uncertainty Analysis The final cross section was defined by: da da LH2 tH2 da DUM -DF(~)()] dO= [dOraw t~fM dflraw X 1 RCOR (C.l5) Corrections for the aluminum background subtraction were typically 3.0 ± 0.3% under reversed flow conditions, and were 0. 7 ± 0.1% after the hydrogen flow had been restored. The final cross sections had three sources of statistical uncertainty: the - 153- elastic cross section of the LH2 target, the cross section in the elastic region measured with the dummy target, and the measured normalization factor. The first two contributions could be handled in a straightforward fashion. However, the third contribution (~DF) had to be treated more carefully because the additional aluminum in the beam path caused the cross section to both increase, due to direct scattering effects, and decrease, due to indirect radiative effects, causing a systematic correlation that reduced the sensitivity of the cross section to the value of DF. This effect can be included by expressing the uncertainty in the final cross section in the form: du] 2 _ [ ~dn H x [ du LH2] 2 ~dn Hraw (RcloR) [ t!il du DUM] 2 + DF tAl DUM~dn Hraw [ t!il du DUM + f3 tAl DUM dn ~DF Hraw ]2 (C.16) 2 where f3 is a parameter that includes the negative correlation effects. From the measured aluminum background and the expressions for the radiative corrections given in Appendix A, the value of f3 can be derived. The direct statistical contribution of ~DF to the elastic cross section can be expressed by: 2 du;du) tH (duDUM/du) ( ~dO dO =~ (~DF) t~fM dO raw dO = (~DF)a1 (C.17) The radiative contribution of ~DF to the uncertainty was estimated to be: (~~~~~~) = (~DF)bln(R!E) = (~DF)a2 (C.18) -154- where the notation and variables are defined in Appendix A, with b ~ 4/3 and R ~ (Eo/ E') 2 . The parameter (3 was then defined by: (C.19) which related the effect of ~DF on the cross section uncertainty relative to the statistical effect alone. The value of (3 ranged from 0 < l/31 < 0.90, depending on the kinematics, etc. It was artificially limited to 0.25 < 1/31 to provide a safely conservative estimate of the uncertainty caused by the measured dummy factor when the two effects nearly cancelled. Other systematic errors were included in the aluminum background subtraction. These were caused by the uncertainty in the amount of material the electrons passed through as they exited the target through the flow guides, and the variation of (Z}AL from run to run. The direction of the hydrogen flow and the geometry of the flow guides limited the uncertainty in the outgoing electron path to be the thickness of the flow guides, which was 0.00254 em. This had a ±0.1% effect on the cross sections. An uncertainty analysis of the (Z} fits described previously limited the typical fluctuations of ~z to ±0.5 em, which had a ±0.15% effect on the cross sections. -155- Appendix D Data Tables D.l Data From This Experiment Eo GeV Bo deg w;ut (ti) (t,) (GeV /c) 2 % % 1.5938 2.4033 3.2375 2.4076 2.8002 3.2501 4.0030 5.4894 6.2371 6.9806 7.4876 2.7962 3.2413 3.7662 4.2416 7.0542 8.2093 3.2513 4.0082 6.2460 7.0743 8.2334 45.5074 27.5635 19.7400 46.6752 37.7562 31.0965 24.1089 16.7996 14.6021 12.9317 12.0007 46.2336 37.3101 30.5311 26.2759 14.5725 12.3574 44.2630 32.7090 18.8138 16.3060 13.7628 0.9730 1.0104 1.0498 1.0134 1.0359 1.0571 1.0980 1.1095 1.1061 1.1024 1.1021 1.0352 1.0741 1.0881 1.1107 1.1015 1.1026 1.0621 1.1063 1.1068 1.1038 1.1065 1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.33 1.33 1.54 1.54 1.54 1.54 