Search for Color Transparency in A(e,e'p) at High Momentum Transfer Citation O'Neill, Thomas Germano (1994) Search for Color Transparency in A(e,e'p) at High Momentum Transfer. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/GBR8-WR49. https://resolver.caltech.edu/CaltechTHESIS:05092013-102156284 Abstract Rates for A(e, e'p) on the nuclei 2 H, C, Fe, and Au have been measured at momentum transfers Q 2 = 1, 3, 5, and 6.8 (GeV fc) 2 . We extract the nuclear transparency T, a measure of the importance of final state interactions (FSI) between the outgoing proton and the recoil nucleus. Some calculations based on perturbative QCD predict an increase in T with momentum transfer, a phenomenon known as Color Transparency. No statistically significant rise is seen in the present experiment. 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: Unknown, Unknown Defense Date: 21 February 1994 Record Number: CaltechTHESIS:05092013-102156284 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05092013-102156284 DOI: 10.7907/GBR8-WR49 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 7689 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 09 May 2013 17:56 Last Modified: 04 Oct 2019 00:00 Thesis Files PDF - Final Version See Usage Policy. 29MB Repository Staff Only: item control page

Search for Color Transparency in A(e, e'p) at High Momentum Transfer Thesis by Thomas G. O'Neill Jn Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1994 (Submitted February 21 , 1994) II To my family and Alice III ACKNOWLEDGEMENTS I would like to thank the members of the NE-18 collaboration for their contributions to making the experiment a success. Special thanks go to Eric Belz, Rolf Ent, Wolfgang Lorenzon, and Naomi Makins for their work in the set up and analysis of the experiment. I also acknowledge the good work of Carl Hudspeth, Bob Eisle, the beam operators, and other SLAC staff members. I am especially grateful to my adviser Brad Filippone. Brad has provided me with countless hours of guidance and discussion, and I have profittcd greatly from his suggestions and also from observing his methods. My graduate tenure was made much easier by the knowledge that my adviser had my best interests in mind. Finally I would like to thank all of the friends who have offered support throughout my graduate studies . I am particularly indebted to my family, who helped me get here, and to Alice Ilouseholdcr, who put up with me once 1 did. IV Abstract Rates for A(e, e'p) on the nuclei 2 H, C, Fe, and Au have been measured at momentum transfers Q2 = 1, 3, 5, and 6.8 (GeV /c) 2 . We extract the nuclear transparency T, a measure of the importance of final state interactions (FSI) between the outgoing proton and the recoil nucleus. Some calculations based on perturbative QCD predict an increase in T with momentum transfer, a phenomenon known as Color Transparency. No statistically significant rise is seen in the present experiment. v Contents I Introduction 1 A 3 B c II 1 The Plane Wave Impulse Approximation . 6 2 Previous A( e, c'p) Data 12 Color Transparency 17 1 Selection of Point Like Configuration l7 2 Color Screening . 21 3 Lifetime of Point Like Configuration 24 4 Color Transparency Experiments 27 Overview of the Experiment 30 Experimental Apparatus 33 A Electron B <'am 33 B Targets 34 c The Magnetic Spectrometers 35 1 1.6 GeV /c Spectrometer 37 2 8 GeV /c Spectrometer 41 D III The A(e,e'p) Reaction. Trigger Electronics . 48 Analysis 53 A Particle Identification 54 B Track Identification 56 c 3-Momentum Reconstruction 60 D Coincidence Identification . 63 E Reconstruction of Em and Pm 68 Ionization Energy Loss . 68 1 VI 2 IV v VI Coulomb Acceleration 69 F Kinematic Settings .. 71 G Efficiency Corrections 76 1 Absorption of Protons in Target and 8 GeV /c Materials 76 2 Summary . . . . . . . . . . . . . . . . . . . . . . . . 78 The Plane Wave Impulse Approximation Calculation 83 A Overview of the Calculation . . . . . . . . . . . . . . . . 84 B The Born-Level Plane Wave Impulse Approximation Cross Section 85 c Kinematics and Luminosity 86 D External Radiation . 87 E Internal Radiation . 91 F Multiple Scattering, Energy Loss, and Coulomb Acceleration 95 G Spectrometer Acceptance Monte Carlos 99 Results 103 A Extraction of S(Em, Jlm) 104 B Measurement of T( Q2 ) 121 C Systematic Errors . . . 130 D Comparison of T( Q2 ) with Theory ] 34 1 The Glauber Predictions . . . . 134 2 The Color Transparency Predictions 142 Conclusions 150 APPENDIXES 153 A Coordinates 153 B Track Purging 156 VII c Dead-Time Corrections 160 D Coincidence Identification 164 1 Scintillator Timing Corrections 170 2 Scintillator Timing Parameters 173 3 1.6 GeV /c Scintillator ADC Malfunctions 177 E The Model Spectral Function 178 F Radiation in the PWIA Calculation 185 1 Effect of External Radiation on Count Rate 185 2 Effect of Internal Radiation on Count Rate 187 3 Internal Radiation in the One-Photon Limit . 188 4 Description of the Modified Equivalent Radiator Approximation. 191 5 Internal Radiation and Kinematics . . . . . . . . . . . . . . . . . 193 Vlll List of Figures 1 Farrar et al. Color Transparency prediction for A(e, e'p). 2 2 Differential cross section for 3 He(e,e') atE= 3.3- 14.7GeV 4 3 Born-LevePH(e,e'p) . . . . . . . 6 4 A(e, e'p) in the Born-level PWIA 8 5 Schematic A( e, e'p) showing proton intial 3-momenLum 9 6 Em spectra from 12 C( e, e'p) at Saclay . . . . . . . . . 13 7 ls and 1p Pm distributions from 12 C( e, c'p) at Saclay 14 8 A(e, e'p) results for the spectroscopic sum rule 15 9 Schematic 1 H( e, e'p) and A( e, e'p) 17 10 Lowest-order elastic scattering . . 18 11 The Feynman mechanism for elastic scattering 19 12 The A+ B ---* C + D exclusive reaction . . 20 13 Constituent scaling in hadron form factors 22 1·1 Total hadron-proton cross section vs. elastic slope parameter 23 15 T measured in A(p, 2p) . . . . . . . . . 28 lG Correlation ofT and dujdt in A(p, 2p) 29 17 Plan view of End Station A . . 30 18 Schematic of experiment NE-l 8 31 19 Exterior view of the 1.6 GeV /c spectrometer 38 20 1.6 GeV /c spectrometer detector hut . . 39 21 Side views of the 1.6 GeV /c detector hut 40 22 Exterior side view of 8 GeV /c spectrometer . 42 23 Magnetic optics of 8 GeV jc spectrometer in the vertical plane 43 24 Magnetic optics of 8 GeV /c spectrometer in the horizontal p lane 44 25 The 8 GeV /c Detector Stack . . . . . . . . . . . . . . . . . . . . 45 IX 26 8 GeV jc scintill ators 46 27 8 GeV lc wire ch ambers 48 28 1.6 GeV Ic trigger electronics 50 29 8 GeV lc scintilla tor electroni cs . 51 30 8 GeVIc trigger electronics . . 52 31 Ty pical track in the 1.6 GeV lc spectrometer 57 32 Typical track in the 8 GeV lc spectrom eter 59 33 Effect of timing corrections for 197 Au at Q2 = 7 (GeV lc) 2 64 34 flLv-e for 12 C at Q 2 = 1 (GeV lc) 2 65 35 Histogram of fltv-e for (Em, Pm) E R 66 36 Histogram of fl.L p-e for (Em,Pm) E :F 67 37 Central Pm for a given Os . . . . . . 72 38 Coverage of (Em, Pm) at Q 2 = 1 (GeV lc) 2 • 74 39 Cover age of (Em, Pm) at Q 2 = 6.8 (GcV lc) 2 75 40 Proton absorpti on efficiency for hyd rogen . 79 41 One-photon external radiation in 1 H( e, c') 88 42 One- photon internal virtual rad iation in 1 H ( e, e') 92 '1:3 One-photon internal bremsstra hlung in 1 H ( c, e') 92 44 Em distribution of 1 H (e,e'p) at Q2 = 1 (GcVIc) 2 95 45 1.6 GeV lc focal plane distributions 100 46 8 GeV lc fo cal plane distribution s 101 ·17 2 1·1 Pm distributi on at Q 2 = 1- 7 (GeV lc) 2 107 48 2 [J Em distribution at Q 49 12 C Em distribution at Q2 = 1 (GeV lc) 2 110 50 12 C Em co unts at Q 2 = 1 (GeV lc? a nd 08 = 43.4° 111 -51 12 C Em distribution at Q 2 = 1- 7 (GeV lc) 2 •. 112 52 Em distribution of 12 C model spectral function 113 2 = 1 7 (GeV lc) 2 . 108 X 53 12 C lp and ls Pm distributions at Q2 = 1 (GeV /c) 2 114 54 12 C Pm distribution at Q2 = 1- 7 (GeV /c) 2 115 55 56 Fe Em distribution at Q2 = 1- 7 (GeV /c) 2 116 56 197 Au E m distribution at Q 2 = 1- 7 (GeV /c) 2 117 57 56 Fe Pm distribution at Q 2 = 1- 7 (GeV /c) 2 118 58 197 Au Pm distribution at Q 2 = 1- 7 (GeV /c) 2 119 59 Nuclear transparency T(Q 2 , 08 ) measured in 2 H(e, e'p) 124 60 Nuclear transparency T(Q 2 ,08 ) measured in 12 C(e,e'p) 125 61 Nuclear transparency T( Q2 , 08 ) measured in 56 Fe( e, e'p) 126 62 Nuclear transparency T(Q 2 , 08 ) measured in 197 Au(e, e'p) 127 63 Nuclear transparency T(Q 2 ) measured in experiment NE-18 128 64 Nucleon-nucleon total and elastic cross sections 135 65 Standard T predictions for 12 C . 137 66 Standard T predictions for 5 ° Fe 138 67 Standard T predictions for 197 Au 139 68 Farrar et a/. C, Au T predi ctions 144 69 J ennings and Miller 1992 C, Au T predictions 14 5 70 Bcnhar et a/. C, i\ u T predictions . 146 71 Fra nkfurt , Strikman, a nd Zhalov C, Fe, Au T predictions 14 7 72 Pl a n view of ex periment NE-18 153 73 Formation of 1.6 GeV /c M cab and Tcab 161 74 Formation of 8 Ge V / c h /3 and h/3,only 162 75 Coin cidence T DC' non-linearitics . 165 76 ] .6 GeV /c coincidence TDC 167 77 8 GcV /c coin cidence TDC 168 78 Coin cidence identification 169 79 Pm di stribution from 58 Ni( e, e'p) at Saclay 184 XI 80 Single-photon Y(Bw, <Pw) at Q 2 = 1 (GeV /c) 2 . 189 81 Single- photon Y (Ow, <Pw) at Q 2 = 1, 7 (GeV /c) 2 190 82 Angular distribution of radiation at Q 2 = 1 (GeV /c) 2 195 83 Recoi l ratio Rat Q2 = 1 (GeV /c) 2 • . . • . . . . . . • 197 XII List of Tables 1 Approximate NE-18 Kinematics 31 2 Width of energy defining slits 33 3 Target characteristics . 35 4 Target-related materials 36 5 Nuclear radii and average Coulomb accelerations . 70 6 Detailed NE-18 kinematics . 71 7 Proton absorption in liquid target materials 78 8 Proton absorption in solid targets 80 9 Proton absorption in 8 GeV / c spectrometer . 81 10 Tracking efficiency and proton absorption corrections 82 11 The (Em, Pm) range R 122 12 Correlation tail correction to the PWIA calculation 126 13 Nuclear transparency T measured in experiment NE-18 129 14 Uncertainties in the PWIA calcu lation 131 15 Experimental systematic errors 132 16 Total systematic errors 132 17 Laboratory coordinates 154 18 Spectrometer (Transport) coordinates 154 19 Focal plane (hut) coordinates 155 20 1.6 GeV /c track purges 157 21 8 GeV /c track purges 158 22 Timing conventions . 164 23 Scintilla tor timing parameters 174 24 Shell-independent parameters of IPSM S(Es, p ) 180 25 Shell-dependent parameters of IPSM S( Es, p ) 181 Xlll 26 Equivalent radiator thicknesses ). . . . . . . . . . . . . . . . . . . . . 193 1 I. INTRODUCTION In 1982 Mueller and Brodsky [1] independently proposed that in wide angle exclusive processes, the ini t ial and final state interactions (lSI and FSl) of hadrons in nuclei would vanish at high energies . This effect, based on arguments using perturbative QCD (pQCD), is called "Color Transparency" (CT) , in reference to the disappearance of the "color" forces between the h adrons and nuclei. The onset of CT with increasing energy can be tested by measu rement of the nuclear transparency T, the ratio of the measured cross section to the cross section expected in the limit of complete CT (i.e., no lSI or FSI). At low energies, T < 1 because of absorption or deflection of the hadrons by lSI and FSI with the nucleus. At energies below the onset of CT, T should be independent of Q 2 (assuming a constant proton-nucleus cross section). As the energy increases and pQCD processes begin to dominate the scattering, however, T shou ld increase to unity. Some recent models of CT (see Figure 1, for example) exhibit significant in creases in T for Q2 as low as 5 (GeV /c) 2 [2- 6] . Measurement of an unambiguous increase in T cou ld indicate the energy scale at which QCD becomes perturbative in excl usive processes. We present measurements of T for the reaction A(e , e'p) on 2 I-I, C, Fe, and Au nuclei at momentum transfers Q 2 = 1, 3, 5, and 6.8 (GeV /c) 2 . The following sections present a general overview of the theoretical and expenmental questions of interest in the current experiment. First, in Section I.A there are descrip t ions of the formalism used for the analysis of A(e, e'p) a nd of the results of previous A( e, e'p) experiments. Following this, in Section I.B there is a discussion of the current theoretical and experimental knowledge of CT. Finally, in Section J. C there is a brief description of the current experiment. 2 0.0~-----------.1 ----------.1 ----------·1 ----------~r 0 2 4 Q2 (GeV /c) 2 6 8 FIG. 1. Farrar et al. [2] T prediction for A(e, e'p) on A= 12 (upper curves) and 200 (lower curves), calculated in the absence of Color Transparency (solid line) or with the naive parton (dashed line) and quantum diffusion (dash-chain line) Color Transparency models [2]. The CT models are described in Sections I.B.3 and V.D. For comparison, the highest Q2 in the present experiment is 6.8 (GeV /c) 2 • 3 LA. The A(e, e'p) React ion Some of the best information about the structure of the nucleus has been derived using lepton-nucleus scattering. Experiments using leptonic probes have several advantages over those using hadronic probes. Leptons couple to the nucleus through the r<.>latively weak electromagnetic interaction, while hadrons also experience the strong force. The electromagnetic interaction is calculable through low-order diagrams of the precision-tested theory of QED. By contrast QCD, the theory of the strong interaction, is not as well-tested and its perturbative expansion docs not converge quickly (or at all) at the energies of interest. Furthermore, the lSI and FSI of a weaklycoupled leptonic probe are much smaller than those of a hadronic probe, minimizing the distortions to the measured structure. Finally, due to the small coupling, the structure is measured throughout the nuclear volume, rather than just at the surface. Early information about nuclear structure was provided by inclusive electronnucleus scattering. The dependence of the A(c, c') cross section on the e nergy loss at intermediate Q 2 exhibits, among oLiwr structure, a broad "quasielastic" peak due to the direct knockout of an individual nucleon from the nucleus (see Figure 2). The width of the peak is the res ult of the distribution of nucleon momenta inside the nucleus. Approximate nucleon momentum distributions can be extracted from the data through, for example, the study of the distribution in the scaling variable y [7]. Semi-exclusive m easurements such a'3 A(e, e'p) a llow a more complete characterization of the scattering. Four-momentum conservation can be used to infer the 'J-momentum of the ejected proton before the interaction (up to distortions due to higher-order couplings- known ac; radiatiw effects- and the proton 's FS1). The distribution in the proton initial energy shows direct evidence of the distinct levels predicted by the nuclear shell model. The simultaneous measurement of the proton initi a l momentum allows the extraction of the momentum distribution not just for 4 4 250 >4) ::::s '-.. s... 200 E 0 =3.258 CeV 3 150 2 4) ..... 100 '-.. m 50 (ll 1 s:: 0 0 100 200 300 400 500 "' in MeV 250 500 750 1000 1250 CJin MeV 0.6 0.10 >4) ::::s 0 .4 fll 0 .2 '-.. s... 4) +> '-.. E 0 ~10.954 GeV 0.08 E 0 oc14 .696 GeV 0.04 m ... ..• •• 0.06 0.02 c 0 .0 0 500 1000 1500 2000 2500 0.00 0 "' in MeV F IG. 2. E I/ 1000 2000 CJin MeV Differential cross section dajd0.jdi/ for 3.3-J4.7GeV and ()k' = 8° (8]. 1s calle d w . 3 3000 He(c,c') at In the figure, the energy loss The data exhibit a bump at I/ -::::= 100, 500, 1100, 2000 MeV [Q"l -::::= 0.2, 0.9.5, 2.1, 3.6 (GeV jc) 2 ] due to quas ielastic nuclea n knock-out. The strengt h at larger v is due to excitation of a nucleon to the ~{1 236) resonance a nd other inelastic c h an nels. At Q2 .2: I (C:('V jc) 2 , the quasie lastic peak disa ppears in to the s ide of the rapidly ris ing b ackground from deep ine lastic scatte ring. 5 the entire nucleus, but for each nuclear shelL Thus the exclusive reaction allows much more accurate tests of the single-particle description of the nucleus and better differentiation between competing models of nuclear structure. 6 !.A.J. The Plane Wave Impulse Approximation The simplest A(e, e'p) reaction is electron-proton elastic scattering. Figure 3 shows the Born-level diagram (with time going from left to right) and establishes the kinematics for this process. The momentum t ransfer is Q2 = -qiJ.q~-' and the 4-momentum transfer is ( 1.1) The circle at the photon-proton vertex, indicating the unspecified details of the coupiing, is suppressed in the following diagrams. The Born-level cross section for this process is: da _(da) -E' (G'i; + rG~ + G2 tan 2 -0) dO E 1+ 2 - dO 2T Mott M T where the reference cross section for a structureless, non-recoiling target is pI ~ = (E ; ' p ') FIG. 3. Born-Level 1H(e,e'p). H e re k 1', k 111 , p1', and p'~L are the 4-momenta of the incoming electron, the outgoing electron, the incoming proton , and the outgoing proton, respectively. (1.2) 7 2 2 du ) a cos f!.. ( dO. Matt - 4£2 sin4 2~. ( 1.3) Here a = e 2 /47r1ic is the fine-structure constant, 0. = (0, ¢) is the laboratory solid angle of the scattered electron, and T = Q2 /411-1 2 . The effect of the structure of the proton is specified by the electric and magnetic form factors, GE(Q 2 ) and GM(Q 2 ), with approximate values given by the dipole form: Q2 ) -2 0.71(GeV/c)2 ' GP "' ( 1 + - - - - E- GpM ~ flp QPE· (1.1) ( 1.5) Here fl p is the magnetic moment of the proton. For nuclear targets (A > 1) at quasielastic kinematics and high energies [Q 2 ~ 1 (GeV /c) 2 ], the Plane Wave Impulse Approximation (PWIA) is expected to dominate the A ( e, e'p) reaction m echanism. The PWIA assumes k, k', p, and p' are undistorted plane waves (no Coulomb distortions, JSI , or FSI) and that the proton detected in the final state absorbed the entire momentum transfer. The Born-level PWIA is shown in Figure 4. In the PWIA , the proton is treated as being essentially free ("quasi-free") before the reaction, except that it is bound with separation energy Es and moving with momentum p: ( 1.6) The initi a l momentum p is t hC' res ult of th e Fermi motion of the nucleons. The separation e nergy Es is the energy it would take to remove the proton from the nucleus to infinity without recoil (i.e., TA-l = 0). In the quasifree knockout reaction the proton loses an additional en ergy TA-l to the recoil of the A - 1 system. In the PWlA, the scattering ejects the proton from the nucleus without ISI or FSI. (The reaction is "quasi-elastic'·.) 8 pI~ = (E ~ ' p ') (MA,O)~ .. ~(M A-l +TA-l' -p) FIG. 4. A(e, e'p) in the Born-level PWIA. The part of the Feynman diagram above the dashed line is equivalent to Figure 3, except that the proton initial 4-momentum pll- is given by Equation 1.6. ln the Born-level PWIA, p~-' can be reconstructed from the experimentally determined 4-momenta: p~-' = p'~-' - q~-'. Thus , Es and p can be measured by the missing energy ( 1. 7) and missing momentum Pm =p' - q ( 1.8) not accounted for in the detected particles [9]. In reality, the relations Em = Es and Pm = p are modified by FSI and radiative (higher-order) effects, which entail the exchange or emission of additional (and unobserved) 4-momentum (d. Equations 4 .5 and 4.6). Even in the absence of these effects, Em and Pm would measure Es and p 9 k FIG. 5. Schematic A(e, e'p) showing effect of proton intial 3-momentum on kinematics. Note the propagation length r v h/Q of the photon is in reality much smaller than the size of the nucleus (which is indicated in gray). only up to the experimental resolution. Thus, the experime ntall y determined quantities Em and P m can only be considered to be approximations to the actual separation e nergy Es and initial m omentum p. Note th a t in place of Pm, some authors [9] refer to the momentum PB = - p m of the recoil A - 1 system. Vve adopt a conven tion where Pm, the magnitude of P m, IS ass igned a positive (negative) sign if p' is at a la rger (sm aller) angle with respect. to the beam in the elect ron scatte rin g pl ane than is q . (For example, Figure 5 shows a kin em atics where Pm < 0. Note this convention is opposite to that used in [9] .) Because of its sepa ration energy Es and Fermi motiou pin the nucleus, the proton is off-shell. De Forest. [10] and others [9] give a prescription for extending the nucleon current. to off-sh ell kinematics in the Born-level PWJA , resu lting in the cross section: 10 (1.9) Here fl k, and fl p, are the solid angles of the outgoing electron and proton, respectively. The additional four degrees of freedom with respect to Equation 1.2 are from the unconstrained proton initial 4-momentum. The nuclear structure is characterized by the spectral fun ction S(Es, p ), the p robabi lity density for findin g a proton with separation energy Es and 3-momentum p. The normalization of S(Es, p) is given by the spectrosco pic sum ru le: (1.10) where Z is the nuclear ch arge. The off-shell electromagnetic interaction is specified by CTep, the square of the electron-proton elastic scattering amplitude. Equation (1.9) illustrates the intuitive idea that the quasie lastic cross section should be the product of the probability S(Es, p) of finding a proton with 4-momentum pfL and the probability CTep of scattering from it (multiplied by a .Jacobian). In the r eaction, the A - 1 system m ay experience a n excitation to any possible bound or unbound state lA- r >. Because the Born- level cross section, Equation 1.9, ignores the fin a l state of the A- I system, it can be considered to be the closure sum over Ill- ]* >. Data. from the Sacla.y 12 C( e, e'p) experiment [11] indicate that the rate is confi ned to low-lying A- 1 states included in Em < 100 MeV (see Section LA ), so th e use of thi s cross section is appropriate. One goal of many A(e, e'p) experiments is comparison of the experimental cross sect.io ns with th ose predi cted using the P\VlA cross S<"ction (or a refin ement thereof) and a single-particle descrip t iou of the nuc lea r structure, given by the Independent P a rt icle Shell Model (IPSM). In the IPSM , the spectral function is the sum of the contribution from each sh ell: (1.11) 11 where N; is the occupation number of orbital i (2::=; N; = Z), L;(Es) is the energy profile due to the finite lifetime of the one-hole state, and <p;(p) is the bound-state wave function. Except for the the least-bound shell, which may be partially filled, N; = 2j; + 1, where j; is the total angular momentum of orbital i. It is customary in the analysis of A(e, e'p) data to calcu late the cp;( p ) in a Woods-Saxon nuclear potential with shell-dependent adjustable parameters (see Appendix E). 12 J.A.2. Previous A(e, e'p) Data The quasielastic proton knockout reaction has been studied for Q 2 up to 0.79 (GeV /c) 2 and nuclei ranging from the lowest masses (2 H, 3 H , 3 He) up to 208 Pb [9,12-14]. The theory outlined in the previous section, with some refinements, has had great success in explaining many features of the observed data. But there still are unresolved issues, such as the failure of the spectroscopic sum rul e, that can be explored by measuring the reaction at higher Q2 . The measured quantities most relevant to the current experiment are the nuclear spectra l function S(Em,Pm) and the spectroscopi c factors. The spectral function is measured using Equation 1.9. First, the counts are binned in Em and Pm using Equations 1.7 and 1.8. Then S(Em,Pm) is extracted by dividing the cross section measured in an (Em, Pm) bin, corrected for experimental acceptances, by <7cp· Distortions due to radiative effects are removed in a generali zation of the radiative correction procedure used in e lastic 1 ll( e, e') scattering. Other distortions from proton FSI and Coulomb distortions of the particle waves arc not removed from the data, so the measured S(Em,Pm) are referred to as "effective" spectral functions. Instead, the data is compared to the prediction in the Distorted Wave Impulse Approximation (DWIA) which includes these effects [9]. The resu lting S(Em,JJm) can be converted into proton energy and momentum distributions by integration over d3 pm and dEm, respectively. The treatment o f radiation and distortions in the current experiment is som ewhat different (see Section V .A). F igure 6 shows Em spectra measured on 12 C at Saclay [11]. The Em acceptance extends up to "' 60 MeV. The data clearly exhibit the presence of tbe lp and 1s energy levels predicted in the nuclear shell model. The narrow 1p peak at 16 MeV exhibits substructure at Em = 18 and 21 MeV clue to the final state of the A -1 = 11 B recoil nucleus. The states have rapidly cleC!·easing relative strengths: the ~ - grou nd 13 1~,---,r--~---.---.-~---,-----, 12C (cc,,..-p)11B 0~ P._: 36 fvkV/c . .. ... -......·.-..... .. . ..... Q z 0 ~ u w <I) 60< P< 1&0 fvkV/c <I) <I) 030 a:: u 0 FIG. 6. Em 60 1J spectra MISSING ENERGYifvkV) from 12 C(e, e'p) at 70 Saclay [11]; (a) 0 ~ Pm ~ 36MeV, (b) 80 ~ Pm ~ 180MeV s tate has 74%, th e ~- (2.12 MeV) excited state has 1 ~3 %, the ~- (5.02 Me V) has 9% , a nd higher excitations have 4% [9] . This indicates that all of the ra t e involves the low-lying A - l stat es which are include d in Em < 100 M e V. The 20-MeV width of the 1s she ll at Em = 38 MeV is due to the s hort lifetime of the 1s hole state. The ide ntifi cation of th ese peaks with the nuclear s hell model states is confirme d by the clear l = 1 and l = 0 nature of the momentum di stributions m easured for the she lls, shown in Figure 7, and the good agreement with the Pm distributions calculated in the DWIA using t he IPSM spectral function (Equ at ion 1.11). The cp;( p ) and L;(Es) used in the lPSM S (Es, p) a re based o n c ha rge distributions from e last ic e lectron -nucleus scattering and separati on e nergies from (p, 2p) [11 ,15]. The good agreement. between the 12 C( e, e'p) data and the DWIA calcul ation demonstrates that t he s ing le-particle model sim ultaneously expla ins the three different d ata sets. Another feature of Figure 7 is that the PWIA a nd DWIA calculation s differ in 14 t 60 15< E< 21 5MeV >- 1- <f) z w 0 30<E<SOMeV 0 so 100 150 200 2 so 300 RECOIL MOMENTUM (MeV/c) PIG. 7. ls and lp Pm distributions from 12 C(e, ~'p) at Saclay [11]. The solid and dashed lines are DWIA and PWIA calculations respectively, with normalizations fit to the data. magnitude and shape. This distortion of the Pm distribution is due to the FSI of the proton, which arc generall y calculated with an optical-model of the proton-nucleus potential [9,13]. The DWIA calculations are sensitive to the scattering kinematics [9] and can be subject to large theoretical uncertainties [13]. The existing A( e, e'p) data have good agreement with the shape of the DWIA/IPSM calcu lation, but are consistently lower in magnitude. The discrepancy is seen for the entire nucleus in the incomplete ("' 70%) saturation of the spectroscopic sum rule (Equation 8), as shown in Figure 8. The discrepancy persists up to A = 208 Pb, the heaviest nucleus measured [13]. The measured fraction of the sum rule strength is closely related to the nuclear transparency T, except that it is corrected for absorption in the proton FSI. Thus, the value expected for standard FSI is unity. By contrast, for the nuclear transparency, the expected value is less than one. The 15 t f• •t4 N :::::w 0~ n "',w + 9 t OWIA ~ PWIA t + ? ? *? oQ " ? ~ 0 0 10 20 JO 40 GO ,0 II. F IG. 8. A(e , e'p) r e s ul ts for t h e s p ect r oscopic s um rule, fro m refe r e n ce [9]. connection between the nuclear transparency T and the m eas u red fraction of the sum rule strength is elabora t ed in Section V. B. Part of the s um rule disc re pa ncy m ay be due to pe r fo rming the integrals over the fin it e ranges of Pm a nd Em imposed by ex pe rimenta l accepta n ces . Ex pe riments with la rger acceptance ranges exhibit sm a lle r discre pa ncies [9] . In particular, o ne must be care fu l to take account of the large "' 15% strengt h a t IPml > PJ (where PJ is t he Fermi mome ntum) due to short-ra nge corre latio ns in the initial state. See Sect ion V.B for the treatment of these issues in the curre nt experiment. Another proble m is un cert a inties in the m odel o f t he proton F SI due to P a uli b locking , m edium modifications, a nd nucleon-nucleon corre la tions [16] . Most of these un cert ainties sho uld be smaller a t Q 2 ;:: 1 (Ge V /c) 2 [17]. A possible ex planation for the re m a ining discrepa ncy is t h a t the virt ua l p hoton can couple to mo re th a n one nucleon because for Q 2 < 0.8 (G eV /c) 2 its wavele ngt h 1icJQ is compara ble t o the nucleon ra dius RN ~ 0 .8 fm . The ex pectati on tha t t he inte raction should be single- particle in nat ure a t higher Q 2 is consistent wit h the observatio n of y scaling at Q 2 ;:: J (GeV /c) 2 [1 8]. However it has been arg ued [14] 16 that the multi-nucleon contribu t ions for 50 ::; Em ;S 100 MeV may increase at higher Q2. It is clear that the A(e, e'p) reaction changes character at Q 2 ~ 1 (GeV /c) 2 . First, it b ecomes more truly a reaction invol ving a single nucleon a nd, as Q 2 increases, the virtual photon resolves not just individual nucleons but a lso individual partons therin. As a conseque nce t here is the onset of the dominance of inelastic scattering in the inclusive A( e, e') reaction, as shown in Figure 2. In addition, treatment of the FSI should simplify at Q 2 ~ 1 (GeV /c) 2 where, due to the shorter de Broglie wave length of the outgoing proton, the FSI are dominated by nucleon-nucleon scattering. In Section V it is shown that a t such high Q 2 , the FSI produce a reduction in the proton flux without much distortion in the shape of S(Em, Pm)· Given the theoretical uncerta inties in the low-Q 2 distortion calculations [13], it is important to verify the low-Q 2 S(Em, Pm) with higher-Q 2 measurements. Furthermore, since th e strong co upling constant a 5 decreases with increasing Q 2 , one expects Q C D to enter the pcrturbative regime. Hence, the internal dynamics and FSI of the struck proton could change dra mati call y a t Q 2 > 1 (GeV /c) 2 , possibly leading to entirely new effects such as Color Transparency. For all of these reasons, it is imperative to test the A( e, e'p) qu asielastic scat tering mechanism a t Q 2 > 1 (GeV /c) 2 . T he curr<'nt ex pe rim ent represents the extension of th ese measurements by nearl y a n order of magnitude to Q 2 = 6.8 (GeV /c) 2 . 17 I.B. Color Transparency The arguments leading to CT involve three assumptions. First, observing an exclusive reaction requires the hadrons to ha.ve fluctuated to a small size- referred to as a point like configuration (PLC)- at the time of the interaction. Second, the PLC experiences a reduced interaction with the nucleus, a phenomenon known as color screen ing. Third , the hadrons remain small while they propagate out of the nucleus. The three assumptions are discussed in the following sections, with special attention paid to the A( e, c'p) reaction. The process is shown in Figure 9 for the reaction A(e,e'p) with A = 1 H and A > 1. /.8.1 . Selection of Point Like Configuration The first assumption can be supported by a simple pQCD model. As Q 2 increases and QCD becomes perturbative, the lowest order diagrams dominate the (quasi) c lastic scattering. Such a diagram is shown in Figure 10 for the case of electronproton e lastic scattering in the Breit frame where q1, = (0, Qz) (choosing z II Q). k k FIG. 9. 