\noindent Polarized inclusive deep--inelastic diffractive scattering is dealt with in a quantum field theoretic approach. The process can be described in the general framework of non--forward scattering processes using the light--cone expansion in the generalized Bjorken region applying the generalized optical theorem. The diffractive structure functions $g_1^{D(3)}$ and $g_2^{D(3)}$ are calculated in the twist--2 approximation and are expressed by diffractive parton distributions, which are derived from pseudo-scalar two--variable operator expectation values. In this approximation the structure functions $g_2^{D(3)}$ is obtained from $g_1^{D(3)}$ by a Wandzura--Wilczek relation similar as for deep inelastic scattering. The evolution equations are given. We also comment on the higher twist contributions in the light--cone expansion. %
p.tex
 
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\begin{document}
\noindent
%
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\begin{flushleft}
DESY 02--011 \hfill
{\tt hep-ph/0202077}\\
January  2002
\end{flushleft}
%
%\setcounter{page}{0}
% 1
%\mbox{}
\vspace*{\fill}
\begin{center}
{\LARGE\bf Polarized Deep Inelastic Diffractive }

\vspace{2mm}
{\LARGE\bf $ep$ Scattering: Operator Approach}
%

\vspace{2cm}
\large
Johannes Bl\"umlein$^a$  and
Dieter Robaschik$^{a,b}$
\\
\vspace{2em}
\normalsize
{\it $^a$~Deutsches Elektronen--Synchrotron, DESY,\\
Platanenallee 6, D--15738 Zeuthen, Germany}
\\
\vspace{2em}
{\it $^b$~Brandenburgische Technische Universit\"at Cottbus, 
Fakult\"at 1,}\\
{\it  PF 101344, D--03013  Cottbus, Germany} \\
%\today
\end{center}
\vspace*{\fill}
%
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\begin{abstract}
\noindent
Polarized inclusive deep--inelastic diffractive scattering is dealt with
in a quantum field theoretic approach. The process can be described in 
the general framework of non--forward scattering processes using the 
light--cone expansion in the generalized Bjorken region applying the
generalized optical theorem. The diffractive structure functions 
$g_1^{D(3)}$ and $g_2^{D(3)}$ are calculated in the twist--2 approximation 
and are expressed by diffractive parton distributions, which are derived 
from pseudo-scalar two--variable operator expectation values. In this
approximation the structure functions $g_2^{D(3)}$ is obtained from 
$g_1^{D(3)}$ by a   Wandzura--Wilczek relation similar as for deep
inelastic scattering. The evolution equations are given. We also comment 
on the higher twist contributions in the light--cone expansion.
%
\end{abstract}
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\vspace*{\fill}
\newpage
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\section{Introduction}
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\label{sec-1}
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\vspace{1mm}
\noindent
Unpolarized deep inelastic diffractive lepton--nucleon scattering was 
observed at the electron--proton collider HERA some years ago~\cite{EXP1}.
In the region of hard diffractive scattering this process is described 
by structure functions which are represented by diffractive parton 
distributions. They depend on two scaling variables
$x$ and $x_{\PP}$ and are different from the parton densities of deep 
inelastic scattering. New diffractive parton densities are expected
to occur in polarized deep inelastic diffractive lepton nucleon 
scattering. They can be measured at potential future polarized $ep$
facilities capable to probe the kinematic range of small $x$~, 
c.f.~\cite{SXPOL}. Dedicated future experimental studies of this process
can reveal the helicity structure of the non--perturbative color--neutral 
exchange of diffractive scattering with respect to the quark and
gluon structure and how the nucleon spin is viewed under a diffractive
exchange.
At short distances the problem can be clearly 
separated into
a part, which can be described within perturbative QCD, and another
part which is thoroughly non--perturbative. In this paper we use the
light--cone expansion to describe the process of polarized diffractive
deep--inelastic scattering similar to a recent study for the unpolarized
case~\cite{BRDIFF}. While the scaling violations of the process can be
calculated within perturbative QCD, the polarized diffractive 
two--variable parton densities are non--perturbative and can be related 
to expectation values of (non--)local operators. Their Mellin-moments 
with respect to the variable $\beta = x/x_{\PP}$ may, in principle, be 
calculated on the lattice and one may try to understand the ratios of 
these moments and those for the related deep--inelastic process w.r.t. 
their scaling violations as being measurable in future experiments.