1.43 1.54 1.54 1.33 1.54 1.54 1.54 1.54 1.33 1.54 1.79 2.06 1.53 1.63 1.75 1.90 2.16 2 .27 2.36 2.40 1.53 1.63 1.74 1.85 2.27 2.38 1.57 1.70 2.08 2.17 2.29 Dint o:nt (6ext) RCOR -0.1637 -0.1680 -0.1699 0.1769 ) .1763 ' -0.1777 -0.1773 -0.1961 -0.2062 -0.2154 -0.2204 -0.1799 -0.1738 -0.1798 -0.1808 -0.2209 -0.2314 -0.1818 -0.1810 -0.2142 -0.2243 -0.2344 0.0179 0.0175 0.0172 0.0244 0.0242 0.0237 0.0234 0.0212 0.0203 0.0197 0.0193 0.0271 0.0275 0.0261 0.0257 0.0213 0.0206 0.0295 0.0287 0.0238 0.0228 0.0221 -0.1147 -0.1307 -0.1442 -0.1100 -0.1161 -0.1233 -0.1313 -0.1500 -0.1625 -0.1831 -0.1895 -0.1079 -0.1104 -0.1173 -0.1273 -0.1786 -0.1831 -0.1073 -0.1156 -0.1596 -0.1721 -0.1768 0.771 0 0.755 1 0.7435 0.7694 0.7650 0.758 1 0.7522 0.7228 0.7061 0.6850 0.6771 0. 7708 0. 7739 0.7628 0.7542 0.6854 0.6749 0.7716 0.7652 0. 7050 0.6886 0.6780 Table D.l -Radiative correction values calculated for the kinematics of this experiment. The number of radiation lengths before and after the scattering ( t; and t1) are approximated in this table because the aluminum normalization parameter (DF) fluctuated from run to run, and each kinematic setting frequently contained many runs (see Appendix C). -156- Q2 € Eo E' 0 tm dcr fl. dcr 1.000 0.692 1.594 1.061 45.221 5.253E+O 4.179E-2 1.000 0.869 2.403 1.871 27.277 1.780E+1 1.618E-1 1.000 0.930 3.238 2.705 19.454 3.950E+1 3.391E-1 2.003 0.635 2.408 1.340 46.389 4.432E-1 4.070E-3 2.003 0.735 2.800 1.733 37.4 70 7.795E-1 5.871E-3 2.003 0.808 3.250 2.183 30.810 1.288E+O 7.869E-3 2.003 0.878 4.003 2.936 23.822 2.422E+O 2.315E-2 2.003 0.938 5.489 4.422 16.513 5.747E+O 1.365E-1 2.003 0.953 6.237 5.170 14.316 8.284E+0 8.403E-2 2.003 0.963 6.981 5.913 12.645 1.109E+1 1.054E-1 2.003 0.968 7.488 6.420 11.714 1.312E+1 3.676E-1 2.497 0.619 2.796 1.466 45.947 1.893E-1 1.719E-3 2.497 0.723 3.241 1.911 37.024 3.369E-1 2.875E-3 2.497 0.800 3.766 2.436 30.245 5.629E-1 3.569E-3 2.497 0.846 4.242 2.911 25.989 8.295E-1 7.724E-3 2.497 0.949 7.054 5.724 14.286 3.612E+0 2.736E-2 2.497 0.963 8.209 6.879 12.071 5.433E+O 9.714E-2 3.007 0.623 3.251 1.649 43.976 9.661E-2 9.355E-4 3.007 0.761 4.008 2.406 32.422 2.195E-1 1.876E-3 3.007 0.910 6.246 4.644 18.527 9.087E-1 2.316E-2 3.007 0.932 7.074 5.472 16.020 1.316E+O 1.376E-2 3.007 0.951 8.233 6.631 13.4 76 2.013E+O 4.797E-2 tm Table D.2 - Cross section values at each kinematic point used to extract the form factors. -157 Time (Days) Q2 (GeV/c) 2 4.25 4.79 5.00 5.08 5.67 5.75 5.79 5.88 5.96 6.17 6.29 6.79 7.13 7.25 7.33 7.42 7.63 7.71 7.83 7.88 8.08 8.96 9.04 9.17 9.21 9.42 10.13 14.33 14.38 14.50 14.54 14.63 14.63 14.96 15.33 16.25 24.25 24.29 26.38 26.42 37.17 37.29 37.42 37.42 37.46 37.54 37.58 37.63 37.63 37.67 37.67 37.67 38.96 1.001 2.501 2.501 2.501 2.000 2.000 1.999 3.001 3.001 3.003 3.002 2.497 2.495 2.495 2.493 2.497 2.498 2.495 2.498 2.497 2.001 1.998 1.996 1.000 0.999 0.999 2.504 3.004 3.005 3.007 3.008 3.008 3.008 3.007 2.004 2.503 2.007 2.009 2.010 2.010 3.011 3.010 3.011 3.013 3.010 3.009 2.509 2.508 2.509 2.509 2.506 2.511 3.008 ~ Nete5 DF tl.DF 4.91 4.83 5.09 5.46 4.00 3.91 3.96 5.88 7.13 6.54 5.51 3.36 6 .01 8.81 15.03 6.92 3.85 5.95 4.71 3.74 5.35 7.23 6.22 4.06 4.85 5.54 5.84 7.87 3 .96 3 .65 3.21 3.80 3.36 3.62 4.01 4.36 4.58 3.14 4.89 5.28 5.49 4.09 3.73 11 .25 3 .02 4.17 4.58 6.27 5.36 4.43 5.27 3 .82 3.68 0.52 1.00 1.29 1.06 0.51 0.71 0 .53 1.61 1.80 1.77 1.78 2.40 2.11 2.51 7.01 1.42 1.33 1.25 2 .04 1.10 0.73 1.32 1.17 0.69 0.75 0.80 0 .70 3.41 1.13 2.15 0.95 1.92 1.55 0.58 0.59 0.64 1.31 1.32 0 .78 0.80 