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6 Chapter 2. Reaction Observables and Kinematics transverse polarization of the absorbed photon, respectively. This separation is well known as the Rosenbluth decomposition [24]. Exclusive reactions from composite systems in which one or more particles are detected in coincidence with the scattered electron, allow to extract additional information about the target system. At present, the availability of high-duty electron facilities make these coincidence experiments involving the analysis of polarization degrees of freedom much easier than in previous times. In discussing A + e → B + e ′ + p processes, one faces the possibility of polarizing the incident and scattered wave, as well as the hadrons in the initial and final channel. In this work we will follow the conventions for the A(⃗e, ⃗e ′ ⃗p)B kinematics and observables introduced by Donnelly and Raskin in Refs. [25, 26]. The four-momenta of the incident and scattered electron are labeled as K µ (ɛ, ⃗ k) and K ′µ (ɛ ′ , k ⃗′ ). The electron momenta ⃗ k and k ⃗′ define the scattering plane. The four-momentum transfer is given by q µ = K µ − K ′µ = K µ A−1 + Kµ f − Kµ A , where K µ A and Kµ A−1 are the four-momenta of the target and residual nucleus, respectively, while K µ f is the four-momentum of the ejected nucleon. Also, qµ = (ω, ⃗q), where the three-momentum transfer ⃗q = ⃗ k − k ⃗′ = ⃗ k A−1 + ⃗ k f − ⃗ k A , and the energy transfer ω = ɛ − ɛ ′ = E A−1 + E f − E A , are defined in the standard manner. The xyz coordinate system is chosen such that the z-axis lies along the momentum transfer ⃗q, the y-axis lies along ⃗ k × k ⃗′ and the x-axis lies in the scattering plane; the reaction plane is then defined by ⃗ k f and ⃗q, as in Fig. 2.1. We now discuss processes in which a polarized electron with helicity h impinges on a nucleus and induces the knockout of a single nucleon, leaving the residual nucleus in a specific discrete state. The Feynman diagram corresponding to this process is shown in Fig. 2.2. The Bjorken-Drell convention [27] for the γ matrices and Dirac spinors is followed. Accordingly, the normalization condition for the Dirac plane waves, characterized by a four-momentum K µ and spin-state S µ , is ū(K µ , S µ )u(K µ , S µ ) = 1 . (2.1) The electron charge is denoted by −e, and the virtual photon is represented by the propagator D F (Q) µν = −g µν /Q 2 , with Q 2 ≡ −q µ q µ ≥ 0. The differential scattering cross section in the laboratory frame can then be written as [25, 26, 27] : dσ = 1 β m e ɛ ∑ if |M fi | 2 m e d 3 k ⃗′ M A−1 d 3 ⃗ kA−1 M f d 3 ⃗ kf ɛ ′ (2π) 3 E A−1 (2π) 3 E f (2π) 3 ×(2π) 4 δ (4) (K µ + K µ A − K′µ − K µ A−1 − Kµ f ), (2.2) where β = | ⃗ k|/ɛ = |⃗v e | and where ∑ corresponds to the appropriate average over initial states and sum over final states as will be discussed below. The corresponding
7 e → y → e x scattering plane K' µ ( ε',k') → µ → K f (E f ,k f) φ f θ f h=- + 1 q µ ( ω,q → ) K µ ( ε',k) → θ e µ → K (E ,k ) A-1 A-1 A-1 → e z reaction plane Figure 2.1 Kinematics for the exclusive A(⃗e, ⃗e ′ ⃗p)B scattering process. The scattering plane is defined by the electron momenta ⃗ k and k ⃗′ , and the xyz coordinate system is chosen such that the z-axis lies in the direction of the momentum transfer ⃗q, the y-axis lies along ⃗ k × k ⃗′ and the x-axis lies in the scattering plane; the reaction plane is then defined by ⃗ k f and ⃗q. The electron helicity h = +1 and h = −1 corresponds to the direction parallel and antiparallel to the electron beam, respectively.
Page 1: De Eikonale Benadering in Dirac-Ge
Page 4 and 5: ii CONTENTS 10 Conclusion and Outlo
Page 6 and 7: 2 Chapter 1. Introduction the stron
Page 8 and 9: 4 Chapter 1. Introduction a wide Q
Page 12 and 13: 8 Chapter 2. Reaction Observables a
Page 14 and 15: 10 Chapter 2. Reaction Observables
Page 17 and 18: Chapter 3 Relativistic Bound State
Page 19 and 20: 3.1. Formalism 15 content to the me
Page 21 and 22: 3.2. Results 17 12 C 16 O proton ne
Page 23 and 24: 3.2. Results 19 have verified that
Page 25 and 26: 3.2. Results 21 where the positive
Page 27: 3.2. Results 23 Figure 3.4 Nucleon
Page 30 and 31: 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
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Page 51 and 52: 5.4. The Glauber Approach 47 ε pp
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Page 55 and 56: 5.4. The Glauber Approach 51 so tha
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Chapter 6 Final State Interactions
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6.1. 16 O(e, e ′ p) 15 N 59 d 5
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6.1. 16 O(e, e ′ p) 15 N 61 ρ [(
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6.1. 16 O(e, e ′ p) 15 N 63 12 C(
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6.1. 16 O(e, e ′ p) 15 N 65 CEA w
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6.1. 16 O(e, e ′ p) 15 N 67 Figur
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6.1. 16 O(e, e ′ p) 15 N 69 0.4 0
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6.2. 12 C(e, e ′ p) 11 B 71 0 -0.
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6.2. 12 C(e, e ′ p) 11 B 73 ρ [(
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6.2. 12 C(e, e ′ p) 11 B 75 P n P
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Chapter 7 Off-Shell Ambiguities A m
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79 momentum region. Since these exc
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81 R L [fm 3 ] R T [fm 3 ] R TT [fm
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83 R L [fm 3 ] R T [fm 3 ] 0.04 0.0
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85 manner, we will from here on onl
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87 R CC1/CC2 R CC1/CC2 R CC1/CC2 R
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89 The relation obtained in Eq. (7.
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91 A LT 1 0.5 0 OMEA (CC2) OMEA (CC
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94 Chapter 8. Relativistic Effects
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Chapter 9 Nuclear Transparency The
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9.1. Theoretical Background 105 Req
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9.1. Theoretical Background 107 few
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9.2. Review of Experimental Results
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9.2. Review of Experimental Results
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9.3. Nuclear Transparency Calculati
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Chapter 10 Conclusion and Outlook I
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123 with increasing Q 2 the uncerta
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Bibliography [1] V. Pandharipande,
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Bibliography 127 [42] J. Kelly, Phy
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Bibliography 129 [85] A. Saha, W. B
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