1 Schematic H(e, e'p) and A(c, e'p). indicating the 3-momenta involved m 18 q = (0, Q) -Q/6 -Q/6 Q/6 Q/6 -Q/6 Q/6 FIG. 10. Lowest-order elastic scattering. To lowest order in a 8 , the scattering involves the minimum possible number of partons {3) and gluon exchanges (2). The scattering is shown in the Breit frame, where the proton momentum is changed in sign but not in magnitude, and v = 0. The quark lines are labeled by the momentum components along Q. For definiteness the quarks are s hown as having typical momentum values (Bjorken x = 1/3) b efore a nd after the collision. (After Jennings and Miller [3].) Unless the struck quark shares the momentum transfer with the other quarks, the proton fragments and the reaction is inelastic. Hence, clastic scattering requires the exchange of two gluons (in the lowest order) of 4-momentum ,....., (0, Qz) and off-shell mass,....., Q. By the uncertainty principle, the gluons have a lifetime of,....., lifcQ and a range of,....., h/Q. Thus an elastic interaction only occu rs if the proton h appens to have fiuduatC'd to a PLC of transverse extent b ;S li/Q before:> the interaction. At large Q 2 , b can be much smaller than R 1., the eq uilibrium radius of the hadron. (Small longitudinal extent is guaranteed by Lorentz contraction. ) Implicit in the simple model is the roughly equal distribution of momentum b etween tlw quarks. In the Fcynman mechanism, however, the proton fluctuates to a state where a lmost all of the momentum is carried by a single parton of 4-momentum 19 ~-------. -Q/2 Q/2 ---------- q = (0, Q) Before After FIG. 11 . The Feynman mechan ism for elastic scattering. As in Figure 10, the scattering is shown in the Bre it frame and the quark lines are labeled by the long itudina l momenta. Crosses indicate the "wee" partons, which carry a lmost none of the longitudinal momentum. '"" (0, - Q /2) (see Figure 11). The momentum is reversed in the scattering, restoring the proton to its initial state without any gluon exchanges. At high Q2 t his diagram is suppressed, however, because the la rge acceleration of the struck quark produces gluon radiation (in analogy with e lectromagnetic radiation from accelerated electrons), making the interaction inela.'Stic. This reasoning leads to the most general argument supporting the formation of a PLC: the radiation of gluons is only suppressed if the color-neutral proton Ouctuates to a PLC with no residua l color fie lds (in analogy with the vanishing of the radiation field of a zero-size electric dipole). This a rgument does not require the exchange of a small number of gluons between the p artons, and therefore can apply a t energies lower than t he onset of pQCD. Unfortunately, it is not known what Q2 is high enough for g luon radi a tion to p revent elastic scattering from non -PLCs. The selection of the PLC may be responsible for the constituent co unting rule scaling behavior predicted by pQ CD and observed in many exclusive processes for Q2 2:, 5(GeVjc) 2 . Consider a reaction invol ving two (possibl y composite) particles in both the initi al and final state, with a total of N constituents before a nd aft e r 20 FIG. 12. The A+B-+ C +D exclusive reaction. Particle X( = A, B, C, or D) has nx elementary constituents (nx = 1 is allowed). The total number of constituents is N = nA + na + nc +no. the interaction (see Figure 12). If the momenta involved are much greater than the particle masses, then constituent counting predicts the cross section scales [19) : !(GeM)/ sin 2 N- 4 (8c M / 2) (Q2 )N-2 (1.12) Heres is the Mandelstam invariant and f(8 c 111 ) is a function of the cente r of mass angle 8 c 111. The relation Q2 = s sin 2 (8cM / 2) is used. In the PLC-dominated mechanism, the factor 1/Q 2 for each additional constituent represents the geometri c probability"' (bj R,.) 2 that its transverse separation from the location of the scatterin g is< b = hjQ. (The factor l / Q 2 arises for the outgoing consti t uents as well because the probability of a PLC fluctu ating back to a full-sized ha dron is the same as th a t of tlw full -sizf'd hadron flu ctua ting to the PLC.) Equ ation 1.12 a ppears to expl a in the scaling o bser ved at 8 c; 111 = 90° and s ,2:: 2 ( GeV /c) 2 for t he reactions 1 H(p,2p) and 2 li (!,p)n (see Figure 16 and reference [20]). For electron-hadron scattering , the scaling results in (1. 13) 21 where Fh(Q 2 ) is the elas tic form factors of a hadron with nh constitue nts. This scaling is obser ved in the d ata of Fig ure 13 for Q2 ~ 5 (Gc V jc) 2 . In t he case of elastic electron-proton scatt ering, relevan t to the current ex periment, the scaling can be seen as the limit of the dipole form (Equa tions 1.4 and 1.5) of the e lectri c and m agnetic form factors: . l 1m Q2~= . cvM / ji p -_ 11m GP _ [0.71 (G eV /c) j2 E Q 2 4 Q2~= ( 1. 14) T h e onset of the scaling beh avior is not conclusive eviden ce of t he onset of pQ C D. T he dispersion sum rule approach a rgues tha t the obser ved scaling is a cciden tal a nd th a t the true scaling does not set in unt il higher Q2 . For example, Bakulov and R adushkin [21] cla im t hat the Feynman mechanism dominates t he pion form factor for va lues of Q 2 up to 10 (Ge V / c ) 2 • I .B .2. Color Screening In Q C D t he color fi eld of a color ne utral object vanis hes as t he size o f the object is redu ced. This occurs because the fields of the indivi du al qua rks a nd gluons cancel each othe r as the size is redu ced , in a nalogy with the electri c di pole in QED. Thus, the interact ion of the PLC with the nucleus vani shes as its tra nsverse exte nt b becomes sm a ll. In pQ CD for sm a ll b, two-gluo n excha nge is the dominan t scatte rin g m echanism a nd the PLC-nucleon cross section is [27]: b2 CTPLC,N ~ CTh,N R~, (1.1 5) where CTh,N is the st andard h adron-nucleon cross section a nd R h is t he rad ius of t he free hadron. The cross section va nish es as b - t 0, in contrast wit h bl ack-disk scattering whe re the cross section is 1r( b2 + R'fv ) - t 1r R'fv. Alt ho ugh t h e t wo-gluon ap proximati on is proba bly not v<'ry accura t e for Q 2 ::; 7(GeV/c) 2 , there is evidence tha t Eq u ation 1.15 is valid over a wider kinem atic 22 ! ~r~ •• r j ---·· .••••• , / ... , ..... ,;•· 1::'" ' · 1(1 • : " ~--- Nvu lf (~r l . ,. r;: i""'"'" " a. l 0 Th 1s E<pe r~ me nt <..? 0 1 > I 4 Q? 1 8 0.3 0 .2 0 (JI . .!... -· 0.4 ~ a.::l! 0 I 4 P r ev10u":> Do to • j .':~: I 1 / f :::~:·"': • I()" 0 -BL • I •• ' • •.~ \l\4"1 11111' 11 1 v·! .', '" 'l 'l -- cz ····.v 11. 0 I 'otul, n 12 8 0 10 20 02 ((>('V/d 30 ( tGeVIc )2 ] FIG . 13. Cons titue nt scaling in hadron form factors. The le ft-hand figure, taken from r e ference [22], ·exhibits approximate dimensional scaling (Equation 1.13) at Q 2 ;?: 5 (Ge\1 jc)2 for the pion, proton, a nd n e utron. More r e cent proton d a ta is s hown in the right hand fi g ure, take n from reference [23] . range t han pQC D. For exam ple, Povh and Hi.ifner [24] independently arrived at a n analogous form throug h t he study o f had ron-prot o n c ross section d ata . They ded uce t he RMS hadron radius Rh from t he e lastic scattering s lope pa rameter, (1.16) where du'f.~fdt is t he d iffe rential elastic cross section and l = Q 2 is the Mandelstam invarian t . T he proton a nd pion radii a re fo und to be R p = J3bppj2 = 0 .82 ± 0.01 fm , R "+ = J3b1rP - R~ = 0.64 ± 0.01 fm , and R7r- = 0. 65 ± 0.01 fm , consist ent with the m easured electromag ne ti c radi i. T hey fi nd t h at t he to ta l cross section is given by (1.17) equ ivale nt to Equa ti on 1.1 .5. The d a t a are consistent with this linear b11 P dependen ce 23 7l .s t, p GO p 30 - n n· 20 K• 10 ~ K- <I> J I <lJ ---,+---, 0 0 5.0 10.0 b [Gev- 2 ) PIG. 14. Data for total cross sections a:,~t vs. the experimental slope parameters b for hadron-proton reactions at c.m. energy .jS::; 16GeV. for Rh between 0.84 (for pp) and 0.20 frn (for J j?/;p). (see Figure 14). Note that the radii relevant to the elastic and total cross sections are the equilibrium radii and are not reduced by the formation of a PLC because the clastic slope parameters are m easure d at t ~ 0 and because the total cross sections do not represent exclusive processes. Direct evidence of color screening has been seen in the Fermilab experiment E665 [25] in the same Q2 range as NE- 18 . The experiment measured A(Jt,ft'p0 ) and A(ft, 1t' <P) production on deuterium , carbon , calcium, and lead in deep-inelastic kine- m atics. In a simple mode l of the scat te ring, the photon fluctuates to a virtual quarkantiquark pair with transve rse extent b "" 1/Q (where Q2 is the muon momentum transfer) before it enters the nucleus, where it is scattered on-shell through a n "elasti c" interaction with a nucleon. The distance travelled by the m eson before obtaining full size is lh ~ fw jc4 mv!:::.m [26] whe re the e ne rgy loss is v ~ 200GeY, the p 0 or <P mass is mv "" 1 GeV /c2 , and the ty pi cal level splitting of quarkonium is m 1• '""'0.71 GeY /c 2 (based on the mass cliiTcrcnces of the p0 or <P excitations). The formation le ngt h is large ""40 fm because the proper tim f' of formation , "" 1ijc3 !:::.m '""'0.2 fm /c, 24 is subject to a large time dilation ,"' l/jc2mv "'200. The data were taken at the Fermilab Tevatron muon beam line with an average beam energy of 470 GeV. The outgoing muon and meson were detected in the E665 spectrometer. Exclusive meson production was selected by requiring the detection of the scattered muon and exactly two oppositely charged hadrons (from the meson decay). Because the recoil nucleon was not detected, kinematic cuts were used to select the incoherent scatterin g off of individual nucleons (describe above) from coherent scattering from the entire nucle us. The data show an increase in T and a decrease in the A-depende nce with increasing Q 2 , consistent with the onset of Color Transparency. Thus, hadron-nucleon reactions exhibit color screening in the Q2 range of NE-18. Note that this does not imply that Color Transparency occurs in A(e, e'p) and A(p, 2p) because for these reactions the PLC would be produced- if at all-by a different mechanism. Furthermore lh in E665 is larger than expected in A( e,· e'p) and A(p, 2p) ex periments at the sam e Q 2 because of the large time dilation. Frankfurt, Mille r, and Strikman [27] find additional s upport for the phenomenon of color screening in the observation of precocious Bjorken scaling in deep inelastic lepton-proton scattering, jet production in deep inelastic lepton-nucleus scattering (which suggests that Equation 1.15 may be valid forb :S 0.25 fm 2 ), and high energy difrractive processes. I.B.3. Lif etime of Point Like Configuration The PLC t akes a finite prop<'r tim<' to fluctuate back to its full size. Because the PLC is moving at momentum Q, time dilation guarantees that at some Q 2 the PLC will be able to propagate out of th e nucleus before returning to full size. The earliest quantitative models for the lifetime of the PLC were advanced by Fa rra r cl al. [2] (see Figure 1). The simplest is the "naive parton " model , which 25 assumes that the quarks expand back to their usual separation at the speed of light, taking a time T ~ R~t/ c. The PLC can propagate through a length (1.18) of the nucleus with reduced cross section, where 1 = E~/ Mh is the time dilation Lorentz factor. If the hadron is a nucleon , Rh = RN ~ 0.8 fm , and the PLC can have a significantly enhanced probability of escaping the nucleus, even for modest values The naive parton model ignores the basic quantum behavior of the system. For example, the virtual gluons and sea quarks can fluctuate back into existence at radii "' Rh on a smaller time scale. The more realistic "quantum diffusion" model assumes the expansion takes a total time given by the average value of the dominant pQCD energy denominator, IT=< li/(E~. - E~) >. Here E~ and E~. are the energy of the hadron and its intermediate state. The key point in this model is that the PLC is not the ground state of the free hadron hamiltonian (which has a much larger radius), and therefore represents a fluctuation to a state with MPLC > l\1h· By the energy-time uncertainty principle, this off-shell state can propagate a distance: he l~t ~ E'PLC - E'h - 2hp' c26_M2' (1.19) where p' is the momentum of the outgoing hadron (and of the PLC) and 6.M 2 M~LC - M~. The right-hand approximation is vctlid for p' / c » 1\1PLC, l\1h. Tl1c authors use 6.M2 = 0.7(GeV/c) 2 , which corresponds toT~ 0.5fm/c. (For easier comparison with the naive parton model result, 1 = E~/ Mh has been used instead of The other model of PLC expansion length commonly used to predict CT effect s, due to Jennings and Miller [3], is closely related. The authors expand the PLC in 26 a hadronic basis, IP LC >= L: Jh* >< h*IP LC >, where the sum is over hadronic excitations Jh* > with the same quantum numbers. Because the PLC is produced by cancellations between the Jh* > wave functions, the propagation distance lh is then given by the phase decoherence distance of the state JP LC(t) >= L: e(i/fi)p',..·rjh* >< h*IP LC(t = 0) >. Here the Jh* > are assumed to have equal energy E~, so p~. = VP + M;- M~.. Assuming the PLC is dominated by the nucleon IN > and a 12 low-lying nucleonic excitation JN1 > of mass Af1 and momentum p~, the decoherence distance is: (1.20) The result is similar to Equation 1.19. The underlying physics is basically equivalent, except that here the intermediate state is assumed to have the same energy as the outgoing proton, and so it is the position-momentum uncertainty principle that applies. One of the largest differences between models of CT in !l(e, e'p) is the value of 6M 2 or M 1 assumed. Larger h (smaller 6M 2 • Atfi) indicate that CT can set in at smaller Q 2 . The CT predictions use 6M 2 = 0.7- 1.1 GeV /c 2 [6] or /111 = 1.23- 1.77 GeV /c2 [3]. Clearly these ranges arc only estimates. A complication to this picture that has apparently been overlooked in the published theories is the possible dependence of M PLC on Q2 . Specifically, as Q 2 increases and b .:S h/Q decreases, the overlap < P LCIN > of the PLC with the ground state also decreases. Therefore, in the hadronic basis , the ov<'rlap < P LCJh* > with highermass states must increase. Thus, the !11 pu· may incrc>asc with Q2 , lessening the PLC lifetime r and the size of the CT-induced transparency increase. 27 I.B.4. Color Transparency Exper·iments Due to the difficulty of performing realistic QCD calculations at the relevant energy scales, there is no theoretical consensus on either the Q2 value at which the production of the PLC begins to dominate elastic scattering or the dominant mass scale M PLC (or M 1 ) of the PLC configuration. Thus, these values must be constrained by experimental searches for the onset of CT. The first experiment to investigate CT was performed by Carroll el al. (28], using simultaneous measurements of A(p, 2p) and H(p, 2p) reaction rates at Brookhaven National Laboratory. To determine T they divided the A(p, 2p) rate by the rate expected in the limit of full CT. This rate was given by the product of the measured proton momentum distribution and the H(p, 2p) rate, corrected for the theoretical variation of the elastic cross section and the difference in the experimental acceptances for t.he two reactions. Their results showed T increasing for incident proton momenta from 4.5 to 9.5 GeV /c, but then decreasing from 9.5 to 13 GeV jc (Figure 15). Because of the subsequent decrease, the increase at lower momenta cannot be taken as an unambiguous signal of CT. Ralston and Pire have noted that the free proton-proton cross section oscillates about the s- 10 behavior predicted by the pQCD constituent counting rule, indicating the interference of a soft process with the pQCD amplitude (29]. If tbe soft process is suppressed in the nuclear environment, this would lead to the observed bump in Tat 9.5 GeV /c [29] (see Figure 16). In contrast, the electron-proton interaction is well-understood and its cross section varies smooth ly as a function of Q 2 . Thus the A(e, e'p) reaction allows an important independent measurement of the Q2 dependence of CT. 28 '" ,I, ,., ,.T,, , ,l, " : .,. . ., f ,-,-r-t'l"T""'1'"""'T' ' I 00 (a Van o u s Nuclc~ • - LJ <> -c 0 50 >-. () c QJ s.... ~ _ 1t-. ...- - - - z.. .- - - ; , . ..ljJ ' r >< - AI 020 t -i / tn c <0 L 0 10 E- 0 .0 5 t¥ 05 >. 0 c 0 - 0 - f*t t t4 . QJ L. "' 0. ·- en : c 12 G<!V/c Q.~ . <0 L. f- 0 .1 >O 0 ~1 I ... •• ''I',,' I''' ,It'' 2 .5 5 7 .5 10 12.5 15 Incident Momentum Gev/c FIG. 15. T m easured in A(p, 2p) [28]. The graphs show (a) the transparency vs. beam momentum for nuclear targets, (b) the trans parency vs. effective beam mome ntum for the aluminum target. The e ffective b eam mome ntum is given by the actual b eam momentum, coupled with the measured fermi momentum along th e b ea m. 29 R,.----------------------. 8 6 (a) 4 2 2 3 4 In s(GeV2 ) 1.0 >u :z. 0 .8 a:: < 0 .6 C/) 0.-! w 0... :z. < a:: f- AE:27 (b) 0 .2 0 .0 2 3 4 ln s(GeV2 ) FIG. 16. Correlation of T and dajdt In A(p, 2p). erence [29], the graphs show (a) From ref- the energy dependence of R 1 (s) = const x s 10 da jdt for high-energy ]J]J elastic scattering at 90° c.m. angle [30] compared to a model including the interference of a soft Lands hoff-scattering process with the pQCD amplitude, (b) the prediction for oscillating transparency for A = 27 compared with the Carroll et al. [28] aluminum data. 30 I.C. Overview of the Experiment Experiment NE-18 measured A(e, e'p) scattering at quasielastic kinematics for A = 2 II, C, Fe, and Au aucl Q2 = 1, 3, 5, and 6.8 (GeV /c) 2 . E last ic II(e, e'p) was also ll!Casurcd as a Lest or systematics. The experiment was performed in End St<ttion A at the Stanford Linear Accelerator (SLAC) using the electron beam from the Nuclear Physics Injector (see Figure 17). Electrons and protons were detected in coincidence by the 1.6 GeV /c and 8 GeV /c magnetic spectrometers, respectively. A schematic of the experiment is shown in Figure 18. The kinematics used in the analysis arc summarized in Table 1. For the nuclear targets, m ul tiple proton angle settings were used in "perpendicular kinematics" to measure the desired range of Pm (see Figure .5 or Section III.F). fZl Quadrupole 1.6 GeV Spectrometer m Magnets Toroids 90° Dipole Magnet Fixed and Moveable Slits Chamber To Beam Dump • 15° Dipole Magnets End Station A 8GeV Spectrometer 2-94 7624A3 FIG . 17. P lan v iew of End Station A . 31 1.6 GeV/c Spectrometer k From NPAS injector, A line To Beam Dump 8 GeV/c Spectrometer Scattering Chamber FIG. 18. Schematic of experiment NE-18 TABLE 1. Approximate NE-18 Kinematics. Here E is the beam energy, E' and 0 1 6 are the momentum and angle seUings of the 1.6 GeV /c spectrometer (used to det.ect electrons), a nd p' a nd Bs a re the momentum and a ngle setti ngs of the 8GeV /c spectrom eter (used to detect protons). The exact kinematics are given in Table 6. Q2 E E' (}16 p' Bs (GeV /c) 2 (GeV) (GeV) (degrees) (GeV /c) (degrees) 1 2.0 1.4 36 1.20 35.0- 54 .G :3 3.2 1..5 48 2.45 27.7- 33.3 5 4.2 1.5 53 3.54 18.5- 22.6 7 5.1 1.5 57 4.49 15.9- 17.3 32 To identify coincidences, the time difference between the electron and proton triggers is converted into a time difference 6.tp-e between the electron and proton at the target by removing the effects of path length through the spectrometer and scintillator response time (Section III.D) . The coincidence rate inferred from the 6.tp-e distribution is corrected for the absorption of outgoing protons in the target and spectrometer materials. The e lectron and proton particle tracks measured in the spectrometers are converted to 3-momentum at the target by using matrix models of the spectrometer optics. These y ield the approximate 4-momentum of the proton before the collision via Equations 1. 7 and 1.8. The spectral function S(Em, Pm) is extracted as discussed earlier (in Section I.A .2). Using model spectral functions fit to this and previous data, the PWIA is used to predict the rate measured in the absence of FSI. We define the nuclear transparency T as the ratio of the measured coincidence rate to the calcu lated PWIA rate. Comparison ofT at Q2 = 1, 3 (GeV /c) 2 (where CT is not expected to occur) wit h the value expected using conventional nuclear physics tests the saturation of the spectroscopic sum rule. Also, the Q2 dependence of T is used to test for the presence of Color Transparency. In the limit of complete CT, T is expected to increase to unity. 33 II. EXPERIMENTAL APPARATUS II.A. Electron Beam The current experiment was performed in End Station A at SLAC usmg the electron beam from the Nuclear Phys ics Injector [31]. The beam energy was E = 2.015- 5.120GeVjc (see Table 6). Average currents were 0.1- 15J.LA, delivered in 12 J.LS pulses at a rate of 120Hz. The beam was transported to End Station A through the A-Line of the beam switch yard. The en ergy spread of the beam is defined by slits following eight momentum-analyzing dipole magnets (see Table 2). Two ferromagnetic toroids roughly 10m upstream of the target measured the beam charge. The toroids were calibrated at least once an hour by passing a known charge through a wire that passed through them. The charge measured by the two TABLE 2. Full width 26E/ E of the energy defining slits during the different experi- mental kinematics (cf. Table 6). An ent ry of"*" in t he target (or a ngle) column indicates that the co rresponding slit width was used for a ll targets (or angles), except for the two exceptions in the second ha lf of t he table. T hese exceptions, which effect the resolution but not the count rates, are ignored in the calcu lation of t he PWJA prediction for the expe rim ent (Section lV .A). Target 1.04,1.16,3.06,3.23 * 1.21 Bs 26E/E (degrees) (%) * 0.3 * 0.2 .') .00 ,5.14,6.77 ,6.85 * * 0.4 1.16 4 em 1 11 43.29 0.2a 5.00 6%C 20.88 0.3 aon ly 1 out of 6 of the run s 34 toroids agreed to within 1%. Beam position and angle on target were controlled by a position monitor and steering magnet closed-loop feedback system (the Beam Control System). The beam size was roughly 0.2 em high and 1 em wide. The beam delivery and monitoring have not ch anged substantially since SLAC experiment NE- ll, described in reference [32]. II.B. Targets The targets, listed in Table 3, were mounted on remotely moveable target ladders in a sealed vacuum scattering chamber. There were two target ladders, one holding the solid targets, and the other the cryogenic and "dummy" targets. Thin aluminum entrance and exit windows, described in Table 4, allowed the beam and scattered particles to pass through with minimum energy loss and multiple scattering. The targets were positioned at normal in cidence to the beam. The standard SLAC cryogenic liquid targets (I II and 2 H) were used. The liquid was contained in thin Aluminum target cells made of Coors aluminum beer can stock. The liquid was circulated at a flow rate of 2 m/s, with pressure and temperature monitored by pressure transducers and platinum resistors. These were used to calculate the target density. The small ( < 1%) contri bu Lion to the coincidence rate by the 0.0762 mm and 0.1219 mm thick upstream and downstream end caps was measured using replica empty cells (the dummy targets) with aluminum walls at the same positions but 9.5 times thicker. In the analysis, the inferred endcap coincidence rate is subtracted from the measured endcap plus liquid ratf'. 35 TABLE 3. Target characteristics. Targets are referred to by their nominal lengths (liquid and dummy targets) or radiator thicknesses (solid targets). Purity of solid targets is effectively 100%. The lengths and thicknesses of the liquid targets are given after applying the expected 0.996 shrinkage factor in cooling to 21 K. The dummy targets are two aluminum sheets of equal thickness at the positions of the endcaps of the liquid targets. Name Material Density Length Thickness (g/cm 3 ) (em) (% r.J.) Purity (%) 4 em 1 H Liquid H2 0.0705 4.011 0.461 99.94 4cm 2H Liquid D2 0.1701 4.011 0.557 99.68 15cm 2H Liquid D2 0.1701 15.680 2.176 99.68 2% c 12c 2.193 0.4097 2.104 6% c 12c 2.193 1.1730 6.024 6% Fe 56 Fe 7.829 0.1 064 6.019 12% Fe 56 Fe 7.829 0.2098 11.87 6% Au 197 Au 19.157 0.0206 6.137 12% Au 197 Au 19.157 0.04013 11.96 4,15cm dummy AI 3004 and 2.72 1.270 1.461 AI 5052 2.68 0.6096 0.691 II.C. The Magnetic Spectrometers Electrons were detected in the 1.6 GeV /c magnetic spectrometer [33] and protons in the 8 GeV jc spectrometer [34]. Both arc vertical bend spectrometers with focusing in momentum and scattering angle, yielding good ('"" 0.1 %) resolution in particle 3-momentum. The optics focus point-to-point from target to nwmentum fo cal plane in the bend plane and parallel-to-point in the hori zontal plane. Scintillator planes in both spectrometers were used to determine the Lime of particle passage, allowing the 36 TABLE 4. Target-related materials. The wire array measures beam position, and is used by the Beam Control System. The length and thickness listed arc averaged over the wires and the gaps between them. The hymen and chamber windows provide the vacuum isolation of the scattering chamber. The endcaps and cell wall are the ends and side of the aluminum can encasing a liquid target. The cell wall thickness includes the contribution of the liquid target flow guide. The mylar insulation provides thermal isolation of the cryogenic liquid targets. The endcaps, cell wall, and insulation do not apply to solid targets. Object Material Density Length Thickness (mm) (% r.l.) Wire Array AI 2.70 0.03990 0.0449 Entrance Hymen AI 2.70 0.02540 0.0286 Upstream Endcapa AI 5052 2.68 0.07620 0.0864 Downstream Endcapa AI 3004 2.72 0.12192 0.1403 Cell Waua AI 3004 2.72 0.12700 0.1461 Ins ulation" Mylar 1.39 0.06350 0.0221 1.6 Chamber Window AI 5052 2.68 0.12700 0.1440 1.6 Air Gap Air 0.00121 1.6 8ntrance Window Mylar 1.39 0.35560 0.1237 8 Chamber Window AI 5052 2.68 0.3048 0.3457 8 Air Gap Air 8 Entrance Window AI 5052 acryogenic liquid targets only 0.00121 2.68 132.588 160.0:2 0.25400 0.0434 0.0524 0.2881 37 selection of coincident electrons and protons. JJ. C. J. 1.6Ge Vj c Spectrometer The 1.6 GeV jc spectrometer, described in more detail elsewhere [33,32], is a 90° vertical bend dipole spectrometer (see Figure 19). It has a maximum momentum of 1.5 GeV jc, a solid angle t:.n ~ 3 m sr , and a momentum bite of ±6%. Note that the quadrupoles used to increase the solid angle for SLAC experiment NE-ll (reference [32]) were not used in the present experiment. After passing through the dipole magnet, e lectrons pass through a 1.5 m long gas Cerenkov counter filled with C0 2 at atmospheric pressure (see Figures 20 and 21) . Following this, they pass through two crossed layers (XD and YD) of 0.5 em t hick, 11.1 em wide plastic scintillators, 3 pairs (A, B, C) of X -Y drift chambers, another hodoscope pair (XU and YU) , and two rows (PA and PB ) of 10.8 em-thick (6.4 radiat ion lengths) lead glass shower counter blocks. Each drift chamber bad two p la nes of 31 X wires and two planes of 21 }·· wires spaced at 2 em intervals . These provided an effective wire spacing of 1 em and a coverage of approximately 40 em by 60 em. For paired wire hits, the drift time inform ation a llowed location of the track to a resolution of around ±0.3 mm. The Cerenkov and shower counte r provide electron ide ntification and pion rejection, a nd the hodoscopes provide trigger and timing information. The scintillators a nd segmented showe r co unter also provide position information used to remove tracking ambiguities. The 1.6 GeV /c spectrometer detector package was the same as in SLAC ex periment NE- ll , except that all of the scintillato rs had photomultiplier tubes (PMT) on both si des for better timing resolution. 38 1.6 GeV Hut Detector Dipole Bending Magnet Fixed and Moveable Slits Scattering ~::::2,­ Chamber Pivot 2-~ 7S<SA6 FIG. 19. Exterior side vtew of the 1.6 GeV /c s p ectrometer, as configured for the present experiment. The momentum a n a lysis is provided b y a 90° dipole bending magnet, with optical properties optimized by the use of rotated entrance and exit pole faces and other pole tip shaping (31] . The detectors a r e above the exit of the dipole magnet, ins ide the concrete shielding hut. 39 Upper Scintillators Lead Glass Shower Counter ~ •tt ~On Chambers Lower Scintillators ~Focal Plane ........... Meters 1.0 Cerenkov Counter 0.5 11-93 0 7545AS FIG. 20 . Side view of 1.6 GeV fc spectrometer detector hut with Y1 6 increasing to the left. Particles are incident from the bottom. The light guides of the YU scintillators, shown bent in the figure, were straightened before the experiment (see text for a description of the detectors). 40 =~I I I I I I jPB IPA Meters 1.0 xu YU CY ex BX BY 0 .5 AY AX XD YO 0 CK CK ) Y,6 ) F IG. 21. Side views of the 1.6 GeV jc detector hut showing the sizes and positions of the detectors to scale. With the exception of the lead glass shower counte r s, the thicknesses of the detectors are not to scale. 41 II. C. 2. 8 G e V/ c Spectrometer The 8 GeV /c is a 30° vertical bend QQDDQ magnetic spectrometer (Q = Quadrupole and D = Dipole, see Figure 22). It is discussed in more detail in reference [20], which describes an experiment that took data with the 8 GeV /c spectrometer in the same configuration (sec also references [32,34]). In order to increase the angular acceptance of the 8 GeV /c spectrometer, the polarities of the first two quadrupoles were reversed from the standard configu ration. In this large acceptance tune, the spectrometer has a maximum momentum of 4.5 GeV /c and a factor of > 4 larger solid angle. See Figures 23 and 24 for raytrace diagrams showing the optics of the large acceptance tune. The detector package also differed significantly from previous experiments {see Figures 25 and 26). The first detector encountered by a proton in the shielded hut was a 3.1 m long gas Cerenkov counter, which was filled with atmospheric Freon 114 (n = 1.00140) at the Q 2 = 5, 7 (GeV /c) 2 kinematics but evacuated at Q 2 = 1, 3 (GeV jc) 2 . Intended to provide 1r+ identification, it was not necessary in the analysis because the pion rate was only ~ 10% of the accidental coin cidence rate (see Section III.A ). Following the Cerenkov were the front layer of plastic scintill ators (SF), ten multiwire proportional chambers, two planes of scintill ator hodoscope (NBS), and two more scintillator layers (SM , SR). The lead g lass shower counter used in previOus experiments was removed. For optimal timing resolution, the scintillators were read out on both sides by PMTs (s<'e Figure 26). The three scintillator planes were refurbished and repositioned before the current experiment. The refurbishment included the replacement of all scintillator material with new Bicron BC-420 fast scinti llator material, and t.he replacement of the PMTs with fast (1.5 ns rise time) Amperex XP2020 phototubes from the decommissioned MARK-III detector at SLAC. 42 8 GeV SPECTROMETER Quadrupole Focusing Magnets Dipole Bending Magnets Target Pivot Platfonn • Bectron - Beam 7545A7 FIG . 22. Exterior side v1ew of 8 GeV /c s p ectrometer, as configured for the present experiment. The momentum analysis is provided by the two dipole magnets, with focussing provided by the three quadrupoles. The detectors are above and to the right of the exit of the dipole magnet, inside the concrete shielding hut. The SF plane wa'5 completely replaced. The eight new scintillators are 0.9525 em t hick, 6 em wide, and 45 em long. The scintilla.tors a.r<' vertically oriented, layered a lon g th e horizontal dir<'dion with 2 mm overlap. Light guides with alternating bend dir<'ct.ions prev<'nt posit.io n con fli ct s between adjacent readout PMTs. Th<' layer in eluded two of the old 21.6 em-wide S F scintilla tors, one on e ither end, to increa'5e the acceptance for SLAC experiment E-140X, which was run just be fore NE-18. The SF layer was placed in a new fr ame and mounted in fro nt. of the wire chambe rs. 43 5 0 15 10 z 20 (m) FIG. 23. Vertical (bend) plane magnetic optics of the 8 GeV fc spectrometer's large acceptance tune. Rays are started at <I> from - 18 to 48 mr in 6 mr increments (bottom through top sets of curves at Z < 2m) and o from - 5% to .5% in 1% increments (bottom through top set of rays at the tilted momentum focal plane at Z ~ 22m). The rays are propagated forward using the optics model described in Section IV.G and are cut off if they intersect an aperture. 25 44 20 0 X - 20 0 5 15 10 z 20 25 (m) FIG. 24. Horizontal plane magnetic optics of the 8 GeV /c spectrometer's large acceptance tune. Rays are started at X from -5 to 5 em in 1 em increments and e = -28 to 28 mr in 4 mr increments (bottom through top set of rays at the angle focal plane at Z:::::::: 21m). See Figure 23 for details . 45 CK SFT1 P2 T3 P4 T5 P6 T7 PB T9 P10 NBS Meters I 0 CK SFT1 P2 T3 P4 T5 P6 T7 I I I 0.5 SM SA SM SA I I 1.0 PB T9 P10 NBS PIG. 25. Top and side views of the 8 GeV /c spectrometer detector stack, to scale (detector thicknesses are exaggerated). The scintillator planes SF, SM, and SR were used to identify charged particles and record their time of passage. The 10 planes of multiwire proportional chamber (shown in gray) were used to measure the particles' focal plane coordinates. In the top view, Y8 (not shown) is out of the page, and in the bottom view X 8 is out of the page. In both views, Zs is to the right, with Z 8 = 0 indicated by the dashed line (see Table 19). Protons enter from the left. Note the stack is inclined at 30° from horizontal. 46 SR SM SF FTG. 26. Schematic diagram of the 8 GeV / c s cintillator timing planes as they would appear from the front of the detector hut, approximately to scale. The horizontal-by-vertical dimensions of the SF, SM and SR planes are 89. 1 e m x 45.0 e m, 101.6 c m X 41.6 c m and 101 .6 e m X 42.4 em, res pectively. 