1.15 0.65 1.23 6.60 0.70 0 .82 0.63 2.03 1.07 0.78 1.22 0.89 1.02 x10 0.930 0.723 0 .723 0.723 0.808 0 .808 0.808 0 .623 0 .623 0.623 0.623 0.619 0 .620 0.620 0 .620 0.619 0.619 0.620 0 .619 0.619 0.735 0.634 0 .635 0.869 0.869 0.692 0 .797 0 .761 0.761 0 .761 0.761 0 .761 0 .761 0.761 0.878 0.846 0.968 0 .968 0 .963 0.963 0.932 0 .932 0.932 0 .932 0.932 0.932 0.949 0 .949 0.949 0 .949 0.949 0 .949 0.910 17.260 5.688 2.650 7.117 13.635 4.329 11.026 3 .372 4.012 3.235 2.068 0.368 0 .878 0.982 0 .182 3.051 1.500 3.200 0.755 2.138 20.904 6.916 6 .772 6 .184 6.751 17.433 13.048 0.293 1.522 0.348 1.728 0.424 0.566 9.945 12.461 13.334 0.867 0 .495 6.058 7.205 1.334 4.044 0 .681 0 .056 1.794 2.045 10.424 0.496 2.108 3.504 1.281 1.821 1.755 Table D.3 - Aluminum normalization factor (DF) for each of the elastic data runs included in the analysis before the hydrogen flow direction was corrected. -158Q2 Time (Days) (GeV/c) 2 39.00 43 .63 45.25 46.96 47.00 60.33 60.42 68.38 2.008 2. 502 2.008 2.007 2.009 3.014 2.514 2.423 ~ Ne/e~ DF .6.DF 2 .879 6.487 7 .162 0 .501 1.385 1.917 3.241 7.360 1.81 1.68 0.86 1.71 1.63 2.17 1.75 1.63 0.45 0.39 0.20 1.07 0.70 0.83 0 .42 0 .36 xlO 0 .953 0 .797 0.953 0 .938 0.938 0.951 0.963 0.808 Table 0.4- Aluminum normalization factor (DF) for each of the elastic data runs included in the analysis after the hydrogen ftow direction was corrected. -159- D.2 Data From Other Experiments (World Fit) Q2 2 Q2 (fm)- 2 0.1558 0.1555 0 .1557 0.1792 0.1794 0.1794 0.1793 0.1794 0.1794 0.1795 0.194 7 0.1947 0.1944 0.2336 0.2339 0.2335 0.2335 0.2728 0.2730 0.2725 0.2721 0.2728 0.2922 0.2916 0.2923 0 .2916 0 .2915 0.3115 0.3113 0.3116 0.3112 0.3113 0.3498 0.3500 0.3503 0.3894 0.3891 0.3897 0.3894 0.3898 0.3893 0.3894 0.4282 0.4287 0.4280 0.4671 0.4672 0.4677 0.4675 0.4674 0.4865 4.00 3.99 4 .00 4.60 4 .61 4.61 4.61 4.61 4.61 4.61 5 .00 5 .00 4 .99 6 .00 6.01 6.00 6.00 7 .01 7 .01 7.00 6 .99 7 .01 7 .50 7.49 7.51 7.49 7.49 8.00 7 .99 8 .00 7.99 8.00 8.98 8.99 9 .00 10.00 9.99 10.01 10.00 10.01 10.00 10.00 11.00 11.01 10.99 12.00 12.00 12.01 12.01 12.00 12.49 (GeV/c) GeV E' GeV deg 0.5590 0.3680 0.2590 0.6030 0.4740 0.3990 0.3510 0.2970 0 .2820 0.2750 0.6900 0.4180 0.2960 0.6900 0.5500 0.4640 0.3310 0.6000 0.5080 0.4340 0.3640 0.3560 0.6240 0.5280 0.4680 0.3990 0.3800 0.6900 0.6470 0.5490 0.3960 0.3870 0.6920 0 .5880 0.4270 0.9000 0.7360 0 .6270 0.5570 0.4790 0.4570 0.4470 0.9500 0.6640 0.4860 0.9500 0.9000 0.7000 0 .5150 0 .5040 0 .7170 0.4760 0.2851 0.1760 0.5075 0.3784 0 .3034 0.2554 0.2014 0.1864 0.1794 0.5863 0.3142 0.1924 0.5655 0.4253 0.3395 0.2066 0.4546 0.3625 0.2888 0.2190 0.2106 0.4683 0.3726 0.3123 0.2436 0.2247 0.5240 0.4811 0 .3829 0.2302 0.2211 0.5056 0.4015 0.2403 0.6925 0.5287 0.4193 0.3495 0.2713 0.2495 0.2395 0.7218 0.4356 0.2579 0.7011 0.6510 0.4508 0.2659 0.2549 0.4577 45.000 75.000 135.000 45.000 60.000 75.000 90.000 120.000 135.000 145.000 40.590 75.000 135.000 45.520 60.000 75.000 135.000 60.000 75.000 95.000 135.000 145.000 60.000 75.000 90.000 120.000 135.000 55.300 60.000 75.000 135.000 145.000 60.000 75.000 135.000 46 .560 60.000 75 .000 90.000 120.000 135.000 145.000 46 .550 75.000 135.000 49.510 53.040 75.000 ") .000 L . 5 .000 75.000 Eo (} ~ diT nb/sr dr7 t:;,.!k.. dO nb/sr t:;,.!k.. d l1 % 0.7362 0.4485 0.0759 0 .7350 0.5880 0.4469 0 .3224 0. 