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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