47 The replacement of the scintillator material reduced the thickness of the SM and SR planes to 1.27 em. The scintillators were 15.24 em wide and 1.016 m long. They were oriented in the horizontal direction, with at least 1 em overlap. The two planes were mounted at the back of the hut, separated by 29 .3 em along the central ray. The repositioning of the three scintillator planes increased the distance between the SF and SR to 386 em. This allowed for particle identification using particle velocity from Time-of-Flight (Section liLA). As with the Cerenkov counter above, this particle ide ntification was not necessary for the analysis. The multiple-wire proportional chambers (MWPC's) and NBS hodoscope are essentially unchanged from SLAC ex pe riment NE-ll [32) . The wire cham bers (MWPC's) were operated in proportional mode, with the ion cascades provided by a gas mixture of 65.75% argon, 30.0% isobutane, 0.25% Freon 13B1 and 3.0% m ethylal. The anode wires were arrayed in planes orthogonal to the central trajectory (the Zs ax is) . The angle of the wires from the }'8 axis, in order of increasing distance into the hut, were 90 , -30, 90, 30, 90, -30, 90, 30, 90 , and - 30° (see Figure 27). The five chambers at 90° , called the "P" chambers, had horizontally-oriented (along X 8 ) wires for m easuring vertical track position and angles. These alternated with the "T" chambers which were used to m easure the hori zontal position and angle. The names P a nd T refer to the sensitiv ity of the cha mbe rs to the proton momentum p' and scattering angle (0~), respectively. The wire spacing in the chambe rs was 2m m , although the effective spacing in the T ch ambers was 4 mm because adjacent wires w<:>rc <:>lectrically connected . The MWPC readout gate was 80 ns wide. The N BS hodoscope consisted of 22 verti cal a nd 20 hori zontal scintill ators, and was used to help remove tracking ambiguities. Each scintillator ha d a single PMT, the signals from which were fed into latches with a 15 ns gate. 48 T1 P2 T3 P4 T5 P6 T7 P8 T9 P10 FIG. 27. Oblique view of the 8 GeV fc wire chambers showing the alternating wire angles. The wires shown are those registering a hit during the event, and a track is shown connecting them. (The apparent variations in the track angle are a rtificially induced by the projection used.) II.D. Trigger Electronics The detection electronics of either spectrometer signaled th e presence of an event by generating a candidate trigger, called a pre trigger. Due to data-acq uisition rate li m itations, only one pretrigger per beam pulse was a llowed to generate a trigger, leading t o the read-out of the spectrometer's ADCs, TDCs, latches, sca lers, and wire chamber information. If either spectrometer generated a trigger during a given beam pulse, both were read out (sec Figures 28 a nd 30). The NE-18 trigger electroni cs were designed under the ass umption of a modest e lectron rate in the 1.6 GeV /c a nd a potentially large rate in the 8 GcV /c. The 1.6 GeV /c triggered on the first electron identified in a beam spill. The 8 GeV /c tri ggered on the first proton follow ing a 1.6 GeV /c tri gger, including true coincidences. 49 In the 1.6 GeV /c spectrometer (see Figure 28), two complimentary electron pretriggers were used to provide maximum efficiency. The module M et,low required a signal in the Cerenkov, as well as at least two of the following: a) a signal in the XU or YU, b) a signal in XD or YD, c) a signal in the PA shower counter greater than a "low" threshold. The pretrigger M et,high required signals in at least three of the four hodoscopes, a PA shower signal greater than a "high" threshold , and a total (PA + PB) shower signal greater than another threshold. The electron pretrigger P et is the OR of these. There was also a prescaled pion pretrigger P 1r,pre used to measure the 1r- background. Both spectrometers shared the random pretrigger P,·an, which was generated periodically to allow the determination of ADC pedestal values in both spectrometers. With the exception of adjustments and recalibrations, the 1.6 GeV /c trigger and readout electronics were essentially unchanged since NE-ll. In the 8 GeV fc (see Figures 29 and 30) the passage of a charged particle is marked by the firing of the proton module, M p,·ot, a 3-way coinciden ce between the three scintillator planes (SF, SM, SR). Candidate coincidence events are ide ntified by the pretrigger P coin, which resulted when Mpr ot, occurred during a 100 ns gate opened by the 1.6 GeV /c P et pretrigger (see Section III.D for more detail). The pretrigger Ptongcoin wa.s similar except with a 3.1 f.LS gate. The purpose of this pretrigger was to record any 8 GeV /c M prot single that followed a 1.6 GeV /c singles trigger in the same beam gate. The module M 213 fired when at least 2 of SF, SM, SR had signals. This was ANDed with the bea.rn gate, prescaled by a factor cp,· e = 2- 8 - 2- 10 , to c reate the pretrigger P 2/3,pre· The prescali ng was necessary so that the high M 2 13 rate (up to 8 per spill) did not create' a high computer dead-time. For the data on the hyd rogen targets the M2/3 rate was lowe r a nd Epre = 1 was used. The P 2 13 ,pre events were used to determine the efficie ncies of the sci ntillator pl a nes. 50 3 ?2 7 10 1AI 0 Coincidence MPulser Logical D Gale Gencralor Complcmenl 0 FIG. 28. The formation of the 1.6 GeV jc trigger. The labelled inputs are: "CK" = discriminator on Cerenkov signal, "PA Low", "PA High" = discriminators (with diffe rent thresholds) on analog sum ofPA signals, "SU" = logic sum of XU and YU discriminators, "SD" = logic sum of XD and YD discriminators, "SC" = logic sum of XU, YU, XD, and YD dis criminators, "SH" = discriminator on analog sum of PA and PB s ig nals . " SPILL" = beam gate, and "CK Veto" = not CK . The beam g ate is include d as a veto to the trigge r module to pre v e nt trigger s in b e tween b e am gates . 51 SF1U SMUL SMD ------' SRUL SRUR SRML SRMR SRDL SRDR FIG. 29. The 8GeV/c SF, SM, and SR signal electronics. The full e lectronics diagram is displayed for only one PMT in each scintillator plane. Only the upper SF detectors contributed to the SF summed signal. For our purposes, the mean timers in the SM e lectronics can effectively be treated as an AND gate (2/2 coincidence module). 52 213 Beam Gate RAN FIG. 30. 8 GeV fc trigger e lectronics. The formation of the pretriggers is described in the text. The 7 fLS-wide gate generator into the trigger veto input a llows only the first pretrigger in an event to cause a trigger. The trigger signal is fanned out to a scaler, TDC starts, ADC gates (30- 90 ns wide), latch gates (5- 80 ns wide), the MWPC readout gate (80 ns wide), the computer interrupt, and the 1.6 GeV /c coincidence TDC. 53 III. ANALYSIS 54 III.A. Particle Identification As discussed in Section III.D , the subtraction of accidentals ensures that the measured coincidence rate includes only electron-proton events. Thus no further particle identification is strictly necessary. However, rejection of contamination by other particle types, es pecially in the 1.6 GeV /c spectrometer, can reduce t he size of the accidental subtraction and of the statistical errors in the m easured S(Em, Pm) and T. The 1r- rate m the 1.6 GeV jc spectrome ter was up to 500 times the c- rate. Rejection of events with a Cerenkov ADC value below channel 60 redu ced the 1fcontamina.tion by a. factor of"' 300. The e- efficiency of this cut was m easured to be 99.93 ± 0.03%, using electron s identified with a t ight cut on the shower co unter. In the runs with high 1r : e ratio, most extra tracks in the 1.6 Ge V j c spectrometer were due to pions. Tn events with multiple tracks and an electromagnetic shower detected in the lead glass a.rray, the track associated with the shower was chosen (see Section lll.B). Events were not rejected if there was no electromagnetic showe r, however , because of the highly variable e- inefficiency (1 % - 5%). Some of t he reasons for the high c- inefficiency were variable gain in the PMT of one of t he counters (PA4) a nd an ADC m a lfunction similar to that of the 1.6 G<>V /c scintillator ADCs described in Appe ndi x D .3 . The two laye rs of the shower counter were offset to reduce the a lignment. of the cracks, but electrons passmg m between counte rs ll1 the first layer (PA) experienced enhanced shower leakage in the second (PB). The cause of the PA4 variab le gain was not identified. The ADC malfunction was cha racteri zed by the appearance of the sam e la rge value in a ll of the ch annels of the :3 ADC modules m easuring the shower counte r signals. As in the case of the scin tillator PMT malfunction, it is thought that these events were caused by problems with the gate input signa l. 55 The 8 GeV /c PROTON module provided minimal particle identification since an SF-SM-SR 3-way scintillator coincidence would fire on any charged particle in the 8 GeV jc spectrometer hut. Tests with additional particle identification by TOF measurements of the velocity in the hut (at Q 2 = 1, 3 (GeV /c) 2 ) and with the Freon gas Cerenkov detector (at Q 2 = 5, 7 (GeV /c) 2 ) demonstrated that the non-proton contamination was always small. (In the Cerenkov counter, a signal indicated the passage of a pion, while there would be no signal for a proton.) Thus, no further particle identification wa.s used in the analysis of the 8 GeV jc data. 5() III.B. Track Identification The drift chamber and wire chamber information in the 1.6 GeV /c and 8 GeV /c spectrometers, respectively, are used to measure the positions and angles of the detected particles at the focal plane. The identification and selection of the particle tracks are performed by modified versions of the existing 1.6 GeV fc and 8 GeV /c tracking software. The 1.6 GeV /c tracking software begins by identifying which wires recorded hits during the readout gate. When a pair of adjacent wires fired and had consistent drift time information, the drift Limes are used to infer the position of the track. For an unpaired wire, the position of the wire is used. The number of unpaired wires was as high as 30-40% in some runs. For a pair with inconsistent drift times, the position is chosen randomly between them. The software then generates candidate tracks by grouping nearly-colinear hit positions in the three chambers. The focal plane coordinates of the tracks are given by linear regression. Because the X and Y wire planes a.re orthogonal, this procedure results in separate, uncoupled tracks in both directions (see Figure 31). In events with multiple tracks, deciding which X track is associated with a given Y track is the task of the track purging software. The purging software is also responsible for identifying and discarding spurious tracks, based on the information provided by the other detectors (see Appendix B for details). The 8 GeV /c tracking software works on similar principles. In this case, there is no drift time information, and the hit position is given by the centroid of adjacent wires that fired during the 80 ns readout gate. (In the P chambers the hit locations arc randomized to ±1/2 wire spacing at no cost to the resolution. This is done to prevent the otherwise discrete nature of the possible track angles from causing spikes in the dYs and <l>s spectra. Such spikes would make the cross section sensitive to small changes in the value of the <P 8 cut used in the analysis.) If a sequence of more 57 PB PA xu YU CY ex BY BX AY AX XD YO CK CK ) ) FIG. 31. Typical track in the 1.6 GeV jc spectrometer. Detectors with signals are colored in . The thin rectangles in the drift chambers indicate hit positions of the particle t rack. Note the presence in the X chambers of hits not associated with the particle track. These m a rk the positions of dead or hot wires. 58 than two adjacent wires fired in a given chamber, the position used is given by the pair with the best x2 for the track. The software identifies candidate tracks in the P chambers first, and then identifies the associated tracks in the T chambers. Because the different T chambers had different wire angles (see Figure 27), hit positions are all converted into X 8 values through the use of the P tracks (see Figure 32). In most cases P and T tracks from different particles are inconsistent, so the ambiguity in matching P and T tracks was much smaller than in matching X and Y tracks in the 1.6 GeV fc. The combined P and T track is required to include hits in 2: 3 P chambers and 2: 3 T chambers, for a total of 2: 7 hits. If no tracks are found, this condition is relaxed to 2: 2 T, 2: 2 P, and 2: 6 total. The purging software is similar to that used for the 1.6 GeV /c, and is detailed in Appendix B. A tracking efficiency correction for each spectrometer accounts for inefficiencies in the chambers and in the tracking routine. For events with no final track in a given spectrometer, the correction is determined by measuring the area of the coincidence peak in the trigger time difference TDC (because tracking information is used to calculate the timing corrections, 6.-tp-e cannot be calculated for these events). The resulting values were 1.01- 1.03 in the 1.6 GeV /c spectrometer and 1.00 in the 8 GeV /c spectrometer, with errors of ±0.01 (sec Tables 10 and 10). The final three purges in either spectrometer are used to choose the final track when more than one track survives the other purges. In either spectrometer, this is usually necessary in ;S 1% of the events, although there are runs where it is necessary for up to ,...._ 10% of the events. It is expected that most of these ca..<>es the software either chooses t be correct track or a. track with (Em, Pm) within the range R used to measure the transparency. Rather than estimate a correction factor for these events, we include an extra systematic error of 1% for each spectrometer. 59 CK SFT1 CKSF T3 P2 T5 P4 T7 P6 SM SR T9 P8 P10 SM SR F1G. 32. Typical track in the 8 GeV /c spectrometer. Overhead and side view of the 8 GeV /c detector hut, with a compressed Zs scale to exaggerate the track angle (compare with Figure 25). Positions of track hits are shown in gray. 60 III.C. 3-Momentum Reconstruction The track focal plane coordinates measured in each spectrometer are converted into 3-momentum at the target by using a third-order matrix model of the spectrometer's magnetic optics. The matrix model was derived using the measured correlations between the momenta and angles in the 1 H(e,e') , 1 H(c ,e'p), and 2 H(e,e'p) reactions. The behavior of the optics at the edge of the acceptance was constrained with data taken at momenta or angles away from the nominal values. The resulting optics model agreed reasonably well with those derived from a fit to a Transport [35] model of the spectrometers (see Section IV . G). In each spectrometer, the momenta are reconstructed using a third-order matrix that maps from the position and angle of a particle track in the hut to the initial 3-momentum at the target. Usually this matrix (or a second-order version) is given by fitting to rays generated with the forw ard Transport model of the spectrometer (see Section JV.G ). F'or this experiment, however , the matrix elements were inferred solely from the kinematic correlations seen in 1 H(e,e') and A(e,e'p) data. These were then compared with the Transport-based matrix as a consistency check. Although some of the correlations were fitted using Minuit, most were fit by eye in order to discount the false correlations and centroid values caused by the rad iative tail, the spread in beam energy, and multiple scaiiering. These effects were the principle sources of uncertainty in the determination of the matrix elements. For example, the radiative tail contributes differently in Pm , Em, and ll\12 (the invariant mass, squared) spectra, and the fitted peak position depends on how close to the elastic value the data are cui. In some 1 H(e, e'p) spectra the effect of the beam energy spread on the correlations in one spectrometer was removed by inferring the beam energy with the other. The 1 H( e,e'p) data provided strong constraints on the matrix elements because 61 the measured centroids for Em and Pm should be 0 exactly, and should also have no correlations with any hut quantity. Additiona l constraints were obtained from the 2 H(e, e'p) data, where Em should reconstruct to the deuteron binding energy (2.2 MeV). Finally, it is worth noting that the determination of most of these matrix elements was performed iteratively, to ensure that any matrix element changes did not destroy good agreement for earlier-determined elements. The first step was optimizing the matri x ele ments used to m easure Ztarg, since this could be done without knowing any other matrix elements. (See Tables 18 and 19 for the coordinate systems used.) This was accomplished using singles data taken on the 15 em dummy cell, which consists of two aluminum targets with known positions and separations. The matrix elements were adjusted to generate two peaks at the correct positions with the best resolution (roughly 4 em in both spectrometers, depending on the angle). Next, the <J> matri x elem ents were determined to an estimated un certainty of 5 mr. The <1> 16 acceptance is limited to ±46.3 mr by slits placed in front of the 1.6 GeV jc dipole. The angular magnification ( <PJX) 16 was chosen to reproduce this width. Th e cl> 8 m atrix elem e nts were adjusted to produce Pm,y = 0 for 1H( e, e'p) scattering. (Here Pm,y is the out-of-pl ane com ponent of P m·) The <I> matrix e lements in both spectrometers were then adjusted to ensure that Pm,y = 0 , independen t of the measured values of cl>16, cl>s. The dispersive (8) matrix elements were then det ermined to an unce rtainty of about 0.2%. The first constraint was that the measured Em in 1 H( e, e'p) and 2 H( e, e'p ) should be 0 and 2.2 MeV, independent of any fo cal plane coord inate. The :3-momenta of the electron and proton are directly (ignoring radi ation) correlated in 1 I-I( e, e'p). Thus , for example, an Em- dXs, 16 correlation m ay be caused by either a (8 JdX . . . )!6 matrix element (where " ... " can be anything) or a (8JdX .. .)8 matrix element. The ambig uity is resolved through the use of 2 H(e, e'p ), which has a much weaker kine- 62 matic correlation between the electron and proton (i.e., IPml > 0 is allowed). The 8 matrix elements were also constrained through the use of 1 ll( c, c') and t H( c, e'p) data taken at non-central momenta, by ensuring that the position of the elastic stripes changed accordingly. For these data., we assume that the spectrometer magnet field configuration is linear with the NMR reading (to better than 0.1%). Finally, the scattering angle ~0 is determined using 1 ll(c, e'p) data where the angle of each spectrometer was stepped in both directions from its central value. (Here "central" refers to the standard kinematics, where 1 H(c, e'p) events arrive at the center of both spectrometers.) For these data, the elastic stripes should be in the correct location (for the electron, this is W 2 = M;), independent of the focal plane coordinates. The measured 6.0 16 and ~0 8 were further constrained by ensuring that Pm = 0 in all of the 1 H( e, e'p) data. The estimated uncertainty in the scattering angles is less than 2 mr. Although the spectrometer magnets were run through a "de-gaussing" cycle whenever their strength was lowered, there were still "'0.1% history-dependent variations in the magnetic field strengths. Hence it wa.<> necessary a.t. each Q2 t.o use 1 II(e,e'p) and 2 H(e, e'p) to re-determine the "offset" matrix elements to remove the effects of these variations on the measurement of Em and Pm. The offset matrix clements are those namely (811), (~011), and (<I>Il)- which do not. depend on the focal plane coordinates. 63 III.D. Coincidence Identification The pretrigger Pcoin resulted when an 8 GeV /c proton M prot occurred during a 100 ns coincidence gate opened by the 1.6 GeV /c electron pretrigger P el· The time difference ~t 8 _ 16 between the two spectrometers' triggers was m easured by the 8 Ge\1 /c coin cidence TDC. This is converted into a time difference ~t p-e be tween the electron and proton at the target by removing the effects of path length through the spectrometer and scintillator response time (see Figure 33). The path length corrections a re made using the Transport model, adjusted to produce the best resolution. The scintillator response times are determined using measurements of the Time-of-Flight b etween scintilla tors for particles of known velocity. The calculation of ~tp- e is detailed in Appendix D. Figure 33, which shows the uncorrected and corrected time differences for one of the worst cases cases, illustrates the utility of this approach. The FWHM of the coincidence peak was "' 0.6 ns (see, for example, Figure 34). The true to accide ntal ratio is l.ypica.lly 2: 10:1 , but at one kinematics is as bad as 1:1. T he rate of true coincidences is dete rmined by subtracting the rate o f acciden- tal s (measured away from the coincidence peak) from the rate within ±2 ns of the coincidence peak , and then correcting for true coincidences outside thi s 4 ns window. For the m eas urem ent of transparency, we need the numbe r of true coin cidences Ntrue in a range R of (Em,Pm) (see Figure 35). Let Np ea k be the number of counts in R with l~t p-el < ~tpeak/2, where ~tpeak = 4 ns is the width of the coin cidence window. The number of true coincidences is given by subtracting the inferred number of accidcnta ls under the peak: (3. 1) For improved determination of the accidental subtraction, the number of accidentals under the peak is cleduced from a large window -20 ns < ~l p- r < -6 ns (of width ~l wm = l 4 ns) where the accidenta l background is Oat. 64 35 30 25 2 20 r::: ;::l 8 15 10 5 0 -10 5 -5 0 Trigger Time Difference (ns) 10 -5 10 35 30 25 2 20 r::: ;::l 8 15 10 5 0 - 10 0 tot p-e (ns) 5 FIG. 33 . Coincidence time for 197 Au at Q 2 = 7 (GeV jc) 2 before (top) and after (bottom) corrections for flight path and scintillator response times. 65 800 rn ....,_;> 600 ~ ;:J 0 u 400 200 o~~~~~~~~~~~~~~~~~~~~~~rrl- -6 -4 -2 0 Llt p-e (ns) 2 4 6 FIG. 34 . .6.tp-e for 12 C at Q2 = 1 (GeV jc) 2 The number of counts in the 6.twin window and the range R can be small (especially in the spectral function measurement described in Section V.A). The determination of the background subtraction is improved by using the number of counts N!:;n in the "full" (Em, Pm) range F ( -100 < Em < 220 MeV and IPm I < 320 MeV /c), and then multiplying by the fraction Nacc/ N?"cc of these that should be in R. For best accuracy, Nacc/ N?"cc is determined using almost the entire 6.tp- e accidental range (see Figure 36). Specifically, N[cc is the number of accidentals in F with l6.tp-el > 5 ns, and Nacc is the number of these inside R. Thus there should be N!:;nNacc/ N?"cc accidentals in the rangeR in the accidental timing window 6.twin - This value, multiplied by 6.tpeak/ 6.twin, is the estimated number of accidentals in the coincidence window 6.tpeak (see Equation 3.1). Whe n calculating the error in the background subtraction, 66 o~~~~~==~~~4L~~L+~--~~4=~~~~==~ -10 -5 ~t 5 0 p-e 10 (ns) FIG. 35. Histogram of 6./p-c for (Em, JJm) E R the correct binomial error m Nacc/ N?'cc must be used when R "' :F because t he n A coincidence identification effi ciency correction of 1.01 5± 0.010 (1.000 ± 0.010 for the 1 I-1 t arget) is applied to correct for t rue coincidences outside of t he ~tpeak t iming wi ndow. This is m easured using a ~tpcak = 8 ns window, a nd verified by compa rison wit h t he number of coincidences calcul ated from t he 8 Ge V /c trigger time differe nce TDC . T h e effi cie ncy includes a correction for the sm a ll numbe r of t rue coinciden ces in th e underflow ch a nnel of the 8 Ge\1 fc coin cidence T D C, as describ ed in A p peudix D 67 8000 7000 (c NFace ) 6000 r:n 5000 -+-' c 4000 ;:::1 0 u 3000 2000 I I< ~t . I >I w1n 1000 -20 -15 -5 (ns) p -e -10 ~t 0 F IG. 36. Histogram of tl.Lp-e for (Em, Pm) E :F 5 10 68 III.E. Reconstruction of Em and Pm TheE, E', and p' measured in the experiment are affected by ionization energy loss and Coulomb acceleration. The PWIA calculation also simulates these effects on an event-by-event basis (see Section JV.A). The measured and PWIA values are corrected for the typical magnitude of these effects in order to improve the resolution of the measurement of Em and Pm· Any bias introduced by the use of typical values is common to the data and the PWIA calculation, and so does not effect the measured transparency. The effects of energy loss and Coulomb acceleration are added to E and subtracted from E' and p' to obtain the correct values at the time of the scattering. The energy loss is calculated using the measured angles and assuming the interaction occurred at the center of the target. The correction for the Coulomb acceleration is the average Coulomb potential energy inside the nucleus (modelled as a uniform sphere of charge) . lli.E.J. Ionization Energy Loss The energy loss is calculated usmg the material thicknesses g1ven 1n Tables 3 and 4. !11 cases of non-normal incidence, t hi cknesses are increased by the appropriate geometrical factors calculated using the measured angles and assuming the interaction occurred at the center of the target. Ionization energy loss in each material is modelled by the most probable value [36,37]: 2 tlEmv = ( 0.15354 MeV~cm ) Az; [In (n~L2c ) + 21n (n~~J + ln (;~) - /3 2 2 2 - o], (3 .2) where Z, A, and t are the charge, mass number, and thickness (length times density p) of the traversed material and /3c is the speed of the traversing particle. The ionization constant I = 21.8 eV for the liquid 1 H and 2 Il targets and 166 eV for Al 69 windows . For other elements, we use I = 16Z0 ·9 eV [38]. We neglect the density effect 6 for protons. For the highly-relativisti c electrons, inserting the asym ptotic form of 8 reduces Equation 3.2 to [36]: 2 ~Emp = ( 0.15354 MeV~cm ) ~t [ 19.2551 + ln (~)] . (3.3) The width ~ = (0.15354 MeV-cm 2 jg)(Zt/A!P) [36] of the Landau tail in each arm is less than 0.2 MeV for all kinemati cs, so usc of the most probable value is quite accurate. (Ideally, the average energy loss should be used instead of the most probable value. The difference is negligible because of the small width~-) The energy loss in each arm is ~Emp ;S 3.5 M eV . lll.E.2. Coulomb Acceleration The Coulomb attraction of the nucleus accelerates the electron before scattering and decelerates it after scattering. The change in energy of the electron is given by the addition or subtraction of the Coulomb potential energy Vc(1·) of the recoil nucle us, modeled as a uniform sphere of radius Ro (given in Table 5). Using an energy scale with Vc(1·) = 0 at r = oo, the potential e nergy inside the nucleus (at 7' < R 0 ) is, \1, r c( ) = _ ( Z - 1 )c 2 (~ _ ~) 2 47rcoRo 2m ' (3.4) where e is the electron charge. The contribution of the stru ck proton to the potential energy is ignored, with negligible error. Note that electron screening is also negligible: inside tlw nudeus, tlw Coulomb potential energy of thP e lectron cloud is less thau 10- 4 that of the nucle us. No correction is applied in the Z = 1 nucle i 1 II a nd 2 H. Insertin g < r 2 > = 3R6/5 for the uniform sphere, one find s the average Coulomb potential energy: < Tv7c ( 1' ) > = - 6- (Z- 1)c .o..__ 2 ___:__ 5 47rEoRo 6 (Z - l)a hc 5 Ro (3.5) 70 TABLE 5. Equivalent uniform nuclear radius Ro from elastic electron-nucleus charge distributions [39], the average Coulomb acceleration from Gquation 3.5, and the fractional effect of the Cou lomb acceleration on the elastic electron-proton Born cross section at the Q 2 = 1 (GeV /c) 2 kinematics. A Ro (fm) < Vc (r·) > (MeV) t::.aja 12C 3.23 0.3 -0.0008 56pe 4.85 8.9 -0.0233 197 Au 6.88 19.6 -0.0505 The resulting values, which are used for the correction to the scattering kinematics, are listed in Table 5. The table a lso lists the effect of the acceleration of the incoming electron on the scattering cross section for the Q2 = 1 (GeV /c) 2 solid target kinematics. The effect becomes smaller as Q 2 increases, and is approximately half as large at Q2 = 7. The effect is relevant because the corrected kin ematics are used to calculate the off-shell cross section aep when weighting the data (see Equation 5.3). 71 III.F. Kinematic Settings The kinematic settings are listed in Table 6. For the nuclear targets, data were taken at near-elastic kinematics, with energies adjusted to allow for the binding energy of the proton in the nucleus. For a fixed q and IPml less than the Fermi momentumpf, the protons scatte r into a. cone of opening angle,....., 2arctan(p1 jq) (see Equation 3.7, below). The coverage of Pm was achieved in nearly perpendicu lar kinematics, by setting t he central p' = q and varying the angle of the 8GeV /c spectrometer. At higher Q 2 the proton cone becomes smaller a nd fewer angle settings are needed. To maximize the cross section (by minimizing 016 ) for a given Q2 , the central 1.6 GeV jc momentum E' was chosen close to its maximum value of 1.5 GeV jc. For the Q 2 c::: 1 ( GeV j c ) 2 d ata, a lower value ("' 1.4) was used to satisfy the constraint ()1 6 > 35° imposed by the extent of the scattering chamber's exit window. Because nuclei h ave S(Em, Pm) = 0 for Em .S 0, best use of the Em acceptance TABLE G. Detailed NE-18 kinematics using definitions of Table 1. Q2 Targets (GeV /e) 2 1.04 2% c 1.04 6% Fe , 12% Au 1 E E' 0!6 p' Os (GeV) (GeV) (degrees) (GeV /c) (d egrees) 2.015 1.39 35.5 1 .20 35, 37.8, 40.6 , 43.4, 46.2, 49.0, 5 1.8, 5 4 .6 43.4, 46.2, 49.0, 5 1 .8, 54.6 1.16 4 em 1-1 1.40 37.3 1.24 43.3 1.21 4 em 2 H 1.36 38.8 1. 28 39.1' 41 .3, 4 3.5, 46.7 3.06 2% C, 6% Fe, 6% Au 1.47 47.7 2.45 27.7, 30.5, 33.3· 3.06 4 em 2 1-1 3.23 •I em 1 1-1 5.00 6% C, 6% Fe, 12% Au 15 e m 5. 14 4 c m 1 11 6.77 6% C, 6% Fe, 1 2% Au 6.85 ac, Fe on ly ac on ly 27.7 4.212 1.47 49.0 2 .49 26.5 53.4 3.54 18.5b , 20.9, 22.6. 54.2 3 .56 19.6 56.6 4.49 15.9•, 16.7, 17.3. 2 11 .5.00 6.77 3.1 88 15 e m 2 4 em 1 19.5 5.120 1.4 7 1-1 1-1 1 5.9 57.0 15.9 72 Pm FIG. 37. Central Pm for a given Ba. For clarity, Bs - Bq is grossly exaggerated compared to the actual experimental kinematics. is made by centering it at Em > 0. From Equation 1.7, the Em value for central kinematics is, (3.6) Here the approximation TA-l = 0 (exact at ]Jm = 0, and still very accurate at IPm I ~Pi) and nearly perpendicu la r kinematics (p' 2 = q2 = Q 2 + 11 2 ) were used. The result is easi ly inverted to allow calcu lation of v from Q2 and the desired Em, central· The Em,central values used in the present experiment were approximately 50, 30, 20, and 10 MeV at Q2 ~ l , 3, 5, 7 (GeV /c) 2 , a lthough Em ,ccnt?·al ~ 10 MeV was used for the 2 ll target at Q2 = 1.21 (GeV /c) 2 . The broader Em acceptance at high Q 2 a llowed the choice of smaller Em,centra/, maximizing the acceptance near the peak of S'(Em, Pm)· Once 11 is given by Equation 3.6, the central proton kinematics are given by p' = q . The limit on the central momentum of the 8 GeV jc, p' < 4.5 (GeV /c) 2 prevented data-taking at Q2 > 6.77 (GcV /c) 2 . Assuming p' = q, the central Pm value for a given 8 GeV /c angle setting is given by (see Figure 37): Pm .ccnlr al = 2p, tan (Os-Oq) · 2 (3.7) 73 To ensure overlapping (Em, Pm) acceptance at consecutive Os settings, one must account for the strong correlation in the Em and Pm acceptances. The two-dimensional (Em, Pm) acceptance is given by plotting the size of the detection volume contributing to each (Em, Pm) bin. This is given by the "phase space" integral: (3.8) The acceptance functions of tbe two spectrometers, A 16 (E',nk') and A 8 (E~,np'), give the probability that the coincidence kinematics being integrated over would be accepted. The acceptance functions are set to zero outside of the acceptance cuts applied to the data (see Section V), and so are technically functions of (8, ,6.8, <I>) as well. Here the distinction between (E', 0) and (8, ,6.8, <I>) is that between the true and the measured momenta. The integration region is over all kinematics with (Em,Pm) inside the bin, and the presence of the acceptance functions insures that only the experimentally relevant kinematics contribute. The integral 1PS is given by a Monte Carlo calculation including the effects of ionization energy loss, multiple scattering, spectrometer acceptances, and spectrometer resolutions. Examples of the (Em,Pm) coverage are given in Figures 38 and 39. 74 __ ____ , _ 300 / /'- - -' I ----- .. : ..- ....- ----- -. /- 200 88 = 88 = 86 = ,..--.... (.) 86 = 100 86 = ""-... >Q) ~ ...._... 0 El 0..-100 -200 -300 -50 0 100 50 E m (MeV) 150 200 FIG. 38. Coverage of (Em, Pm) at Q 2 = 1 (GeV/ c) 2 for each carbon Os setting, indicated by contours whe re the acceptance I p s reaches 10% of its maximum value. The (}8 = 40.6 and (}8 = 43.4 contours are s haded in to emphasize the continuity of acceptance across Pm = 0, where the volume integral df,, = p?,, dpm vanishes. 75 3oo~~~~~~~~~~~~~~~~~~~~~ . .. ~ - . - . - -.- - : -~- .. ~ - - . - - : -_ ·- ! :_ - - - - - - - - -_ - -' . - - - . . .-. 