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58 Chapter 6. Final State Interactions and the Eikonal Approximation
In this experiment, the separated 16 O(e, e ′ p) structure functions and differential
cross section were measured at Q 2 = 0.8 (GeV/c) 2 and ω = 0.439 GeV for missing
(or, initial) proton momenta p m = | ⃗ k f − ⃗q| below 355 MeV/c. The variation in
missing momentum was achieved by varying the detection angle of the ejected proton
with respect to the direction of the momentum transfer (“quasi-perpendicular
kinematics”). The measured cross sections for knockout from the 1p 1/2 and 1p 3/2
levels are displayed in Fig. 6.1 along with the predictions of our calculations.
The major objective of this chapter is to compare the various schemes for the
description of the final state interactions. To that purpose, we will compare the
A(e, e ′ p) predictions for the various frameworks at specific kinematics. More in particular,
we will present results in the Relativistic Plane Wave Impulse Approximation
(RPWIA) and three different versions of the eikonal method :
1. a Consistent Eikonal Approximation (CEA) in which the eikonal phase is calculated
from the relativistic scalar-vector potential that determines also the
bound states
2. an Optical Model Eikonal Approximation (OMEA) in which the eikonal phase
is calculated from the optical potentials as they are derived from global fits to
elastic proton-nucleus data
3. the Relativistic Multiple-Scattering Glauber Approximation (RMSGA) as it
was outlined in Sec. 5.4
All results presented in this chapter were obtained with the standard dipole form
for the electromagnetic form factor. In Fig. 6.1 16 O(e, e ′ p) results are displayed for
all the different models that were sketched above. The spectroscopic factors were
determined by performing a χ 2 fit to the complete data set and are summarized in
Table 6.1. Inspecting this table, it becomes clear that in utilizing the CC1 form of
the hadron-photon vertex one reaches a far better description of the data than what
is obtained with the CC2 version. The spectroscopic factors, on the other hand, do
not substantially depend on the choice with respect to the photon-hadron vertex.
6.1.1 The Self-Consistent Approach
Fig. 6.1 clearly illustrates that the CEA predicts cross sections that do not follow
the experimentally determined trend in the p m dependence. Nevertheless, we will
discuss the CEA results in some detail, as they do provide insight in the general
trends set by the eikonal approach to calculating nucleon scattering states [87].
At low missing momenta, the eikonal results provide a fair description of the p m
dependence of the data. As a comparison, the results of a relativistic plane wave
calculation in the impulse approximation (RPWIA) are also displayed. Through
comparing the plane wave and the eikonal calculations, thereby keeping all other
ingredients of the calculations identical, one can evaluate how the eikonal method