In this paper we describe the process of polarized deep--inelastic
diffractive scattering, which is a non--forward process in its hadronic
variables, at large space--like momentum transfer $q^2$. In this approach
there is no need to refer to any specific mechanism of color--singlet
exchange. It is completely sufficient to select the process by a 
rapidity gap between the final state proton and the other diffractively  
produced hadrons, which is sufficiently large. 
The operator formulation allows straightforwardly the
description of also higher twist operators in the light cone expansion,
which is potentially more involved in other scenarios~\cite{FACT},
to which we agree on the level of twist--2.

We firstly derive the  Lorentz--structure of the process for the
general kinematics, before we specify to the case of 
$-t = -(p_2 - p_1)^2, M^2 << -q^2$ which is
often met in experiment. The 
diffractive parton densities are derived on the level of the twist--2 
operators. In this approximation the scattering cross sections are 
described by two structure functions $g_1^{D(3)}(x,Q^2,x_\PP)$ and
$g_2^{D(3)}(x,Q^2,x_\PP)$ for pure electromagnetic scattering\footnote{
The exchange of electro--weak gauge bosons requires at least five 
structure functions \cite{BK}. QED radiative corrections to the process 
were given in \cite{PRAD}.}. Also in the present case it turns out that 
the structure functions are related by the Wandzura--Wilczek 
relation~\cite{WW}. Analogously to the unpolarized case, 
Ref.~\cite{BRDIFF}, the anomalous dimensions ruling the evolution of
the polarized diffractive parton densities turn out to be those for
deep--inelastic forward scattering.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lorentz Structure}
\vspace*{-5mm}\noindent
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\label{sec-2}
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\vspace{1mm}
\noindent
The process of deep--inelastic diffractive scattering is described by the
diagram Figure~1.
The differential scattering cross section for single--photon 
exchange is given by
%-----------------------------------------------------------------------
\begin{equation}
\label{eqD1}
d^5 \sigma_{\rm diffr}
= \frac{1}{2(s-M^2)} \frac{1}{4} dPS^{(3)} \sum_{\rm spins}
\frac{e^4}{Q^2} L_{\mu\nu} W^{\mu\nu}~.
\end{equation}
%-----------------------------------------------------------------------
Here $s=(p_1+l)^2$ is the cms energy of the process squared and $M$ 
denotes the nucleon mass.

\vspace*{0.8cm}
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\Text(-40,130)[]{$l'$}
\Text(-90,75)[]{$q$}
\Text(-110,-10)[]{$p_1$}
\Text(10,-10)[]{$p_2$}
\Text(10,110)[]{$M_X$}
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\vspace*{1mm}
\noindent
\begin{center}
{\sf Figure~1:~The virtual photon-hadron amplitude for 
diffractive $ep$ scattering} 
\end{center}
The phase space $dPS^{(3)}$ depends on five variables
since one 
final state mass varies. They can be chosen as 
Bjorken~$x= Q^2/(W^2+Q^2-M^2)$, the photon virtuality $Q^2=-q^2$, 
$t = (p_1 - p_2)^2$, a variable describing the non--forwardness w.r.t.
the incoming proton direction,
%-----------------------------------------------------------------------
\begin{equation}
\label{eqV1}
x_{\PP} = - \frac{2\eta}{1-\eta} = 
\frac{Q^2 + M_X^2 - t}{Q^2 + W^2 -M^2} \geq  x~,
\end{equation}
%-----------------------------------------------------------------------
demanding $M_X^2 > t$ and where
%-----------------------------------------------------------------------
\begin{equation}
\label{eqV2}
\eta = \frac{q.(p_2 - p_1)}{q.(p_2+p_1)}~\epsilon~\left[-1,\frac{-x}{2-x}
\right]~,
\end{equation}
%-----------------------------------------------------------------------
and $\Phi$ the angle between the lepton plane $\pvec_1 \times \lvec$ and
the hadron plane $\pvec_1 \times \pvec_2$,
%-----------------------------------------------------------------------
\begin{equation}
\label{eqV3}
\cos \Phi = \frac{(\pvec_1 \times \lvec).(\pvec_1 \times  \pvec_2)}
                 {|\pvec_1 \times \lvec ||\pvec_1 \times  \pvec_2|}~.
\end{equation}
%-----------------------------------------------------------------------
$W^2 = (p_1+q)^2$ and  $M_X^2 = (p_1+q-p_2)^2$ denote the hadronic
mass squared and the square of the diffractive mass, respectively.
The process of hard diffractive scattering is characterized by a large
rapidity--gap of the order $\Delta y \sim \ln(1/x_{\PP})$ \cite{COL}.
