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\section{\usemenu{slacpub7064::context::slacpub7064006}{Conclusions}}\label{section::slacpub7064006}
A semiclassical approach has been developed for both inclusive and
diffractive structure functions at small $x$.
In this kinematic regime the momentum transfer to the proton,
needed to produce a diffractive state with invariant mass $M=O(Q)$,
is very small. Hence, it should be possible to describe deep inelastic
scattering as quarkantiquark pair production in a classical colour
field representing the proton. We have carried out such a
calculation based on a high
energy expansion for quark and antiquark wave functions in the presence
of a colour background field, which is assumed to be sufficiently smooth and
localized within some typical hadronic size. In the high energy expansion
corrections two orders beyond the leading eikonal approximation had to be
considered to obtain the complete leading order result for the cross section.
No expansion in the strong coupling constant $\alpha_s$ has been used.
Instead of gluons, triangular averages of the colour field are the
basic entities of the calculation.
The final result for the inclusive structure functions has been obtained
neglecting all terms suppressed by $x,\,\Lambda/Q$ or $\Lambda/M$. In
this limit $F_2$ and $F_L$ can be expressed in terms of two constants,
the actual value of which depends on the details of the proton field.
The inclusive structure functions are obtained in the limit $x\to 0$,
for which no unitarity problem exists. $F_2$ grows
logarithmically with $Q^2$, whereas the longitudinal structure function $F_L$
is independent of $Q^2$. This can be traced back to the
suppression of asymmetric configurations, with one relatively soft and one
hard quark. It is also interesting to observe, that the enhancement
of $F_2$ at large $Q^2$ is due to a term linear
in the field strength, i.e. this effect would survive an expansion in
$\alpha_s$.
The diffractive structure functions have been calculated from the
coloursinglet contribution to the above quarkantiquark pair production
cross section. Since the interaction with the proton is generally very soft,
it is natural to expect the appearance of a large rapidity gap event whenever
a colour neutral pair has been created. The coloursinglet projection in the
final state leads to the vanishing of the longitudinal structure function,
$F_L^D=0$, an effect connected with the already mentioned suppression of
asymmetric configurations.
Similar to the inclusive structure function, $F_2^D$ can also be expressed
in terms of two field dependent constants, given by integrals over certain
nonabelian phase factors testing the proton field. $F_2^D$ has no
$Q^2$dependence and is given by some simple function of $\beta$ multiplied
with $\xi^{1}$. The probably most important result of this paper
is the leading twist
behaviour of the diffractive structure function $F_2^D$. The exact
ratio of diffractive and inclusive crosssections depends on constants,
sensitive to the details of the field, which have not been obtained
explicitly. At high $Q^2$ diffraction is found to be suppressed by a factor
of log$\,Q^2$ with respect to the total cross section.
In the last section we have argued that the difference between the $x$slope
of $F_2$ and the $\xi$slope of $F_2^D$ by one unit
is a generic feature of a large class of models.
This relation between $F_2$ and $F_2^D$ directly tests
the hypothesis of a common mechanism for diffractive and
ordinary deep inelastic scattering, with
the colour state of the proton remnant being responsible for the distinction
between the two event classes. The present calculation shows the possibility
of soft interactions adjusting the colour of the produced quark pair
as well as the
proton remnant to be in a singlet, resulting in leading twist
diffraction. In this sense the semiclassical approach, although quite
different from the partonic models \cite{12,14}, supports the
idea of soft interactions being responsible for diffraction in
deep inelastic scattering at small $x$.
Finally, it has to be mentioned that several important problems
require further study. First, the very applicability of the
semiclassical approach to deep inelastic scattering
has not been established rigorously. Second, even if this could be done, it
would be necessary to integrate over the different field
configurations forming the proton. Together with a group
theoretical analysis, this could provide more information about the
field dependent functions introduced above. Another important issue,
not considered here, concerns hard gluon radiation in the process of
pair creation. We expect that this effect will significantly modify
the inclusive structure function at small $x$ as well as the
$\beta$dependence of the diffractive structure function.\\[.1cm]
We would like to thank J. Bartels, M. Beneke, S. J. Brodsky, B. L. Ioffe,
E. Levin, M. L\"uscher and O. Nachtmann for valuable discussions
and comments. A.H. has been supported by the
Feodor Lynen Program of the Alexander von Humboldt Foundation.
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