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%\title{Low $x_{bj}$ DIS, QCD and Generalized Vector 
%Dominance\footnote{Presented at the 7th International Workshop on Deep 
%Inelastic Scatterind and QCD, DESY-Zeuthen, April 19-23, 1999}}
%       
%\author{D. Schildknecht\address{Department of Physics,
%        University of Bielefeld, \\
%        P.O. Box 10 01 31, 33501 Bielefeld, Germany}}%
        
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\hfill \begin{minipage}[t]{4cm}BI-TP 99/14\\
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\vspace*{2cm}
\begin{center}
{\Large\bf Low $x_{bj}$ DIS, QCD and Generalized Vector Dominance}\footnote{
Presented at the 7th International Workshop on Deep 
Inelastic Scatterind and QCD, DESY-Zeuthen, April 19-23, 1999}
\end{center}
\vspace*{0.3cm}
\begin{center}
D.\ Schildknecht
\\[.3cm]
University of Bielefeld, Department of Theoretical Physics, \\[.3cm]
33501 Bielefeld, Germany
\end{center}
\vspace*{3cm}
\section*{Abstract}
We give a brief overview on the present status of Generalized Vector 
Dominance as appplied to vector-meson production and the total 
photoabsorption cross section in the region of small $x_{bj}$. 
We comment on how GVD originates from QCD notions such as color transparency.
\vfill
%June 1994 \hfill

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\section{THE BASIC QUESTION}

Concerning DIS at low values of the scaling variable, $x_{bj}$, a basic
question has been around for about thirty years \cite{Cornell}: when
does the virtual photon behave hadronlike, is it when $Q^2 \to 0$ or is it
when $x_{bj} \to 0$, but $Q^2$ fixed and arbitrarily large? Here,
``hadronlike'' behaviour includes the transition of the (virtual) photon
to (massive) $q \bar q$ states and their subsequent diffractive forward
scattering from the proton, in generalization of the role of the low-lying
vector mesons in photoproduction. There is qualitative experimental
evidence for this picture of generalized vector dominance (GVD)\cite{Sakurai}
at low $x_{bj}$ and large $Q^2$,
\looseness -1
\begin{itemize}
\itemsep -2pt
\item[i)] the existence of high-mass diffractive production discovered at 
HERA \cite{H1},
\item[ii)] the similarity in shape (thrust, sphericity) \cite{Zeus} of the
states diffractively produced in DIS and the ones produced in $e^+e^-$
annihilation,
\item[iii)] the persistence of shadowing in $\gamma^* A$ collisions for
$x_{bj} \to 0$ at fixed $Q^2 >> 0$ \cite{Achman}.
\end{itemize}
Quantitatively, one starts \cite{Sakurai} from the mass dispersion relation for
$\sigma_T (W^2, Q^2)$,
\beq
\sigma_T  = \int dm^2 \int dm^{\prime 2} {{\rho_T (W^2, m^2, 
m^{\prime 2}) m^2 m^{\prime 2}} \over {(Q^2 + m^2) (Q^2 + m^{\prime 2})}},
\eeq
and its generalization to the longitudinal photon absorption cross section,
$\sigma_L$, where the spectral weight function is related to the product
of the $\gamma^* q \bar q$ transition (in the initial and the final state
in the forward Compton amplitude) and the imaginary part of the $q \bar q$
proton forward scattering amplitude. Frequently, the diagonal
approximation, $\rho \sim \delta (m^{\prime 2} - m^2)$, is adopted
that requires
$\sigma_{q \bar q p} \sim 1/m^2_{q \bar q}$ to obtain scaling
for $\sigma_T$.

\section{DIAGONAL GVD}

Lack of space does not permit me to reproduce the phenomenologically
successful representation of $\sigma_{\gamma^*p} (W^2, Q^2)$ at low $x_{bj}$,
including photoproduction, by GVD. I have to refer to ref. \cite{6}.
The diagonal approximation, nevertheless, cannot be the full story.
After all, diffraction dissociation
exists in hadron reactions, and there is no particular reason in a 
gluon-exchange picture that would forbid different masses, $m_{q \bar q}
\not= m^\prime_{q \bar q}$, for ingoing and outgoing $q \bar q$ states in
the forward Compton amplitude.