1369 0.0755 0.0452 0.7760 0.4459 0.0752 0.7271 0.5845 0.4433 0.0745 0.5820 0.4408 0.2804 0 .0738 0.0441 0 .5807 0.4396 0.3159 0.1334 0.0734 0.6260 0.5795 0.4383 0.0731 0.0437 0.5771 0.4358 0.0724 0.7086 0 .5746 0.4333 0.3105 0 .1305 0 .0717 0 .0428 0.7067 0.4309 0.0711 0.6749 0 .6393 0.4285 0.0704 0.0420 0.4273 3.5720E+02 9.9500E+01 2.2700E+01 2.6950E+02 1.3020E+02 7.7000E+Ol 4.8500E+Ol 2.4200E+01 1.9800E+01 1.6950E+01 2.9572E+02 7 .3610E+01 1.8440E+01 1.7714E+02 8.6020E+01 4.9320E+01 1.2900E+01 6.4220E+01 3.7520E+01 1.9280E+01 9.7200E+OO 9.1100E+OO 5.5800E+01 3.2410E+01 2.0650E+01 1.0580E+01 8.8500E+OO 5.6370E+01 4.6820E+01 2.8050E+01 7.7900E+OO 7.4500E+OO 3.7120E+01 2.2240E+01 6.4900E+OO 5.9430E+01 3.0820E+01 1.7800E+01 1.1890E+01 6.2700E+OO 5.1500E+OO 4.7300E+OO 4.8330E+01 1.4510E+01 4.3500E+OO 3.3410E+01 2.7780E+01 1.1780E+01 3.5500E+OO 3.2700E+OO 1.0440E+01 1.3931E+01 3.9800E+OO 8.8530£-01 1.1049E+01 5.8590E+OO 3 .0800E+OO 2.3765E+OO 1.2100E+OO 7.9200E-01 6.7800E-01 1.4490E+01 3.6805E+OO 9 .2200E-01 8.8570E+OO 3.5268E+OO 2.5153E+OO 5.1600E-01 2.6330E+OO 1.5008E+OO 9 .6400E-01 3.8880E-01 4.5550E-01 2.2878E+OO 1.3288E+OO 1.0532E+OO 5.2900E-01 3.5400E-01 2.8185E+OO 2.3410E+OO 1.0940E+OO 3.1160E-01 3.7250E-01 1.8931E+OO 8.8960E-01 2 .5960E-01 2.9121E+OO 1.2328E+OO 7.2980E-01 5.8261E-01 3.1350E-01 2.0600E-01 2.3650E-01 2.4165E+OO 7.1099E-01 1.7400E-01 1.7039E+OO 1.3890E+OO 4.7120E-01 1.7750E-01 1.6350E-01 4.1760E-01 3.9 4.0 3.9 4.1 4.5 4.0 4.9 5.0 4.0 4.0 '1. 9 5.0 5.0 5.0 4.1 5.1 4.0 4.1 4.0 5.0 4.0 5.0 4.1 4.1 5.1 5.0 4.0 5 .0 5.0 3.9 4.0 5 .0 5.1 4.0 4.0 4.9 4.0 4.1 4.9 5.0 4.0 5.0 5.0 4.9 4.0 5. 1 5.0 4.0 5.0 5.0 4.0 Table 0.5- Cross sections and kinematics for Q~ $ 0 .5 (GeV /c)~ from Janssens, et al. , [34]. -160Q2 Eo E' () 2 (fm) - 2 GeV GeV deg 0 .5061 0.5066 0.5064 0.5451 0.5452 0.5453 0.5445 0.5456 0.5840 0.5837 0 .5833 0 .6230 0.6229 0.6228 0 .6230 0.6226 0 .6232 0 .6618 0.6613 0.6624 0 .6809 0.7009 0.7005 0 .7006 0 .7012 0.7013 0 .7398 0.7391 0 .7404 0.7782 0 .7787 0 .7792 0 .7787 0.7784 0 .7795 0.8567 0.8567 0 .8571 0.8559 1.0120 1.0899 1.1686 13.00 13.01 13.00 14.00 14.00 14.00 13.98 14.01 15.00 14.99 14.98 16.00 16.00 15.99 16.00 15.99 16.01 16.99 16.98 17.Ql 17.49 18.00 17.99 17.99 18.Ql 18.Ql 19.00 18.98 19.01 19.99 20 .00 20.01 20.00 19.99 20.02 22.00 22.00 22.01 21.98 25.99 27.99 30.01 0.9500 0.7350 0.5430 0.9500 0.9000 0.7690 0.5700 0.5590 0.9500 0.8020 0 .5970 0.9500 0.9000 0.8350 0.8020 0.6240 0.6120 0 .9500 0.8670 0.6510 0.8830 0.9500 0 .8990 0.8640 0.6770 0.6640 0 .9500 0.9300 0 .7030 0.9610 0.9500 0.9250 0.9000 0.7280 0.7150 1.0220 0.9500 0.7790 0.7640 0.8620 0.9100 0.9580 0.6803 0.4650 0.2732 0.6595 0.6095 0.4784 0.2798 0.2683 0.6388 0.4910 0.2862 0 .6180 0.5680 0.5031 0.4700 0.2922 0.2799 0.5974 0.5146 0.2980 0 .5202 0.5765 0.5257 0.4907 0.3034 0.2903 0.5558 0.5361 0.3085 0.5463 0.5350 0 .5097 0.4850 0 .3132 0.2996 0 .5655 0.4935 0.3223 0.3079 0.3227 0 .3292 0.3353 52.520 75.000 135.000 55.600 59.800 75 .000 135.000 145.000 58.750 75.000 135.000 62.000 67 .000 75.000 80 .000 135.000 145.000 65.360 75.000 135.000 75 .000 68 .890 75.000 80.000 135.000 145.000 72 .580 75.000 135.000 75.000 76.470 80.000 83.800 135.000 145.000 75.000 85.050 135.000 145.000 145.000 145.000 145.000 Q2 (GeV /c) ~ 0.6424 0.4261 0.0698 0.6090 0.5670 0.4238 0.0692 0.0413 0 .5751 0.4215 0.0686 0.5406 0.4923 0.4192 0.3763 0.0680 0.0405 0.5057 0.4169 0.0674 0.4158 0.4699 0.4146 0 .3720 0 .0668 0 .0398 0.4339 0.4124 0 .0662 0.4102 0 .3974 0 .3677 0.3371 0.0657 0.0391 0.4058 0.3235 0.0645 0.0385 0.0372 0.0366 0.0360 da ~da ~dq nb/sr nb/sr % 2.4420E+01 9.4100E+OO 2.9000E+OO 1.8330E+01 1.4290E+01 7.8700E+OO 2.4700E+OO 2.3700E+OO 1.3370E+01 6 .6400E+OO 2.1400E+OO 9.2800E+OO 