As we will show below it is {\it this property}, which is sufficient for 
our treatment below and no reference to a special kind of a 
non--perturbative color--neutral exchange is not needed.\footnote{
Indeed, the literature offers a large host of different pomeron 
models, c.f.~\cite{POM},
to describe these processes. The fact that many of the descriptions yield
similar results at equally large rapidity gaps and the same kinematic 
variables supports our observation.}

Unpolarized deep inelastic diffractive scattering was considered in a
previous paper~\cite{BRDIFF} in detail. Here we focus on the polarized 
part only, which can be measured in terms of a polarization asymmetry
%-----------------------------------------------------------------------
\begin{equation}
\label{eqP1}
A(x,Q^2,x_{\PP},S_\mu) = \frac{d^5 \sigma(S_{\mu}) - d^5 \sigma(-S_\mu)}
                              {d^5 \sigma(S_{\mu}) + d^5 \sigma(-S_\mu)}~.
\end{equation}
%-----------------------------------------------------------------------
$S_\mu$ is the spin vector of the incoming proton with $S = S_1$ and
$S.p_1 =0$.
Since the cross sections are linear functions in the initial--state
state proton spin--vector, the denominator projects on the even and the 
numerator on the odd part in $S_\mu$. 

We consider the case of single photon exchange, which is projected by the
polarized contribution
%-----------------------------------------------------------------------
\begin{equation}
\label{eqPLE}
L_{\mu\nu}^{\rm pol} = 2i \varepsilon_{\mu\nu\rho\sigma} l^\rho q^\sigma
\end{equation}
%-----------------------------------------------------------------------
to the leptonic tensor. Since the electromagnetic current is conserved,
the strong interactions conserve parity and are even under 
time--reversal,\footnote{Here we disregard potential contributions due to
strong CP--violation~\cite{THOOFT}, because of the smallness of the 
$\theta$--parameter, $|\theta| < 3 \cdot 10^{-9}$~\cite{THETA}.} 
and the hadronic tensor has to be hermitic due to Eq.~(\ref{eqPLE}), the
following relations hold~\cite{TM}~:
%-----------------------------------------------------------------------
%\begin{equation}
\begin{alignat}{3}
\label{eqP2A}
{\sf Current~conservation:}&~~q^{\mu}~W_{\mu\nu}(q,p_1,S_1,p_2,S_2)
  &=&
W_{\mu\nu}(q,p_1,S_1,p_2,S_2)~q^{\nu} = 0~,\\
{\sf P~~invariance:}&~~W_{\mu\nu}(\overline{q},%
\overline{p}_1,-\overline{S}_1,\overline{p}_2,%
-\overline{S}_2)  &=&
W^{\mu\nu}(q,p_1,S_1,p_2,S_2)~, \\
\label{eqP2C}
{\sf T~~invariance:}&~~W_{\mu\nu}(\overline{q},%
\overline{p}_1,\overline{S}_1,\overline{p}_2,\overline{S}_2)   &=&
\left[W^{\mu\nu}(q,p_1,S_1,p_2,S_2)\right]^*, \\
\label{eqP2B}
{\sf Hermiticity:}&~~W_{\mu\nu}(q,p_1,S_1,p_2,S_2)
 &=&
\left[W_{\nu\mu}(q,p_1,S_1,p_2,S_2)\right]^*,
\end{alignat}
%\end{equation}
%-----------------------------------------------------------------------
with $\overline{a}_\mu = a^{\mu}$. Constructing the hadronic tensor we 
seek a structure which is linear in the initial proton spin. 
Upon noting that
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP3}
\varepsilon^{\mu\nu\alpha\beta} = - \varepsilon_{\mu\nu\alpha\beta}
\end{eqnarray}
%-----------------------------------------------------------------------
the spin  pseudovector  $S_{1\mu}$ has to occur together
with the Levi--Civita pseudo--tensor.
