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%% section 5 On the relation between $F_2$ and $F_2^D$ [slacpub7064005 in slacpub7064005: slacpub7064006]
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\section{\usemenu{slacpub7064::context::slacpub7064005}{On the relation between $F_2$ and $F_2^D$}}\label{section::slacpub7064005}
The above results are similar to those obtained from the simple partonic
picture of \cite{12}, as far as the relation between the inclusive
structure function $F_2$ and the diffractive structure function $F_2^D$ is
concerned. In both models $F_2^D$ is related to $F_2$ by some constant
connected with the coloursinglet requirement for the produced
quarkantiquark pair, and in both cases the slope in $x$ of $F_2$
is larger by one unit than the slope of $F_2^D$ in $\xi$.
This relation between the slopes of the two structure functions has first been
proposed in \cite{11} based on a picture of diffraction as
scattering on wee parton lumps inside the proton. In the present
section we will argue, that there is a rather general class of models for
diffractive deep inelastic scattering,
which should reproduce the above relation.
Since in small$x$ events the kinematically required momentum transfer to the
protontarget is relatively small, it is natural to think of these events
in terms of a virtual photon crosssection $\sigma_T(\gamma^*p\to p'X)$.
Here $p'$ is the proton remnant, which, in this picture, can be separated
from the produced massive state $X$ before hadronization. Assume
that, with $X$ being in a coloursinglet state, no hadronic activity develops
between $X$ and $p'$, and a rapidity gap event occurs. Now, neglecting the
longitudinal contribution, the structure functions read
\begin{eqnarray}
F_2(x,Q^2)&=&\frac{Q^2}{\pi e^2}\int\limits_x^1\frac{d\sigma_T(\gamma^*p\to
p'X)}{d\xi}d\xi\nonumber\\ \\
F_2^D(x,Q^2,\xi)&=&\frac{Q^2}{\pi e^2}\cdot\frac{d\sigma_T(\gamma^*p\to p'X)}
{d\xi}\cdot{\cal P}\, ,\nonumber
\end{eqnarray}
where the probability for the proton remnant to be in a coloursinglet state
is given by the factor ${\cal P}$. This results in the relation
\beq
F_2(x,Q^2)=\int\limits_x^1d\xi F_2^D(x,Q^2,\xi)\cdot{\cal P}^{1}=x
\int\limits_x^1d\beta\,\beta^{2}F_2^D(x,Q^2,\xi)\cdot{\cal P}^{1}\, .
\eeq
We now assume that the $\xi$dependence of $F_2^D$ factorizes,
\beq
F_2^D(x,Q^2,\xi)=\xi^{n}\hat{F}_2^D(\beta,Q^2)\, ,
\eeq
with some number $n>1$. This is consistent with data in a
wide region of $\xi$ (cf. \cite{1,2}). Dropping terms suppressed in the
limit $x\to 0$, the relation between diffractive and inclusive structure
function takes the form
\beq
F_2(x,Q^2)=x^{1n}\int\limits_0^1d\beta\,\beta^{n2}\hat{F}_2^D(\beta,Q^2)
\cdot{\cal P}^{1}\, .\label{f2f2d}
\eeq
At small $x$ the photon energy $q_0$ in the proton rest frame is much larger
than any other scale in the problem. Therefore it is natural to consider
the limit $q_0\to\infty$ and to try to understand the presence of a nonzero
probability factor ${\cal P}$ in this limit. Assuming that this small$x$
limit for ${\cal P}$ has already been reached in the present measurements,
${\cal P}$ is a function of $\beta$ and $Q^2$ only.
In this case Eq. (\docLink{slacpub7064005.tcx}[f2f2d]{100})
implies that the $x$slope of $F_2$ and the $\xi$slope $F_2^D$ differ
by one unit. It is interesting to observe that the result of this rather hand
waving argument appears to agree well with the data on $F_2^D$
\cite{1,2} and $F_2$ \cite{24,25} at small $x$.
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