7.7100E+OO 5.5400E+OO 5.1600E+OO 1.7800E+OO 1.6400E+OO 7.1600E+OO 4.9800E+OO 1.6300E+OO 4.7900E+OO 5 .6500E+OO 4.4400E+OO 3.6400E+OO 1.2900E+OO 1.1900E+OO 4.1600E+OO 4.1500E+OO 1.1800E+OO 3.4900E+OO 3.4000E+OO 2.5900E+OO 2.3200E+OO 1.1500E+OO 9.0300E-01 2.8190E+OO 1.8250E+OO 8.8200E-01 7.7700E-01 4.3200E-01 3.2500E-01 2.4900E-Ol 1.1966E+OO 3.8581E-01 1.4500E-01 9.1650E-01 7.4308E-01 3.2267E-01 1.2350E-01 1.1850E-01 6.5513E-01 3.3864E-01 1.0700E-01 4.6400E-01 3.9321E-01 2.6592E-01 2.5800E-01 8.9000E-02 8.2000E-02 3.5800E-01 2.4402E-01 8.1500E-02 1.9639E-01 2.7685E-01 2.2200E-01 1.7472E-01 6.4500E-02 5.9500E-02 2.1216E-01 2.5315E-01 5.9000E-02 1.7450E-01 1.6320E-01 1.1396E-01 1.1600E-01 5.7500E-02 4.5150E-02 1.3813E-01 1.0950E-01 4.4100E-02 3.8850E-02 2.3328E-02 2.2750E-02 1.9920E-02 4.9 4.1 5.0 5.0 5.2 4.1 5.0 5.0 4.9 5.1 5.0 5.0 5.1 4.8 5.0 5.0 5 .0 5.0 4.9 5.0 4.1 4.9 5.0 4.8 5.0 5.0 5.1 6.1 5.0 5.0 4.8 4.4 5.0 5.0 5.0 4.9 6.0 5.0 5.0 5.4 7.0 8.0 an Table 0.6 - Cross sections and kinematics for Q~ ~ 0.5 (GeV /c)~ from Janssens, et al., from reference (34]. an d1'! -161Q2 (GeV/c) 2 Q2 (fm)- 2 Eo E' 8 GeV GeV deg 0.3894 0 .7788 0.7788 0 .7788 1.1682 1.1682 1.5575 1.7522 1.9469 2.9204 4.0886 10.00 20.00 20.00 20.00 30.00 30.00 40.00 45.00 50.00 75.00 105.00 3.6850 5.2740 4.4330 3.6400 5.9070 4.4360 5.2780 5.2450 5.8880 5.8830 5.8860 3.4775 4.8590 4.0180 3.2250 5.2845 3.8135 4.4480 4.3113 4.8505 4.3268 3.7073 10.000 10.001 12.003 14.799 11.101 15.100 14.799 16.002 15.003 19.502 24.999 ~ 0.9833 0.9816 0.9737 0.9604 0.9755 0.9553 0.9536 0.9441 0.9489 0.9025 0.8248 ;m dt1 t:.. d t1 nb/sr nb/sr ;m % 1.7600E+03 3.3900E+02 2.2760E+02 1.4620E+02 8.7500E+01 4.3700E+01 1.8000E+01 1.0240E+01 8.1300E+OO 9.5200E-01 1.1300E-01 5.2800E+01 1.0170E+01 6.8280E+OO 4.3860E+OO 2.6250E+OO 1.3110E+OO 5.4000E-01 3.0720E-01 2.4390E-01 3.8080E-02 5.6500E-03 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 4.0 5. 0 ;m 6. du Table D. 7 - Cross sections and kinematics from Bartel, 1966, et al., from reference (37] q2 (GeV /c) 2 q2 (fm)-2 Eo E' 8 GeV GeV deg 1.9469 2.9204 9.5594 50.00 75.00 245.50 1.7666 2.3716 6.1281 0.7291 0.8153 1.0340 75.870 75.830 75.780 I! 0.3464 0.3106 0.1819 6. du nb/sr t:.. dt1 di"i nb/sr % 1.4100E-01 2.7600E-02 6.0400E-05 6.0630E-03 1.1868E-03 8.5164E-06 4.3 4.3 14.1 dt1 ;m ;m Table D.S - Cross sections and kinematics from Albrecht, et al., from reference (4) (GeV /c) 2 q2 (fm)-2 GeV 0 .9990 0 .9988 0.9987 1.4987 1.4989 1.4985 1.9983 1.9985 1.9986 1.9977 2.5011 2.5047 2.5061 2.4969 2.5011 2.4983 2.4992 2.4969 2.4977 3 .7447 3 .7438 3.7452 25.66 25.65 25.65 38.49 38.49 38.48 51.32 51.33 51.33 51.30 64.23 64.32 64.36 64 .12 64 .23 64.16 64.18 64.12 64.14 96.17 96 .15 96.18 3.9960 3.2960 2.9980 6.1970 3.2960 2.9980 6.1970 3.9960 3.2960 2.9980 7.9090 5.2530 3.8020 3.2940 7 .9090 6.1970 3.9960 3.2960 2.9980 9.9980 7.9110 3.9960 q2 Eo E' GeV 3.4636 2.7638 2.4658 5.3984 2.4973 2.1994 5.1321 2.9310 2.2310 1.9334 6.5762 3.9183 2.4665 1.9634 6 . 576~ 4.8657 2.6642 1.9654 1.6670 8.0025 5 .9159 2.0002 8 I! deg 15.440 19.060 21.180 12.150 24.640 27.580 14.400 23.840 30.220 34.140 12.590 20.090 29.960 36.200 12.590 16.550 28.040 36.170 41.400 12.420 16.260 40.030 0.9549 0.9325 0.9177 0.9687 0.8803 0.8534 0.9524 0.8774 0.8140 0.7719 0.9600 0.9030 0.8032 0.7325 0.9600 0 .9326 0.8243 0.7329 0.6720 0 .9534 0.9223 0.6462 ~ t:..