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9.1. Theoretical Background 107
few GeV energy range.
In brief, CT can only exist if one assumes that at high momentum transfer small
color singlet objects (PLC’s) are produced, which experience little interaction with
the surrounding nucleons. This system expands and interacts as it moves through
the nucleus.
To implement CT effects in the description of (e, e ′ p) processes, it is mandatory
to adjust the interaction matrix elements. In the energy regime of interest here,
the expansion distance, or coherence length, is still rather small. In an optical
potential model one has that the imaginary part of the optical potential U opt is
roughly proportional to the total proton-nucleon cross section U opt ≃ −ıσ tot ρ(r),
where ρ(r) is the nuclear density. Only the imaginary part of the optical potential is
retained here. Indeed, at high energies the real part of the nucleon-nucleon forward
scattering amplitude is small. Equivalently, from Eq. (5.38) it is clear that in a
Glauber formalism, the magnitude of the final-state interactions is proportional
to the total cross section. Since a point-like configuration cannot be treated as a
nucleon, it becomes inaccurate to use the proton-nucleon cross sections as a measure
for the interactions that the PLC is subject to in the medium. The simplest way
to implement color transparency is to replace the free proton-nucleon cross section
by a new quantity σeff
P LC
, that describes the interaction of the PLC with the
medium. This effective cross section should take into account both the suppresion
of interaction in the point where the PLC is produced and the restoration of soft
final state interactions with the nucleons as it moves through the nuclear medium.
The need to include this expansion was recognized by Farrar et al. [109], who argued
that the square of the transverse size is approximately proportional to the distance
travelled from the point where the PLC is formed. Thus, the cross section σ that
appears in the optical potential, or, equivalently, in the Glauber profile function, is
replaced by one that grows as the ejectile moves in the z direction [3, 4] :
σpN tot
{[ 〈
σ eff
z n 2
P LC = k 2 〉 (
σtot
T
pN +
l c Q 2 1 − z ) ] }
θ(l c − z) + θ(z − l c )
l c
. (9.3)
Here, z is the distance moved by the expanding color singlet along the trajectory
from its point of formation, n is the number of constituents in the proton (i.e. n =
3), and k 2 T is the average transverse momentum of the proton’s constituents [k2 T ≃
(0.35GeV/c) 2 ]. The linear dependence of the cross section follows from analyses of
perturbative Feynman diagrams. Moreover, it is commonly assumed that the size of
the object’s configuration decreases inversely proportional with Q 2 . This assumption
is legitimized by the fact that this reproduces the correct Q 2 dependence of the
transverse size of nucleons found in realistic models of the nucleon form factor. The
coherence length l c depends upon the squared mass difference of the initial PLC and