200 --- 100 C) ~ >Q) 0 ::E ..._... s ~ -100 -200 -300 -50 0 50 100 (MeV) Em 150 FIG. 39. Coverage of (Em, Pm) at Q 2 = 6.8 (GeV /c) 2 for the two extreme carbon angle settings, ()8 = 15.9 (solid line) and 17.3 (dotted line). 200 76 III.G. Efficiency Corrections The count rates used in the extraction of S(Em, Pm) and T( Q 2 ) must be corrected for the electron-proton coincidence detection inefficiency. The correction includes contributions from the tracking inefficiency in both spectrometers, the coincidence timing inefficiency, and the absorption of protons in the target and 8 GeV /c spectrometer materials. For convenience, the tracking inefficiency is defined to include a ll single-arm particle identification inefficiencies, such as trigger inefficie ncies and, in the 1.6 GeV /c spectrometer, the ineffici ency of the cut on the Cerenkov ADC. Most of the efficiency corrections are derived in other sections. This section describes the correction for the absorption of protons and summarizes a ll of the efficiency results. IJJ.G.l. Absorption of Protons in Tar-get and 8GeVjc Mater·ials Protons arc absorbed and deflected in strong interactions with the nuclei of the material in between the struck nucleus and the back of the 8 GeV /c detector stack. Because the proton trigger requires an SF-SM-SR coincidence, the proton is not detected unless it reaches the rear (SR) scintillator plane. The proton absorption correction is derived from the measured thicknesses of the intervening materials and the known proton-nucleus cross sections. For the hydrogen target, the correction is consistent with the ratio of 1 H(e,e'p) to 1 ll(c,e') counts. The probability of absorption of a proton in a thickness X (= length x density) of material is (3.9) where >. is the proton mean free path. Because of the strong forward peaking of lhe elastic cross section, many elastic scatterings do not remove the proton from the detector volume. This is especially true for elastic scatterings in the detector hut. 77 Thus, the correction used is intermediate between the two extremes given by A = Ac and A = A1 . Here Ac an d A1 are the m ean fr ee paths between collisions (elastic or inelastic) and between inelastic interactions, resp ecti vely. The values used are from [38] . Tables 7 through 10 detai l the calculation of the absorption correction. The aver age a bsorption is given ( t.o high accuracy) by the value for a scattering at target center, x = y = z = 0, into the center of the 8GeV/c solid angle, (O,¢)p' = (Os,O). The thicknesses of the target -related materials traversed by the proton (Tables 7 and 8) varies with 08 . The resulting variation in the absorption is negligible ( < 0.3%), however , because at every kinematics, either the absorption in the target is small (;S 1%) or the maximum range of 08 is small (;S 1°). Thus, a typical value of 08 (44, 28, 19, and 16°) is used to calculate A at each Q2 [1 , 3, 5, 7 (GeV /c) 2 ]. The total absorption is calculated as 1 - fli (l -A;), where (1 - A;) is the transmission through each a bsorbing m aterial i . The total A1 and Ac differ by 3-4%. lie nee, the average of the two is use d for the absorption correction, with a fractional systematic error of ±2%. As a consistency check, the transmission for the hydrogen t arget is measured directly as the ratio of the number of coin ciden ces (corrected for the 8 GeV /c tracking effi ciency a nd dead Lime contri bution and also for the coincidence timing effi cien cy) to the number of 1.6 GeV jc electron singles. Radiative tail events are discarded with a cut on the elastic stripe in the 1.6 GeV jc, ensuring near-exact kinematic correlation of the e lect ron a nd proton . (Note t hat if tlw e ffect of proton intern a l bremsstra hlung on the electron kinematics is small enough, the cut may not exclude a ll radiative tail events. However, the total fraction of events outside the 8 GeV jc momentum acceptan ce is estimated to be< 0.1 %.) Then the (8, ~0,<I>) illumina tion of the 8 GeV /c spectrometer is restricted to the centr a l, unit-acceptance, region through cuts on (~ 0 , <I> ) in the 1.6 GeV /c spectrom eter. Proton absor ption is indicated by 78 TABLE 7. Proton absorption in liquid target materials . Target 4cm 1 H 4cm 2 H Density AJ Ac Ql length thickness A1 Ac (g/cm 3 ) (g/cm 2 ) (g/cm 2 ) (GeV jc) 2 (em) (g/cm 2 ) (%) (%) 50.8 43.3 1 2 .79 0.197 0.39 0.45 3 2.27 0.160 0.31 0.37 5 2.12 0.150 0.29 0.34 7 2.09 0.147 0.29 0.34 1 2.79 0.474 0.86 1.03 3 2.27 0.386 0.70 0.84 5 8.29 1.410 2.55 3.04 7 8.16 1.387 2.50 2.99 1 0.169 0.461 0.43 0.65 3 0.138 0.376 0.35 0.53 5 0.129 0.351 0.33 0.50 7 0.127 0.345 0.32 0.49 1- 7 0.006 0.009 0.01 0.01 0.0705 0.1701 54.7 45.7 15cm 2 H AI Endcap Mylar 2.72 1.39 106.4 60.2 70.6 85.7 Insulation 1 H( e, c') events that did not result in an 8 GeV jc track. Note that the measurement does not depend on the efficiency of the 1.6 GeV jc spectrometer. The measured values are consistent with those predicted using Equation 3.9 (see Figure 40). !II. G.2. Summary This section summarizes efficiency corrections applied in the analysis of S(Em, Pm) and T( Q2 ). For each correction, the associated systematic error is also given. The trackin g efficiency corrections given in Table ] 0 are described in Section III.B. The 79 0 .95 0.90 0.85 FIG. 40. Proton absorption efficiency for hydrogen. The data points are measured as the efficiency-corrected ratio of 1 H (e, e'p) coincidences to 1.6 GeV / c 1 H( e, e') singles with a kinematic cut to restrict the 8 GeV /c illumination. The inner error bars are statistical only and the outer error bars include the systematic errors in quadrature. The two curves are the transmission calculated using the inelastic {upper curve) and the total {lower curve) nuclear scattering lengths AJ and >. c . 80 TABLE 8 . Proton absorption in solid targets. Density AJ >.c Q2 length thickn ess A1 Ac (g/cm 3 ) (g/cm 2 ) (g/cm 2 ) (GeV /c) 2 (em) (g/cm 2 ) (%) (%) 2.193 86 .3 60.2 1 0.285 0.625 0.72 1.03 3 0.232 0.509 0.59 0.84 5 0.620 1.360 1.56 2.23 7 0.610 1.338 1.54 2.20 1 0.0740 0.579 0.44 0.70 3 0.0603 0.472 0.36 0.57 5 0.0563 0.441 0.33 0.53 7 0.0553 0.433 0.33 0.52 1 0.0279 0.534 0.28 0.47 6% Au 3 0.0117 0.223 0.12 0.20 12% Au 5 0.0212 0.407 0.21 0.36 12% Au 7 0.0209 0.400 0.21 0.35 Target 2% c 6%C 6% Fe 12% Au 7.829 19.157 131.9 190.4 82.8 113.7 proton absorption correction is derived in Section III.G.l. The coincidence identification efficiency is discu ssed in Section l lJ .D. 81 TABLE 9. Proton abso rption in 8 Ge V / c s p ec trometer Absorber Material C hamber Exit• and 8GeV /c Density le ngth >.c AJ 2 2 Ac (%) (%) 0.15 0.14 0.21 0.08 0.22 0.21 0.31 (g/cm ) (g/ cm AI 5052 2.68 106.4 70.6 0.056 AI 2.70 106 .4 70.6 ) (g/cm ) thickness A1 3 (em) (g/cm 2 ) E ntrance Windows Cer enkov E ntrance and Exit Windows Cer enkov Mirror Lucite 1.2 83.6 59.2 0 .64 0 .77 0 .92 1.29 Cer enkov Gasb C2Cl2 ~"• 0.0049 106 70.6 3 10 1.52 1.4 2 2.13 Wire C hambe rs F'e fo il 7.9 13 1.9 82.8 0.026 0.21 0.16 0.25 SF scin t illators Polysty•·cne 1.03 82.0 58.4 0.95 0.98 1.19 1.66 NBS scintilla t ors Polystyrene 1.03 82.0 58.4 1. 27 1.31 1.58 2.22 SM scintilla t or s Polysty•·cne 1.03 82.0 58.4 1.27 1.3 1 1.58 2.22 0.001205 90.0 62.0 410 0.49 0.5'1 0 .79 Ai1· Gaps Ai•· Totalc fo r Q2 = 1, 3 ( GeV /c ) 2 6. 16 8.64 Totalc for Q2 = 5, 7 (GeV fc) 2 7.50 10.58 ascattering chamber exit included in table f o r com pacLness. bQ 2 = 5, 7 (GeV /c) 2 only csolid targets (does not include a ddition a l 0.01 from myl a r ins ulation) S2 TABLE 10. Tracking efficiency and proton absorption corrections. The 1.6 GeV jc and 8 GeV /c tracking efficiencies and the proton coincidence identification have an error of ±0.01 each. The fractional error on the proton absorption is ±0.02. Q2 Tracking Coincidence Proton A (GeV /c) 2 1.6GeV/c 8GeV jc Identification Absorption lH 1, 3 1.00 1.00 1.00 1.09 5, 7 1.00 1.00 1.00 1.11 1, 3 1.01 1.00 1.015 1.09 5 1.01 1.00 1.015 1.14 7 1.02 1.00 1.015 1.14 1.01 1.00 1.015 1.09 3 1.02 1.00 1.01.5 1.09 5, 7 1.03 1.00 1.015 1.13 1 1.01 1.00 1.015 1.08 3 1.02 1.00 1.015 1.08 5, 7 1.03 1.00 1.015 1.11 2H 12c 5617c, 197 Au 83 IV. T H E PLANE W AVE IMPULSE APPROXIMATION CALCULATIO N Extraction of the spectral function and the nuclear transparency are performed assuming the scattering can be modelled in the Plane Wave Impulse Approximation with an Independent Particle Shell Model S(Es, p). The transition from the Bornlevel cross section (Equation 1.9) to higher orders is made through the addition of the effects of internal radiation (see Section IV.E). This allows direct comparison in both the transparency and spectral function measurements between the PWIA calculation and the data. The present calculation improves on simple the PWIA/IPSM approximation through the addition of effects due to Coulomb distortions of the electron waves (Section IV.F). The calculation also includes effects due to the experimental apparatus such as external radiation, ionization energy loss, multiple scattering, spectrometer acceptances, and spectrometer resolutions. It is worth discussing higher-order corrections to the PWIA that are not included in the calculation. These effects are difficult to calculate but are expected to be small for the low-Q 2 kinematics of previous A( e, e'p) experiments, and so were not corrected for. Fortunately, they are even smaller for Q2 ;::: 1 (G0V /<Y. Dispersion corrections and exchange terms involve the coupling of the virtual photon to the entire A or A -1 nucleus, and so must scale like the electron-nucleus elastic form factor. Due to the large extent of the nucleus, the form factor drops rapidly for Q2 ;::: 1 (GeV /c) 2 . Here dispersion corrections refer to two-photon exchange processes (""' a 4 ) leaving the nucleus in an excited state before the knockout of the detected proton [9). Exchange terms refer to one-photon processes where the virtual photon knocks out the A - 1 system instead of the detected proton [9]. This is further suppressed at high Q2 because it involves nucleons with initial momentum ""' q, where the spectral function has extremely little strength. 84 Multi-nucleon processes are expected to decrease for Q2 > 1 (GeV /c) 2 due to the decreasing wavelength of the virtual photon (see Section l.A .2) . However, Lourie et al. argue that the enhancement in the measured rate at 50 < Em :S 100 MeV due to multi-nucleon knockout may be increased at high Q2 (14]. The issue can be resolved through a comparison of the PWIA/IPSM calculation and the data (see Section V.A). In meson exchange currents, the photon couples to a virtual meson (produced by the nuclear strong interaction) or to a nucleon undergoing a strong interaction with another nucleon (dynamic correlations). This includes isobar contributions, where the nucleon is excited to a virtual resonance. For inclusive A( e, e') , these contributions peak in the dip region and are :S 4% at quasielastic kinematics even for Q2 ~ 0.18 (GeV /c) 2 (9]. IV.A. Overview of the Calculation The PWIA calculation is performed by a Monte Carlo simulation that begins with events generated according to Equation (1.9) and includes the effects of internal radiation. Coulomb distortion, and the experimental apparatus. The first step in the calculation is the degradation of the energy of the incoming electron by ioni zation energy loss and radiation (internal and external). Internal radiation is calculated in the equivalent radiator approximation , modified to exactly reproduce the theoretical E' (40] and Em (41] dependences. Trial values of scat- tered electron and proton kinematics are chosen, and accepted with a probability proportional to Equation 1.9. The outgoing electron and proton energies are then degraded by radiation and ionization energy loss. The outgoing electron and proton a.re then traced through the magnets of the 1.6 and 8 Ge V j c spectrometers using the Transport-based models of the spectrometers. If the models predict both particles will be accepted, their momenta are reconstructed and recorded. The calculation is 85 described in more detail below. IV.B. The Born-Level Plane Wave Impulse Approximation Cross Section This section describes the values of aep and S(Es, p ) used to evaluate Equation 1.9, the A( e,e'p) cross section in the Born-level PWIA. In principle, one would need to know the off-shell nuclear current in order to calculate aep· At present, however, this is not possible and there is some ambiguity in the choice of aep · We use the De Forest prescription, a ep = afc [10], which accounts for the differences iu the on- and off-shell kinematics and satisfies 4-momentum and current conservation. Other forms for O"ep, including the on-shell value, have been tested, with little(~ 2%) effect on the measured T. For the hydrogen values of p = 0 and Es = 0, afc reduces to the free-nucleon cross section (Equation 1.2), divided by the recoil factor E' IE and flux term E' I M [JO]. The full e lastic cross section results from integrating Equation (1.9) over the 4-dimensiona.l delta function in the hydrogen spectral function. For the S (Es , p) of 1 II and 2 ll we use a 4-dimensional delta-function and the Bonn potential momentum-space wave function [42], respectively. ro r the solid targets we usc the Independent Particle Shell Model (Equation 1.11). ln thi s case, L;(Es) is a Lorent zia n energy profile (with a cut-off at the one-proton removal energy, see Appendi x E) and <p;(p) is the bound-state wave function in a Woods-Saxon nuclear potential with shell-dependent parameters. The Lorentzian and Woods-Saxon paranwtcrs, based on from fits to spectral functions extracted from previous A( e, e'p) experiments [9], are given in Tables 24 and 25. The carbon spectra l fun ctio n is taken from the Saclay 1 2 C(c., e'p) experiment [11] shown in Figures 6 and 7. Since there is uo A(e, e'p) data on iron and gold, their spectral functions are based on the S(Em,JJm) measured a t Saclay on 58 Ni [11] and at NJKIIEF on 208 Pb [13]. The momentum dis- 86 tributions for Fe and Au are modified to improve agreement with the distributions measured in the current experiment. For the proton elastic form factors G'f:: and G~ used in afc, we assume the dipole (Equation 1.4) and Gari-Kriimplemann [43] forms, respectively, as suggested by SLAC experiment NE-ll [44]. The form factor model-dependence is estimated to be ::; 2% by comparing elastic cross section calculated with different form factor parameterizations . IV.C. Kinematics and Luminosity The calculation generates events in a region of dE'dD,k,dE~dD,P, with 6-fold volume Vkin· The region includes the entire acceptance range of the spectrometers. When radiation is being simulated, the region's upper E' and E~ boundaries are increased by Wmax (see Section IV.D) to allow the outgoing particles to radiate down into the ::;pectrometer's momentum acceptance. The program generates trial events with a flat distribution in Vkin· An event is accepted with a probability given by d6 ajdE'dD,k'dE~dD,P, (Equation 1.9) divided by a constant value amax, so the accepted events are distributed accord ing to the quasielastic cross section. II ere a max 2:: the maximum value of ~a j dE' dD,k,dE~dD,P, in Vkin· The integrated luminosity LpwJA = f Ldt of the calculation is needed to convert to count rates. lt is most easily found by realizing that the trial events arc effectively generated with the cross section a max (i.e., if the quasielastic cross section were constant at amax, every trial event would pass the cross section test). Thus, the integrated luminosity is simply: r .t....PWJA = Ntrial VkinUmax where Ntrial is the number of trial kinematics. ( 4.1 ) 87 To mcrease computing speed this procedure is modified for all targets except 197 Au. For hydrogen, the spectral function is a delta function in energy and 3- momentum, which is impractical to simulate. The calculation circumvents this issue by generating events according to the e lastic cross section, Equation 1.2, instead of the quasielastic cross section. For the nuclei 2 H, C, Fe, which are highly peaked in Es, it is convenient to generate events in Es instead of E'. In this case the relevant cross section is (cf. Equation 1.9): (4.2) where 1 dE' ) ( - dEs = 1+ p · k' VM~-1 + p2. (4.3) The calculation performs the Es integral directly by choosing a shell i (weighted by Ni) and then sampling Es according to L,(E.,). Proper weighting is then given by an accept-reject test with probability proportional to (cf. Equation 1.11): ( 4.4) IV.D. External Radiation The emission of real (bremsstrahlung) photons in external radiation (Figure 41) changes the kinematics of the detected particles. In a single arm experiment, this process distributes the counts along a radiative tail in E', and can significantly lower the count rate by moving counts to below the spectrometer momentum acceptance. In a coincidence experiment one can observe not just the magnitude, but a lso the direction of the radiation. In the PWIA, the 4-momentum pJJ. of the initial proton equals the missing 4-momentum p~, =k'JJ. + p'JJ.- kJJ.. But if real photons are emitted, there is an unknown additional m issing 4-momentum wJJ. in the final state: pll- + kJJ. = 88 p FIG . 4 1. One-photon external radiation m 1 H (e, e' ). The incoming and outgoing electrons experience bremsstrahlung interactions in the electromagnetic fields of target nuclei (indicated by gray circles). Because of the large distances separating the interactions and the high Q 2 of the electron-proton scattering, the three diagrams shown are incoherent. p'J.L + k J.L + w J.L . Thus, p~ = pJ.L- wJ.L and the m easured missing energy and m om ent um a re now : (4 .5) Pm = P - w . ( 4.6) T his lead s t o the observed tail s in t he Em-Pm pl ane. ln the present ex periment t he tails can move either k' or p ' o utside the spectro met er acceptances. It is t herefore n ecessary for the calculation to include the effects of radi a tion on the kinem a ti cs of the det ected par t icles. (Note t hat t he distribu t io n of the photo ns them selves is irrelevant , except through their effect on t he detected kinem a tics.) T he following t wo sections de t ail how th is is accomplished . Ext ernal radiation refers to t he emission of bremsstrahlung photons by the incoming a nd o utgoing electrons in the fields of the target nuclei (see F igure 4 ] ). The m ore m assive ou tgoing p roton is subject t o much smaller accelera tions, a nd emi ts 89 a negligible amount of radiation. R. A. Early [45] solved numerically, in the complete screening approximation, for the probability h(w, t) that an electron of energy E would radiate an energy w in passing through a thickness of t radiation lengths. Note that h(w, t) is the probability of emitting any number of photons, with energy totalling w. The present calculation uses Y. S. Tsai's fit [46], 1 ( w ) bt bt ( w ) h(w, t)dw = f( 1 + bt) E w </> E dw ( 4. 7) (4.8) with the t values given in Tables 3 and 4 (multiplied by geometric factors for nonorthogonal incidence). Tsai's fit is below Early's result by a fraction that varies between about t/10 and t/5 as w varies between 0 and 0.8E for t < 0.1 [45]. For example, the deviation at w ~ 0 is ,...., 1% for a t = 10% radiator. The discrepancy increases for w > 0.8E , but this is always outside t.he experimental acceptance (see the discussion of Wmax, below). The resulting error in the calculated count rate is dominated by the t/10 deviation at w « E. As a result of this discrepancy, Equation 4.7 does not integrate to 1 (the probability is not normalized): l EI o b( w t ' ) dw = 2 [ 8+7bt+8(bt) ] 1 - t 10 r (1 + &t) 4 ( 1 + bt) (2 + bt ) "' I · 1 (4.9) This is a small effect ( < 2%) and the total radiator thickness t does not vary much for events inside t.he acceptance. Thus the calculated rate is simply renormalized through division by Equation 4.9 evaluated for an event at the center of the target and at central kinematics. After this correction the deviation of the total PWIA count rate from the Early result is estimated to be < 0.2%. Equation 4. 7 is valid in the Born approximation, where each electron-nucleus collision releases at most one photon. For a heavy nucleus like 197 Au, with Za ,...., 1, 90 diagrams with multiple photons also become important. However, we will argue in Appendix F .3 that the equation , which includes the effects of multiple collisions, approximately incorporates the effects of diagrams with multiple photons. In the t -+ 0 limit , one recovers ¢(w/ E), the shape of the bremsstrahlung spectrum from a single collision: 1- 1 ]b bt (w,t)dw = w ¢ (w) E dw. (4.10) The quantity bin Equation 4.7 is given by [46]: b=~{1+~[ Z+1 ]} 3 12 ZLrad + L~ad (4.11) where Z is the charge of the external radiator nucleus. We use the Thomas-FermiMoliere model for Lrad and L~ ad [46]: ln (184. 15Z- lf 3 ) ( 4.12) . 94..., z-2/3) . L ,rad -- I11 ( 11 ( 4.13) Lrad = The resulting values of b agree with those calculated using Tsai 's recommended values of Lrad a nd L~ ad [46) to better than 0.1 % for Hydrogen , and almost exactly for 12 C a nd heavier nuclei. Because Equation 4. 7 becomes inaccurate at large w, the PWIA calculation only includes events with w less than a chosen cut-off w = Wmax < 0.8E. This is justified because events with su ch large w a re outside of the Em cuts used in the a nalysis (see Equation 4.5 a nd Section V.B). For nuclear targets, Wmax = 200MeV. For 1 11 , where t h e (e , e') cross section is also of interest, la rger values of Wmax a re used , ensuring that the radiative ta il is computed correctly beyond the momentum acceptance of the 1.6GeV/c spectrometer. For Q2 = l ,3,5,7GeV/c2 , t he 1 H values are Wmax = 300,700, 1200 , 1300 MeV jc. 91 The photons are emitted nearly parallel to the electron momentum, with an angular spread of 6.()w = n<e/E < 0.4mr for electron energies of E ;::: 1.4GeV. The deflection of the electron , roughly w6.0w/ E, is .:S 0.05 mr for the photon energies of interest , w < 200 M eV. Thus, Lhe calculation uses the angle peaking approximation, where the radiation changes the magnitude but not the direction of the electron. IV .E. Internal Radiation The Born-level PWIA describes Lhe e lectron-proton interaction as the exchange of a single virtual photon (see Figures 3 a.nd 4). Higher-order diagrams, involving multiple photons, are referred to as intern al ra diati ve corrections. The diagrams involving a single radiative photon, shown for H( e, e') in Figures 42 and 43, have been calculated by Mo and Tsai [40]. The diagrams in Figure 42 describe an elastic scattering, with the exch ange of a vi1'ittal rad iative photon. These graphs are simply a.dditiona.l contribut,ions to the elastic cross section , in creasing it by rv 10% without cha nging the kinematics. The diagrams in Figure 43 describe an in elas tic scatte ring, with the emission of a 1·cal photon. This process, called internal bremsstrahlung, produces radiati ve tails as discussed at the beginning of Section IV.D. ln Lhe spirit of the PWlA , for the radi ative correction to the qttasielastic A(e, e'p) cross section (Equation 1.9) the correction for 1 H( e,e'p) elastic scattering at the same E and () is used. The recoil nucleus is Lreated as a spectator , which does not pa rti ci pate in the high-Q 2 scattering and therefore is not involved in the exchange or radiation of ha rd photons. In particular, recall from Section l.A.1 that the PWIA quasielastic cross section , EquaLion 1.9, implicitly includes a closure sum over the final stat e of the A - 1 system. The sum includes inelastic channels, such as internal bremsstr a hlung from the recoil nucleus. The Coulomb attraction of the recoil nucleus, involving the exchange of soft photons , is treated in Section IV .F. 92 e p Vacuum Polarization Vertex Correction Two-Photon Exchange FIG. 42. One-photon internal virtual radiation in 1 H(e, e'). Internal radiation diagrams involving an additional virtual photon, coherent with the electron-proton scattering. FIG. 43. One-photon interna l bremsstrahlung in 1 H (e, e') . Diagrams involving the emission of a single real photon, coherent with the electron-proton scattering. 93 Mo and Tsai (40] calculated the count rate in an H( e, e')p experiment for E~1 ~E' < E' < E~ 1 assuming the emission of a single radiati ve photon. (This is the "one-photon" approximation, correct to order a\ see Equation F8) . Exponentiation approximates the effect of higher-order diagrams to all orders [40]: ( 4.14) Here the Born-level (order a 2 ) cross section (de7/dD)Born is given by Equation 1.2. The correction factor 8 is given by Mo and Tsai [40], Equation II.6 , with the addition of the Schwinger correction and vacuum polarization from quark and heavy lepton vacuum polarization loops [47]. These additions are necessary for consistency with SLAC NE-11, on which the present calculation's form factors are based. The 1/w character of the internal bremsstrahlung leads (see Appendix F.3) to the following D.E' dependence: ( 4.15) where N E' and AE' are functions of E and fh, only. Mo and Tsai give a formula for 8s, which is valid for D.E' « E'/(1 + 2E/Mp) [48]. In an A( e, e'p) experiment, one is also interested in the number of counts within the experimental range of Em· Wasson et al. [41] have repeated the Mo and Tsai work for the case of coincidence kinematics. In this case, one finds the analogous result: ( 4.16) One finds that the exponent As, = A£' at all kinematics. This is because 8E' and 6Em are both integrals of Equation F7, except with different integration regions. The present calculation simulates the internal radiation through the equivalent radiator approxim ation, modified to exactly reproduce the above E' and Em cut dependences. The standard equivalent radiator approximation simulates the internal 94 bremssLrahlung by passing the incoming and outgoing electron through two imaginary exLernal radiaLors, both wit,h bt = ;..EQ: ( 4.17) Not,e t,hat for interna l bremsstrahlung b and t are separately meaningless ( cf. Equation 4.7). The value is typically ;..EQ ""3.5 (see Table 26). The present calculation uses this method to simulate the effects of internal radiation on the count rates and kinemat,ics of A(e, e'p). There are two refinements, however. First, as the scaLtering energy increases, a few percent of the radiation becomes peaked near the scattered proi,on direction [41]. (The large acceleration of the proton in the scattering begins to overcome the suppression of radiation by its high mass.) Thus, the scattered proton is also passed through an equivalent external radiator, with bt values between 0.00037 [at Q2 = 1 (GeV /c) 2 ] and 0.00524 [Q 2 = 7 (GeV /c) 2 ] (see Table 26). Second, the three equivalent radiators have differenL thicknesses. (The equality of the incoming and outgoing electron radiators in the standard equivalent radiator approximation is only valid in the limit of no target recoil [41].) The thicknesses are adjusted to simultaneously reproduce the theoretical cut dependences in both E' and Em: i PWIA GO" 1..\, = N E' ( !J.E ) E ( dD. ) D.E' (ddD. ) 0" , (4 .1 8) Born ( 4.19) Doth conditions must be satisfied becaus<' events are simultaneously subject to the 6.E' range given by the 1.6 GeV /c momentum acceptance and the !J.Em range applied in the analysis. The selection of the).. values used in the calculation is described in Appendix F.4. Figure 44 demonstrates the success of the equivalent radiator technique. 95 10 3 rn +-> ~ ;:::::1 0 u 10 2 2 lOl~~~~~~~~~~~~~~~~~~~~~nn~-r -25 0 25 50 Em (MeV) FIG. 44. Em distribution of 1 H(e, e'p) at Q 2 75 100 1 (GeV /c) 2 • Com- parison of data (points with error bars) and calculation (histogram). Error bars are statistical only. To reproduce the internal radiative tails, the equivalent radiator approximation must correctly simulate the effect of the radiation on k , k', and p'. This is discussed in Appendix F.5. Appendix F.3 illuminates som e iss ues s urrounding internal radiation through the u se of th e one-photon approximation. IV.F. Multiple Scattering, Energy Loss, and Coulomb Acceleration The calculation simulates multiple scattering, energy loss, and Coulomb acceleration based on the position and a ngles of the scattering in each event. After the calculation, the reaction kinematics a re then corrected for the average value of these 96 effects (Section III.E). Multiple scattering and energy loss calculations are performed using the material thicknesses given in Tables 3 and 4. The location of the scattering is chosen uniformly along the length of the target and inside the transverse extent of the beam spot. In cases of non-normal incidence, the thicknesses are increased by the appropriate geometrical factors. Multiple scattering of the incoming and outgoing particles in the target and spectrometer materials is simulated in the gaussian approximation, which is accurate for 98% of the multiple scattering distribution (38]. The approximation is sufficient, given the small value of the RMS angular defiection orms < 5 mr in each arm. Ioni zation energy loss in each material is accounted for by subtraction of the most probable value 6.Emp from E, E', and E~. Values are given by Equation 3.3 for electrons and Equation 3.2 for protons. The width ~ < 0.2 MeV , so it is ignored. Changes in 6.Emp due to vari ations in the location and angles of the scattering are more important, and a rc correctly simulatf'd. The Cou lomb attraction results from radiative diagrams, neglected in SectioniV.E. where the electron exchanges soft photons with the recoil A - l system. In a Complete Distorted Wave Impulse Approximation model of the scattering, the attraction gives rise to the Coulomb distortion of the t = ±(X) plane wave fronts. The main effect of the Coulomb acceleration is a change in the scattering cross section caused by the change in scattering kinematics. Note the potential used to calculate the IPSM wave function <p;( p ) includes the Cou lomb potential en<'rgy of the A - 1 system (see Appendix E). Thus, the Coulomb acceleration of the struck proton is already accounted for in the separation energy Es The calculation simulates the change in the energy of the electron by the addition or subtraction of the Coulomb potential energy of the recoil nucleus at the point of the interaction (Equation 3.4). The position of the scattering is chosen randomly 97 inside the nucleus, which is modelled as a uniform sphere. This procedure neglects the localization of the charge in discrete nucleons and the shell-dependence of the radial wave function. These approximations are well justified, however. The model does not include hard interactions with h/Q ,$ 1 fm, which resolve individual nucleons. However, since the elastic cross section is strongly peaked at small Q 2 , these interactions have vanishing probability. At small Q 2 , E' ~ E and the proton structure is not resolved, so the cross section (Equation 1.2) reduces to the Mott cross section (Equation 1.3): (4.20) Here the small angle approximation and Q2 ~ E 2 02 have been used. The cross section for Q > Qmin = Ti/1 fm is then: r"' <YQ>Qm,, = 27f Jc. 2 47fa ~ (dO") dD. sin(O)dO ~ Q = 0.00067 fm\ 2 . Qnun/L or 6. 7 pb. ( 4.21) nun The probability is greatest for 197 Au , with density PA = 3A/41f ~ = 0.144fm - 3 and thickness"" fJARo = 1 rrn- 2 : O"Q>Qm,, PARo ~ 0.00067. Hence the localization of charge has an insignificant effect. The error introdu ced by choosing r from a uniform sphere instead of each shell's radial wave function is greatest by far in 197 Au, which has both the largest Z/ Ro and the largest variation in < 1 2 > between shells. The smallest and largest values are < r"i., > = 6.6 fm 2 and < 1ih 1112 > = 35.4 fm 2 , while the uniform sphere has < 7' 2 >= 28.4 fm 2 . The error in < Vc(1) > is greatest for the 1s shell: 3.8 MeV. For the incoming electron, this would produce a 1% chang<' in the elastic cross section at Q2 = 1 (GeV /c) 2 (or less at higher Q2 ). Since the majority of shells have < 1 'l > much closer to < 1 2 >, the error in the transparency is estimated to be < 0.2%. For finite impact parameters 7'j_ , the Coulomb interaction also produces small deflections in the particle trajectories. Because the deflection is small it can be 98 approximated by integrating the transverse force along the undeflected t rajectory (the s traight line approximation ). The deflection is less than or order of that for tangential trajectories with r 1. = Ro: t:lp1. = - 1 {Jc jo d1· F1. = - -Zo.h -. - oo 11 {Jr 1. (4.22) The integral is shown for the incoming electron. The outgoing electron a nd proion experien ce deflections of the sam e magnitude. The largest value occurs in Au, t:lp1. = - 16.5 MeV jf]c. The deflection may be thought of as another contribution to the ex perimental Pm resolution. Inclusion of the deflection did not have a statistically significant ( < 0.5%) effect on the PWIA rate at the Q2 = 1 (GeV /c) 2 Au kinematics. This kinematics should be the most sensitive because the smaller experimental mome nta result in larger ("' 10 mr) angular deflections. The width of the PWIA mome ntum distribution was increased negligibly (1.3%) by including the deflections. Because of its sm all effect, the deflection was not included in the PWIA calculation used in the present a nal ysis. On the other hand , the Coulomb d eflection produces an observable in crease in the PWIA count rate by bending non-impacting trajectories into the nucleus, an effect that is iHcluded in the PWIA calculation. The increase is easily calcula ted by conserva tion of angula r momentum for rays that just graze the nucle us. The angular momentum of the incom ing electron at 1'JJ = -oo is r 1.k and at r11 = 0 is Ro[k- Vc( Ro)]. Equating these, one finds 1·1. = Ro(1 - Vc(Ro)/ k] > Ro. The count rai<' scales as r}_, so it is inc reased by a fa ctor (1 - Vc (Ro)/k]2. Th e increase is largest (1.016) for Au a t Q2 = ] (GeV fc) 2 . 