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68 Chapter 6. Final State Interactions and the Eikonal Approximation
defined by the following set of unit vectors and is shown in Fig. 6.5
⃗ l =
⃗ kf
| ⃗ k f |
⃗n = ⃗q × ⃗ k f
|⃗q × ⃗ k f |
⃗t = ⃗n × ⃗ l (6.4)
Note that for coplanar kinematics ⃗n determines the y-axis of the reference frame.
The escaping nucleon polarization observables can be determined through measuring
ratios. The induced polarization can be addressed with unpolarized electrons (i =
n,l,t)
P i = σ(si N =↑) − σ(si N =↓)
σ(s i N =↑) + , (6.5)
σ(si N
=↓)
whereas the polarization transfer also requires polarized electron beams (i = n,l,t)
P ′
i = [σ+ (s i N =↑) − σ− (s i N =↑)] − [σ+ (s i N =↓) − σ− (s i N =↓)]
[σ + (s i N =↑) + σ− (s i N =↑)] + [σ+ (s i N =↓) − σ− (s i , (6.6)
N
=↓)]
where s i N = ↑ (↓) denotes that the ejected hadron is spin-polarized in the positive
(negative) i direction (i = n,l,t) and where the plus (minus) sign in σ ± denotes the
helicity h = ±1 of the electron impinging on the target nucleus. σ ± (s i N ) is then
a shorthand notation for the differential cross section for an electrodisintegration
process initiated by an electron with helicity h = ±1 and for which the ejectile is
detected with a spin polarization characterized by s i N . One distinct advantage of the
polarization observables is that, unlike the response functions, they are independent
of the applied spectroscopic factors, which cancel out in a natural way.
In Fig. 6.6, we have plotted the calculations for the P l
′ and P t ′ observables for
the 16 O(⃗e, e ′ ⃗p) experiment of Ref. [15]. We note that the P n ′ is identically zero in
the kinematics considered here. These polarization observables are expected to be
rather insensitive to final-state interactions. This is confirmed by our calculations.
For the P l ′ observable the Glauber predictions are almost indistinguisable from the
RPWIA ones. The OMEA prediction follows somewhat more closely the trends
set by the data. At contrast, both methods predict equivalent trends for the P t
′
observable, and there’s no real preferred calculation. Both methods follow the trend
set by the RPWIA curve, reflecting the fact that the P t ′ is indeed rather insensitive
to the effects of final state interactions.
The polarization observables can be combined to investigate nucleon form factors.
For the free nucleon, the polarization transfer can be written in terms of the form