~ Ll du nb/sr nb/sr ;m % 6.5930E+01 4.0650E+01 3.3080E+01 3.3070E+01 6.3500E+OO 4.8180E+OO 8.3000E+OO 2.4140E+OO 1.3340E+OO 9.9900E-01 4.7080E+OO 1.5380E+OO 5.6500E-01 3.5320E-01 4.7770E+OO 2.5660E+OO 6.8800E-01 3.4750E-01 2.4690E-01 9.7300E-01 5.1120E-01 4.7100E-02 9.2000E-Ol 6.1000E-01 5.1000E-01 4.9000E-01 l.lOOOE-01 7 .5000E-02 1.4000E-01 3 .9000E-02 3.2000E-02 1.7000E-02 6.7000E-02 2 .6000E-02 l.OOOOE-02 6 .5000E-03 6 .6000E-02 5.2000E-02 1.2000E-02 6.5000E-03 4.5000E-03 1.4000E-02 8.9000E-03 9.2000E-04 1.4 1.5 1.5 1.5 1.7 1.6 1.7 1.6 2.4 1.7 1.4 1.7 1.8 1.8 1.4 2.0 1.7 1.9 1.8 1.4 1.7 2.0 Table D.9 - Cross sections and kinematics from Litt, et al., from reference [40) -162Q2 Q2 (fm)-2 Eo E' 8 2 GeV GeV deg 0.2726 0 .3894 0.5841 0 .7788 1.1682 1.1682 1.7522 1.7522 2.7257 2.9204 3 .5045 3.8939 4.4779 5 .0620 5.8408 7.00 10.00 15.00 20.00 30.00 30.00 45.00 45.00 70 .00 75 .00 90.00 100.00 115.00 130.00 150.00 1.5777 1.9035 2.3617 2.7570 3.4389 3.4389 4.3069 4.3069 5.5000 6 .0000 6.0000 5.5000 6.0000 6.0000 6.0000 1.4324 1.6960 2.0505 2.3420 2.8164 2.8164 3.3732 3.3732 4.0475 4.4438 4.1325 3.4250 3.6138 3 .3025 2.8875 20.000 20.000 20 .000 20.000 20.000 20 .000 20.000 20.000 20.153 19.050 21.670 26 .280 26 .268 29 .277 33.754 (GeV/c) ~ 0.9372 0.9354 0.9324 0.9294 0.9235 0.9235 0.9148 0.9148 0.8993 0.9066 0.8725 0.8133 0.8017 0.7504 0.6714 da t::. da em t::. da nb/sr nb/sr % 7.6750E+02 4.1190E+02 1.5640E+02 7.5950E+01 2.3980E+01 2.3040E+01 6.0560E+OO 5.7920E+OO 1.2090E+OO 9.5800E-01 3 .6400E-01 1.3500E-01 7.2500E-02 2.9800E-02 1.0400E-02 1.6118E+01 9.0618E+OO 4.0664E+OO 1.7469E+OO 1.0791E+OO 6.6816E-01 2.7858E-01 1.9114E-01 6.2868E-02 7.7598E-02 1.6744E-02 7 .5600E-03 4.0600E-03 2.0562E-03 9 .9840E-04 2.1 2.2 2.6 2.3 4.5 2.9 4.6 3.3 5.2 8.1 4.6 5.6 5 .6 6.9 9.6 em Table 0 .10 - Cross sections and kinematics from Goitein, et al., from reference (42] em -163Q2 Q2 Eo E' 8 2 (fm)-2 GeV GeV deg 0.0780 0.1172 0.1194 0.1952 0 .1945 0.3122 0.3903 0.3891 0.3892 0.3892 0 .3890 0 .3895 0.3894 0.5072 0.5847 0 .5842 0 .5844 0 .5841 0 .5843 0.5788 0 .5840 0 .5843 0 .5844 0.5837 0.5845 0.5844 0.5846 0.5834 0 .7790 0.7784 0 .7791 0 .7791 0 .7792 0.7783 0.9729 0 .9737 0.9739 0.9731 0.9740 1.1683 1.1679 1.1667 1.1678 1.3631 1.3621 1.3626 1.3626 1.5576 1.5571 1.5576 1.7527 1.7524 1.7522 1.9461 2.00 3.01 3.07 5.01 5.00 8.02 10.02 9.99 10.00 10.00 9.99 10.00 10.00 13.03 15.02 15.00 15.01 15.00 15.01 14.86 15.00 15.01 15.01 14.99 15.01 15.01 15.01 14.98 20.01 19.99 20.01 20.01 20.01 19.99 24.99 25.00 25.01 24.99 25.01 30.00 29.99 29 .96 29.99 35.01 34.98 34.99 34.99 40.00 39.99 40.00 45.01 45.00 45.00 49 .98 0.6600 0.8150 0.6050 1.0640 0.7840 1.3640 1.5370 1.2490 1.2310 1.1420 0.8480 0.6960 0 .5560 1.7700 1.9120 1.6290 1.5400 1.5220 1.4310 1.1710 1.0720 1.0420 0.8920 0.8860 0.7180 0.7170 0.6470 0.6450 1.7890 1.6830 1.3920 1.0640 0.8650 0.7840 1.9130 1.7180 1.2240 1.0030 0.9150 1.9100 1.3750 1.1350 1.0400 1.7900 1.5120 1.2630 1.1600 1.6610 1.3890 1.2820 1.7890 1.5110 1.3970 1.6320 0.6184 0.7525 0.5414 0.9600 0.6803 1.1977 1.3290 1.0416 1.0236 0 .9346 0 .6407 0.4884 0.3485 1.4997 1.6004 1.3177 1.2286 1.2107 1.1196 0.8626 0.7608 0.7306 0 .5806 0.5749 0.4065 0.4056 0.3354 0.3341 1.3739 1.2682 0.9768 0.6488 0.4498 0.3693 1.3945 1.1992 0.7050 0.4844 0.3960 1.2874 0.7526 0 .5133 0.4177 1.0636 0.7862 0.5369 0.4339 0 .8310 0.5593 0.4520 0.8550 0.5772 0.4633 0.5949 25.250 25.250 35.150 25.250 35.150 25.250 25.250 31.740 32.270 35.150 50.060 64.720 90.270 25.250 25.250 30.240 32.270 32.700 35.150 44.480 50.060 51 .960 64.170 64.720 90.080 90.270 110.300 110.720 32.700 35.150 44.480 64 .170 90.080 110.130 35.150 40.210 64.170 90 .080 110.130 40.320 64.170 90.080 