99 IV.G. Spectrometer Acceptance Monte Carlos The PWIA calculation requires models of the spedrometers in order to correctly simulate the experimental acceptances and resolutions. Monte Carlo models from earlier experiments [32,49] were modified to account for changes in the apparatus . In the Monte Carlos, the spedrometer optics are modelled with second-order Transport matrices [35] based on surveyed positions, wire-float measurements, and previous data. The 8 GeV /c optics model is not as well-constrained as the 1.6 GeV /c model because this was one of the earliest experiments using the 8 GeV /c in the largeacceptance tune, which is more sensitive to the strengths of the first two quadrupoles. The Monte Carlo uses beam-pipe aperture checks at strategic locations in the magnets to determine the spectrometer acceptance. The effeds of multiple scattering, tracking resolution, and detector extents are also included. Inverse matrix elements, allowing conversion from focal plane coordinates to target. coordinates, were fit to "data" generated by 1he Monte Carlo. The inverse matrix elements were varied to minimize the RMS difference between the reconstruded (resolution-blurred) and generated (actual) target coordinates. In previous SLAC experiments, these matrix elements were used to reconstrud the data. In the current experiment, the data reconstrudion matrix elements were derived independently through the use of kinematic correlations in A(e, c'p) (Section Ill. C). The data-based and Transport-based reconstrudion matrices agreed very well for the 1.6 GeV fc spectrometer, but not as well for the 8 GeV /c. The inconsistencies indicate that the 8 GeV /c Transport model is not a complete description of the magnetic optics. The inconsistency is best demonstrated by comparison of the focal plane distributious of the data with those generated by the PWIA calculation (which includes the Monte Carlo model of the spectrometer). In the 1.6 GeV /c spectrometer, these distributions are consistent with each other up 100 .3000 2000 2500 1500 2000 c"' 1500 c 1000 u u Ill " " 0 0 1000 500 500 -20 -10 0 ' 10 20 0 -30 30 -20 - 10 0 x,. (em) Y,. (em) 2 4 em H 1.6 GeV Singles at 0 = 1 10 20 30 2000 1200 ~~~~~~~~~~~~~~~~ 1000 1500 800 c"':> 600 "' §0 1000 0 u u 400 500 200 o +---~~~~~~~~~~~---+ -160 -120 -80 -40 0 40 80 120 160 0:< 06 (mr) 0 ~~~~~~~~~~~--~~~~ -50 -40 -30 -20 -10 0 10 20 30 40 50 OY,. (mr) FIG. 45. Distributions of the 1.6 GeV / c spectrometer focal 2 plane coordinates for the 1 H(e, e) reaction at Q 2 = 1 (GeV /c) • The points are the data and the histogram is the PWIA prediction for the same integrated luminosity. to shifls, which arc accounted for with matrix element offsets (sec Figure 45). The spikes in the Y16 spectrum are due to discrete track positions for unpaired wires. The discrepancy in dY16 is acceptable because dY16 is not important in the reconstruction. The other coord inates X 16 , dX 16 , and Y16 dominate the measurement of 5, ci>, and ~0 respectively. In the SGeV/c spectrom eter, one can see good agreement (up to shifts) in X8, dX8, and Y8 (Figure 46). 2.5% too narrow. However, the calculation's dY8 distribution is roughly This disagreement is due to a problem with the optics model and not with the aperture sizes, because the height of the other distributions (hence 101 1600 4000 1400 1200 3000 1000 c"' 800 "' ~0 2000 :l 0 u u 600 400 1000 200 0 -30 -20 -10 0 10 20 0 -30 30 -20 -10 X8 (em) 0 10 20 30 Y 8 (em) 4 em H 8 GeV Coinci.jences at 0 2 2000 1200~~~~~~~~~~~~~~~~ 1000 1500 800 "' "' ~ u u §0 1000 0 600 400 500 200 0 ~~~~~~~~~~-T~~~~--f -30 -20 -10 0 C·:<, lmr\ 10 20 30 -20 -10 0 DY 8 (mr) 10 20 30 FIG. 46. Distributions of the 8 GeV /c spectrometer focal plane coordinates for the 1 H( e, e'p) reaction at Q2 = 1 (GeV /c) 2 • The points are the data and the histogram is the PWIA prediction for the same integrated luminosity. the total acceptance) are correct and the disagreement persists when the 8 GeV /c illumination is restricted by placing cuts on the electron kinematics in the 1.6 GeV /c. The coordinates X 8 and dX8 are sensitive to 60, Y8 to 8, and dYs to <I>. (The width of Y8 is determined not by the acceptance, but by the extent of the e lastic stripe. T he small disagreement in widths is probably due to too much tracking-resolution broadening in the spectrometer Monte Carlo.) In order to account for the limitations m the 8 GeV / c Transport model, we introduce several correction factors. The Transport-based inverse matrix elements are reconstructed from the spectrometer Monte Carlo focal plane coordinates. This way, the resolution of the Monte 102 Carlo is not artificially degraded by differences b etween the forward and reverse optics models. In the 8 GeV /c, the Monte Carlo reconstru ction matrices are subjected to two ad hoc modifications. The resulting inconsistency between th e forward and reverse Monte Carlo matrix elem ents is necessary in order to compensate for the dY8 inconsist ency described above and for the presence of a large third-order term in the 6.0 data reconstruction matrix. The dY8 inconsistency is ha ndled by the use of a modified <I> reconstruction: <].)~ C,new = l.0 25 <]_)~C. (4.23) Furthermore, the 6.0 reconstruction matrix elements are re placed by those derived from kin ematic correlations in the data (Section III. C) : (4.24) Without this change, the presence of the third-order term in the data recon stru ction but not in the Monte Carlo reconstruction creates a 3-4% dependence of the H (e, e'p) cross section on th e 6.08 cuL. These modifications and the un certainty in their origin a rc th e main source of the 5% systematic error in the understandin g of th e acceptances. 103 V.RESULTS Extraction of the spectral function S(Em, Pm) and the nuclear transparency Tare performed assuming the scattering can be modelled by the calculation discussed in Section IV, based on the Plane Wave Impu lse Approximation with an Independent Particle Shell Model S(Es, p ). The calculation improves on the PWIA in several ways, but it does not include any effects from Initial and Final State Interactions. In Section V.A the measured S(Em,Pm) are compared with tbe results of the PWIA calculation in order to demonstrate the success of the PWIA and to check the input IPSM S(Es, p ). The good agreement of the data and the calculation then justifies the extraction of T as the ratio of the data and PWIA rates. For IPml < 250 MeV /c the FSI are modelled in the analysis as producing a uniform suppression of the count rate by the measured value ofT. Monte Carlo calculations of the experiment, including the FSI in a sem i-classical cascade model using the free nucleon-nucleon cross section, indicate that the PSI distortions to the measured S(Em,Pm) arc :S 5% for IPml < }JJ [50]. This is because the average angular deflection in elastic NN interactions is .2:: 5°, much larger than the spectrometer acceptances. (In fact the deflection is,...__ 20° at Q 2 = 1 (GeV jc) 2 , where the elastic cross section is largest.) The large deflection is due to the large average t of the elastic differential cross section and to Pauli blocking of small-t collisions. Note that inelastic events occur at Em > m1r and arc not accepted. In extracting S(Em,Pm) and T(Q 2 ), the acceptance of both spectrometers was restricted to 181 < 5%, 16.01 < 15 mr, and I<PI < 40 mr where the acceptance function s A16(E',Ok') and A 8 (E~,np') are large. 104 V.A. Extraction of S(Em, Pm) In principle, the spectral function S( Es , p) could be extracted by dividing the measured six-fold differential cross section by cr~c (Equation 1.9). In practice, finite measurement statistics do not allow binning the data in six dimensions. An alternate approach involves binning the data in on ly Em and Pm and dividing by the rate that would be measured if S(Em. Pm) were equal to one. Specifically, in the PWIA , the number of true coincidences measured m a bin during a run of integrated luminosity £ = J Ldt is: Ntrue(Em, Pm) = £1 dE' dfh,dE~df2p,[p' E~cr~c S(Em, Pm)]A16(E', Ok' )As(E~, Op' ). E-m,Pm (5.1) Here the term in brackets is the differential cross section, Equation 1.9. The integraLion region is over all kinematics w ith (Em, Pm) inside the bin. If the value of S(Em,Pm) is approximately constant across the bin, it can be pulled out of the integral. Also , du e to the large ra nge of kinemati cs that can contribute to the sam e (Em , Pm) bin , the term p'E~cr~c can vary by factors of two in the integral. Contributions of these variations to the statistical error in S(Em, Pm) is minimized by applying this t erm as a weight to the data counts . That is, the relevant values of thi s term are not the average ove r all possible kinematics, but the values actually obser ved for each event. Then the spectra l function can be extracted as: S(E 'm, ) 1 " ] _ _ 1_Nweighted Pm '::: ---;;---L..., 'E' cc ~'fps true · 1 J..- PS true P p(T1 J..- (5 .2) Here N1~~~htcd is given be low and the phase space integr al lps is given by Equation 3.8. ln principle, the inverse of the acceptance functions could also be applied directly as weights to the data, further reducing the statistical error. This is not done because the increased weighting of counts with small acceptances would amplify the systematic errors due to mismatch between the model and actual acceptances. 105 For a measurement of the spectral function, Equation 3.1 is used with a range R corresponding to a single (Em, Pm) bin. The weight is applied to counts in R when calculating Ntrue , Npeak, and Na cc : Nweighted = ""' ~ counts P 1 'E'palcc · (5.3) H is not necessary to weight counts when calculating Nt;;n and N/:cc· Because of the small bin size, the unweighted Nacc « N:Cc and binomial weighting is not used in the calculation of the statistical error. Equation 5.2 can be used to extract a value of S( Em, Pm) at one Os setting. In the (Em, Pm) bins where more than one 08 can be used, the best statistics are obtained by combining the S(Em, Pm) measurements with weights given by the counting statistics. Unfortunately, direct averaging of S(Em,Pm) values calculated independently at each 08 is made difficult by the unknown error in a measurement with zero counts. Furthermore, in bins with low statistics, weighted averages bias the result to artificially low values because of underestimation of the statistical errors in fluctuations to small values. This difficulty is avoided and the correct weighting is achieved by simply summing the numerator and the denominator of Equation 5.2 over 08 before dividing. Radiative effects in the data. produce distortions in the measured spectral functions (Equations 4.5 and 4.6), which are difficult to remove. Currently, radiative effects have not been removed from the measured S(Em,Pm), which therefore do not represent the true (unradia.ted) S(Es,P) described in Section I.A.l. Fortunately, the S(Em,Pm) data can still be used to verify the validity of the PWIA/IPSM model and therefore, of the transparency analysis. When comparing the measured values with the model spectral function, radiative effects are instead added to the model spectral function as described earlier in Section IV.A. In the following graphs, the comparison is usually shown for S(Em, Pm) integrated 106 over the Pm or Em ranges (listed in Table 11) used to measure the transparency. For example, Figure 47 shows the deuteron momentum distribution n(pm), obtained by integrating the S(Em,Pm) measured at Q2 = 1- 7 (GeV /c) 2 over dEm for -30 < Em < 100 MeV. The error bars shown in this and the following graphs are statistical only. For the systematic errors, see Section V.C. Also shown is a histogram giving the integral of the S(Em , Pm) extracted from the PWIA Independent Particle Shell Model prediction, calculated using the Bonn potential wave function (42]. In this and the following graphs, the PWIA result is scaled by the measured transparency (see Section V.B). The statistics of the PWIA S(Em,Pm) are generally 2: 10 times higher than the data, so its error bars are not shown. The graph exhibits, at all Q2 , good agreement over several orders of magnitude between the data and the PWIA prediction. Figure 48 shows the deuteron Em distribution P(Em), obtained by integrating the S(Em, Pm) measured at Q2 = 1- 7 (GeV /c) 2 over d3 pm = 27rP?ndPm for - 170 MeV /c < Pm < 170 MeV jc. In the absence of radiation and resolution, the Em distribution would be a delta function at 2.2 MeV, the deuteron binding energy. The difference in the heights and widths of the data and PWIA peaks is due to imperfect simu lation of the spectrometer resolutions in the PWIA calculation . One sees that the resolution degrades with increasing Q2 due to the larger momenta involved in the measurement of Em and Pm· The peak is followed by a tail due to internal and external radiation. Because S(Em,Pm) is highly peaked at Pm ~ 0 (see Figure 47), the tails satisfy IPmI rv Em (see Equations 4.5 and 4.6). Thus, even though the radiative strength dies rapidly with increasing Em, the p;,, weighting causes the tail height to look almost constant. The absence of strength at Em 2: 120 MeV is due to the lack of Em acceptance there. Figure 49 shows the carbon P(Em) measured at Q2 = 1 (GeV /c) 2 for 0 < Pm < 250 MeV /c. The range extends beyond the fermi momentum PJ rv 220 MeV jc, so 107 10-~~~~~~~~~~~~~~~~r -200 -100 0 100 Pm (MeV/c) 200 300 10-~ ~~~~~~~~~~~~~~~rr -200 -100 0 100 200 300 pm (MeV /c) Q2 = 6 .8 (GeV/c) 2 10 - ~ ~~~~~~~~~~~~~~~r -200 - 100 0 100 p m (MeV/c ) 200 300 10- 1T~~~~~~~~~~~~~~rr 1 -200 - 100 0 100 200 p m (MeV /c) F IG. 47. 2 H Pm distribution at Q2 = 1-7 (GeV /c) 2 • Points witherror bars are the integral over -30 < Em < 100 MeV of S(Em, Pm) extracted from the data and the histogram is the integral of S(Em, Pm) extracted from the PWIA calculation, scaled by the measured transparency. The e rror bars s hown in this and the following graphs are statistical only. 300 108 60 60 50 50 f'40 f'40 >Q) >Q) 2-30 2-30 ..-..ll ..-..ll l'il l'il P::' 20 P::' 20 10 10 0 -50 0 50 100 Em (MeV) 150 0 -50 200 0 60 50 100 Em (MeV) Q2 50 50 f'40 f'40 >Q) 150 = 6.8 (GeV/c) 2 >Q) 2-30 2-3o ..-..ll ..-..ll l'il l'il P::' 20 P::' 20 10 10 0 -50 0 50 100 Em (MeV) FIG. 48. 2 H Em 150 0~~~~~~~~~-t 200 distribution -50 at Q2 0 50 100 Em (MeV) 150 1-7 (GeV /c) 2 for IPml < 170MeVjc. The histogram is the PWIA result, scaled by the transparency. 200 109 most of the spectral function strength is included. The narrow peak at Em ~ 17 MeV due to the carbon 1p nuclear shell and the broad peak at Em ~ 38 MeV due to the 1s shell seen in this figure are more clearly exhibited in Figure 50 because of finer binning. This figure shows the data counts binned in Em for the Q 2 = 1 (GeV /c) 2 , 08 = 43.4° kinematics. In Figure 49 a sh ift of -2.5 MeV is applied to the PWIA result . This is done to cancel disagreement with the data on the position of the lp peak, apparently produced by changes in offset matrix elements (the position used in the PWIA is known to better than 1 MeV). The shift is not applied in Figure 51, which shows the measured Em distribution at each value of Q2 • As Q 2 increases, the peaks become broader and the relative height of the radiative tail increases due to the larger momenta involved. One observes that the 1s shell in the PWIA has more strength at low Em than does the data. Previous A( e, e'p) experiments have indicated that the energy distribution of the nuclear shells are skewed to high Es (see Figure 6) but for sim plicity the IPSM spectral function uses a symmetric Lorentzian. The asymmetry is not due to multi-nucleon effects (such as correlation s) at large Em because it is observed even in the valence orbitals of 208 Pb . Fortunately, the discrepancies in the data and PWJA Em distributions due to this and the Em shift mentioned above do not bias the measurement of transparency, which is largely insensitive to the details of the energy distribution. This is because P(Em) is small at the edges of the 0 <Em < 100M eV range used to measure T and because the range includes the majority of the spectral function strength. In fact, the PWJA Independent Particle Shell Model input spectral function has very little strength at Em 2: 60 MeV (Figure 52). The high-Em strength in the 12 C PWIA/IPSM curves is caused by the radiative tails of the single-particle strength at E m :S 60 MeV. Note the radiative effects are more pronounced than in low- Q2 measurements because E, E', and E~ are large compared to Em. The good 110 60 50 ;::-' 40 I > Q) 230 ,..-._ ~ s 0::20 10 0 -50 FIG. 49. 0 12 C 50 100 E m (MeV) Em distribution at Q2 150 200 1 (GeV /c) 2 for 0 < Pm < 250 MeV j c. The two curves are as in Figure 4 7. agreement of the data and PWIA/IPSM Em distributions for Em 2:, 60 M eV indicates that there is not a great deal of stren gth from multi-nucleon processes (e.g., dynamic correlations) in the measured range of Pm, in spite of the concerns mentioned in reference [14]. Presumably these effects are more important at Pm > Pi· As was the case for the nucleus 2 II, the importance of the radiative tail is enhanced in the P(Em) distribution because of the p;. weighting in the integral over Pm· The significance of the correlation tail at high Pm and Em is also increased. The si ngle-particle nature of the observed strength is dramatically confirmed by the separate momentum distributions of the 1p and 1s shells, shown in Figure 53. These were obtained by integrating the S(Em, Pm) measured at Q2 = 1 (GeV /c) 2 over 111 70 60 50 2 40 ~ ;:J 0 u 30 20 10 0 -50 0 50 100 150 200 E m (MeV) FIG. 50. 12 C Em counts at Q 2 = 1 (GeV jc) 2 and ()8 = 43.4 ° (no Pm cut). Background from accidental coincidences has not been subtracted. The prominence of the ls shell peak is enhanced by the low values of missing momenta accepted at this angle (IPml ;S 100 Me V /c). - 20 <Em< 25MeV and 35 < Em< 70MeV , respectively (after the -2.5MeV shift in the PWIA Em)· The data clearly exhibits the p and s shell behavior present in the PWIA / IPSM calculation: p eaks at IPmI ~ 100 MeV /c and Pm ~ 0, respectively. Th e gap in th e l.s momen t um distribution a t Pm ~ 0 is due to th e absen ce of m easure m ent phase sp ace there (d3 pm = 2n-p'!, dpm ~ 0 for small Pm)· Note that to reduce syst em a tic errors a t the edge of the (Em, Pm) accep tance, bins with ips < 10% of its maximum value at each Oa are not included in the integrals over the S(Em, Pm) of solid target s. This exclusion is no t practiced for 2 1-I , which is hi ghly p eaked at Pm = 0. 112 50 50 = 1 (GeV/c)2 Q2 40 Q2 = 5 (GeV/c)2 40 ......... ......... i i s s ;El:2o ........... ~ ~ 30 ~ 30 ......... 8 20 p.,. p.,. 10 10 0 -50 0 50 100 Em {MeV) 50 150 0 -50 200 0 50 100 Em {MeV) 50 Q2 = 3 (GeV/c)2 40 Q2 150 200 = 6.8 (GeV /c}2 40 ......... i ~ 30 ......... i ~ 30 s 0 ........... ......... ......... 8 20 ~8;20 ~ p.,. p.,. 10 10 0 - 50 0 50 100 Em {MeV) FIG. 5 1. 12 C Em 150 200 distribution 0~~~~~~~~~ -50 at Q2 0 50 100 Em {MeV) 150 1- 7 (Ge V /c) 2 for 0 < Pm < 250 MeV / c . The two curves are as in Figure 4 7. 200 113 140 120 .... > 100 <J) s 80 40 20 o~~~~~~~~~~~~nn~~Hn~~~~~..r -50 0 50 100 150 200 E rn (MeV) FIG. 52. Em distribution for 1 2 C at Q 2 = 1 (GeV /c) 2 from the PWIA calculation with radiative effects disabled. The total 12 C momentum distribution, shown in Figure 54, is obtained by integrating S(Em, Pm) over -30 < Em < 100 MeV. The data exhibit good agreement with the PWIA/IPSM at all Pm and Q2 . One observes that, due to time constraints during the experiment, the distribution was not as well-measured at Pm < 0. The data on 56 Fe and 197Au were taken with fewer statistics, but also indicate the success of the single-particle PWIA description. In Figures 55 and 56, one observes that the Em resolution is not sufficient to distinguish the more-closely spaced energy levels of Fe and Au. But the PWIA/IPSM adequately describes the general shape of the Em distributions. At Q2 ;::=: 5 (GeV /c) 2 in both nuclei, the statistics are so bad that the data can hardly be called a measurement of the spectral function . 114 M M ...-.. :> Q) .....-.. :> Q) C.!> C.!> -.........10' -.........10' 0 ..__., 0 ..__., ...-.. ...-.. 8 0. ..__., 8 0. ..__., ~ ~ 10°~rrn~~rrrr~~rrrrnn~~rrrr~~ -250 -150 -50 50 150 250 10o~~~rrrr~~~~~~~~~rrr -250 -150 -50 p m {MeV /c) 50 150 250 p m {MeV /c) FIG. 53. 12 C (a) lp and (b) ls Pm distributions at Q 2 = 1 (GeV /c) 2 for -20 < Em < 25 MeV and 35 < Em < 70 MeV respectively. The points and histogram are as in Figure 4 7 . The Fe and Au Pm distributions, presented in Figures 57 and 58, also indicate the continued success of the PWIA/IPSM description. It is important to note, however, that the momentum distributions used in the PWIA/IPSM calcu lat ion is na rrower than those used to fit A(e, e'p) measurements at low Q2 . The apparent discrepancy in the measured Pm widths is probably due to uncertainties at low Q 2 in the correction s for proton FSI and Coulomb distortions (see Appendix E). In conclusion, one observes excellent overall agreement between the data and the PWIA/IPSM model. The agr<'emc>nt indicates that the> single-particle PWIA description of the scattering describ<"'s the dominant contribution to the scattering. The observc>d disagreements do not greatly influence the transparency measurement and could be removed by refining the IP SM model. The S(Em,Pm) of Fe and Au are measurcd for the first time, with errors comparable to the pr<"vious, low-Q 2 measurements 11 5 10 3 3 Q 2 = 1 (GeV/c) 10 2 2 Q2 = 5 (GeV/c) 2 102 10 ~ ~ > > Q) Q) C!l "10 C!l 1 "101 ~ ~ ..-..s ..-..s p. p. ~ 10° ~ 10° 1 10- ~~~~mrr~mnmnrrmrrmrrmnm~mrrmr -300 -200 -100 0 100 200 300 400 Pm (MeV/c) Q2 = 6.8 (GeV/c) 2 >Q) C!l "10 0 1 ........ l0- ~mn~~rrm~rrmwmnm~~~~~~ - 300 - 200-100 0 100 200 300 400 Pm (MeV /c) 1 FIG . 54. 12 C Pm distribution 10- ~~~mrrmn~mmNmrrmrrmrrm~~mrr~ 1 - 300-200-100 0 100 200 300 400 pm (MeV/c) at Q2 1- 7 (GeV /c) 2 - 30 < E m < 100 MeV. The two curves are as in Figure 47. for 116 80 ~ I ~ 60 ~ 20 20 a~~~~~~~~~~~~~~~ 0 50 100 Em (MeV} 150 -50 200 0 50 100 Em (MeV} Q2 Q2 = 3 (GeV /c)" 80 200 = 6.8 (GeV/c)2 80 ~ I ~ 150 ~ I ~ c..'J 60 - ~ 20 60 20 0~~~~~~~~~~~~~~~ o~~~~~~~~~~~~~~~ - 50 -50 0 FIG. 55. 50 100 Em (MeV} 56 Fe Em 150 200 distribution at Q2 0 50 100 Em (MeV} 150 1-7 (GeV /c) 2 for 0 < Pm < 250MeV jc. The two curves are as in Figure 47. 200 117 125 125 tl 7'100 > 7'100 > (!) (!) 2- 75 2- 75 ,...__a £::!_ 0.. 25 50 25 0~~~~~~~~~~ 0 50 100 Em (MeV) Q2 150 -50 200 0 50 100 Em (MeV) Q2 = 3 (GeV/c 2 125 150 200 = 6.8 (GeV/c) 2 125 7'100 > (!) 2- 75 ,....._ rx:l 8 0::: 50 25 25 a~~~~~~~~~~~~~~~ o~~~~~~Hn~Hrl~~~~~~ -50 - 50 0 50 100 Em (MeV) FlG. 56. 197 Au 150 Em 200 distribution at Q2 0 50 100 Em (MeV) 150 1- 7 (G eV jc) 2 for 0 < Pm < 2 10 Me V jc. The two curves are as in Figure 47. 200 118 10-~~~~mmmm~~~mm~~rrM~~nr -200 -100 0 100 200 p m (MeV /c) Q2 300 400 = 3 (GeV/c) 0 100 200 Pm (MeV/c) PTG. 57. 56 Fe Pm -200 -100 0 100 200 300 400 pm (MeV/c) 2 10-~~~~~~~mm~~~mmmm~~~ -200 -100 10-~~~~mmmm~~~mm~~mm~mm~ 300 400 Q2 = 6 .8 (GeV/c) 10- ~riW~~~~~~~~~mm~~rrr 1 -200 -100 0 100 200 300 pm (MeV /c) distribution 2 at Q2 1- 7 (GeV /c) 2 - 30 < Em< lOOMeV. The two curves are as in Figure 47. for 400 119 ,t, Q2 = 5 (GeV/c)Z r t1 If J- 1 10°~~~mm~ntt~~~~~mM~~~ -200 -100 0 100 200 pm (MeV /c) 300 400 ;u 1:- ~ 10° 'I -200 -100 'I 'I' 'I 0 100 200 pm (MeV /c) Q2 = 3 (GeV /c) 2 Q2 ~l I .r' 300 = 6 .8 (GeV/c) 400 2 f~~ ~ l r- ~ 10°~~~~~~mm~~~~~~~~ -200 -100 0 100 200 pm (MeV /c) FIG. 58. 1 97 300 400 Au Pm distribution 10° - 200 - 100 0 100 200 300 pm (MeV /c) at Q2 1- 7 (GeV /c) 2 for - 30 < Em< lOO MeV. The two curves are as in Figure 47. 400 120 of nearby nuclei. Due to the higher momenta. involved, the current measurements ha.ve poorer (Em, Pm) resolution tha.n the low-Q 2 da.ta.. Interestingly, the momentum distributions of Fe a.nd Au a.ppea.r to be narrower tha.n observed in the low-Q 2 experiments for nearby nuclei. As expected, the da.ta. show no evidence of distortion due to the FSI of the proton (excluding the re-norma.liza.tion due to absorption) or of significant multi-nucleon effects a.t IPml < 250 MeV /c. The success of the PWIA/IPSM description of the scattering justifies the use of this model in the calculation of the nuclear transparency T. 121 V.B. Measurement of T(Q 2 ) At low Q 2 the fraction of saturation of the spectroscopic sum rule (Equation 1.10) was given by the integral of the measured S(Em, Pm), (5.4) divided by z. As discussed in Section I.A.2, one shortcoming of this approach is that S(Em, JJm) is only m easured in a finite range of (Em, JJm), so the integral over the measured region is artificially low. Ideally, a correction would be appl ied based on a mode l for the amount of the Zef 1 strength outside of the experimental acceptance. In addition to this model dependence, measurement of S(Em,Pm) also assumes the validity of the PWIA cross section (see Equation 5.2). (Although in the low-Q 2 experiments, the approximation is improved by dividing Zef f by the prediction of the Distorted Wave Impulse Approximation instead of Z.) For the current experinwnt , the usc ofT= ZeJJ/Z would be further complicated by the presence of radiative effects in the m easured S(Em, Pm) · The presence of internal bremsstrahlung and internal virtual photons changes both the distribution and the magnitude of S(Em, Pm) · Except through the use of a full radiative correction procedure, the correction for these effects would also depend on the model of S(Es, p ). Use of Equation 5.4 would generate additional problems common to measurements which require binning the data. First, the statistical error is increased by unequal weighting of counts. The presence of d 3 pm = 47rp~dpm in the Zeff integral m akes the S(Em, Pm) measurement es peciall y sensiti ve to the values at high Pm, where the statistical errors are large. The method is likely to be very unreliable at Q2 2:: 5 (GeV /c) 2 for the Fe and Au nuclei, where the statistics are especially poor. Second, the biases introduced by reconstruction errors (offsets and resolution) are a mplified by the b inning. For example, the width of the Pm distribution increases 122 TABLE 11. The (Em,Pm) range n Targets Em,rn.in Em,max Pm,min Pm,max lH, 2H -30MeV lOOMeV -170 MeV /c 170MeV jc C, Fe -30MeV lOOMeV OMeVjc 250MeV /c Au -30MeV lOOMeV OMeV/c 210MeV /c with Q 2 due to the degraded resolution. Thus, the p~,-wcighted integral also increases, mimicking the onset of CT. Third, errors in the acceptance function are amplified by the binning since the uncertainty in A 16 A 8 at the edge of the acceptance is larger than the error in the product of the spectrometers' solid angles. In light of the preceding discussion, the technique used in the current experiment IS division of the measured rate in a large (Em,Pm) range R (given in Table 11) by the rate calculated in the PWIA/IPSM. Although still dependent on the model for S(Es, p ), the PWIA calculation is more accurate than the standard PWIA used in Equation 5.4. Furthermore, the method maximizes the use of the data statistics by weighting the counts equally and is sensitive only to the broad structure of the spectral function and acceptance functions. The extraction ofT only uses data where the acceptance of the spectrometers and the shape of the spectral function are well described. In order to maximize statistics and minimize sensitivity to details of the spectra.! function model, the range R is chosen to be as large as possible without including (Em,Pm) bins where lps is< 10% of its maximum value for a ll 08 . Rejecting events with Em ~ 140 MeV ~ m 7r ensures that no inelastic processes have occurred. Use of IPml < 250 MeV /c for the solid targets minimizes the contribution of the correlation tail. At each kinematic setting, the measured transparency is: (5.5) 123 Here Ntrue (calculated usmg Equation 3.1) and NpwJA are the measured and calculated number of coincidences for the experimental integrated luminosity £. The T(Q 2 , 08 ) are shown in Figures 59 through 62 for each kinematics. In most cases, the measured T( Q2 , 08 ) are consistent within errors at each Q2 • They are combined in a weighted average: (5.6) The rise in T(Q 2 ,08 ) with increasing 08 in the 2 H(e, e'p), Q 2 =I (GeV/c) 2 data IS probably due to a Pm shift introduced by imperfect offset matrix elements. This data is very sensitive to such shifts because it is highly peaked in Pm· The effect of the shift averages out when combining the angles, so there should be no degradation of the error in T( Q 2 ) (in particular, the measurement is dominated by the (}8 = 41.3° kinematics, which is centered at Pm = 0 and so is not sensitive to the shift). The IPSM spectral function does not include the effects of short-range nuclear correlations, which move strength to IPml > Pi· The measured T must be corrected by the ratio of J Sd3 pdEs for the IPSM and correlated spectral functions, integrated over the (Em, Pm) rangeR: 1 (5.7) Ccorrel = { dEs(PpS(Es, p ). 1correl JR The correction assumes the correlations produce a uniform suppression of the uncorrelated spectral function strength at IPml < 250 MeV /c. This strength appears at larger IPmI but, since this region is excluded from the transparency measurement , the form docs not need to be specified. Due to uncertainties in llw amou nt of correlated spectral function strength at large Em, the integral over the rangeR is averaged with the integral over all Em: { 3 , . f correl = 0.5 ( }-,. dEsd pScorrei(Es, P) + R ioo dEs 1Pm,max d pScorrei(Es , 3 0 Pm,m.1n p) ) . (5.8) 124 1.4 = 1 (GeV/c) = 3 (GeV /c) = 5 (GeV /c) = 7 (GeV/c)2 2 2 1.2 2 1.0 0.8 E-c 0.6 0.4 0.2 0.0 0 10 20 40 50 60 PIG. 59. Nuclear transparency T(Q 2 • 88 ) measured in 2 H( e, e'p). values with multiple 08 kinematic settings. The inner error bars are statistical only and the outer error bars give the total error (statistical plus systematic, combined in quadrature). For 1 2 C, the integral over R is calculated using Scarret(Es, p ) calcul ated for 12 C [52] and the other integral is calculated using Scarret(Es, p ) calculated for 16 0 [53]. For Fe and Au the integrals were calculated using a correlated nuclear m a tter spectral function corrected for finite nucleus effects (54]. The fraction of strength a t Em > 100 MeV for this spectra l function is 21%, which is quite possibly an overestimate. Vonderfecht el al. [55] claim that the fraction should only be 13%. This discre pancy motivates the use of Equation 5.8. The resulting correction factors are given in Table 12. The quoted errors are based on the diffe rence between t he two integrals in Eq uation 5.8. It is expected that the error in the correlation correction will be red uced through 125 1.0 0 X 0.8 • 0 Q2 Qz Qz Qz = 1 (GeV /c) = 3 (GeV/c) = 5 {GeV/c) = 7 (GeV /c) 0.6 ~ 2 2 2 2 f t t1! f~ ~ ~ ff 0.4 0.2 PIG . 