The most general asymmetric  hadronic tensor, which
obeys Eqs.~(\ref{eqP2A}--\ref{eqP2B}),
is\footnote{A sub-set of this structure based
on $p,q$ and $S$ was considered in Ref.~\cite{DJT}.}
%-----------------------------------------------------------------------
\begin{alignat}{10}
\label{eqH2}
W_{\mu\nu} &=&~~
i \left[ \hat{p}_{1\mu} \hat{p}_{2\nu}-
\hat{p}_{1\nu} \hat{p}_{2\mu} \right] \varepsilon_{p_1,p_2,q,S} 
&\frac{W_1}{M^6}&~~
                &+&~~
i \left[ \hat{p}_{1\mu} \varepsilon_{\nu S p_1 q}
     - \hat{p}_{1\nu} \varepsilon_{\mu S p_1 q} \right] 
&\frac{W_2}{M^4}&
\nonumber \\ &+&~~
i \left[ \hat{p}_{2\mu} \varepsilon_{\nu S p_1 q}
     - \hat{p}_{2\nu} \varepsilon_{\mu S p_1 q} \right]
&\frac{W_3}{M^4}&~~
                &+&~~
i \left[ \hat{p}_{1\mu} \varepsilon_{\nu S p_2 q}
     - \hat{p}_{1\nu} \varepsilon_{\mu S p_2 q} \right]
&\frac{W_4}{M^4}&
\nonumber \\ &+&~~
i \left[ \hat{p}_{2\mu} \varepsilon_{\nu S p_2 q}
     - \hat{p}_{2\nu} \varepsilon_{\mu S p_2 q} \right]
&\frac{W_5}{M^4}&~~
                &+&~~
i \left[ \hat{p}_{1\mu} \hat{\varepsilon}_{\nu p_1 p_2 S}
     - \hat{p}_{1\nu} \hat{\varepsilon}_{\mu p_1 p_2 S} \right]
&\frac{W_6}{M^4}&
\nonumber\\ &+&~~ 
i \left[ \hat{p}_{2\mu} \hat{\varepsilon}_{\nu p_1 p_2 S}
     - \hat{p}_{2\nu} \hat{\varepsilon}_{\mu p_1 p_2 S} \right]
&\frac{W_7}{M^4}&~~
                &+&~~  i \varepsilon_{\mu \nu q S}
&\frac{W_8}{M^2}&~.
\end{alignat}
%-----------------------------------------------------------------------
It is constructed out of the four--vectors $q,p_1,p_2$ and $S =S_1$.
Terms with a genuine structure $\propto M^2/q^2$ are not considered.
Here we use the abbreviations
%-----------------------------------------------------------------------
\begin{eqnarray}
\hat{V}_\mu &=& V_\mu - q_\mu \frac{q.V}{q^2}~, \\
\hat{\varepsilon}_{\mu v_1 v_2 v_3}            &=&
    {\varepsilon}_{\mu v_1 v_2 v_3}            -
    {\varepsilon}_{q v_1 v_2 v_3} \frac{q_\mu}{q^2}~,  \\
\tilde{\varepsilon}_{\mu \nu v_1 v_2}            &=&
    {\varepsilon}_{\mu \nu v_1 v_2}            -
    {\varepsilon}_{q \nu v_1 v_2} \frac{q_\mu}{q^2}
  - {\varepsilon}_{\mu q v_1 v_2} \frac{q_\nu}{q^2}~.
\end{eqnarray}
%-----------------------------------------------------------------------
The Schouten--relation~\cite{SCHOUT} in either of the forms
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP6}
X_\mu \varepsilon_{\nu\rho\sigma\tau} &=&
X_\nu \varepsilon_{\mu\rho\sigma\tau} +
X_\rho \varepsilon_{\nu\mu\sigma\tau} +
X_\sigma \varepsilon_{\nu\rho\mu\tau} +  
X_\tau \varepsilon_{\nu\rho\sigma\mu}  \\
g_{\lambda\mu} \varepsilon_{\nu\rho\sigma\tau} &=&
g_{\lambda\nu} \varepsilon_{\mu\rho\sigma\tau} +
g_{\lambda\rho} \varepsilon_{\nu\mu\sigma\tau} +
g_{\lambda\sigma} \varepsilon_{\nu\rho\mu\tau} +  
g_{\lambda\tau} \varepsilon_{\nu\rho\sigma\mu}
\end{eqnarray}
%-----------------------------------------------------------------------
is  used to eliminate other possible structures. Particularly, the
spin vector $S_\mu$ may always be contracted with the Levi--Civita
symbol, along with it it has to occur due to parity conservation.
Because $S.p_1 = 0$ two other structures are eliminated using
%-----------------------------------------------------------------------
\begin{eqnarray}
q.p_1 \tilde{\varepsilon}_{\mu \nu S p_1}
&=& p_1.p_1 \varepsilon_{\nu \mu q S}
- \left[\hat{p}_{1\mu}\varepsilon_{\nu p_1 q S} - \hat{p}_{1\nu}
\varepsilon_{\mu p_1 q S}\right]\\
q.p_1 \tilde{\varepsilon}_{\mu \nu S p_2}
&=& p_1.p_2 \varepsilon_{\nu \mu q S}
- \left[\hat{p}_{1\mu}\varepsilon_{\nu p_2 q S} - \hat{p}_{1\nu}
\varepsilon_{\mu p_2 q S}\right]~.