110.130 50.060 64.720 90.270 110.720 64.170 90.130 110.130 64 .720 90 .270 110.720 90.130 (GeV /c) da b. da ~ m nb/sr nb/sr 0 .9070 0.9061 0.8282 0.9042 0.8253 0.9015 0.8997 0 .8478 0.8433 0.8178 0.6738 0.5286 0.3084 0.8970 0.8953 0.8545 0.8367 0.8329 0 .8104 0.7198 0.6630 0.6436 0.5218 0.5165 0.2995 0.2982 0.1721 0.1700 0.8263 0.8032 0.7100 0.5102 0.2899 0.1665 0.7961 0.7451 0.4991 0.2809 0.1605 0.7358 0.4886 0.2725 0.1549 0.6231 0.4731 0.2632 0.1468 0.4687 0.2566 0.1447 0.4540 0.2485 0 .1375 0.2428 3.2800£+03 1.8570£+03 8.6300£+02 8.4100£+02 4.0000£+02 3.6100£+02 2.2850£+02 1.3280£+02 1.3100£+02 1.0800E+02 4.6030£+01 2.5290£+01 1.1780£+01 1.2860£+02 9.0100£+01 6.1400£+01 5.1600£+01 4.9360£+01 4.2570£+01 2.4120£+01 1.7870£+01 1.6770£+01 1.0030£+01 9.7300£+00 4.5500£+00 4.5360E+OO 2.9460£+00 2.9900£+00 2.3360£+01 1.9810£+01 1.1380£+01 4.5000£+00 2.1360£+00 1.3950E+OO 1.0540£+01 7.8800£+00 2.4300£+00 1.1070£+00 6 .9500E-01 4.4490£+00 1.4130£+00 6.1500E-01 4.1400E-01 1.4750£+00 8.1500E-01 3.6200E-01 2.3020E-01 5.4000E-01 2.2060E-01 1.4840E-Ol 3.2500E-01 1.5060E-01 9.1500E-02 9.2100E-02 9.9000£+01 5.5000£+01 2.6000£+01 2.6000£+01 1.2000£+01 9.0000£+00 4.6000£+00 2.6000£+00 2.6000£+00 2.2000E+OO 9.4000£-01 4.1000£-01 2.3000£-01 2.6000£+00 1.8000E+OO 1.2000E+OO l.OOOOE+OO 9.9000£-01 8.5000£-01 4.8000E-01 3.6000E-01 3.4000£-01 2.0000£-01 2.0000£-01 l.OOOOE-01 9.0000£-02 6.8000E-02 6.1000£-02 4.8000£-01 5.9000£-01 2.3000£-01 l.lOOOE-01 4.7000£-02 3.0000£-02 2 .1000£-01 1.7000£-01 5 .6000E-02 2 .8000£-02 2.4000£-02 1.8000£-01 3.1000£-02 1.6000E-02 2.1000£-02 4.1000E-02 2.9000E-02 1.2000£-02 8.8000£-03 2.0000£-02 9.2000£-03 5.0000£-03 l.OOOOE-02 6.1000£-03 5.5000E-03 4.9000E-03 Table D.ll - Cross sections and kinematics from Berger, et al., from reference [44) m b. da dii % 3.0 3.0 3.0 3.1 3.0 2.5 2.0 2.0 2.0 2.0 2.0 1.6 2.0 2 .0 2.0 2.0 1.9 2.0 2.0 2.0 2.0 2.0 2.0 2.1 2.2 2.0 2.3 2.0 2.1 3.0 2.0 2.4 2.2 2.2 2.0 2.2 2 .3 2 .5 3.5 4.0 2.2 2.6 5.1 2.8 3.6 3.3 3.8 3.7 4.2 3.4 3.1 4.1 6.0 5.3 -164- Q2 Eo E' 8 2 (fm)- 2 GeV GeV deg 0.2726 0.2726 0.3894 0 .5841 0 .5841 0.7788 0.7671 1.1253 1.7522 7 .00 7 .00 10.00 15.00 15.00 20.00 19.70 28.90 45.00 0.4489 0.4489 0.5570 0.7180 0.7180 0.8651 1.1051 1.1077 1.5975 0.3036 0.3036 0 .3495 0.4068 0.4068 0 .4501 0.6963 0.5080 0.6637 90.000 90.000 90.000 90.000 90.000 90.000 59.900 90.000 80.000 Q2 (GeV/c) ~ 0.3170 0.3170 0.3105 0.3002 0.3002 0.2905 0.5529 0.2748 0.3217 dl7 D. dl7 ~ dd nb/sr nb/sr % 2.3660£+01 2.5240£+01 1.2020£+01 4.8720E+OO 4.8060E+OO 2.1560E+OO 5.9570E+OO 7.0200E-01 2.0320E-01 7.3346E-01 7.8244E-01 3.7262E-01 2.0462E-01 1.6340E-01 7.7616E-02 2.0850E-01 3.3696E-02 1.0770E-02 3. 1 3. 1 3.1 4.2 3.4 3.6 3.5 4.8 5.3 m an m Table 0.12 - Cross sections and kinematics from Price, et al., from reference [48) Q2 (GeV/c) 2 Q2 (fm)-2 Eo E' 8 GeV GeV deg 0 .6700 0.6701 1.0002 1.0021 1.1690 1.1690 1.1673 1.5005 1.5007 1.7505 1.7508 1.7525 2.0002 2.0006 2.0006 2.3306 2.3307 2.3356 3.0004 3.0003 3.0003 17.21 17.21 25 .69 25.74 30.02 30.02 29.98 38.54 38.54 44.96 44.96 45.01 51.37 51.38 51.38 59.85 59.86 59.98 77.05 77.05 77.05 4.0980 0 .8047 5.0580 1.0480 5.2870 1.1370 1.1620 6.2730 1.3830 6.5610 1.5120 1.5440 6 .3600 1.6990 1.6990 6.5020 1.9010 1.9040 6.4980 2.3000 2.3000 3.7410 0 .4476 4.5250 0 .5140 4.6641 0 .5141 0.5400 5.4734 0.5833 5.6282 0.5790 0.6101 5 .2941 0.6329 0.6329 5.2600 0.6590 0.6594 4.8991 0.7012 0.7012 12.000 86.000 12.000 86 .000 12.500 90.000 86.000 12.000 86.000 12.500 90.000 86.000 14.000 86.000 86.000 15.000 86.000 