60. Nuclear transparency T(Q 2 , 08 ) meas ured in 1 2 C (e, e'p). further study of available correlated spectral functions. Based on compan sons of tra ns parency calcul ated with different IPSM spectral functions, the Q2 dependent variations associated with the correlation correction a re on ly half as la rge as the quoted error in Ccorrel· Thus, most of the error in Ccorrel docs not affect the sear ch for a Q2 dependence in T( Q2 ). The correla tion correction assumes the absence of CT , which suppresses t he Initial State Interactions of the proton a nd could therefore redu ce the e ffect of the short-range correlations. However, if the onset of CT we re to suppress the correlations, it would bias the m easured T to artificia ll y high values. This phenomenon would therefore inc1·ease the sensiti vity of the transparency m easurem ent to the onset of CT. The degree of reduction of the correlation tail depends on the relative time scales for the 126 1.0 D X • 0 .8 0 0.6 GeV /c) 2 3 (GeV /c) 2 5 (GeV /c) 2 (GeV /c) 2 I !t $$I E-c 0.4 ~ ~ *~ ~ 0.2 FIG. 61. Nuclear transparency T(Q 2 , 88 ) measured in 56 Fe(e, e'p). TABLE 12. Correlation tail correction to the PWIA calculation A lH, 2H Ccorrel 1.00 12c 1.11 ± 0.03 s6Fe 1.26 ± 0.08 197 Au 1.32 ± 0.08 127 1.0 0.8 0 X • 0 Q2 Q2 Q2 Q2 = 1 (GeV /c) 2 = 3 (GeV/c)2 = 5 (GeV/cY2 = 7 (GeV/c) 0.6 0.4 0.2 FIG. 62. Nuclear transparency T(Q 2 , 88 ) measure d in 197 Au(e, c'p) . formation of a P oint Like Configuration (tPLc) and the occurrence of a short-range corre lation (tsrc). In the PLC rest fram e, th e formation t ime lPLC should be t he same as the expansion time /T ~ Lh/c. T h e n aive pat-ton model gives th e largest value, T = Rpjc. Time dilat ion from the Fermi motion is negligible, and tPLC :S 0 .8 fm jc. The correlation time scale tsrc can be estima ted as the crossing t ime of two nucleons' ha rd cores, roughly 1 fm di vided by the relative velocity (0.21c for a typical relative momentum rv 200 MeV /c): isrc ~ 5 fm. Thus, it appears that the correlation occurs before the form ation of the PLC an d p ersists until the electroll-proton interaction. The resulting t ra nsparen cy values are given in Fig ure 63 a nd Table 13. Values a re not ide nti cal to those published in reference [56], where these results are averaged with valu es from an independent a nalysis [57] that was performed as a consisten cy 128 I 1.2 1.0 I i 1 ~ ~ f fT :p _l ( tH 2H 0.8 E-- 0.6 - 0.4 - 0.2 0.0 0 r- ~ $ * ' t* * ! ' I I I 2 4 6 Q2 (GeV /cf ~ c I Fe - - Au r- 8 FIG. G3. Nuclear transparency T(Q 2 ) Measured m Experiment NE-18. The inner error bars are statistical only and the outer error bars give the total error (statistical plus systematic, combined in quadrature). check. The 1 ll res ults are consist ent with the expected T = 1 (no Ji'Sl). In ad dition , the m eas ured inclusive H( e, c') cross sections agree with the exclusive II( e, e'p) values to within 2%. The m easured transparency is insensitive ( < 2%) to large changes (~ 40 MeV Ill E, a nd ~ 50 MeV /c in Pm) in t.he range R , indicatin g once again the successful d escription of th e data by the PWIA / IPSM model and th e a bsen ce of large multinucleon effects at Em?:, 60MeV. The nuclear targets show decreasing proton transmission for increasing A, as expectcd . The T(Q 2 ) measured on 2 ll varies between about 0.9 and 0.95 , indi cating the 129 TABLE 13. Nuclear transparency T Measured m Experiment NE-18, with statistical and systematic errors. A T(Q2) Systematic Q 2 = 1 (GeV /c) 2 Q 2 = 3 (GeV jc) 2 Q 2 = 5 (GeV /c) 2 Q 2 = 7 (GeV jc) 2 Error 1I-I 0.967 ± 0.006 0.990 ± 0.013 1.042 ± 0.025 1.028 ± 0.041 0.066 2H 0.889 ± 0.014 0.894 ± 0.028 0.941 ± 0.028 0.949 ± 0.037 0.067 12C 0.637 ± 0.018 0.614 ± 0.026 0.598 ± 0.031 0.663 ± 0.037 0.080 56 Fe 0.506 ± 0.018 0.401 ± 0.023 0.409 ± 0.037 0.425 ± 0.036 0.097 197Au 0.372 ± 0.024 0.2-14 ± 0.023 0.236 ± 0.036 0.329 ± 0.056 0.097 presence of weak FSI effects. For Fe, and Au one observes an enhanced transparency at Q 2 = 1 (GeV /c) 2 . This is at least partially due to the smaller nucleon-nucleon inelastic cross section at momenta-::::= 1 GeV /c [6]. The effect is smaller in C, and cannot be resolved within the error bars. Color Transparency is expected to produce an increase in T with increasing Q 2 for the nuclear targets. ln testing for this increase, the Q 2 = J (GeV /c) 2 data should not be used because of its enhanced T. None of the nuclei exhibit an increase larger than !.he error bars in range Q2 = 3- 7 (GeV /c) 2 , so the measured transparency is consistent wit.h the absence of CT. The solid target data can be used to rule out some models of CT, as will be discussed in Section V.D. [The 2 H data cannot be used to test for the presence of CT, since the errors are consistent with T2H(7 (GeV /c) 2 ) = 1 (full CT) a.nd 12 11 (Q 2 ) =constant (no CT).] 130 V.C. Systematic Errors Systematic errors in the calculation of S(Em,JJm) and T(Q 2 ) are caused by uncertainties such as those in the efficiency corrections, the kinematics, the luminosity, and the PWIA calculation. These errors are discussed throughout the text, but are summarized here for convenience. The systematic errors are indicated in the graphs of T(Q 2 ), but not in those of S(Em,Pm). The errors in the PWIA calculation are given in Table 14. The PWIA calcu lation of Ips used in the extraction of S(Em , JJm) has a unit spectral function, no radiative effects, and no correlation tail correction . In this case the third, forth, and fifth lines of the table do not apply. The measurement of T( Q 2 ) involves the full PWIA calculation, and all five error sources are relevant. The error on the radiation is an upper limit derived by comparing transparency values for different radiation length targets. The error in the correlation tail correction is given in Section V.B and the remaining errors are discussed in Section IV. B. The uncertainty in the 5( Es, p) energy distributions is characterized by two spectral function models with Es distributions based on shell locations from Ilartree-Fock ca.lculations or from A(e, e'p) and A(p, 2p) measurements at low Q2 (see Appendix E). The Ilartree-Fock Es distributions agree with the current data, and the low Q2 distribution should be considered as an extreme model to test the sensitivity to the Es distribution. The transparency calculated using the two models disagrees by 4% in Fe and 5% in Au, but the Hartree-Fock Es distribution is probably more accurate than the disagreement between the models. The systematic uncertainty in T(Q 2 ) is therefore estimated to be < 2%. The insensitivity of the transparency measurement to rearrangements of the strength in Em is due to the large range of Em used in the measuren1ent. Note the systematic error on the model S( Es, p) does not include the model- 131 dependence of the p distributions. The Fe and Au transparency are sensitive to the width of the P\t\!IA/IPSM spectral function in Pm, since there is still moderate strength at the high Pm cut. This model-dependence is mostly independent of Q2 (see the discussion following Equation 5.8) and so does not effect the search for Color T ransparency presented in this paper. Note that the IPSM momentum distributions used in the analysis do not a.g ree with those measured in nearby nuclei at low Q2 (for details, see Appendix E). If these spectral functions are used instead , the measured Fe and Au transparen cies increase by a factor of"' 1.05 and "' l .15 respectively. Table 15 lists the systematic errors from other sources (i.e., the apparatus and analysis procedures). These errors are common to any quantity derived from the measured A(e,e'p) cross section. The total errors in the S(Em,Pm) and T(Q 2 ) measurements are given in Table 16. One of the main sources of error is the 5% uncertainty in the experimental acceptance. The uncertainty is greatest for the 8 GeV /c spectrometer, which was run in the TABLE 14. Fractional uncertainties in the PWIA calculation. The first two rows apply to both the S(Em, Pm) and T(Q 2 ) measurements. The following three rows apply to t he measurement of T(Q 2 ) only. Source JH 2H 12c 56Fe, 197 Au Off-shell aep 0.00 0.01 0.02 0.02 G~ and G~ 0.02 0.02 0.02 0.02 Model S(Es, P) 0.00 0.005 0.02 0.02 Radiation 0.02 0.02 0.03 0.03 Correlation ta.i Ia 0.000 0.000 0.027 0.061 Total S(Em, Pm) 0.020 0.022 0.028 0.028 Total T(Q 2 ) 0.028 0.030 0.053 0.068 aError is almost entirely independent of Q 2 . 132 TABLE 15. Experimental systematic errors Notes Source Value Kinematic uncertainties 0.01 Beam charge 0.005 Dead times 0.003 Appendix C Proton absorption 0.02 Table 10 1.6 GeV jc track ID 0.01 Table 10 1.6 GeV jc multiple tracks 0.01 Section Ill.B 8 GeV jc track ID 0.01 Table 10 8 GeV jc multiple tracks 0.01 Section IIJ.B Coincidence identification 0.01 Table 10 Spectrometer Acceptances 0.05 Section IV .G Target thickness 0.007 Maximum value a Total 0.060 aLiquid targets 0.007; solid targets 0.002 TABLE 16. Total systematic errors. The final column indicates the error in the Q 2 dependence of T(Q 2 ), obtained by subtracting the Q2 -indepcndent portion of the error in the correlation correction. 56Fe, 197 Au Sou IT<' S"(Em , Pm) 0.063 0.064 0.072 0.083 T(Q2) 0.066 0.067 0.080 0.097 Q 2 -dependent T(Q 2 ) 0.066 0.067 0.076 0.081 133 previously untested large acceptance tune. In the current experiment, the acceptance functions of the spectrometers were constrained m.osLly with elastic electron-proton scattering, which provides limited illumination. Use of deep inelastic scattering to provide nearly structureless illumination of the spectrometers could help constrain the acceptance functions at the edges of the spectrometers. Experiment E-140X, which ran immediately before NE-18, took some deep inelastic scattering data using the large acceptance tune of the 8 GeV fc spectrometer. Further study of the acceptance may reduce the acceptance systematic. Another large uncertainty is the width of the momentum distributions of Fe and Au, which seem to be narrower then observed at low Q 2 . High statistics, high resolutions S(Em, Prn) measurements at Q 2 > 1 (GeV /c) 2 would be valuable to confirm this apparent disagreement and to fine-tune the PWIA/IPSM model spectral functions. 134 V.D. Comparison of T(Q 2 ) with T heory V. D . l . The Glauber Predictions Before considering predictions including the Color Transparency effect, it is worth comparing the data with the predictions of standard nu clear physics, generically referred to as the Glauber prediction. The followin g sem iclassical model of the FSI of the outgoing proton illustrates the necessary considerations in a simple way. The model is developed to predict T(Q 2 ) for the Q 2 2:: 3 (GeV /c) 2 data. In this energy regime, due to the small ( < 0.1 fm) de Broglie wavelength of the proton, the FSI can be treated as interactions with the individual nucleons in the recoiling A - 1 system. Furthermore the inelastic and total proton-nucleon cross sections are approximately constant (within about ±5mb) at O"in ~ 26mb and O"tot ~ 43mb (see Figure 64). Almost all elasti c nucleon-nucleon interactions knock t h e proton outside of the exrerimental acceptance [50], so t he total pN cross section is most relevant. The rroba.bility that a proton stru ck at position r = (.~:,y,.::) escapes the nucleus without interacting is, -1 00 P (r) = exp ( dz'O"tot PA-I(r')), (5.9) where the :: ax1s IS parallel to p' and PA-l (r' ) is the nu cleon numbe r density of the recoil A- 1 system at the position r ' = (:r,y,z' ) on the proton's path. The transparency is this probability, averaged over the position of the struck proton: Tcla ssical = A1 J rpA(r)P(r). d3 (5.10) Here PA( r ) is the number density in the nucleus A. For simplicity we take A and A- 1 to be uniform spheres of r adius Ro given in Table 5. The resulting proton mean free path,).= 1/(PA-JO"tot), is 2.984, 2.021, a nd 1.61 8fm respectively for C, Fe, Au. Substituting 1'j_ = ,j.T 2 + y2, one finds: 135 400 I - 1 1 1 1 I nr - - . · - · 1 - 1- T n I I f - -I -- , - - 1 1 1 lTC(- -~ - -, -rn .,,1-- - ~ -· l -, oln,i I 10' Utotal .D .rlNI .....,._----..:... .4 E l; t, ·" •.I', I"'l i'l I .'"' 1 I I Ue(astic 101 ' 5 I _. I I I I I II I J I 1111 . _ ,_ , I .. I _ I 10 0 10-1 10 1 J- I . I • 10 I I I I Ill 7 IIJJI 1 . 10' 10' (GeV/c) Pbeam 1. 9 I I.J.Jll I I I I I I I Ll I 2 3 4 5 6 7 10 J I 20 I I I I 30 40 50 I I 100 Ecm (GeV) I I I 1 I II j I I I I II I I I I " I I I , .. · · ·.;, ·-.· -:I.,_. I • l~l I II I I I . I I "f Uc!aslic I I I I I I ill Pbeam 3 19 I I 3 4 5 6 II I I II I I I I I ; 10' 10' (G e V/c ) 5 4 I I ' ... , I 10' 2 9 i ! ~ Utolal 6 7 B 910 I I ,I I 7 B 910 I I 10 I I 20 30 40 30 I I 40 50 60 Ecm (G eV) FlG . 64. Nucleon-nucleon total and e lastic cross sections vs. laboratory beam momentum Pbeam and total center-of-mass energy Ecrr" from [38] . 136 (5.11) P(r)=exp[(JR& -7·1-z)/A]. This factors into a z-dependent piece and a r 1. -dependent piece, and the 3-dimensional integration can be performed in two steps: integration over dz with limits±)R5- 1-}_ and integration over d2 r1. = 2nTJ.dr 1.· The result is: _ 3 TclaSStcal - A 1 A 1 A A 4 ( Ro ) [ 1 - 2 ( Ro ) 2 + 2 ( Ro ) ( 2 + Ro ) e l _ 2 Ro;" . (5.12) The resulting transparency predictions for C, Fe, and Au are 0.52, 0.29, and 0.17 , respectively. These are 15- 30% below the values measured at Q 2 = 3 and 5 (GeV /c) 2 , which is not surprising in light of the semiclassical approximation and the grossly simplified model of the nuclear density. The nuclei Fe and Au approach the limit Ro » A, where the expression simplifies to Tclassical c::: 3A/4Ro. This is easily understood by noting that escaping protons come from a region with z within >. of the nuclear surface. The transparPncy is the volume of this region, 7f R5A, divided by the volume of the entire nucleus, 47r R6/3. The data are compared to more sophisticated models of the transparency in Figures 65 through 67. Because the predictions vary so much (up to 40%), the differences in their assumptions will be discussed in some detail. Farrar et al. (2] used the semiclassical approximation with a 101 = 40mb and a more realistic Woods-Saxon shape for the nuclear density, and obtained T uniformly higher than the above estimates. In general, the other calculations are performed usmg some variation on the Glauber multiple scattering theory (60]. As in the semiclassical model above, the struck proton propagates without disto1tion between interactions with individual nucleons, but now the calculation is (ideally) performed quantum-mechanically. For example, Jennings and Miller (3,4] calculate the transparency in the eikonal approximation, assuming that any FSI res ulting in excitation of the A - 1 system 137 1 1 .8~--------~~--------~1 ---------~ --------~ -------,r 0 .7 - C Data L -- -_--~----_--_-_:-_ - -----,-+ - -_ - - ----.. E-< .6 - :.::::·(ccA)r ==''=====I=====+===:==-==== Farrar --~--=-=_;-~~~-__:: ~ -:=-~ ~ _--- -- -;: ::: .5 - - - -\ - - - - - - - - - - - - - Benhar (GA) t- FSZ '93 .4 ~--------·~---------~~--------r-l--------~----------t- 0 2 4 6 2 Q (GeV /c) 8 2 10 FIG. 65. Standard T predictions for 12 C. Curves are from refere nces [2-4,6,5,58 ,59]. prevents detection of the proton (similar to the use of <Itot instead of <Iin in the semiclassical calculation above). To lowest order in the Glauber expansion , the wave function of th e outgoing proton is [3]: i p ·r 'lj;( r ) = ( e1r)3/ 2 ex p 2 ( - <7;ot ;= z ) d::'PA-1 (r' ) . (5.13) Ilere th e use of <7totf2 instead of <Itot indicates that the a ttenuation of the amplitude is just the square root o f the attenuation of the probability given by Equation 5.11. The amplitude for different initi al positions z of the stru ck proton are allowed to interfere, y ielding: (5.14) .Je nnin gs and Miller also calculated th e transparen cy using the sem iclassical a pprox imation and found agreement with the eikonal to within a few percent [3]. They used 138 .60 ,_______~------~------~------~------r <> Fe Data .55 .50 E-< .45 .40 .35 \ \ \ ···y······· +--------1--------T------ KYS ,______ · · · · · · · · · · · · Nikolaev · · · · · · · · · · · · · · · · · Benhar (CGA) --- FSZ '93 =================================== Farrar • - - - ---:-= ---~ ~- · - · - · - · - - - - - - JM '91 Benhar (GA} .30 ,_------.-------.-------.-------.-------r 0 2 8 10 FIG. 66. Standard T predictions for 56 Fe. Curves are from refe rences [2,3,6,5,58,59]. <Tt ot =40mb and the Woods-Saxon (Ferm i) form for the densities. Use of the total cross section is not strictly correct. Kohama et al. (58] argue that because the Em acceptance includes the most important excited states of the A - 1, it is more appropriate to apply a closure sum over the state of the A - 1 system. (See, for example, the discussion following Equation 1.10.) In a G lauber model neglecting the A - 1 recoil (justified for large A) and nucleon-nucleon correla tions, they find a form for the d3 p'-integrated transparency identical to Equation 5.10 except with <Tin in pla ce of o-101 . The connection is understood by noting that the classi cal calculation effectively performs the closure sum on the A - 1 system by ig noring its excitations . Furthermore, a m occurs because protons undergoing only elastic inte ractions are included in the d3 p' integraL Unfortun ately, the integral over d3 p' is not well-justified in the current experiment because the finite p ' acceptance excludes most 139 1 .40 ~----.---~'--------~'---------L-'--------L--------r • A~ Data .35 - t- \ j"' .30 - -\--- = .25 - · · -\- -------- - - - -- - --·\ ...-- -1 \ 1- _.__ ----,~c=--------t=~~~ KYS, JM '92 ~JI'/;. Benhar (CGA) Nikolaev FSZ '93 1- ----- ===~===F======f======= Farrar Benhar (GA} t- .20 - . 15 4---------.,---------.-,--------.-,--------,.-------~l- 0 2 4 6 Q2 (GeV /c} 2 FIG . 67. Standard T predictions for 197 Au. 8 10 Curves are from references [2- 4,6,5,58,59]. elastically-scattered protons. This is the reason why a-tot was used in the semiclassical derivation. Note that it is difficulL to estimate the error made in performing the cf3p' integral because if it is not performed, t he result is not simply Equation 5.10. The truth probably lies somewhere between the eikona l a nd the closure approach. Kohama et al. use a;n =28mb a nd the Woods-Saxon form for the de nsities [58] . Because this analysis includes excitations of the A- 1 system (or eq uivalentl y, because am < atot), the resulting transparency is higher than the J ennings and Mille r value . B enh ar el al. [5] also begin wit h Equation 5.13, but apparently do not allow interference between the different initial positions. They publish two curves, with the lower val ues (GA = Glauber Approximation) calculated as : (5.15) which is clearly eq ui valent to the classical expression, Equation 5.10. They u se nuclear 140 density distributions obtained by fitting elastic electron-nucleus scattering data and o-101 = 43.3mb. Their upper curve (CGA = Correlated Glauber Approximation) includes the suppression of nuclear density near the initial position r due to nucleonnucleon correlations (i.e., other nucleons are excluded from the immediate region by the hard core of the nucleon-nucleon potential). They find the correlations enhance the transparency by 20 30%, but Kohama et a/. [58] and Nikolaev et al. [59] argue that the effect is much smaller. Nikolaev et al. (59] independently arrived at the same transparency equation as Kohama et al. through the use of the same closure and d3 p'-integration arguments. Like Benhar et al., they then included the correlation of the other nucleons with the struck proton but found that the effect was reduced by r~j(r~ + bpN) ~ 1/3 by the finite range of the pN interaction (here 1·c ~ 0.5 fm is the standard deviation of their assumed gaussian 2-nucleon correlation function and bpN ~ 0.5 fm 2 is the pN elastic slope parameter) . Crudely speaking, the blurring introduced by the finite range "fills in" the correlation hole. They also included the 2-body correlations of the spectator nucleons and found that these raised the transparency, because the correlation reduces the shadowing of one spectator nucleon by the other. The two correlation effects parLly cancel each other, for a net transparency increase of only 2 3% (59]. Nikolaev ct al. use CTin =32m b and a rermi nuclear density. The smaller value of CTin used in Kohama et al. is at least partly responsible for their larger transparency prediction. Frankfurt, Strikma.n, and Zhalov were the first to publish a. prediction for A(e, e'p) that took the variation of the pN cross section into account, leading to a Q 2 -dependent Glauber prediction (6]. They used a more general version of the eikonal approx- imation, with the distortion modelled by an optical model potential (with phase shift and absorption) rather than just by an absorbing cross section ( cf. Equa- t.ion 5.14). For o-101 they used 26mb at p' ~ 1 GeV jc, increasing to the maximum 141 of 43mb at p' :::::= 2 Ge V / c, and slowly decreasing and becoming constant at 40mb at p' » 3 GeV /c. The value at p' :::::= 1 GeV jc is 4mb less than the free-nucleon value, based on observed disagreements between Glauber calculations and the pA cross sections measured at momenta of 1 GeV /c. (Such a correction is not applied at higher Q 2 because the Glauber approximation is expected to be more accurate at higher energy.) This model does a good job of reproducing the measured transparency drop between Q 2 = 1 and 3 (GeV jc) 2 . The main conclusions from the graphs of the Glauber predictions are: 1) the transparency expected using standard nuclear physics is uncertain by "' ±20%, and 2) the Q 2 ~ 3 (GeV jc) 2 data is within the range of the theories, and 3) the enhanced transparency at Q 2 = 1 (GeV jc)2 is at least partly due to the lower pN cross section. Conclusion (1) indicates that the average magnitude ofT cannot be used to test for the presence of CT. In comparing with any of the CT predictions, one can easily imagine improving (or destroying) the agreement with the data by a 20- 40% renorma.lization of the theory. Currently, the only fair test of the presence of CT is in the Q2 dependence of the data. If an accurate theoretical consensus on the G lauber prediction arises, then much more stringent tests of CT will be possible. Currently the main uncertainties in the Glauber prediction are the treatment of elastic interactions (i.e., whether or not to include the excit<'d states of the A- 1 system) and the treatment of nucleon-nucleon correlations. Conclusion (2) indicates that the present data may not be subject to the incomplete saturation of the spectroscopic sum rule observed in A(e, e'p) at Q 2 < 1 (GeV /c) 2 . Given the current dispersion in the theoretical predictions, this question also cannot be resolved with certainty. But it is suggestive that the Q 2 ~ 3 (GeV jc) 2 data generally lie at the high end of the Glauber predictions. As discussed in Section I. A .2, there were many reasons to think that the problem wou ld not persist in the current experiment: use of a large measurement range R, the use of a correction 142 for strength in the correlation tail, the increased single-particle nature of the reaction at Q2 ~ 1 (GeV /c) 2 , and the simpler theoretical description of t he proton FSI. This simpler description should allow for decreased uncertainty in the FSI predictions. The significance of conclusion (3) is that the various Glauber calculations can be tested more rigorously if the authors repeat their work taking the change in cross section into account. Then the high-statistics Q 2 = 1 (GeV /c) 2 data can be used to set the highly uncertain normalization of the predictions. Thus, the entire data set provides an excellent opportunity for the refinement of the Glauber predictions. V.D.2. The Color Transpar·ency Predictions This section includes compansons between the data a nd models of the transparency with and without the Color Transparency effect, as presented in several different calculations. As described in the previous section, the uncertainty in the Glauber predictions hampers use of the data to discriminat<' between the vari ous mode ls. Because of this, the interpretation of these comparisons is necessarily limited. Also, for models that do not include the enhancement of the transparency at Q2 = 1 (GeV /c) 2 by the decreased pN cross section, the compari son is restricted to t.he data at Q 2 ~ 3 (GeV /c) 2 . In calculating the effect of CT, the general approach has been to use a. reduced value l7cf 1 for the pN cross section in the Glauber model transparency calculations (presented in the previous section). The physics content of t he CT models is in the Q2 and time dependencc of the reduced cross section, basC'd on the considerations discussed in Section LB . Figure 68 shows the predictions of Farrar el al. [2] in the nai ve parton a nd quantum diffusion models discussed in Section l.B .3. They assume that the transverse extent of the Point Like Configuration is b""' (n < kz > 112 )J4/Q 2 at the time of the interaction 143 and that it propagates a distance lh while expanding back to the normal proton size. Here n = 3 is the numbe r and < kt > 1 12 -::= 0.35 GeV lc is the RM S transverse momentum of the constituents of the proton. Using aef f -::= C7tot b 2 I R; (Equation 1.15), they arrive at the following effective cross section during th e expansion of the PLC: (5. 16) where 8 = (z'- z )llh is the distance z' -:: traveled sin ce t he interaction in units of lh. The te rm in brackets, which represents ( bl R~t) 2 du rin g the expansion , increases from the initial value at 8 = 0 to 1 at 8 = 1. The expa nsion length for the naive parton mode l (" = 2) is lh = /TC -::= (E~IMP)Rp, where T is the expansion time in the PLC rest fram e and 1 = E~IM11 is the time dilation factor. In the naive parton model , b increases linearly with t ime due to free ex pansion of the partons. For the quantum diffusion model ("' = 1 ), the expansion length is lh -::= 2hp' I c 2 6.M 2 , where 6.M 2 = 0.7GeV 2 Ic4 . (See Section I.B.3 for more details.) As can be seen in Figure 68, th e qu antum diffusion model is consistent with the C and Au d ata at Q2 ;::: 3 (GeV lc) 2 . However , the data clearly rules out. t.he nai ve patton model. Jennings and Miller [3] treat. t.he PLC as being the s uperposition of st.at.es 1n a hadronic basis (see Section l.B.3). They argue t.hat. t.he PLC-nucleus FSI a re soft and can only connect low-lying excitations wit.h a proton in t.he final st.a t.e. They assume that the prot.on final sLate is dominat.ed by cont.ributions from t.he nucleon IN > and a low-lying nucleonic excit.at.ion IN1 > of mass M 1 • The aut. hors consider the two states to have the same ene rgy E~ but. different. momenta p' and p; = Jp' 2 + l\IJ';c2 - M?c 2 -::= p'- (M? - M 1;)c3 I(2E~). For clefinit.eness, t.hey model t.he t.raiJ sverse wave functions with the ground state a nd first excit.ed staLe of a two-d imensional harmonic oscillat.or, (5.17) and 144 I 1.0 - - I - 0 Naive Parton - Quantum Diffusion - - No Color Transparency • 0.8 0.6 - -- - - I c Data Au Data - -- - - - -- -- - -- -- - -!-~ - -r-- - -l- i -- - ~ c- - I 0.4 - 0.2 - -- -~ - - - --- - --- - - - - - -- - - -- -- -.- - --.- J - - + 0.0 I f- Au - - 1- I I 2 0 - 6 8 FIG. 68. Farrar et al. [2] C, Au T predictions (5.1 8) where 1·.1 is the transverse coordinate (b =< Ti > 112 ) . Introducing the associated operator 7~.t, one observes that this choice of wave functions results in the conven ient relation: (5.19) In calculating the effective cross section U e f 1 ~ Utot b 2 / R; one must include the longi- tudinal part e•P'=' of the wave function: (J cf f = (J tot [ 1 ~ Utot =~ ei(p'-p; )(z'-z)] [1 - ei(p'-p;)(z'-z)]. (5.20) (5.21) Here the longitudinal wave functions were assumed to be in phase at the time of scattering (z ' = z), giving Ueff = 0. As the PLC propagates, it expands to full size 145 1.0 ~-----------L-'----------L-----------~'----------~r 1 Sharp Cutoff g(M/) M 1 = 1.44 GeV/c2 Power Law g(Mx2 ) 0.8 - D C Data • Au Data Glauber 1- 0.6 0.4 - 0.2 - * 0.0~----------,,-----------.1 ----------.-,----------r 0 2 4 2 Q (GeV/c) 2 6 8 FIG. 69. Jennings and Miller 1992 [4] C, Au T predictions and full a 10 t because the phase decoherence of the IN > and IN1 > state destroys th<' cancellation of the Pi expectation value. This is the model used to calculate the M 1 = 1.44 GeV fc 2 curve in Figure 69. In 1992 the authors generalized the technique to include contributions from more than two hadron states, with the decreasing importance of higher-mass resonances characteri zed by a function g(MJ.:. ), where Mx is the resonance mass [4] . The function is given by a sharp cutoff g(M'J.:.) = 0.60(Mz - M'J.:.) or by a power law g(Ml) = (M / M x )!3 (see Figure 69) . Here M z ~ 2.2 GeV 2 /c4 and fJ ranges from 2.4 to 4.0. Thcs<' values ar<' constrained using th<' m easured proton cl iffractive dissociation and <'lcctron deep inelastic scattering cross sections. Any of these models can be made consistent with the data by renormalization within the theoretical uncertainty of the Glauber approximation. The Benhar et a!. [5] calculations (Figu re 70) arc performed 111 the Correlated 146 1.0 4-----------J___________ r L __ _ _ _ _ _ _ _ _ _L __ _ _ _ _ _ _ _ _ _ CGA CGA + CT 0 0 • 0.8 0.6 C Data Fe Data Au Data ------- -----c------ --- --- 0.4 0.2 0.0,_--------.---------.---------.---------r 6 8 2 0 FIG. 70. Benhar et al. [5] C, Au T predictions Glauber Approximation (CGA, which includes nucleon-nucleon correlations) . For t.he CT curves, they use the Farrar el al. quantum diffusion model (Equation 5.16 with "' = 1 and .6.M 2 = 0. 7 GeV 2 fc 4 ). Within the uncertainties in the Glauber approx im ation , the data cannot rule out either CGA + CT' or CGA . The CT predictions of Frankfurt, Strikm an a.nd Zha.lov [6] in Figure 71 were calculated using their Glauber model and the Farrar et al. quantum diffusion model of a ef f (Equation 5.16 with "' = 1 and .6.M 2 labelled in the figure). The label "CSE" r<:>f<:>rs to the inclusion of the Color Scr<:><'ning Effect, which causes suppression of t,he PLC in the nuclear environment,: th<' PLC has a smaller interadion with nearby nucl<'ons and therefore a smaller binding energy. Hence, it is energetically unfavorable a.nd, for Q 2 > Q5, is suppressed by the factor: (5.22) 147 1.0~----------~----------~----------L-----------r .t.M 2 =0.7 CSE + .t.M 2 = 0.7 Glauber D 0 • C Data Fe Data Au Data 0.8 c 0.6 Fe 0.4 Au 0.2 0.04---------~--------~--------.---------r 0 2 6 8 FIG. 71. Frankfurt, Strikman, and Zhalov [6] C, Fe, Au T predictions Here E A ::::: 8 MeV is the mean binding energy, ~E ::::: 0.6 GeV ba'3ed on the EMC effecl, and Q6::::: 2 (GeV /c) 2 based on the SLAC 2 H(e, e') data at x 2:: 1 and large Q 2 [6]. Comparing the calculations to the Q2 2:: 3 (GeV /c) 2 data, one again observes that none of the curves can be ruled out within the accuracy of the Glauber approximation. However, unlike the other predictions, these include the effecl of the variations in CTtot and should a lso reproduce the Q2 = 1 (GeV /c) 2 data. One observes that, regardless of the renormalizalion of the authors' Glauber prediction, the ~M 2 = 0. 7 and C S E + ~1\1 2 = 0.7 curves are inconsistent at the two sigma level with either the Q 2 = 1 (GeV /c) 2 or the Q 2 = 5 (GeV /c) 2 Au data. The disagreement of t hese curves with the data indicate either a failure of the CT model or an underestimate in the drop in the Glauber value from Q2 = 1 (GeV /c) 2 to Q2 = 3 (GeV /c) 2 . In summary, the data has ruled out the naive patton Color Transparency model 1·18 of Farrar et al. [2] and both of t he Frankfurt , Strikman , and Zhalov [6) 197 Au( e, e'p ) CT predi ctions. It is import a nt t o realize that t he exclusion of the la tter resul ts was m ad e possible only through the use of the Q2 = 1 ( GeV /c )2 d at a, which serves to fix the ve ry uncertain normalization of the Glauber model. Because of this, t he fa ilure of the predictions could be evide nce of shortcomings in the Gla uber model a t Q 2 = 1 ( Ge V f c) 2 ra ther t han the CT m odel. Since the Frankfurt , Strikman , a nd Zha lov calculations were the only predi ctions with a realistic treatment of the FSI at Q2 = 1 (GeV /c) 2 , it was the only theory th a t could be subject to this m ore ri gorous t est. If the other calculations we re improved to include the variation in <7tot with Q2 , it is possible that other CT predictions could be ruled out as well. As a n alternative, a more rigorous test of the CT predictions would also be allowed by the reduction of the uncertainty in the Glauber predictions. Finally, it is worth considering the genera l st ate of t he theory predictions. Clearly no ne of the CT predictions re present a reali sti c Q C D dy na mi cal mode l of the e lectronproton sca.Lt.ering and subsequent Final St at e Interactions. The m ode ls a re in st ead la rgely ad hoc and crudely justified, an d so contain a large degree of adjust a bili t.y (e. g., through the m ass scales 6.M 2 and M 1 ) . This is underscored by the continua l di scovery a nd controversy over ne w effects (e.g., t he cha rge screening effect, nucleonnucleon correlations, a nd the suppression of t he effect of nucleon-nucleon correla ti ons t hrough the spectator effect and t he fini te range of t he nucleon-nucleon interaction) that a re argued to produce la rge cha nges in the measured va lues ofT . Thus, while the presen ce o f a strong Q 2 d epen dence in a n experiment. like NE- 18 would be evide nce of Color Tra nsparency, the abse nce t hereof does not ru le out CT. However, because t h e CT effect is expected to increase with Q2 , its a bsence in a hig her Q2 ex peri men t would be more convincing. A d ecisive t est of the phenomeno n of Color Transpa rency in A (e, e'p ) awaits eit he r a truly realistic predi ction of T( Q2 ), the meas urement of a Q2 dependence larger tha n 149 the experimental errors, or the measurement of no Q2 dependence m an accurate experime nt at much higher Q2 . 