\end{eqnarray}
%-----------------------------------------------------------------------
The structure functions $W_i$ are real functions and are given by
%-----------------------------------------------------------------------
\begin{equation}
\label{eqD3}
W_i = W_i(x,Q^2,x_{\PP},t)~.
\end{equation}
%-----------------------------------------------------------------------

Let us consider the limit in which target masses can be neglected
and $t$ is very small. In this case the proton momenta become
proportional: $p_2 = z p_1$ with,
%-----------------------------------------------------------------------
\begin{equation}
\label{eqD4}
z = 1 - x_{\PP} =  \frac{1 + \eta}{1 - \eta}~.
\end{equation}
%-----------------------------------------------------------------------
Correspondingly the hadronic tensor simplifies to
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqD5}
W_{\mu\nu} &=& i \varepsilon_{\mu \nu q S} \frac{W_8}{M^2}
+ i \left[\hat{p}_{1\mu} \varepsilon_{\nu S p_1 q}
-       \hat{p}_{1\nu} \varepsilon_{\mu S p_1 q}\right] \frac{W_9}{M^4}~,
\end{eqnarray}
%-----------------------------------------------------------------------
and contains only two structure functions,
where
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqD6}
W_9 = W_2 + (1-x_{\PP}) \left[W_3 + W_4 \right]
+ (1-x_{\PP})^2 W_5~.
\end{eqnarray}
%-----------------------------------------------------------------------
One may wish to re-write Eq.~(\ref{eqD5}) further into the form which is
similar to that used in polarized deep--inelastic scattering.
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqD5A}
W_{\mu\nu} &=& i \varepsilon_{\mu \nu \lambda \sigma} \frac{q^\lambda
S^\sigma}{p_1.q}
g_1(x,Q^2,x_{\PP})
+ i  \varepsilon_{\mu \nu \lambda \sigma}
\frac{q^{\lambda}(p_1.q S^\sigma - S.q p_1^\sigma)}{(p_1.q)^2}
g_2(x,Q^2,x_{\PP})~.
\end{eqnarray}
%-----------------------------------------------------------------------
This again is achieved by using the Schouten relation~Eq.~(\ref{eqP6})
noting that 
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSCH1}
\hat{p}_{1\mu} \varepsilon_{\nu S p_1 q} - \hat{p}_{1\nu}
\varepsilon_{\mu S p_1 q} = - \frac{(S.q)(q.p_1)}{q^2}
\varepsilon_{\mu \nu q p_1} + \frac{(q.p_1)^2}{q^2} 
\varepsilon_{\mu \nu q S}~.
\end{eqnarray}
%-----------------------------------------------------------------------
The relation between the structure functions
$W_{8,9}$ and $g_{1,2}$ is~:
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqg1}
g_1 &=& \frac{q.p_1}{M^2} W_8 \\
\label{eqg2}
g_2 &=& \frac{(q.p_1)^3}{q^2 M^4} W_9
\end{eqnarray}
%-----------------------------------------------------------------------

Due to the dependence of the structure functions on $x_{\PP}$ or 
$\eta$,~Eq.~(\ref{eqV1}), the process is {\it non--forward} w.r.t. the
protons, although the algebraic structure of the hadronic tensor
is the same as in the forward case. Finally the generalized 
Bjorken--limit is carried out, 
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqBL}
2 p_1.q = 2M \nu \rightarrow \infty,~~~~~p_2.q \rightarrow \infty,~~~~~
Q^2 \rightarrow \infty~~~{\rm with}~~x~~{\rm and}~~x_{\PP} =~{\rm fixed}~,
\end{eqnarray}
%-----------------------------------------------------------------------
which leads to (\ref{eqD5A}) using (\ref{eqg1},\ref{eqg2}).
For the scattering cross sections we consider the cases of longitudinal 
and transverse target polarization for which the initial state hadron
spin vectors are given by
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSPI}
S_{\parallel} &=& (0,0,0,M)\\
S_{\perp} &=& M(0,\cos\gamma,\sin\gamma,0)~,
\end{eqnarray}
%-----------------------------------------------------------------------
and $\gamma$ the spin direction in the plane orthogonal to the 3--momentum
$\vec{p}_1$.
In the limit $p_2 = zp_1$ and $M^2,t = 0$ the $\Phi-$integral becomes 
trivial in the case of longitudinal nucleon polarization, while it is
kept as differential variable for transverse polarization.
