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78 Chapter 7. Off-Shell Ambiguities Relative Deviation 5 4 3 2 1 0 CC2/CC2 CC1/CC2 p 3/2 DON/CC2 WT/CC2 -1 5 Relative Deviation 4 3 2 1 0 p 1/2 -1 -500 -400 -300 -200 -100 0 100 200 300 400 500 p m [MeV/c] Figure 7.1 The sensitivity of relativistic eikonal calculations for the 16 O(e, e ′ p) experiment of Ref. [15]. Plotted are the ratios of the differential cross sections relative to the cross section obtained with the CC2 current operator.
79 momentum region. Since these excited states are expected to be largely off-shell, it is only natural that the off-shell ambiguities manifest themselves more strongly in this region of phase space. One can estimate the off-shellness of a particle as follows. The actual energy transfer to the nucleon ω is determined by the electron kinematics. Assuming that the struck nucleon was on its mass shell, it would have an energy E on equal to (⃗p 2 m + M 2 ) 1/2 . The energy transfer ω ′ which one would have in that case is given by ω ′ = E f − E on , (7.1) where E f is the escaping particle’s energy. The difference, δω δω = ω − ω ′ (7.2) is a measure for the off-shellness and a quantity which grows with increasing p m . Figs. 7.2 and 7.3 show the predictions for the structure functions that contribute to the differential cross sections shown in Fig. 7.1. The results are obtained within the framework of the relativistic eikonal model and utilize the EDAI 16 O optical potentials of Ref. [56]. Among the infinite number of possible recipes for the off-shell proton-electron coupling we have selected four that are frequently used in literature. They are the commonly used CC1 (Eq. (4.2)) and CC2 (Eq. (4.3)) current operators, the operator proposed by Donnelly et al. (Eq. (4.10)), and the operator that was constructed via the Ward-Takahashi identity (Eq. (4.11)). Current conservation was imposed by either modifying the longitudinal component of the vector current operator (hereafter denoted as the “J0 method”), or by modifying the charge operator (hereafter denoted as the “J3 method”), along the lines of Eqs. (4.5) and (4.7), respectively. Note that for the operator of Eq. (4.10), both methods yield the same results, since, by construction, this operator is current conserving, regardless of the method adopted to compute the initial and final wave functions. Turning one’s attention to the results of Figs. 7.2 and 7.3, one immediately observes that the calculated (e, e ′ p) observables are far from independent from the choices made with regard to the electron-proton coupling. We first look at the differences between the J0 and the J3 method. Obviously, the transverse R T and R T T structure functions are insensitive to whether the J0 or J3 scheme is adopted, as they only involve a modification of the purely longitudinal and the charge component. Looking at the R L response function, we see that the CC1 current operator, for example, produces results in the J3 method that are much bigger than the ones in the J0 method. A similar observation applies to the “WT”operator. For the R L response all current operators produce comparable results in the J0 scheme. The deviations become sizeable though in the J3 scheme, favoring the J z → (ω/q)J 0 substitution, as it turns out that the purely longitudinal channel is then insensitive to the choice of the adopted operator. In the transverse responses R T and R T T , all couplings but the CC1 one produce the same results. The CC1 results are identical in the J0 and J3 scheme. A response
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De Eikonale Benadering in Dirac-Ge
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ii CONTENTS 10 Conclusion and Outlo
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2 Chapter 1. Introduction the stron
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4 Chapter 1. Introduction a wide Q
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6 Chapter 2. Reaction Observables a
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8 Chapter 2. Reaction Observables a
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10 Chapter 2. Reaction Observables
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Chapter 3 Relativistic Bound State
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3.1. Formalism 15 content to the me
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3.2. Results 17 12 C 16 O proton ne
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3.2. Results 19 have verified that
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3.2. Results 21 where the positive
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3.2. Results 23 Figure 3.4 Nucleon
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26 Chapter 4. Off-Shell Electron-Pr
Page 32 and 33: 28 Chapter 4. Off-Shell Electron-Pr
Page 35 and 36: Chapter 5 The Eikonal Final State I
Page 37 and 38: 5.2. A Consistent Approach to Final
Page 39 and 40: 5.2. A Consistent Approach to Final
Page 41 and 42: 5.3. Optical Potentials and the Eik
Page 43 and 44: 5.3. Optical Potentials and the Eik
Page 45 and 46: 5.4. The Glauber Approach 41 where
Page 47 and 48: 5.4. The Glauber Approach 43 Figure
Page 49 and 50: 5.4. The Glauber Approach 45 shape
Page 51 and 52: 5.4. The Glauber Approach 47 ε pp
Page 53 and 54: 5.4. The Glauber Approach 49 1250 1
Page 55 and 56: 5.4. The Glauber Approach 51 so tha
Page 57 and 58: 5.4. The Glauber Approach 53 1.2 1
Page 59 and 60: 5.5. Higher Order Eikonal Correctio
Page 61 and 62: Chapter 6 Final State Interactions
Page 63 and 64: 6.1. 16 O(e, e ′ p) 15 N 59 d 5
Page 65 and 66: 6.1. 16 O(e, e ′ p) 15 N 61 ρ [(
Page 67 and 68: 6.1. 16 O(e, e ′ p) 15 N 63 12 C(
Page 69 and 70: 6.1. 16 O(e, e ′ p) 15 N 65 CEA w
Page 71 and 72: 6.1. 16 O(e, e ′ p) 15 N 67 Figur
Page 73 and 74: 6.1. 16 O(e, e ′ p) 15 N 69 0.4 0
Page 75 and 76: 6.2. 12 C(e, e ′ p) 11 B 71 0 -0.
Page 77 and 78: 6.2. 12 C(e, e ′ p) 11 B 73 ρ [(
Page 79 and 80: 6.2. 12 C(e, e ′ p) 11 B 75 P n P
Page 81: Chapter 7 Off-Shell Ambiguities A m
Page 85 and 86: 81 R L [fm 3 ] R T [fm 3 ] R TT [fm
Page 87 and 88: 83 R L [fm 3 ] R T [fm 3 ] 0.04 0.0
Page 89 and 90: 85 manner, we will from here on onl
Page 91 and 92: 87 R CC1/CC2 R CC1/CC2 R CC1/CC2 R
Page 93 and 94: 89 The relation obtained in Eq. (7.
Page 95: 91 A LT 1 0.5 0 OMEA (CC2) OMEA (CC
Page 98 and 99: 94 Chapter 8. Relativistic Effects
Page 107 and 108: Chapter 9 Nuclear Transparency The
Page 109 and 110: 9.1. Theoretical Background 105 Req
Page 111 and 112: 9.1. Theoretical Background 107 few
Page 113 and 114: 9.2. Review of Experimental Results
Page 115 and 116: 9.2. Review of Experimental Results
Page 117 and 118: 9.3. Nuclear Transparency Calculati
Page 125 and 126: Chapter 10 Conclusion and Outlook I
Page 127 and 128: 123 with increasing Q 2 the uncerta
Page 129 and 130: Bibliography [1] V. Pandharipande,
Page 131 and 132: Bibliography 127 [42] J. Kelly, Phy
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Bibliography 129 [85] A. Saha, W. B
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