150 VI. CONCLUSIONS The A(e, e'p) reaction has been measured at quasielastic kinematics for A = 1 H, 2 H, 2 L C, 56 Fe, and 197 Au and Q2 = 1, 3, 5, and 6.8 (GeV /c) 2 . This represents an increase by an order of magnitude of the range of Q2 rneasured in A(e, e'p) experiments. The effective spectral functions S(Em,Pm) extracted from the data exhibit the shell structure expected in the nuclear s hell model. The shapes of the measured S(Em, Pm) agree well with those predicted using an Independent Particle Shell Model spectral function input and the Plane \"'ave Impulse Approximation. The agreement confirms the expected single-particle nature of the quasielastic knockout reaction at Q 2 ~ 1 (GeV /c) 2 . There is no evidence of significant multi-particle effects at Pm <Pi and Em < m1f. The nuclear transparency T is extracted as the ratio of the measured rate to the rate expected in the absence of Final State Interactions, which is calculated in the PWlA/IPSM model. Thus, T is a measure of the absorption and deflection of the protons by the FSI. The Tat Q2 = l, 3 (GeV /c) 2 is compared with the expectation using the standard nuclear physics model of the FSI (called the Glauber model) in order to test the saturation of the spectroscopic sum rule. The data is within the range of the Glauber calculations, indicating full saturation of the sum rule within the ±20% uncertainty in the Glauber predictions. This is in contrast to the situation at low Q 2 , where there is only "'70% saturation of the spectroscopic sum rule. The data can also be used to better constrain the Glauber predictions. htturc experiments at the Continuous Electron Beam Pacility (CEBAF) should result in higher-statistics and higher-resolution measurements of S(Em, JJm) at this Q 2 range. These are particularly interesting in light of the apparent disagreement of the Fe and Au momentum distributions measured in the current experiment with those measured for nearby nuclei at low Q2 . It is important to accurately test the earlier 151 results with measurements at Q 2 ,<: 1 (GeV /c) 2 , where the model of the FSI simplifies to independent proton-nucleon collisions and where the distortion to the shape of S(Em, Pm) is minimized by the large angular deflection in e lastic interactions and by the decreasing value of the elastic proton-nucleon cross section. Although S(Em,Pm) measurements at much higher Q 2 would have even smaller distortions due to elastic FSI, tbey would be hampered by the degraded (Em,Pm) resolution and larger radiative effects resulting from the increased experimental momenta. Because of this, it is likely that the optimal value for a precise measurement of S(Em, Pm) is within the Q2 range of the present experiment. Measurements of the nuclear transparency in A(p, 2p) have indi cated the possible onset of the Color Transparency effect [28]. The limitations in the interpretation of these data are intrinsic to the A(p, 2p) reaction, so it is important to examine the issue with the theoretically simpler and better-understood A(e, e'p) reaction. Within the total errors, the T measured in the present experiment does not exhibit the increa.'3e with Q 2 that. would indicate the onset of Color Transparency. The data rule out three calculations ofT( Q 2 ) which include Color Transparency effects. The use of the data to rule out additional Color Transparency models is inhibited by the uncertainties in the Glauber predictions. More stringent limits on the Color Transparency models would result by reducing the theoretical uncertainty in the Glauber predictions. As an alternative, stricter limits could be obtained by normalizing the Glauber predictions to the T observed at Q2 = 1 (GeV jc) 2 . At present this is not possible for the majority of Color Transparency models because they do not account for the reduced protonnuckon cross section at. Q2 = 1 (GcV jc) 2 . l\Iany of the systematic errors in the current analysis are due to fundamental limitations in the understanding of the A(e, e'p) reaction and to the increased difficulties inherent in the measurement of coincidence reactions. H would be difficult for future experiments to greatly improve the systematic error in T . Thus, the chances for ob- 152 servation of the Color Transparency effect in A(e, e'p) seem best at higher Q 2 , where the models predict larger changes in T. 153 A PPENDIX A: C OORDINATES A schematic of the experirnent is shown in Figure 72 with 3-momenta corresponding to the Q 2 = 1 (GeV /c) 2 , 08 = 35° carbon kinematics. The figure indicates the coordinate systems used in this document, which are defined in Tables 17-19. 1.6 GeV/c Spectrometer k From NPAS injector, A line To Beam Dump Scattering Chamber Spectrometer F lG. 72. P lan view of experime n t NE- 18. The coordinates y a n d z16 a r e ver t ical (out of the page, as ind icated b y t h e bulls-eyes). Due to the tilt of t h e 8 GeV jc hut, Y8 a nd Z 8 are i ncline d a t 30° to t h e vertical and t he h orizontal, res p ectively. 154 TABLE 17. Laboratory coordinates. The angle variables n, B, and </J generally have a subscript giving which 3-momentum they refer to. If not, they refer to the direction of the scattered electron (i.e.) n = nk'' etc.). Name Description X Horizontal (the 1.6 GeV jc is at x > 0 and the 8 GeV /c is at x > 0) y Vertical, pointing up z Downstream along the beam line ()' <P Polar angles n Solid angle, n = (8, ¢) Central angle of 1.6 Ge V / c (electron) s pectrometer Central angle of 8 GeY jc (proton) spectrometer TABLE l 8. Spectrometer (Transport [35]) coordinates. Coordinates are often accompanied by a. subscript 16 or 8 to specify which s pectrometer they refer to. Name Description Fractional deviation of mom e ntum from central value Angle in the hori zontal (xz) plane with respect to the beam line (minus central value) Angle in vertical plane; large r values are up Ztarg z of scattering, a.s measured by one of the spectrometers 155 TABLE 19. Focal plane (hut) coordinates Coordinates at the "focal plane" of the detector hut. The focal plane (Z = 0) is orthogonal to the optical axis and does not necessari ly correspond exactly to the momentum or angle focal planes of the spectrometers, which can be tilted. For convenience, the coordinates (X, Y, Z) 8 are given in a left-handed system. Name Description In dis persi ve direction (increasing mome ntum) Orthogonal to X 16 , Z 16 (right handed) Along the optical axis; note zl6 II y Orthogonal to Y8 , Z 8 (left handed) In the negative dispersive direction Zs Along the optical axis dXs,16 Shorthand for (dX /dZ)s,l6 (track slope) dYs,I6 Shorthand for (dX/dZ)s ,I6 156 APPENDIX B: TRACK PURGING The track identification algorithms described in Section III.B often result in the identification of more than one track in an event. The purpose of the track purging software is to remove this ambiguity through the use of each spectrometer's positionsensitive detector elements. If a given track fails any one in the series of tests, it is discarded. The tests are applied either as "purges" or as "cuts." Purges are not applied unless at least one track would su rvive. Cuts are app lied regardless of whether or not they reject, all of the tracks. If all of the tracks in either spectrometer are cut, then the event is discarded. Such events are accounted for in the tracking efficiency correction, described below. The purges and cuts are adjusted to minimize the number of true coincidence events rejected, as measured with the coincidence timing peak. In the 1.6 GeV fc spectrometer th is was accomplished with spot checks of t he data. In the 8 GeV /c each purge was adjusted to produce .:S 0.1% coincidence rejection in eve7'Y run, even if it is applied as a cut. In Tables 20 and 21 and below, we describe the track purging tests. The purpose of the 1.6 GeV fc and 8 GeV /c fiducial cuts is to ensure that the tracks arc restricted to a region of constant detector efficiency, thus avoiding positiondependent biases. In the 8 GeV fc the test restricted tracks to at least 1 em inside the (X, Y) 8 extents of the SF, NBS, and SR planes. In the J .6 GeV /c, the cuts are: ldX16I < 0.25 rad, IXvacl < l 00 em , IXPA I < JOO em, (Bl) ldY16I < 0.20rad, !Yvacl < l5cm, i}i"AI < lOOcm, (B2) where (X, Y)vac is the track position projected back to the exit of the dipole and (X, Y)PA is the position projected to the first shower counter layer. The cut on IYvacl is used to throw out tracks where the electron suffered a gra~ing interaction with the 157 TABLE 20. Purge tests applied to tracks in the 1.6 GeV jc to remove tracking ambiguities. Name Type Description Fiducial Cut Track is in detector volume Transv. Scint. Cut Track points at scintillators that fired Shower Track Purge Track points at an electromagnetic shower in the segmented lead glass detector Long. Scint. Purge Track points at location inferred from scintillator PMT time difference Reconstruction Cut Track inside loose (8, .6.0, ~)cuts Nearby Purge Decides between multiple nearby tracks, assumes th ey are due to same particle \2 Purge Hits are co-linear i\lost Wires Purge Take track with most wires l\ Iost. Pairs Purge Ta ke track with most pairs B('st >. 2 Purge Take track with bes t x2 beam pipe inside the dipole. Studies with 1 H( e, e'p) demo11 strate that these events entered th e hut with randomi zed momenta and lowe r e ne rgi es: they are broadly distributed in E m and Pm, they appeared at artificially low b..tp-e (and are therefore slower than they should b e), and they deposited less e nergy in the shower counter. It is important to discard these because the acceptance mode l of the spectrometer assumes that. an e lectron is lost if it hit the beam pipe. The x2 t est s are used to discard spurious tracks t.hat arise whe n the track identification software connects up unrela ted chamber hits. The transverse and longitudinal scintillator test s use th e pos ition a nd timing information of PMTs th a t fired to limit the (X, Y)l6 ,B position of the detected particle. The reconstruction Lest is used to 158 TABLE 21. Purge tests applied in the 8 GeV j c to remove tracking ambiguities. i':ame Type Description \2 Purge Hits are co-linear Fiducial Cut Track is in detector volume Scintillator Purge Like Transv. and Long. Scint test in 1.6 GeV /c NBS Purge Track points at NBS counter that fired Combined Purge Track passes a restrictive scintillator OR a res trictive NBS test Xs- dXs Purge Track satisfies observed correlation Reconstruction Cut Track inside loose (8, ~e , <I>) cuts Ztarg Purge Track reconstr ucts back to target Most Chambers Purge Take track with most chambers Best x 2 Purge Take track with best x2 Best ..,'{s- dXs Purge Take track with best correlation throw out tracks clearly outside the spectrometer acceptance. The last three purges 111 both spectrometers are used to select only one track out of any that r emained. The 1.6 GeV /c software also contains a purge on the "shower track" value. The \'a.luc is the gain-matched sum of the sig nal heights in the shower counter blocks within ,1 em (the approximate width of an electromagnetic shower) of the track. Electrons arc identified by large shower track values and pions, the main source of multiple tracks, by small values. The 1.6 GeV jc software also makes a series of purges on nearby tracks, assuming they are caused by the same particle, to decide which of them is best. It is safe to Lake only one of several nearby tracks because choosing the \\'rong one only res ults in worse resolution for that event. In the 8 GeV /c, the scintillator tests are supplemented by the analogous NBS test and a lso the new combined test. In the combined Lest, the track is required lo pass restrictive versions of either the scintillator OR the NBS tests. While the 159 individual inefficiencies of these restrictive tests is 1-5%, the combined inefficiency is ;S 0.1 %. Similarly, the 8 GeV /c reconstruction test is supplemented by a purge on Ztarg to throw out particles that could not have come from the target. The purge on the X 8 - dX8 correlation, produced by the 8 GeV jc spectrometer magnetic optics, is largely redundant with the Ztarg purge. It might seem that the final purge, on the best X 8 - dX8 correlation, would never be necessary because it is preceded by a purge on best \_' 2 . However, due to the discrete nature of the track angle measurement, two tracks that are parallel within the wire chamber resolution could have the same x2 . 160 APPENDIX C: DEAD-TIME CORRECTIONS The measured coincidence rate is corrected for the computer dead-time .fcoin, giving the number of events missed because the computer could read out only one event per beam pulse. The dead-Lime is inferred from scalers m easuring the rates of the various p retriggers. The computer dead-time cannot gener a lly be calcu lated as simply the ratio of pretriggers to triggers because some of the pretriggers have temporal correlations. We will use the conventions of Section IJ.D , where "M " re fers to a module and "P" refers to a pretrigger. We a lso adopt new notations whereby "T;,retrig" refers to a trigger resulting from the pretrigger pretrig, and "I: M " is a scaler counting the number of times module M fired. The computer dead-time correction for electrons in the 1.6 GeV fc spectrometer is given by the total number of electrons divided by the number of electrons giving triggers. These values a.re given by t.he scalers L M cab (corrected for hardware deadtime, below) and l:Tcab, see Figure 73. The M cab scale rs are used instead of P et scale rs because of their smalle r 1r- contamination. This is necessary because the correction is applied to the measured number of t1·ue electrons (in the coincidence peak), not to the measured number of Tet triggers (wh ich includes 1r- contamination). In the 8 GeY fc, the computer dead-Lime is more subtle. When the rate of M 2 ; 3 signals is high, the dead-time can different dramatically depending on the status of t he beam gate prescaler. For a prescaling fraction of Epre (typically < 2- 8 ), P 2 ; 3 ,pre prdriggers were blocked during 1 - cprc of the beam gates. During these beam gates only the pretriggers P coin, Ptongcoin, and Pran could occur. Since these a lways occur in concert with a 1.6 GeY fc pretrigger (Pet or Pran), the computer dead-time for Lbese beam bursts is already accounted for by L M cab/ L Tcab· In the remaining fraction Cpr e of the beam spills, every M 2 ; 3 signal generates a 161 PRETRIGGER 2/2 1/1 CAB40,TRIG 40ns CAB40 CK PALOW 3/3 1/ 1 PB 20 ns 60ns CAB20 CAB60 1/1 eons CAB80 FIG. 73. Formation (labelled CAB20- CAB80) and the of fcab 1.6 GeV /c Mcab (labelled CAB40,TRIG) sig- nals. The module M cab requires signals in the Cerenkov and both shower counter layers (PA, PB), and has therefore has a lower rrcontamination than Pel· The module fcab is vetoed by the inverted trigger signal, and therefore indicates a Pe1 that came in concert with a trigger. See Figure 28 for the origins of the input s ignals. pretrigger P2/3,p•·e· The scaler L: P 2 ; 3 ,pre (L: T2;3,prc) counts th e number of these p retriggers (t riggers). (See Figure 74.) In fact, these scalers count all possible pretriggcrs (triggers) that occur during the prescaled beam gates, except for the extremely small fraction caused by 'Pran· This is because 'Pcoin and Plongcom both require Mpr ot, which is a subset of M 2 ; 3 . The number of triggers due solely to t he pretrigger 'P2; 3 is measu red by the scaler L: T2; 3 ,only (sec Figure 74). During p rescaled beam gates, the fraction of triggers due to 'Pcoin, Plongcoin, and 'Pran is 1 - L: T2; 3 ,only/ L: T2; 3 ,pre· The prescnc<? o f t h e pretrigger P 2 ; 3 ,pre has not prevented the firing of these triggers , and their dead-time is again correctly given by L: M ead L: Tcab· In the remaining fraction L T2;3,only/ L: T2;3,pre of the triggers, due only to the pretrigger P 2; 3 ,pre, the additional 1G2 2/2r 8 GeV/c TRIGGER 213,PRE 2/3 TRIG 8 GeV!c TRIGGER 2/2 LONG COIN 1/2 RANDOM 2/3,0NLY FIG. 74. Formation of the 8 GeV /c T2; 3 (labelled 2/3,TRIG) and T2; 3 ,only (labelled 2/3,0NLY) signals . See Figure 30 for the origin of the input signals. 8 GeV jc dead-time is given by the pretrigger to trigger ratio for the prescaled beam gates, 2:: P2/3 ,pre/ 2:: 'T;; 3 ,pre· Thus the dead-time correction for the prescaled beam gates is: fvre = 2:: M cab [ L Tcab (l _LL J;_/ 3,only ) + L J;_/3,only L P2/3,pre] . J;_/3,pre L J;_/3,pre L J;_/3,pre (Cl) The coincidence live time (1/dead-time) for both classes of beam gate is the weighted average of JP-,.~ and the live time for non-prescale beam gates. The total coi ncidence dead-time is therefore: r ) coin= L M cab [( L Tcab 1- ) Epre + (1- .........,~='-"- (C2) For the large majority of runs, this formula agrees to within 1% with the dead-time expected from the measured rates and Poisson statistics. The disagreement is almost certainly due to the statistical error in the expected rate, which was not calculated. Equation C2 should be accurate to much better than 1%, even though it is only an approximate form , The approximations are assuming a statistical distribution of events between the prescaled and non-prescaled beam gates (negligible error for the 163 two cases used, Epre = 1 and Epre « 1) and neglecting the (tiny) fraction of Pran triggers. The dead-time ! coin is usually < 20%. In the above, the signals Mcab and P 2 ; 3 ,pre can occur more than once a beam burst and must be corrected for hardware dead-time, wherein the finite width of input logic gates (typically T = 20 ns) prevents the scalers from distinguishing two nearly simultaneous firings. The measured rate Wscaler is reduced from the true rate w by the probability of simultaneous firing: Ws ca ler = we-WT ~ w(1 - wr). (C3) The linear approximation never under-estimates the hardware dead-time by more than 0.04% in the 8 GeV jc and 0.08% in the 1.6 GeV jc. The scalers used to calculate the computer dead-time are measured using T = 20, 40, 60,80 ns. The measured scaler counts vs. T shows a linear relationship for a ll but T = 20 ns, probably because of double pulsing. Thus, only the T > 20 ns scalers ca.n be used to perform the linear extrapolation to the true counts at T = 0. The total error of the dead time corrections is estimated to be < 0.3%, based on comparisons of the measured dead times with values expected using Poisson statisti cs. 164 APPENDIX D: COINCIDENCE IDENTIFICATION The time difference 6t8 _ 16 = t 8 - t 16 between the triggers was measured by two TDCs, a 50 p s/channel TDC started by the 8 GeV /c coincidence trigger (T16-s) and a 100 ps/channel TDC started by the 1.6 GeV jc trigger (T8 _ 16)· (See Table 22 for the naming conventions used to refer to times.) Figure 75 shows non -linear response in both TDCs for small va lues, and describes how they were linearized for the analysis. During the Q 2 = 7 (GcV /c) 2 data-taking, the stop signal to the 1.6 GeV jc TDC became errati c, causing the TDC to time out. The 8 GeV jc TDC was used at all kinematics to identify coincidences. TABLE 22. Timing conventions, including t imes commonly referred to in t he discussion of coincidence identification. Many statements apply equally to t he 1.6 GeV /c and the 8 GeV /c s pectrometers. In these cases the 8 GeV /c spectrometer will be taken as an example. Constant time offsets are generally s uppressed, but s hould be viewed as being carried a.long. For our purposes, all of these co nstants can be t reated as being absorbed in the C of Equation 04. Name Description 1.6 GeV /c trigger time (leading edge) 8 GeV /c trigger time tim e of electron at target tp time of proton at target ljpB particle time at spectrom eter focal plane, 8 CcV /c ljpl6 particle t.ime at spect,rom eter focal plane, l.6 GeV /c tPMT time discriminator fires and stops PMT's TDC /).ta - b ta - Ta - b TDC measurement of /).ta - b (b = 8 or 16 starts the TDC and a stops it) tb, where a, bare two of th e s ubsc ripts above 165 800 700 600 ,....-..._ 500 ~ ro ..c ~400 CX) I ~300 f--4 200 100 0 0 100 300 200 400 T 8 _ 16 (chan) FIG. 75 . Coincidence TDC non-linearities. Comparison of the two coincidence TDC values for 1/100 of the Q2 = 1 (G eV /c) 2 carbon data. The plus signs are the raw values, and the circles show the linearized values (when different). non-linearly for small values. Both raw TDCs behave The 8 GeV fc TDC Tl6- 8 is fit as having three slopes (for values < 45, 45-130, and 2:: 130) and the 1.6 GeV fc TDC Ts- l6 is fit as having two slopes (for values < 40 and 2:: 40). The linearized values, obtained by applying a piece- wise linear correction derived from the fit slopes, are used in the analysis and all other graphs. 166 The signal delay lines were adjusted so coincidence proton pre-triggers would arrive, on average, about 20 ns into the gate (see Figure 76) . This time was made as short as possible to minimize dead-time caused by non-coincidence 8 GeV /c pretriggers occurring during the gate before true coincidence protons. The time was not made shorter to prevent loss of true coincidences due to the "' ±5 ns width of the 6ts- r6 coincidence peak. Variations in path length through the spectrometer and scintillator response time are responsible for the "' ±5 ns width. The value 6t 8 _ 16 was converted into a time difference 6tp-e between the proton and electron at the target by removing these effects (see Figure 78 and Equation D4). The path length corrections were made in each spectrometer using the Z matrix elements of the reverse Transport model, adjusted to produce the best timing resolution. The matrix elements specify the path length Z from the target to the spectrometer focal plane. The time it takes the particle to reach the focal plane is (Dl) where Z is the path length . Here the particle velocity (38 c is inferred from the measured momentum p8 and the assumed mass /11!8 : (D2) For (e, c'p) the masses were M16 =me and Ms = ./11/ P. Errors in 6tp-fp8 (or 6t e-fp16) for particks of mass =f:. 1\18 (M16 ) do not effect t.hc measured coincid<>nce rate becaus<> these particles are a lways accidentals and are therefore uniformly distributed in 6tp-f· The scintillator response times 6t 8_ 1P8 were determined using measurements of the T ime-of-Flight (TOF) between scintillators for particles of known velocity (see Appendix 0.2). Combining this with the path length correction 6ts-jps, one obtains the time it takes the particle to cause a trigger: 167 T 8 _ 16 (chan) 0 200 0 20 400 600 800 1000 40 60 80 100 3000 2500 2000 UJ -+-' ~ ;j 0 1500 u 1000 500 0 -20 6 t 8 _ 16 (ns) FIG. 76. Time difference ~t8 _ 1 6 between the 1.6 GeV /c elec- tron trigger and the 8 GeV jc COIN trigger, as measured by the 1.6 GeV /c coincidence TDC Ts- l 6 · The peak at ~t8-l6 :::::::: 20 ns is due to true (e, e'p) coincidences, which occur 20 ns after the opening of the coincidence gate. The peak at ~t 8 _ 1 6 = 0 is due to accidentals occurring at the beginning of the coincidence gate. For these events, the time of the coincidence trigger was determined by the the 1.6 GeV jc electron pre-trigger instead of by the 8 GeV jc proton pre-trigge r. This occurred when the 18 ns wide 8 GeV /c proton pre- trigger was already true when the 1.6 GeV jc electron pre-trigger occurred . For these events T 16 _ g is still a valid measurement of ~t8- 1 6 · 168 T 16 _ 6 (chan) 1000 500 0 -500 -1000 60 80 3000 2500 2000 U) .,_; ~ ;:::::1 1500 0 u 1000 500 0 -20 FIG . 77. 0 ~' 8- l G 20 40 ~tB-16 (ns) as measured by the 8 GeV /c TDC T 1 6 - 8 · 100 For this TDC a smaller value corresponds to a Large r time difference between the triggers (the top and bottom axes run in opposite directions). The peak at T 16 _ 8 = 0 is due to underflow of the TDC for accidentals with ~£ 8 _ 1 6 > 34 ns. For these events, T 16-B is not a valid measurement of ~t 8_ 1 6 • 169 3000 2500 ~ ~ 2000 ;j 0 u 1500 1000 500 o~~~~~~~~~~~~~~~~~~~~~~~ -10 -5 0 ~tp-e (ns) 5 10 PIG . 78 . Time difference 6.1 7'-e between the electron and proton at target, calculated from the data in Figure 77 by removing the effects of path length through the spectrometer and scintillator response time. The peak at 6.t7'-e = 0 is due to true (e, e'p) coincideuces. The shaded region are the data at T 16 _ 8 = 0, where TDC underflow prevents a valid measurement of 6.t 8 _ 16 (see Figure 77). (D3) True coincidences are identified by calcul a ting the diffe rence be tween th e times of the electron and proton at t arget : (D4) 170 The 1.6 a nd 8 GeV /c trigger delay line lengths were varied a.'3 Q2 was changed to account for the variation in proton fli gh t time. The constant C was determined at each Q 2 by centering the coincidence peak at ~tp-e = 0. To prevent contamination by the tails of the coincidence peak, the accidental r ate must be measured using events with l~t p-el > 5 ns . The rate is dete rmined using only the data in the range -20 ns < ~tp-e < -6 ns (~twin = 14 ns, below). The measurement could be extended to more negative ~lp -e, but this would provide minimal improvement to the statistics of the accidenta l subtraction . The shaded events in Figure 78 prevent determination of the accidental rate using the data at ~ip-e > 0. To prevent contamination by events with incorrect time determinations, the number of coincidences is determined without the 8 GeV jc TDC underflow events (the shaded region in Figure 78). A furth er efficiency factor is applied to correct for true coin cidences in the underflow channel. This correction is determined by calculating ~lp-e with the 1.6 GeV /c TDC, which does not have the underflow problem. The 8 GeV /c TDC underflow problem could have been prevented by increasing the delay of the 1.6 GeV /c trigger signa.! into the TDC. During the d esign of the experiment, it was incorrectly thought that doing so would increase the dead-time. It would not have, in fact, because the T D C gave time differen ces between triggers a.nd the e lectroni cs guaranteed that on ly one 8 GeV /c a nd one ] .6 GeV jc trigger could occur during a. beam burst. Thus there were no signals com peting to start or stop the TDC. D.l. Scintillator Timing Corrections A PMT's TDC measured Tn.n -s, the time the PMT fired with respect to the trigger. The ns/ chan scale of each TDC channel wa.s measured using a. pulser. We 171 convert to the trigger response time, flt8-fp8 = fltPMT-fp8- TPMT-8 , (D5) needed in Equation D3 by correction for the scintillator response time fltPMT -fp8· This is calculated based on the particle tracking information: fltPMT-fp8 = Cn.n + ZP!v1TJ1 + dX 2 + dY 2 I f3sc- flLivscint P..jmax(APMT, Ao). (D6) The constant CPMT represents cable lengths and PMT response times that can vary from PMT to PMT. The term ZPMTJl + dX 2 + dY 2 I f3 8c is the particle drift time from the focal plane (Z8 = 0) to the scintillator (Z8 = ZPMT)· (Note that for ZPMT < 0 this time can be negative.) The factor J1 + dX 2 + dY 2 represents the small ( < 0.1 %) increase in drift length due to Lbe track hut angles dX and dY. Within a constant, -flLivscint is the Lime it Lakes the scintillation light travelling at velocity Vscint to propagate the distance Ll2- flL along the scintillator from the particle track to the PMT. Here L is the length of the scintillator and flL = X8 or }8 is the track coordinate in the longitudinal direction of the scintillator. The pulse height correction is - P..jmax(AnrT, A 0 ), which corrects for the fact that the discriminator that stops the TPMT- 8 TDC fires earlier for large PMT pulses than for small ones. Here APMT is the value of the PMT's area-integrating ADC after pedestal subtraction. The procedure for determining the values of the adjustable parameters Vscint, CPMT, P, and Ao is described in Appendix 0.2. Each PMT that fired and is also on the particle track can be considered to provide a nearly independent measurement of flt 8 _ 1P8 . In the 1.6 GeV lc most events had 8 measurements of flt 16 - fpl6 (two per plane, because the scintillators are read out on both sides) and in the 8 Ge VIc most provided at least 6 measurements of flt 8_ fp 8 (two 172 per plane, or more for tracks in the overlap regions of the SM and SR). The TOF software averages together the D..ts-JpB measurements for all PMTs satisfying: 1) the X 8 , Ys of the track at Z 8 = ZPMT are inside the extent of the PMT's scintillator, 2) both PMTs on the scintillator fired, and measured the same value of D..ts -fp8 within a tolerance. The tolerance was 0.8 (SF), 1.0 (SM,SR), or 1.5 ns (all 1.6 GeV /c scinti llators). Condition (1) guards against the inclusion of scintillator hits not associated with the particle track and which are likely to have occurred at a different time. The X 8 and Y8 extent of each scintillator was determined using the track coordinates at Z 8 = Znn when the scintillator PMTs fired, coupled with the known width and length. If no scintillator satisfying directly on the track had a PMT that fired the~, to allow for tracking resolution, condition ( 1) is relaxed. At this point the track must point within 1 em (0.5 em in the 1.6 GeV /c) of the scintillator for the PMTs to be included in the D..ts-JpB measure ment. Calculation of D..ts-fpB requires good TDC and ADC values. !:<or the purposes of the TOF software, a PMT is considered to haYe fired if 2 < Tnn -s < 1500 chan (TDC finite but not t imed out) and APMT > 30chan (at least a small pulse). (For the 1.6 GeV jc spectrometer the ranges are 2 < TPMr- 16 < 600 chan and APMT > 30 chan. The 8 GeV /c TDCs had 0.05 nsf cha n resolution and the 1.6 GeV /c TDCs had 0.1 ns/chan resolution.) The two-PMT req uirement in condition (2) was motivated by studies of events where the PMTs' D..ts- fpB values did not agree within rPsolution. Most of the D..ls-JpB outliers were from scintillators where either: a) only one PMT fired or b) two PMTs fired but disagreed on D..ts- fpB· Either ca.se indicates an increased probability that the PMT firing wa.s not associated with the particle track. In case (a) the PMT firing may have been spurious or due to a neutral particle. In case (b) the scintillator was probably hit by a particle at a different longitudinal position than the track. After applying condition (2) the fraction of events 173 with 6.ts-fpB outliers was less than 1%. D.2. Scintillator Timing Parameters As m entioned above, the parameters Vscint, CPMT, P, and Ao were required for each PMT to make the scintillator timing corrections (Equation D6). The pulse height correction was not very sensitive to A 0 and a value of A 0 = 50 chan was adopted for a ll PMTs. The parameters were fit using measurements of t he Time-ofFlight between scintillators for (e, e'p) events and assuming the velocity f3sc given in Equation D2. In order to prevent biasing of the fits, only loose cuts on 6.tp-e were used to identify the coincidences. The fitting data. came from kinematics with good 6.tp-e true-to-accidental ratio to ensure minimal contamination by particles of m ass In add ition, it proved con venient (see Appendix D .3) to m easure the attenuation length >. and the ADC value A 1 for a particle passing t hrough the center of the sci nti llator by t he followin g fit: A PMT = A 1e -t::.Lj>. . (D7) In this case no time differences needed to be calculated. The parameters were fit for each PMT independently. Due to the uniformity of the scintilla tors an d PMTs, the fits generally gave consistent values of Vs cint, P, >.and A 1 for the PMTs in a given scintillator plane. For these parameters, adopting the global values given in Table 23 y ielded adeq uate timing resolution. This is because small uncorrela.ted errors in the corrections for indi vidual PMT's averaged out over the la rge (2: 6) number of 6.Ls - fpB measurements for each event. It was fa r more importa nt to have accurate, individual values of CPMT, because of the significan t differences in cable le ngths and PMT response times. Best timing resolution was 174 TABLE 23. Timing parameter values used to make the scintillator timing corrections m Equation D6. The scintillator thickness is included to demonstrate its effect on Vscint and .A. The inferior condition of the 1.6GeV /c scintillator material results in lower values of Vscint and). than for SM and SR in the 8GeV /c. For comparison with Vscint, the speed of light in vacuum is c = 30.0cm/ns. The value of Ao =50 chan is used for all scintillator planes. Rough values for A1 and ). in the 8 GeV /c are given for comparison, but are not used in the data analysis. The reference planes are those used in fitting the coefficients for the fit plane. The values of a given are rough indications of the timing resolution of each plane, as inferred from the scatter in the difference of 6.ts -Jp8 between pairs of planes. Fit Thick. Vscint p Al ). Reference a Plane em cm/ns ns/chan 112 chan em Planes ns SF 0.9525 13.7 0.19 400 90 SM,SR 0.13 SM 1.27 14.3 0.09 220 250 SF 0.16 SR 1.27 14.3 0.09 190 250 SF 0.19 YD 1.27 13.6 0.12 460 .50 xu 0.21 XD 1.27 13.6 0.10 460 .so YU 0.22 YU 1.27 13.6 0.15 460 50 XD 0.26 xu 1.27 13.6 0.12 460 !50 YD 0.26 obtained by refitting the 8 GeV fc CPMT at each Q2 , because of changes in {38 and the energy loss rate (and therefore APMT) with e nergy. Rathe r than make a global fit of all of the Vscint, CPMT, and P simultaneously, we used an iterative procedure that allowed each value to be fit independently. (The global fit was considered to be more likely to be unstable and more prone to correlations between the different corrections. For example, the pulse height correction, coupled with the attenuation length of the scintillator material, could give a false value for Vscint.) For the starting values of Vscint and P we used parameters measured 175 in a cosmic ray telescope before the experiment. The offsets CPMT were fit parameters and did not need to be specified . In this procedure each parameter was measured for one PMT at a time by comparison with the timing information from referen ce scintillators in different planes. The events used in the fitting were those where a reference scintillator 7' satisfied conditions (1) and (2), above and where the PMT being fit fired and satisfied condition (1). For fitting, 6.tr = 6.ts-JpB was calculated a.s the average of the value given by the PMTs on reference scintill ator r. To improve the statistics and protect against systematic errors in 6.tr each correction was fit against several reference scintillators simultaneously. Incorrect values of Vscint and P in the reference plane could introduce position-correlated biases that could mimic additional Vscint and P dependence in th e fit plane. To remove the correlation between 6.L in the fit and reference scintillators, the reference scintillators were chosen to be orthogonal to the fit scintillator and separated by at least 80 em (see Table 23). When measuring Vscint. the value 6.tf it = 6.ts -fp8 for the PMT being fit was calculated, €'Xcept th at 6.L was set to zero in Equation D6 to disable the Vscintdependent correction. Then the fit 6.tfi t - 6.t r = D..Lfvscint + D..Cr versus 6.L was performed, with fit parameters Vscint and 6.Cr. Here 6.Cr = cjit- Cr is th e difference between the offsets of the fit PMT and reference scintillator 7' . Note that this fit is only used to obtain Vscint, and the values of CPMT res ulting from this ftt were not adopted (see below). Inside a scintillator, most light propagates to the PMT through a series of internal reflection s and the propagation speed Vscmt is less than c divided by the index of refraction of the scintillator. In thicker scintillators light takes a more direct pat h, and Vscint is higher. The age of the 1.6 GeV /c scintillator material probably reduced Vscint because crazed patches could block more direct propagation paths. (See Table 23.) Note that when the 6.ls-Jp8 average is calculated , the correction 6.L/vscint for 176 the two PMTs on a given scintillator cancel. Thus the value of Vscint only effects the determination of whether the two PMTs give consistent timing information (condition 2, above). A similar fit was performed to extract P. In this case 6tfit was calculated with P set to zero in Equation D6 to disable the pulse height correction. The fit was 6tfit- 6t.,. = 6C.,.