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48 Chapter 5. The Eikonal Final State
5.4.2 Connection between the Profile Function and the NN Potential
In this section we sketch a procedure which allows to construct the nucleon-nucleon
potential directly from the profile function of Eq. (5.38), which in itself was derived
from the scattering amplitudes [14, 72]. We assume that the scattering process is
dominated by the central, spin-independent amplitude.
From Eq. (5.30) we have that
and from Eq. (5.31) that
χ(k f , ⃗ b) = 1 ı log(1 − Γ(k f , ⃗ b)) , (5.39)
χ(k f , ⃗ b) = − 1 ∫ +∞
U(
2k ⃗ b, z)dz = − 1 ∫ +∞
V NN (
f v ⃗ b, z)dz , (5.40)
f
−∞
where V NN is the nucleon-nucleon potential and v f denotes the proton’s velocity.
By using the Abel integration equation [73] to invert Eq. (5.40), we obtain
V NN (r) = v f
π
= v f
π
= v f
π
=
1 d
r dr
1 d
r dr
1 d
r dr
v f
ıπβ 2 pN
∫ ∞
r
∫ ∞
r
∫ ∞
0
∫ ∞
0
−∞
χ(b)
bdb√ b 2 − r 2
log(1 − Γ(b))
bdb
ı √ b 2 − r 2
dy 1 ı log(1 − Γ(y2 + r 2 ))
Γ(0)e −(y2 +r 2 )/2βpN
2
dy
, (5.41)
1 − Γ(0)e −(y2 +r 2 )/2βpN
2
where y = √ b 2 − r 2 , and where we have introduced the notation
Γ(0) = σtot pN (1 − iɛ pN)
4πβpN
2 . (5.42)
A word of caution is in order when trying to extract potentials directly from
the scattering observables. In Fig. 5.10 we have plotted the proton-proton potential
calculated with the aid of Eq. (5.41) for a 2 GeV proton-proton collision. We have
assumed a value of 47.5 mb for the total pp cross section and a value of −0.10 for
the ratio of the real to imaginary part of the forward elastic scattering amplitude.
The three calculations present results for three different slope parameters. As can be
seen, relatively small changes in the slope parameters induce considerable shifts in
the obtained potentials. This exercise also illustrates the role of the slope parameter
as a measure for the range in which the nucleon-nucleon interaction takes place.
The smaller the slope parameter the longer the range in which the nucleon-nucleon
interaction takes place. This concurs with the fact that the slope parameter increases
with increasing energies, and, hence, smaller distance scales are probed.