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52 Chapter 5. The Eikonal Final State
where ρ i denotes the density distribution of the nucleon with quantumnumbers i.
One thus arrives at an expression where the profile functions are folded with the
nucleonic densities.
In deriving Eq. (5.52) we have supposed an independent-particle behaviour. A
more realistic description needs the introduction of nucleon-nucleon correlations.
This can now simply be adopted by the substitution
ρ i (⃗r i ) → ρ i (⃗r i )g(⃗r − ⃗r i ) , (5.53)
where g(⃗r − ⃗r i ) is the (central) correlation function. The important role played by
these short-range correlations has been extensively discussed in literature [8, 74, 75,
76, 77]. We will address these correlations further in Sec. 9, when discussing nuclear
transparencies, as they appear to play a significant role there. We use the correlation
function proposed by Gearheart and Dickhoff [78, 79] as plotted in Fig. 5.12. This
choice is based on the fact that this correlation function was shown to produce a
favorable agreement with 12 C(e, e ′ p) data [80]. Also, in comparison with other model
predictions for the central correlation function, the one obtained by Gearhart and
Dickhoff can be classified between the categories of “hard” (with a core at short
internucleonic distances) and “soft” (characterized by a finite probability to observe
nucleon pairs at very short internucleonic distances) correlations.
5.4.4 Relativistic Extension of the Glauber Approximation
In this section a relativistic generalization of the non-relativistic Glauber framework
will be outlined. We start from the scattering amplitude [57]
F ss ′( ⃗ k i , ⃗ k f , E) = − M 2π < Φ ⃗ ki ,s |(βV s + V v )|ψ (+)
⃗ kf ,s ′ > , (5.54)
with a relativistic scattered wave of the type
√ [
ψ (+)
⃗ = E + M 1
kf ,s ′ 1
2M E+M+V s−V v
⃗σ · ⃗p
and the free Dirac solution
√ [
Φ ⃗ki ,s = E + M
2M
]
]
1
1
E+M ⃗σ · ⃗k i
After some straightforward algebraic manipulations one obtains
where
∫
F ( ⃗ k i , ⃗ k f , E) = −iK
e ı⃗ k f ·⃗r e ıS(⃗r) χ 1
2 m s ′ , (5.55)
e ı⃗ k i·⃗r χ 1 . (5.56)
ms
2
d ⃗ b
2π ei⃗q·⃗b (e iχ(⃗ b) − 1) , (5.57)
F ss ′( ⃗ k i , ⃗ k f , E) =< χ s ′|F ( ⃗ k i , ⃗ k f , E)|χ s > . (5.58)