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqD7}
\frac{d^3 \sigma_{\rm diffr}(\lambda,\pm S_{\parallel})}
{dx dQ^2 d x_{\PP}} &=&  \mp 4 \pi s \lambda
\frac{\alpha^2}{Q^4} \Biggl[
y \left(2 - y - \frac{2xyM^2}{s}\right) x g_1(x,Q^2,x_\PP)
\nonumber\\ & &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- 4 x y \frac{M^2}{s} g_2(x,Q^2,x_\PP) \Biggr]   \\
\frac{d^4 \sigma_{\rm diffr}(\lambda,\pm S_{\perp})}
{dx dQ^2 d \Phi d x_{\PP}} &=& \mp 4  s \lambda \sqrt{\frac{M^2}{s}}
\frac{\alpha^2}{Q^4}
\sqrt{xy\left[1-y-\frac{xyM^2}{s}\right]} \cos(\gamma - \Phi)
\nonumber\\ & &~~~~~~~~~~~~~~~~~~~~\times
\left[yx g_1(x,Q^2,x_\PP) + 2x g_2(x,Q^2,x_\PP)\right]~,
\end{eqnarray}
%-----------------------------------------------------------------------
where $y = q.p_1/l.p_1$ and $\lambda$ denotes the degree of longitudinal
lepton polarization.\footnote{In the case of longitudinal nucleon
polarization polarized diffractive scattering was discussed neglecting
the contribution due to the structure function $g_2$ in \cite{BARY}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Compton Amplitude}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\setcounter{equation}{0}
\label{sec-3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1mm}
\noindent
We first consider the operator given by the renormalized and 
time--ordered product of two electromagnetic currents
%-----------------------------------------------------------------------
\begin{eqnarray}
\widehat{T}_{\mu\nu}(x) &=&
RT \left[J_\mu\left(\frac{x}{2}\right)J_\nu\left(-\frac{x}{2}\right) S 
\right] \nonumber\\
&=&
 -e^2 \frac{\tilde x^\lambda}{2 \pi^2 (x^2-i\epsilon)^2}
 RT
 \left[
\overline{\psi}
\left(\frac{\tilde x}{2}\right)
\gamma^\mu \gamma^\lambda \gamma^\nu \psi
\left(-\frac{\tilde x}{2}\right)
- \overline{\psi}
\left(-\frac{\tilde x}{2}\right)
\gamma^\mu \gamma^\lambda \gamma^\nu \psi
\left(\frac{\tilde x}{2}\right)
\right] S
\end{eqnarray}
%-----------------------------------------------------------------------
Here, $\tilde x$ denotes a light--like vector corresponding to $x$,
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{xtil}
\tilde x = x + \frac{\zeta}{\zeta^2}\left[ \sqrt{x.\zeta^2 - x^2 \zeta^2}
- x.\zeta\right]~,
\end{eqnarray}
%-----------------------------------------------------------------------
and $\zeta$ is a subsidiary vector. Following Refs.~\cite{BGR,BR} the
operator $\widehat{T}_{\mu\nu}$ can be expressed in terms of a vector
and an axial--vector operator decomposing
%-----------------------------------------------------------------------
\begin{eqnarray}
 \gamma_\mu \gamma_\lambda \gamma_\nu = \left[g_{\mu\lambda} g_{\nu\rho}
 + g_{\nu\lambda} g_{\mu\rho} - g_{\mu\nu} g_{\lambda\rho} \right]
 \gamma^\rho - i \varepsilon_{\mu\nu\lambda\rho} \gamma^5 \gamma^\rho~.
\end{eqnarray}
%-----------------------------------------------------------------------
    We will consider only the contribution of the latter one, since this
yields the polarized part,
%-----------------------------------------------------------------------
\begin{eqnarray}
 \widehat{T}_{\mu\nu}^{\rm pol}(x)  =
  i e^2 \frac{\tilde x^\lambda}{2 \pi^2 (x^2-i\epsilon)^2}
  \varepsilon_{\mu\nu\lambda\sigma}
O_5^\sigma  \left(\frac{\tilde x}{2}, -\frac{\tilde x}{2}\right)~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $\varepsilon_{\mu\nu\lambda\sigma}$ the Levi--Civita symbol.
The  bilocal axial--vector light--ray operator is
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{oo5}
O^{\alpha}_5\left(\frac{\tilde x}{2},-\frac{\tilde x}{2}\right)
&=&
\frac{i}{2} RT
\left[\overline{\psi}\left(\frac{\tilde x}{2}\right)
\gamma_5\gamma^\alpha\psi\left(-\frac{\tilde x}{2}\right)
+ \overline{\psi}\left(-\frac{\tilde x}{2}\right)
\gamma_5\gamma^\alpha\psi\left(\frac{\tilde x}{2}\right)\right]S~.