86.000 17.660 86.000 86.000 dd ~ 0.9744 0.3257 0.9724 0.3092 0.9690 0.2729 0.3016 0.9695 0.2873 0.9653 0.2504 0.2774 0.9549 0.2683 0 .2683 0.9455 0.2571 0.2569 0.9180 0.2369 0.2369 D. dd an nb/sr di"i nb/sr 3.3730£+02 3.4240£+00 1.2130£+02 1.1290E+OO 6.8200£+01 6.2040E-01 6.7800E-01 3.3710E+01 2.8390E-01 1.7370£+01 1.4470E-01 1.5990E-01 8.4420E+OO 9.4410E-02 9.5610E-02 4.0480E+OO 5.2850E-02 4.9540E-02 1.0070E+OO 1.6980E-02 1.7030E-02 6.4087E+OO 9.2448E-02 2.5473E+OO 3.2741E-02 1.2958E+OO 1.6751E-02 1.6950E-02 6.7420E-01 9.6526E-03 4.1688E-01 4 .6304E-03 3 .9975E-03 1.7728E-01 2 .6435E-03 2 .2946E-03 1.0525E-01 1.6912E-03 1.2880E-03 2.1147E-02 6.6222E-04 4.5981E-04 D. dd dO % 1.9 2.7 2.1 2.9 1.9 2.7 2.5 2. 0 3.4 2.4 3.2 2.5 2.1 2.8 2.4 2.6 3.2 2.6 2. 1 3.9 2.7 Table D.13 - Cross sections and kinematics from Bartel, 1973, et al., from reference [3) Q2 (GeV /c) 2 (fm)-2 GeV E GeV deg 0 .9990 1.4987 1.9983 2 .5011 3 .7557 5.0751 9.9839 25.66 38.49 51.32 64.23 96.45 130.34 256.40 3.9960 6 .1970 6 .1970 7.9090 9 .9930 10.7000 13.3300 3.4636 5.3984 5.1321 6.5762 7.9916 7.9955 8.0097 15.440 12.150 14.400 12.590 12.450 13.990 17.590 Q2 Eo 8 ~ 0 .9549 0 .9687 0 .9524 0 .9600 0 .9531 0.9315 0.8449 Mi nb/sr 6 .5900£+01 3 .3060E+01 8 .2900E+OO 4 .7550E+OO 9.4800E-01 1.8560E-01 3.6200E-03 Table 0.14 - Cross sections and kinematics from Kirk, et al., from reference [13). Only selected Q 2 values are included. /::),.~ nb/sr 1.2000E+OO 5.9000E-01 1.7000E-01 7.6000E-02 1.9000E-02 4.0000E-03 1.4000E-04 D. dd di'i % 1.8 1.8 2.1 1.6 2.0 2.2 3.9 - 1652 (fm)- 2 Eo GeV E' GeV deg 2 .8830 3.6460 5 .0280 5 .0080 5.0320 7.3340 9 .6560 74.04 93.63 129.13 128.61 129.23 188.35 247.98 5.4830 5.4830 7.6400 6.6760 5.5070 9 .6250 11.4700 3.9467 3.5401 4 .9606 4.0073 2.8255 5.7168 6.3244 21.031 25.031 20.987 24.988 33.039 21.036 21.022 Q2 (GeV/c) Q2 8 ~ 0 .8886 0.8329 0.8572 0 .8079 0 .7006 0.8247 0 .7951 t::. ;m d<T nb/sr u ;m d<T nb/sr 7.9400E-01 1.9200E-01 6 .9200E-02 4 .5100E-02 2 .0400E-02 1.0800E-02 2.5100E-03 2.7500E-02 7.2100E-03 2.2200E-03 1.5700E-03 7.6200E-04 4.5000E-04 l.OOOOE-04 3.5 3.8 3.2 3.5 3.7 4.2 4.0 dd t::. d<T t::. d <T nb/sr nb/sr dn % 2.4967E+OO 1.4206E+OO 6.3371E-01 3.5682E-01 2.9407E-01 2.1422E-01 1.8051E-01 1.4337E-01 9.7431E-02 6.8208E-02 4.8225E-02 4.4398E-02 2.3288E-02 1.0190E-02 8.2186E-03 6.0238E-03 3.5357E-03 5.5809E-03 2.6420E-03 1.8611E-03 1.2449E-03 9.2033E-04 1.8 1.6 1.6 2.3 2.0 1.7 3.1 1.8 1.9 1.8 1.9 dd an % Table D.15 - Cross sections and kinematics from Sill, et al., from reference (50]. Only selected Q 2 values are included. Q2 (GeV /c) 0.4900 0.6200 0.8300 1.0100 1.0800 1.1700 1.2300 1.3100 1.4600 1.6000 1.7500 (fm)- 2 Eo GeV E' GeV deg 12.58 15.92 21.32 25.94 27.74 30.05 31.59 33.64 37.49 41.09 44.94 0.5041 0.5921 0 .7275 0.8391 0.8817 0.9360 0.9719 1.0194 1.1076 1.1890 1.2755 0.2430 0.2618 0.2852 0.3009 0.3062 0.3125 0.3164 0.3213 0.3296 0.3364 0.3430 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 Q2 2 8 ~ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 m Table D .16 - Cross sections and kinematics from Katramatou, et al., from reference (54] di"i -166- References [1] J. D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, (1964) [2] F. Halzen, A. D. Martin, Quarks and Leptons, (1984) [3] W. Bartel, et al., Nuclear Physics 58, 429 (1973) [4] W. Albrecht, et al., Phys. Rev. Lett. 18, 1014 (1967) [5] SLAC proposal E140, A Proposal to Measure the x and Q 2 Dependence of R=o-L/ o-r (1984) [6] S. 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