Download Thesis in Pdf Format - Theoretical Nuclear Physics and ...

62 Chapter 6. Final State Interactions and the Eikonal Approximation
(
+ Q2 Q
2
q 2 q 2 + θ ) 1/2 ( tan2 e
Q
2
w I cos φ +
2
q 2 cos2 φ + tan 2 θ ) ⎤
e
w S
⎦ (6.2)
2
where the conventions of Sec. 2 were adopted. The structure functions are explicitly
given by
w C =
w T =
]
1
[(E + E f ) 2 (F1 2 + Q2
4E f E
4m 2 κ2 F2 2 ) − q 2 (F 1 + κF 2 ) 2
Q 2
2EE f
(F 1 + κF 2 ) 2 ,
w S = k2 f sin2 θ
(F1 2 + Q2
EE f 4M 2 κ2 F2 2 ) ,
w I = − k f sin θ
(E + E f )(F1 2 + Q2
EE f
4M 2 κ2 F2 2 ) , (6.3)
where E = (( ⃗ k f − ⃗q) 2 + M 2 ) 1/2 and Q 2 = q 2 − ω 2 , with ω = E f − E.
For the results of Fig. 6.2 we considered in-plane kinematics at a fixed value of
the outgoing proton momentum [k f = 1 GeV/c] and an initial electron energy of 2.4
GeV. The variation in missing momentum was achieved by changing the momentum
transfer q. For recoil angles θ = 0 ◦ (“parallel kinematics”) the eikonal calculations
do not exhibit an unrealistic behaviour up to p m = 0.5 GeV/c, which is the highest
momentum considered here. With increasing recoil angles, and consequently,
growing “transverse” components in the missing momenta the unrealistic behaviour
of the eikonal results becomes manifest. Accordingly, the accuracy of the eikonal
method based on the small-angle approximation of Eq. (5.13) can only be guaranteed
for proton knockout in a small cone about the momentum transfer. A similar
quantitative behaviour as a function of the recoil angle to what is observed in Fig. 6.2
was reported in Ref. [7] for d(e, e ′ p)n cross sections determined in a Glauber framework.
We conclude this discussion with remarking that the eikonal method does
not exclude situations with high initial (or, missing) momenta, it only requires that
the perpendicular component of the ejectile’s momentum ⃗ k f is sufficiently small. It
speaks for itself that such conditions are best fulfilled as one approaches parallel
kinematics.
At first sight, this observation puts serious constraints on the applicability of the
Glauber method, which is based on the eikonal approximation, for modelling the final
state interactions in high-energy (e, e ′ p) reactions from nuclei. However, as we noted
before, this consistent framework does use purely real scalar and vector potentials.
More realistic scattering potentials demand an imaginary part that accounts for the
inelastic channels that are open during the reaction process. The Glauber approach
effectively includes these inelastic channels and on these grounds one may expect
,