\end{eqnarray}
%----------------------------------------------------------------------
The polarized part of the Compton operator 
$\widehat{T}^{\rm pol}_{\mu\nu}$ is related to the diffractive scattering
cross section using Mueller's generalized optical theorem~\cite{AHM} 
(Figure~2), which moves the final state proton into an initial state 
anti-proton.

\vspace*{7mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{picture}(200,100)(0,0)
\setlength{\unitlength}{0.2mm}
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\Line(-107,102)(-107,-2)
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\ArrowLine(-100,0)(-70,30)
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\ArrowLine(-30,70)(0,100)
\SetWidth{1.5}
\CCirc(-50,50){26}{Black}{Yellow}
\setlength{\unitlength}{1pt}
\Text(15,107)[]{\large $2$}
\Text(50,50)[]{\large $=~~{\rm Disc}$}
\Text(-105,115)[]{$q$}
\Text(-105,-15)[]{$p_1$}
\Text(5,-15)[]{$p_2$}
\Text(75,115)[]{$q$}
\Text(75,-15)[]{$p_1$}
\Text(95,-15)[]{$p_2$}
\Text(60,35)[]{$X$}
\Text(175,20)[]{$X$}
\Text(275,115)[]{$q$}
\Text(275,-15)[]{$p_1$}
\Text(255,-15)[]{$p_2$}
\setlength{\unitlength}{0.2mm}
\SetWidth{1.5}
\Photon(80,100)(110,70){5}{5}
\ArrowLine(80,0)(110,30)
\ArrowLine(130,30)(100,0)
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\CCirc(220,50){26}{Black}{Yellow}
\Photon(240,70)(270,100){5}{5}
\end{picture}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{10mm}\noindent
\begin{center}
{\sf Figure~2:~A. Mueller's optical theorem.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The polarized part of the Compton amplitude is obtained as the 
expectation value
%-----------------------------------------------------------------------
\begin{eqnarray}
T^{\rm pol}_{\mu\nu}(x) 
= \langle p_1,S_1,-p_2,S_2|\widehat{T}_{\mu\nu}|p_1,S_1,-p_2,S_2\rangle~,
\end{eqnarray}
%-----------------------------------------------------------------------
which is   forward w.r.t. to the direction defined by the state
$\langle p_1,-p_2|$.
The twist--2 contributions to the expectation values of the operator
(\ref{oo5})  is obtained
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSCA}
\lefteqn{\hspace*{-13.5cm}
\langle p_1,S_1,-p_2,S_2|O^{A,\mu}_{5}(\kappa_+ \xx,\kappa_-\xx)|
p_1,S_1,-p_2,S_2\rangle = } \nonumber\\ \left.  \hspace*{3cm}
\int_0^1 d\lambda
\partial^{\mu}_x
\langle p_1,S_1,-p_2,S_2|O_{5}^A(\lambda \kappa_+ x,\lambda \kappa_-x)
|p_1,S_1,-p_2,S_2\rangle \right|_{x=\xx}
\end{eqnarray}
%-----------------------------------------------------------------------
as partial derivative  of the expectation values of
%-----------------------------------------------------------------------
\begin{eqnarray}
O^A_5(\kappa_+x,\kappa_-x)
= x^\alpha O^A_{5,\alpha}(\kappa_+x,\kappa_-x)~,
\end{eqnarray}
%-----------------------------------------------------------------------
the corresponding            pseudo-scalar operator.  The index $A = q,G$
labels the quark-- or gluon operators, cf.~\cite{BGR}. From now on we
keep only the spin vector of the initial--state proton and sum over
that of the final--state proton.

The pseudo-scalar twist--2 quark operator matrix element has the 
following representation\footnote{For parameterizations
  of similar hadronic
matrix elements see e.g.~\cite{PARA}.} due to the overall symmetry in $x$
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSCAM}
\langle p_1,S_1,-p_2|O^q(\kappa_+ x,\kappa_- x)|p_1,S_1,-p_2
\rangle =
xS
\!\!\!         \int \!\!\!
Dz ~\e^{-i \kappa_- x p_z} {f}^q_5(z_+,z_-)
\end{eqnarray}
%-----------------------------------------------------------------------
with $S \equiv S_1$, $\kappa_- =1/2$ and
where all the trace--terms were subtracted, see
\cite{BGR,TRAC}.
