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%% subsection 4.2 Averages over the proton colour field [slac-pub-7064-0-0-4-2 in slac-pub-7064-0-0-4: slac-pub-7064-0-0-4-3]
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\subsection{\usemenu{slac-pub-7064::context::slac-pub-7064-0-0-4-2}{Averages over the proton colour field}}\label{subsection::slac-pub-7064-0-0-4-2}
All the information on the photon-proton interaction is contained in
the functions $\tilde{f}(\Delta)_{ab},\,\tilde{f}'(\Delta)_{ab}$ and
$\tilde{h}(\Delta)_{ab}$, which occur in Eqs. (\docLink{slac-pub-7064-0-0-4.tcx}[smm]{68})
and (\docLink{slac-pub-7064-0-0-4.tcx}[s00]{69}). As discussed in Sect. 3.2,
the 3-dimensional Fourier transforms with respect to $x_\p$ and $x_\perp$,
as defined in Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[cs]{58}), are needed for
the cross sections and the related structure functions. In this section
several general features of these functions will be discussed, which
will allow us to evaluate the inclusive and diffractive structure functions
in terms of several unknown constants. Our main assumptions are that
the field strength $G_{\mu\nu}(x)$ vanishes outside a region
of size $\sim 1/\Lambda$, and that it varies smoothly on a scale $\Lambda$.
Consider first the dependence on the transverse coordinates $x_\perp$. For
$f(x)_{ab}$, $a\neq b$, to be non-zero, at least one of
the two fermion lines of Fig. 1 has to pass through the region with non-zero
field, which has the transverse size $\sim 1/\Lambda$. Hence, integration over
the transverse coordinates can be expected to yield
\beq
\int d^2x_\perp e^{-ix_\perp\Delta_\perp}
f(x_\p,x_\perp;\Delta_+,\Delta_\perp,\alpha)_{ab}
\simeq \frac{\pi}{\Lambda^2}\exp(-\frac{\Delta_\perp^2}{4\Lambda^2})
f_\p(x_\p;\Delta_+,\alpha)_{ab}\, .
\label{fpar}
\eeq
This relation becomes exact for $f(x;\Delta,\alpha)=f_\p(x_\p;\Delta_+,\alpha)
\exp(-x_\perp^2\Lambda^2)$, but its qualitative features can be expected
to hold in general. The case $a=b$ can be treated completely analogous, after
replacing $f_{aa}$ by $f_{aa}-1$. This is possible since the constant does
not contribute to the Fourier transform for $\Delta_+\neq 0$.
According to eqs. (\docLink{slac-pub-7064-0-0-2.tcx}[stokes]{22}) and (\docLink{slac-pub-7064-0-0-2.tcx}[stokes2]{23}),
given at the end of Sect. 2, the function
$f_{ab}$ integrates the colour field strength in the double shaded
area of Fig. 1. Outside the shaded area the Wilson lines of quark
and antiquark may be connected by a space-like Wilson line in the
quark-antiquark plane, yielding a Wilson triangle.
The light-like vector $a^\mu$ essentially points along
the light-cone axis $x_+ = x^0 +x^3$ = const., and the space-like vector
$b^\mu$ may be chosen orthogonal to the longitudinal axis.
The component of the field strength tensor, which is integrated over the
double shaded area, reads
\beq
G_{+,2\ ab}\ = E_{2\ ab}\ - \ B_{1\ ab}\, ,
\eeq
where we have chosen the 1-axis to be perpendicular to the quark-antiquark
plane, $G_{0i}=E_i$ and $G_{ij}=-\epsilon_{ijk}B_k$.
For small areas the integral $f_{ab}$ is proportional to the area.
Since we are considering invariant masses $M^2 = O(Q^2)$, the opening
angle $\theta$ is always small. Hence, $f_\p$ rises linearly in $x_\p$
and $\theta$ in the range $0