- PJmax(AP!v!T, Ao). The need for the pulse height correction can be seen by a simple model. Scintillator pulses are asymmetric, with fast rise times (rv IOns) and long decay times("-' 20ns). The leading edge, however, is approximately Lorentzian shaped: Fort < to : P(t) ~ ( kAPMT ) f / t- t 0 2 + 2 4 (D8) Here t 0 is the time of maximum value, r is the FWHM of the peak, and k expresses the conversion between the ADC value and the peak height. Note that k varies slightly with variations in the pulse shape because the ADCs are area-integrating rather than pulse-height sampling. The PMT's TDC TPMT-s is stopped by the signal of a discriminator set at a level D. In the above approximation, this occurs at the time: tPMT =to- JkAPMT D , - f2/4. (D9) The form of the correction actually used was only approximately equal to this (EquaLion D6), but gave sufficient resolution. Better resolution could have been obtained by the correction -PJmax(O, APMT- A 0 ), equivalent to the above. Rough fits to this form gave the following values of (P, A 0 ): (1.08,129) for SF, (0.73,107) for SM and SR, and ( -1.04,312). These correspond to rise times ,.,._, f'/2 = Pffo of 12, 8, aud 18 ns, respectively. These values are not intended to represent accurate measurements, but are merely to indicate the reasonableness of the model. The values of CPMT used were determined separately after all of the other parameters were finalized. This was done by averaging 6tfit- 6t.,., where in this case 177 !::.tfit is given by Equation DG with CPMT = 0. Of course the fit only determines the difference in the offsets and the values arc undetermin ed up to a common offset of all PMTs . The common offset a ffects only the choice of C in Equation D4, and is therefore chosen arbitrarily. D.3. 1.6 GeV jc Scintillator ADC Malfunctions Generally, the TDC and ADC of a scintill ator PMT agree on whether or not the PMT fired. However, two classes of events were found for which the ADC values for the 1.6 GeV jc PMTs do not correspond to the TDC values. Fortunately both classes were marked by strong patterns in the ADC values so these events could be identified based on the ADC values alone. When the scintillator response time (A ppendix D.1) was calculated for these events, the ADC value was replaced with an approximate value inferred fro m Equation D7 (if the TDC fired). After this correction. t h e I::.Lp-e distribution for these eYents was nearly as good as for events where the J\D Cs functioned correct ly. Even if this w<>re not so, the coincidence timing efficiency correction would ensure that these events were counted correctly. The scintillator ADCs were conta ined in three differe nt LeCroy 2249 modules. These malfunctioned simulta neously, even though all other modules in the Camac c rate continued to function correctly. lienee, the malfunction was probably due to the discriminator (or cable) providing the gate to these three modules . In the first class of events, possibly due to a too-wide gate signal, every cha nnel in a given module gave the same large value (J 185. J 997, or 1114) to within a few channels. In the second class of events, possibl y due t o a. missing gate signa l, som e channels would read pedestal or below and others would exhibit more complicat ed but reproducible patterns. 178 A P PEND IX E: THE M O DEL SPECTRAL FUNCTIO N The nuclear transparency T is determined by division of the measured A(e, e'p) rate by that predicted in the PWlA calculation. The nuclear structure used in the calculation is specified by the model of the nuclear spectral function S(Es, p ). The model S(Es, p) is calculated in the Independent Particle Shell Model approximation, with parameters adjusted to reproducf' data from elastic electron-nucleus scattering and low-Q 2 A(e, e'p) and A(p, 2p). This section specifies the parameters used for the nucl<>i C, Fe, and Au. The S(Es , p) of 1 II and 2 Il arc discussed in Section IV.B. The IPSM spectral function used in the PWIA calculation is given in Equation 1.11. In the calculation, the energy profile of each nuclear shell i is given by a Lorentzian L;(Es), which characterizes the energy width ri due to the finite lifetime of the one-hole state: L'(E ) = 2_ s I 7f f;/2 (Es- E, )2 + r?;1. (E1) This formula must be modified, how<>vcr, si nce the separation energy cannot be less tha.n the one-proton removal energy of the nucleus, E,·em = Mp+MA-l -MA (here MA and M.11- 1 arc the masses of the initial and recoil nuclei). Therefore the Lorentzian L~(Es) is cut off below Erem and rescalf'd to ensure that the spectroscopic s um rul e will still lw satisfied: L;(Es) = L'(E oo L'(E i s)/ f Erem i s )dEs, { 0, (E2) The momentum-space wave fun ctions <p,( p) are the solutions to the bound-state Schrodingcr equation with potential: V(r) = - Vof(r) + Vso - h ( m.,.c ) 2 2 df - -l ·s + Vc(r). 1' dr (E3) The parameter Vo is the depth of a Woods-Saxon well with radius Ro and diffuseness a: 179 (E4) where (E5) The parameter Vso is the strength of the Thomas spin-orbit term, and Vc(r) is the Coulomb potential energy of a uniform sphere of radius Rc = 1'c(A - 1) 113 • The <p;(p) are obtained as Fourier-transforms of the wave functions 'f(r) calculated by the program described in reference [61]. Elton and Swift [15] found that for Ca. nuclei and heavier, it was necessary to account for the non-locality of the nucleon-nucleon potential. Rather than calcu- late the wave function 'fNL(r) in a full non-local potential, it is sufficient to correct the wave function '!/.;L(r) from the local potential, Equation E3, in an effective mass approximation [61 ,51]: (E6) The resulting 'fNL(r) is then renorma.lized. For (3 > 0 (Fe, Au) , the non-locality correction reduces the wave function at r ~ Ro (where .f(r) is large) and increases it at large T. Consequently, the width of the momentum distribution l<p;(p)l 2 is reduced by the non-locality correction. For (3 = 0 (C), the local-potential result is unchanged. The parameters used to calculate L;(Es) and cp;(p) are listed in Tables 24 and 25. The <p;(p) parameters for 12 C were derived by Elton and Swift [15] using chargedistributions from elastic electron-nucleus scattering and well depths from A(p, 2p). These were verified (and V0 (1s) fine-tuned ) with 12 C(c, c'p) at Saclay [11,9], which provided what is probably the highest-quality S(Em, Pm) measurement of 12 C (see Figures 6 and 7). The parameters for L;(Es) of 12 C are also based on the Sa.clay [11,9] experiment. The Is shell experimental energy distribution is not sym metric about the peak, but is 180 TABLE 24. Shell-independent parameters of IPSM S(Es, p ) Nucleus a (fm) 12c (fm) /3 0.55 1.3 0.0 56Fe 0.60 1.3 0.85 197 Au 0.65 1.2 0.85 1"c skewed to higher Em. Since we approximate this with a nearly symmetric Lorentzian , it is best to use for E 1 s the measured centroid at 38.1 ± 0.3 MeV rather than the peak at 35.6 ± 0.3 MeV . Because the 1p peak is so close to the one-proton removal energy Erem = 15.96 MeV, Li(Es) is also highly asymmetric and the peak is more appropriate than the centroid. The PWIA/IPSM uses E 1P = 16.2MeV, based on inspection of the Saclay data and also on peak positions reported in other experiments [9]. The FWHM measured in the Saclay experiment were f 1s = 21.4 MeV [9] and f 1 p ~ 2.5 (estimated from the graph in reference [9]). The PWlA/lPSM uses f 1 s = 20 MeV and f1p = 5 MeV (which is effectively f 1 p ~ 2.5 MeV due to the E,·em cut-off). Note that th<:' transparency is insensitive ( < 2%) to uncertainties in the energy distribution . The <pi( P) parameters for 56 Fe are based on those measured for the nearby nucleus 58 Ni in A(e, e'p) at Saclay [11,9]. The 58 Ni-based momentum distributions are wider than those m easured in the present ex periment, and have to be modified to produce agreement. The first modification is the use of the non-locality correction f3 = 0.85 , which is recommended by Elton and Swift [15] for A 2: 40. The second modification is t.he use of r 0 = 1.3 fm instead of 1.26fm. The parameters for the L;(Es) of 56 Fe are also based on the Saclay 58 Ni( e, e'p ) [11,9] experiment, with E; values corrected for the 2 MeV difference in the Erem of 58 Ni (8.17 MeV) and 56 Fe (10.18 MeV). The experiment did not resolve the individual j states of the 1p and Jd orbitals, so the magnitude of the spin-orbit splitting was de- termined by the Woods-Saxon calculation. Due to the overlap of energy distributions 181 TABLE 25. Shell-dependent parameters of IPSM S(£5 , p). The two sets of V0 , E;, and r; values presented for the deeper shells of Fe and Au are given by Hartree-Fock (smaller values) calculations and A(e, e'p) data (larger values). The Hartree-Fock values are used in the analysis, with the data values as a check on the sensitivity to the Es distribution . E; (MeV) f; (MeV) 38.1 20.0 16.2 5 50/64 19/21 40 39.5/49.5 16/19 1.3 40 32.1/42.1 13/17 58.2 1.3 23.5 27.2 10 1d3/2 58.2 1.3 23.5 16.7 3.2 2s 52.7 1.3 16.7 3.2 117;2 52.7 1.3 11.3 3 ls 71.9/93.2 1.31 46/65 18/21 1p 71.3/80.7 1.31 41/50 17/19 ld 67.8/76.3 1.31 32/40 13/18 2s 66 .5 1.31 28 11 1/ 65.1 1.31 22.9 9.4 2p 63.2 1.31 17.2 7.9 1g 63.7 1.272 12.5 6.0 2d5/2 62.0 1.317 G 8.3 3.7 1hn /2 67.0 1.288 6 7.7 4.0 2d3/2 57.2 1.361 6 6.2 3 Nucleus i Vo (MeV) r·o (fm) 12C l s t/2 66.0 1.36 1P3/2 55.0 1.36 1s 80.7 1.3 1P3/2 69.0 1.3 1pl/2 69.0 1ds;2 56 Fe 197 Au Vso (MeV) 9 13.8 182 and uncertainties in the distortion corrections, the shell structure is somewhat ambiguous for the deepest orbitals. For these orbitals, Hartree- Fock calculations on 58 Ni [62] provide a more accurate indication of the actual shell energies. The Lorentzian widths are calculated according to the Brown and Rho [63] parameteri zation of f ; data for A < 58 [63] : 2 MeV)(E; - EF):___ (24__ f ; = _...;__ __:_. .:. . ____ (500 Me V 2 ) + (E;- EF ) 2 · (E7) For Fe, EF = 8 MeV is used. The ~;( p) parameters for 197 Au are modified versions of those measured for the nearby nucle us 208 Pb in A( e, e'p) by Quint at NIKHEF [13]. There was some modeld ependence in To due to uncertainties in the calculation of the distortions in the DWIA , but an improved understanding of the distortions indicates that one should use the ( = 0.5 column of Quint 's Table 4.5 [64]. Multiplication of the m easured 7'o by 1.11 is necessa ry to produce agreement with the Pm distributions m easured in the cu rrent experi me nt. The 208 Pb experiment did not m easure r 0 for the deeper she lls; they a r<> assigned the typical value of 1.31 fm based on the valence shells . T he Vo values are chosen to reproduce the separation energies E;. The valu<>s of the other parameters are from the discussion on page 15 of reference [13] or from private communication [64]. Most of the 197 Au separation energy centroids E; and widths f ; a re taken from Quint's Table 4.9 (or, for the 2s shell , from Figure 4.6), adjusted for the 2.2 MeV difference in the Er Pm of 208 Pb (8.01 8 MeV) a nd 197 Au (5.77MeV ). The energy distributions of llH" decpcst.-ly ing energy levels (Is, lp, ld) were not m easured. For these we use two models , one based on Hartree-Fock calculations on 208 Pb [62,13] and the other based on extrapolating the 1s, 1p, and ld levels measured in (p, 2p) and (e, e'p) [65] to Z = 79. The ext rapola tion from Z ~ 30 to Z = 79 is probably accurate to no better than ±1 0 MeV for lp and ld a nd ±20 MeV for l s. The I'; measured by 183 Quint agree well with values calculated using Equation E7, so the equation is used to calculate the unmeasured (Is, lp, ld, 2s) widths. The }e and Au spectr al function models are subj ect to two ambiguities mentioned above: the need for the reduction in the width of the momentum distribution and the disagreement between Hartree-Fock calculations and A(e, e'p) [and A(p, 2p)] measurements on the Es position of the deepest orbitals. Both are probably related to uncertainties in the correction of the measured distributions for t he effects of proton FSI and Coulomb distortions. All indi cations are t hat the present experiment is not as sensitive to these effects. In fact distortions are not necessary to produce agreement between the PWIA calculation and the data. Furthermore, the Fe total momentum distribution cannot be considered to be well-constrained by the previous low-statistics measurement of only the outermost valence orbital (see Figure 79). The ambiguity in the Es of the deepest orbitals is resolved by investigating the Fe and Au Em distributions measured at Q 2 = 1 (GeV /c) 2 and IPml < ] 00 MeV /c, which arc dominated by the l s and 2s orbital s. The positions of the peaks agree with the PWIA calculation when the Hartree-Fock Es distribution is used, but not when the \·alues based on low Q2 measurements are used. Thus, the Ilartree-Fock Es values are more correct, while the low Q2 values provide an alternative spectral fun ction model used to characterize the model-dependence o f the PWIA calculation . To summarize, the deep-shell Es values and total momentum distributions measured in the current experiment are apparently inconsistent with those measured at low Q 2 in A(c, e'p ) and A(p, 2p). The discrepancy is probably due to errors in the distortion corrections at low Q2 . 184 .., !....... ~ .. > lJ ~~. >t- if) z w a3 250 F IG. 79. Pm dis tribution from 7 ~ Hm tion . 58 Ni( e,e'p) a t Saclay [11] for < 9.5 MeV. The s olid line repre se nts the DWIA calcula- 185 APPENDIX F: RADIATION IN THE PWIA CALCULATION In Section IV.E the equivalent radiator a pproximation is adjusted to reproduce the theoretical counting rates for H(e, e'p) . Sections F.l and F.2 in this Appendix fo cus on the calculation of the counting rates in the eq uivalent radi ator a pproximation . Most readers o f these sections will probably o nly be interested in the results. Appendi x F . l details the calculation for radiation from two a rms, specifically in the case of external radiation. Appendix F.2 uses this result to perform the calculation for three-arm radiation in the case of internal radiation. Appendix F.3 discusses radiation in the one-photon limit, and helps explain why the modified equivalent radiator approximation is so successful. Appendix F.4 describes how the parameters of the eq uivalent radiator approximation are chosen. Appendix F.5 demonstrates the good agreement of the calculation and the data on the effects of radiation on the reaction kinematics. F.l. Effect of External Radiation on Count Rate ln the PWIA calculation , t he electron e mits radiation w a nd w' before and after the col lision, distributed according to Equation 4.7 . The final electron energy is B' = E;1 - wj Rk - w' where £~1 is the energy for elasti c scattering (w = w' = 0). Because of the proton recoil, Rk > 1 and radiation by the incoming electron produces smalle r changes in E' than radiation by the outgoing electron. For small w, Rk ~ (dE/dE')w=o = (E/ E') 2 . T h e cross section for an electron Lo elastically scatter from E Lo E' = E~ 1 - !::c.E' when there are t; and t f radiation lengths of material before and after scattering is: (Fl) 186 Here diJ( E) I dD. is the Born cross section for an electron of energy E scattering into the solid angle n, given by Equation 1.2. Since Ib(w, t) is strongly peaked at w = 0, the double integral can be approximated as the sum of two double integrals, one near w = 0 and the other near w' = 0 (the energy peaking approximation): (F2) (F3) We have used the fact that h(w, t;) and h(w', t1) dominate thew and w' dependence of the respective integrands by treating the other, slowly-varying, terms as being constant. The first term in Equation F3 is simply the probability that the outgoing electron radiates l::l.E' and the incoming electron radiates less than Rkl::l.E'. The second term is similar, with the incoming electron dominating the radiation instead. In sum, the measured E' tail is the sum of two tails corresponding to photons emitted in two different directions. The remaining integrals of Ib a.re dominated by the contribution near w and w' = 0, so we can approximate </;(wl E) ~ 1 to find: d2 1J(E, E') dD.dE' (F4) where we have combined the gamma functions r(l + bt;)f(1 + btl) from h(w, ti) and l b(w',tJ) into r(1 + bti + bt 1 ). This~ 0.2% change is required so that in a. trivial scattering process ( diJ I dO. Equation 4.7 with t = ti + t1. 1, Rk = 1, l::l.E' = w) Equation F4 reduces to 187 F .2. Effect of Internal Radiation on Count Rate In the PWIA calculation, radiation by the proton affects E', even though the radiation occurs "after" the scattering. Radiation of massless photons changes the proton's energy and momentum eq ua ll y. So in order to leave the proton on-shell after the radiation, it must be left off-shell by the scattering. The amount of proton radiation is decided befoTe the scattering, and E' is chosen to leave the proton the correct. amount off-shell. The quant ity wP = Rp6.E' specifics the amount of energy the proton arm must radiate to make E' = E~ 1 - 6.E' (in the absence of other r adiation) . For small 6.E', f4 = (Mp + 2Esin 2 (0/2))/(E~- p'). The equivalent radiator three-tail cross section is found in a generalization of the energy peaking approximation used to find Equation F4. Here the integra l is split. into three pieces, one for each arm dominating the total 6.E'. The probability Ib for the other two arms to emit w ~ 0 is simply Equation F4 with du jdf! = 1 a n d th e appropriate values of Rk , R p and 1. The resu lting cross section is: (F5) The integral of the above over small 6.E' is easily found by setting <P(w/ E) = 1 and du / df! con stant: du)IJW I A = .., 1 (Rk6.E' ) ·'k ( df! 6,.E' I (1 + ).k + ).k' + >.,,) B (6.-.£')>-e (f46.E') >.v' du E. E~ 1 E~ df! ( ) (F6) 188 F .3. Internal Rad iation in the One-Photon Limit This section discusses internal radiation in the single- photon approximation in order to h elp ex plain the success of the modified equivalent radiator approximation described in Section IV.E. We begin with the angula r distribution Y(Ow, ¢w) of ll(e, e'p) rad iation in the one photon exchange approximation a nd the soft photon (low w) limit: 2 dsO" w _ d_n _d..,.. 3 w- (F7) The angular distribution Y(Ow, <f>w), which is calculated from the diagrams in Figure 43, is not a function of the magnitude w [40). The result is shown in Figures 80 and 81 for two of the NE-18 kinematics . As in the case of ext ernal radiation, most of the radiation is emitted nearly parallel to k or k'. (In fact Y (Ow, <f>w) diverges in the k and k' directions, although the int.0gra l over dnw is finite.) Futhermore, the energy dependence of the radiation in a given direction is 1/w, t he sam e as for external rad ia tion in the one-photon limit (Equation 4.10). Thus it is natural to simulate the angular distribution and en ergy dependence of the internal radiation by the equi valent radiator m ethod . T he count rate in an H( e, e' )p experiment for £~1 - !:J.E' < E' < E~1 can be calculated by the integral of the inelastic diagrams (Equation F7), plus the elastic diagrams (Figures 3 and 42). The ln w infrared divergence of the 1/w bremsstrahlung integral is cancelled by the soft-photon divergence in the virtual photon graphs, y ielding [40): (F8) The correction factor DE' is a function of the scattering kinematics and the E' acceptancc. It cont ains a cut-independent piece, due to elastic diagrams with hard v irtual photons (Figure 42), and a cut-dependent p iece describing the counts lost outside t he 189 --_----- without Proton 10 - ~~~~~~~~~~~~~~~~~~~~~~-r -200 -100 100 200 0 5 e (.) (degrees) FJG. 80. Single-photon angular distribution Y(Bw, <Pw) of H(e, e'p) internal bremsstrahlung at Q2 = 1 (GeV /c) 2 • The curves assume the emission of a single soft photon, and are valid in the soft photon (low w) approximation. The dashed line shows the angular distribution when diagrams involving radiation by the proton are omitted. 190 10 1 II 10° ,..-... II II II I I I I I I 10-l I l-< rn "'--"" ,..-... 10-2 Q2 = 7 3 3 CJ:) "'--"" \ / -e- \ / / 10- 3 / / / b 10- 4 \ Q2 1 8 0 (degrees) 10-5 -200 -100 (.) 100 200 FIG. 81. Single-photon T(Bw, <Pw) at Q2 = 1, 7 (GeV fc) 2 acceptance due to internal bremsstrahlung (Figure 43). For the 8E' given in Mo and Tsai , Equation F8 is only valid for tiE' « E ' /(1 + 2E/Mp) [48]. Above we noted that the single-photon expressions for internal and external radiation, Equations 4.10 and F7 had the same 1/w dependence. Agreement is maintained in the infinite-photon limit because the coherent multiple-photon exponentiation in Equation F6 serves the same role as the incohe1·ent multiple-collision factor wbt in Equation 4.7. To sec this, recall Equation 4.16, e6 = Nw·\ (we use w = tiEm and drop the Em subscripts for compactness). The tail height (divided by the Born cross section) is de6 jdw = N>..w>--t. For>.. = bl , this h as the same w dependence as Equation 4. 7, the multi-collision form for extern al radiabon. (For simplicity we compare with the one-tail version instead of the three-tail version, Equation F5. The number 191 of tails does not effect thew dependence because Y(Bw, cfJw) is not a function of w.) Now consider the single-photon version of the above: 5 = .\In w + ln N, with tail height d5 j dw = .\jw. This is just the single-collision form for external radiation, Equation 4.10. Thus the conversion from the single-photon to the coherent multiplephoton form is mathematically identical to the conversion to the incoherent multiplecollision form. From this we conclude that, to the accuracy of the exponential approximation, the PWIA calculation correctly includes multiple-photon internal radiation. This in spite of the fact that the calculation uses the multiple-collision form for external radiation. Furthermore, we conclude that if exponentiation approximates the higherorder diagrams of cxtemal radiation as well, then the Equation 4. 7 is approximately correct to all orders of a. F.4. Description of the Modified Equivalent Radiator Approximation This section describes the selection method for the parameters of the eqUJvalent radiator approximation, discussed in Section IV .E, used to simulate internal bremsstrahlung in the PWIA calculation. In order to reproduce the theoretical E' and Em cut dependences, the equiva.lent radiator approximation must reproduce the theoretical values of NE', NEm and ).E, (see Equations 4.18 and 4.19). Conveniently the equivalent radiator cut dependence, given by Equation F6, has the same tlE' dependence as the theory ["-' (tlE')>.E'], so long as the calculation's ;.Pw 1 A a.re chosen so that: ).PWIA k + "'k' ,PWIA + ,PWIA _ , "'r' - "'E'· (F9) Multiplication of the cross section by the proper normalization factor, allows the calculation to satisfy Equation 4.18 at small tlE'. The normalization factor represents the contribution of the hard virtual photon diagrams in Figure 42. 192 The Em cut dependence is given by Equation F6 with Re = Rp = 1 (because of Equation 4.5). Thus, the calculation simultaneously satisfies Equations 4.18 and 4.19 at all (small) values of !:::.E' and !::.Em by using: (FlO) Reproducing the theoretical Ns,, Nsm and A£' places three conditions on the four unknowns ()..k , )..k,, Ap', and the normalization). The theoretical integral of the cross section over another observable (for instance E~) could provide a fourth condition (NE~) and remove the remaining ambiguity. However, the PWIA calculation is insensitive at the< 0.5% level to the ratio of )..k to Ap'· For definiteness, we choose the value >.._t" / >..;;'( given by the Wasson et al. [41] 3-tail peaking approximation: j, W tAk (Fll) ,PWJA _ 'IV ""P' -.tAp' f (F12) ,PWIA _ Ak - where the fraction .ft varies between 0.88 and 0.89: (F1:3) The )..Pw 1 A values resulting from these prescriptions are listed in Table 26. The equivalent radiator parameters are evaluated for central kinematics, and are not adjusted for the kinematics of each event. The errors produced by neglecting variations in E and Ok' are negligible ( < 0.04% in the normalization and < 0.0012 in the).. values) . Wasson el al. [41] obtained the 3-tail peaking approximation )..w used in Equation Fl3 by integrating the radiation's angular distribution over the peaks in the k , k', and p' directions: (F14) 193 TABLE 26. Comparison of the equivalent radiator thicknesses >. for the 1 H kinematics. Values ;_EQ are from Equation 4.17, >. 111 are from Equations P14- F15, and ;_PWIA are from Equations F9- F13. Q2 ;_EQ >.}; >.);';' >-:; ;_PWIA k >,PWIA k' >,PWIA (GeV /c2 ) (%) (%) (%) (%) (%) (%) (%) 1.1538 3.322 3.936 3.767 0.042 3.502 3.614 0.037 3.2233 3.561 4.149 3.790 0.326 3.652 4.282 0.287 5.1459 3.669 4.279 3.790 0.485 3.786 4.619 0.429 6.8496 3.736 4.369 3.790 0.590 3.883 4.836 0.524 In (4k' m~ 2 >.~ = ;a [ >. ~ = ~ P 7r ) - 1 ] [In (E~ + p') - 2] . E~- p' p' (F15) (F16) This ckscription , while an improvement over the standard method, does not satisfy Equations 4.18 and 4.1 9. Due to approximations in the calculation of 8, Equations 4.14 and 4.16 are not valid for for 6.E' ~ E'/(1 + 2E/Mp) [48]. The approximations neglect two effects, which are present in the modified eq uivalent radiator technique: da(E - Re6.E')/df2 > da( E)jdD., and ¢(w/ E) < 1. Thus, the PWIA calculation maintains good agreement (< 0.5%) with exact calculations of the radiation for large 6.£'. (The comparison can be done in the one-photon limit.) F.5. Internal Radiation and Kinematics As discussed at the beginning of Section IV .D, events can a lso be cut by the finite momentum byte and solid angle of the 8 GeV /c spectrometer. Thus the PWIA calculation should correctly include the effect of the radiation on the kinematics of the 194 outgoing particles. The equivalent radiator procedure used in the calculation implicitly makes two assumptions a bout the effect of internal bremsstrahlung on kinem atics: 1) photons can be treated as being emitted exactly parallel to the outgoing particles (the angle peaking approximation) , and 2) photons emitted along one of the outgoing particles were effectively emitted by tha t particle. We touched on the angle peaking ap proximation shortly after Equation F7, but wish to evaluate it more carefully here. In the single-photon limit, most of the radiation occurs within 6,.() = (m.e/ E) 112 of the k direction and (me/ E') 1 12 of the k' direction [40). For E , E' 2:: 1.4 GeV and w < 20 MeV /c, 6,.() < 19 mr. The d e- flection of the electron is less than 0.3 mr, small enough to ignore. But for the few events with w ~ W max = 200 Mev jc, the deflection is 3 mr. Even though the average deflection is zero, this could conceivably have a small effect on the count rate. In Figure 82 one finds, however, that t he calculation does a good job of reproducing the observed widths of t he peaks in {) w for these w > 20 MeV /c events. This is because, in multi- photon brem sstrahlung, t he simultaneous emission of photons from a ll three arms produces most of the peak width. (Resolution broadening is not ver y important in Figure 82. This was checked by comparing with the same graph for w > 40,60 MeV jc.) The events at {) ~ 15 mr are the result of radiation by the elec- tron before and after the scattering. The incoherent addition of the r adi ation before and aft er the scattering in the equivalent radi ator approximation underpredicts the strength given by the coherent interference of the corresponding radiati ve di agr am s . The missing strength is less than 10% of the counts at 20 MeV /c < w < 200 MeV /cLhat is, less than 1% of the total counts. Since in only a fraction of these events would the exact photon a ngle make the difference between the outgoing particles being inside or outside of t he experimenta l acceptance, the error is insignificant. The validity of assumption (2), that photons emitted along one of the outgoing particles were emitted by that particle, is demonstrated in F igure 83. He re we see 195 250 e 200 e' 150 ..._; rn ~ ;j 0 u 100 50 0 ~~~~~~~~~~~~~~~~~~~~~~~ -50 -25 0 25 50 75 19 w (degrees) FIG. 82. Comparison between the H(e, e'p) data and PWIA prediction distributions for the angle {)w = Lan- 1 (Pm ,x/ Pm,z) at Q2 = 1 (GeV jc) 2 • Note that {)w is the projected angle of the radiation in the horizontal plane rather than the spherical coordinate Bw· To reduce the e ffects of the finite resolution, only events with Em > 20 MeV are displayed. The peaks from radia- tion directed along k and k' are clearly visible at {)w = (Jk = 0 and 196 that radiation in the k' direction has R ~ Rk' = 1, and radiation in the k direction has R ~ Re· That is, radiation along k' has the same effect on the kinematics as radiation by the outgoing electron (and analogously for the incoming electron). In the figure, the cross over from the {)w < 15° to {)w > 15° occurs at R ~ 1.4 for both data and calculation. The calculation's underestimate of events at {) ~ 15 mr, discussed above, maps here to an underestimate at R ~ 1.4. In conclusion , the equivalent radiator procedure used in the PWIA calculation simulates the effect of internal radiation on the measured counting rates to good accuracy, as required for the extraction of the nuclear transparency. 197 e' 400 rn ...,; ~ 300 ;j 0 u 200 100 0 2 1 4 3 E m /6E' FIG. 83 . Comparison between the H( e, e'p) data and the PWIA calculation for the recoil ratio R = E J D.. 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