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Chapter 2. Relativistic Eikonal A(p, pN) Formalism 21In the relativistic Hartree approximation, real potentials are used to calculate the boundstatewave functions. If the scattering states are derived using the same potentials, only elasticrescattering contributions are taken into account. In general, however, a fraction of the strengthfrom the incident beam is removed from the elastic channel into unobserved inelastic ones likethe production of an intermediate delta resonance or pion. This local absorption is commonlyimplemented by means of complex or optical potentials obtained by fitting nucleon-nucleusscattering data. In the numerical calculations, we employed the optical potential of van Oers etal. [9] to describe scattering off 4 He and the global S − V parametrizations of Cooper et al. [46]for other target nuclei. Hereafter, the A(p, 2p) calculations which adopt Eq. (2.55) as a startingbasis are labeled the relativistic optical model eikonal approximation (ROMEA).2.3 Relativistic Multiple-Scattering Glauber ApproximationIn the ROMEA approach, like in the DWIA approaches, all the IFSI effects are parametrized interms of mean-field like optical potentials, i.e., the IFSI are seen as a scattering of the nucleonwith the residual nucleus as a whole. As the energy increases, shorter distances are probedand the scattering with the individual nucleons becomes more relevant. For proton kineticenergies T p ≥ 1 GeV, the predominantly inelastic and diffractive character of the underlyingelementary proton-nucleon scattering cross sections makes the Glauber approach [62–64] morenatural. This method reestablishes the link between proton-nucleus interactions and the elementaryproton-proton and proton-neutron scattering. It essentially relies on the eikonal, or,equivalently, the small-angle approximation and the assumption of consecutive cumulativescatterings of a fast nucleon on a composite target containing “frozen” point scatterers (nucleons).In Glauber theory, the scattering wave function is related directly to the nucleon-nucleonelastic scattering data through the introduction of a profile function. This method has a longtradition of successful results in high-energy proton-nucleus elastic scattering (T p > 500 MeV)[68,79], and has also been shown to be reliable in describing A(e, e ′ p) processes at high energies[57, 59, 80–88]. The relativistic Glauber formalism exposed here was originally developed todescribe A(e, e ′ p) observables [57, 59].2.3.1 Nucleon-Nucleon ScatteringWe start our derivations of a relativistic Glauber formalism by considering a nucleon-nucleonscattering process governed by a local Lorentz scalar and vector potential V s (r) and V v (r). Inthe EA, the scattering amplitude reads [70]F msm ′ (⃗ ks i , ⃗ k f , E) = − M 〈NΨ (+)〉2π⃗ kf ,m ′ ∣ (βV ∣s + V v ) ∣Φ ⃗ki ,mss, (2.56)with Ψ (+)⃗ kf ,m ′ sthe relativistic scattered state of Eq. (2.51) and Φ ⃗ki ,m sthe free Dirac solution (2.8).Note that in this section, V s (r) and V v (r) represent nucleon-nucleon potentials; whereas in