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9.3. Nuclear Transparency Calculations 113
9.3 Nuclear Transparency Calculations
In this section we will study the nuclear transparency in the quasi-elastic 12 C(e, e ′ p)
reaction in a wide Q 2 range of 0.3 ≤ Q 2 ≤ 20 (GeV/c) 2 . We will compare the results
of our theoretical predictions with experimental results that were recently obtained
at the Stanford Linear Accelerator (SLAC) [98, 99] and Jefferson Lab (JLAB) [115].
The results of our nuclear transparency calculations in 12 C are contained in
Fig. 9.5. We have performed calculations within the Glauber framework and the
eikonal model with the optical potentials of Ref. [56]. We remind the reader that
the optical potential model is only applicable up to values of T p ∼ 1 GeV. As for
the Glauber results, we have also performed calculations that included the effect
of short-range correlations. Each of these calculations was done with the CC1 and
CC2 current operator.
The measurements of the differential cross section in Refs. [98, 99, 115] were
performed in a limited region of the available phase space, dictated by the requirement
that quasi-elastic conditions are met. Analogously, we have constrained our
calculations to the same portion of phase space. In general, the experimental results
are reported in terms of the experimental transparency T exp , defined as
∫
∆
T exp =
3 k d⃗ k ∫ ∆E dE S exp( ⃗ k, E)
c A
∫∆ 3 k
∫∆E dE S P W IA( ⃗ k, E) . (9.6)
The A-dependent factor c A renormalizes the PWIA results to take corrections induced
by SRC into account. This factor grows with increasing mass number A. For
the 12 C(e, e ′ p) a correction factor of 0.901 ± 0.024 was adopted by the authors of
Refs. [98, 99, 115]. As the implementation of SRC can be done in numerous ways,
we have removed this factor from the data of Refs. [98, 99, 115], as in Eq. (9.5).
These “corrected” data points are shown in Fig. 9.5. Although the JLAB data
points suggest somewhat smaller nuclear transparencies than the SLAC data, they
are consistent with each other.
In line with Eq. (9.5) we have calculated the theoretical nuclear transparency
T theo , according to
∫
∆
T theo =
3 k d⃗ k ∫ ∆E dE S theo( ⃗ k, E)
∫
∆ 3 k
∫∆E dE S P W IA( ⃗ k, E) . (9.7)
From the results contained in Fig. 9.5 it is clear that in a particular scheme, the transparencies
computed with the CC1 current operator converge to those obtained with
the CC2 form of the electron-proton coupling. Accordingly, the theoretical transparencies
are relatively free of ambiguities with respect to off-shell prescriptions in
the Q 2 > 3 − 4 (GeV/c) 2 range. For these high Q 2 values the difference between the
CC1 and CC2 predictions drop to percent levels. The oscillations in our theoretical
curves at high Q 2 reflect the energy dependencies in the pp and pn scattering data