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%\caption{Biologically treated effluents (mg/l)}
%\label{tab:effluents}
%\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}rrrr}
%\hline
%                 & \multicolumn{2}{l}{Pilot plant}
%                 & \multicolumn{2}{l}{Full scale plant} \\
%\cline{2-3} \cline{4-5}
%                 & \multicolumn{1}{r}{Influent}
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%                 & \multicolumn{1}{r}{Influent}
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%\hline
%Total cyanide    & $ 6.5$ & $0.35$ & $  2.0$ & $  0.30$ \\
%Method-C cyanide & $ 4.1$ & $0.05$ &         & $  0.02$ \\
%Thiocyanide      & $60.0$ & $1.0?$ & $ 50.0$ & $ <0.10$ \\
%Ammonia          & $ 6.0$ & $0.50$ &         & $  0.10$ \\
%Copper           & $ 1.0$ & $0.04$ & $  1.0$ & $  0.05$ \\
%Suspended solids &        &        &         & $<10.0?$ \\
%\hline
%\multicolumn{5}{@{}p{120mm}}{Reprinted from: G.M. Ritcey,
%                             Tailings Management,
%                             Elsevier, Amsterdam, 1989, p. 635.}
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\section{OFF-DIAGONAL GVD IN VECTOR-MESON PRODUCTION}
 
Reformulating and extending the off-diagonal GVD ansatz \cite{8} 
for elastic vector
meson production, recent work \cite{Schuler} by Schuler, Surrow and myself
yields a satisfactory representation of the transverse cross section
and the longitudinal-to-transverse ratio, $R$, for elastic $\rho^0, \phi$
and $J/{\rm Psi}$-production \cite{Schuler}. The theoretical 
prediction for $\sigma_{T, \gamma^*p \to Vp}$ is based on
%
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\caption{GVD in $\gamma^*p \to \phi~ p$ \cite{Schuler}.}
\end{figure}
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\caption{GVD in $\gamma^*p \to J/{\rm Psi}~ p$.}
\end{figure}
%
\beq
\sigma_{T, \gamma^* p \to V p} = {{m^4_{V,T}} \over
{(Q^2 + m^2_{V,T})^2}} \sigma_{\gamma p \to V p} (W^2).
\eeq
I refer to ref.\cite{Schuler} for the prediction for $R$. The inclusion of
off-diagonal transitions with destructive interference yields $m^2_{V,T}
< m^2_V$, where $m_V$ stands for the mass of the vector meson being
produced. As an example, in fig. 1, I show $\phi$ production. The curves
(2-par. fit) are based on $m^2_{\phi,T} = 0.40 m^2_\phi$ and 
$\sigma_{\gamma p \to \phi p} = 1.0 \mu b$. The theoretical curves for
$J/{\rm Psi}$ production in fig. 2 were obtained by the replacement $m^2_\phi
\to m^2_{J/{\rm Psi}}$ and $\sigma_{\gamma p \to \phi p} \to \sigma_{\gamma p
\to J/{\rm Psi}~ p}$.


%and $R_\phi \equiv \sigma_{L, \gamma^* p \to \phi p}/\sigma_T, \gamma^*
%p \to \phi p$
%\begin{eqnarray*}
%R_\phi & \to & {4 \over 9} {{Q^2} \over {m^2_{\phi, L}}},~~{\rm for}
%Q^2 \to 0 \\
%& \to & {{\pi^2} \over 4} {{m^4_{\phi, L}} \over {m^4_{\phi, T}}},
%~~{\rm for} Q^2 \to \infty.
%\end{eqnarray*}
%
%We refer to ref.~\cite{Schuler} for the non-asymptotic form of $R$.



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\section{OFF-DIAGONAL GVD FROM QCD}

This is work in progress in collaboration with Cvetic and Shoshi \cite{Cvetic}.
Starting from the QCD notion of color transparency \cite{Nikolaev}
and an impact-parameter representation for $\sigma^{tot}_{\gamma^* p}$,
we obtain a representation for $\sigma_{\gamma^*p}^{tot}$ of the form (1).
The spectral weight function turns out to be much like the one conjectured
a long time ago \cite{12}.
Color transparency, as
fulfilled in a two-gluon exchange ansatz, provides the destructive 
interference necessary \cite{12} for convergence and scaling in (1),
thus resolving what has sometimes been 
called \cite{Bjorken} the ``Gribov paradox''.

\begin{thebibliography}{99}
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\end{thebibliography}
\end{document}