${f}^A_5(z_+,z_-)$ denotes                              the scalar
two--variable distribution amplitudes and the measure $Dz$ is
%------------------------------------------------------------------------
\begin{eqnarray}
\label{Dz}
Dz = d z_+ d z_-
\theta(1 + z_+ + z_-) \theta(1 + z_+ - z_-)
\theta(1 - z_+ + z_-) \theta(1 - z_+ - z_-)~.
\end{eqnarray}
%------------------------------------------------------------------------
Here, we decomposed the vector $p_z$ as
%-----------------------------------------------------------------------
\begin{eqnarray}
p_z = p_- z_- + p_+ z_+ =  p_- \vartheta + \pi_- z_+~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $z_{1,2}$ momentum fractions along $p_{1,2}$ and $p_{\pm} = p_2 \pm
p_1$, $z_\pm = (z_2 \pm z_1)/2$ and
%-----------------------------------------------------------------------
\begin{eqnarray}
\vartheta = z_- + \frac{1}{\eta} z_+,~~~~~~~~~~~~~\pi_-
= p_+ - \frac{1}{\eta} p_-~,
\end{eqnarray}
%-----------------------------------------------------------------------
with       $q.\pi_- \equiv 0$. In the limit         $M^2, t \sim 0$, in
which we work from now on, the vector $\pi_-$ even vanishes.

The Fourier--transform of the Compton amplitude is given by~\cite{BR}
%-----------------------------------------------------------------------
\begin{eqnarray}
T_{\mu\nu}^{\rm pol}(p_1,p_2,S,q)
&=&   \int d^4x \e^{iqx} T_{\mu\nu}(x) \nonumber\\
&=& 4i \varepsilon_{\mu\nu\lambda\sigma} \int Dz \left[
\frac{Q_z^{\lambda} S^{\sigma}}{Q^2_z + i\varepsilon} - \frac{1}{2}
\frac{p_z^{\sigma} S^{\lambda}}{Q^2_z + i\varepsilon}%
+ \frac{Q_z.S}{(Q^2_z+i\varepsilon)^2}
p_z^{\sigma} Q_z^{\lambda}\right] F_5(z_+,z_-),
\end{eqnarray}
%-----------------------------------------------------------------------
with $Q_z = q - p_z/2$ and
%-----------------------------------------------------------------------
\begin{eqnarray}
\overline{u}(p_1) \gamma_5 \gamma_\lambda u(p_1) = 2 S_\lambda~.
\end{eqnarray}
%-----------------------------------------------------------------------
The function $F_5(z_+,z_-)$ is related to the
polarized distribution function $f_5(z_+,z_-)$ by
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{vecval}
F_5(z_+,z_-) = \int_0^1 \frac{d\lambda}{\lambda^2}
f_5\left(\frac{z_+}{\lambda},\frac{z_-}{\lambda}\right)
\theta(\lambda-|z_+|)\theta(\lambda-|z_-|)~.
\end{eqnarray}
%-----------------------------------------------------------------------
We re--write the denominators by
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqDENO}
\frac{1}{Q^2_z+i\varepsilon} = - \frac{1}{qp_-}\frac{1}{(\vartheta
- 2\beta  + i \varepsilon)}~,
\end{eqnarray}
%-----------------------------------------------------------------------
defining
%-----------------------------------------------------------------------
\begin{eqnarray}
\beta =  \frac{x}{x_{\PP}} = \frac{q^2}{2q.p_-}~.
\end{eqnarray}
%-----------------------------------------------------------------------
The conservation of the electromagnetic current is easily seen
%-----------------------------------------------------------------------
\begin{eqnarray}
q^{\mu}T_{\mu\nu}(p_1,p_2,S,q)  =
q^\nu T_{\mu\nu}(p_1,p_2,S,q)~.
\end{eqnarray}
%-----------------------------------------------------------------------
It follows because the contraction with $q^{\alpha}$ leads to 
Levi--Civita symbols being contract with the same 4--vector.
By
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqFF}
\widehat{F}(\vartheta,\eta) = \int Dz F(z_+,z_-) 
\delta(\vartheta - z_- -z_+/\eta) = \int_\vartheta^{-{\rm sign}(\vartheta)
/\eta} \frac{dz}{z} \widehat{f}(z,\eta)
\end{eqnarray}
%-----------------------------------------------------------------------
we change to the variable $\vartheta$, the main momentum fraction in
the subsequent representation. Eq.~(\ref{eqFF}) is the pre--form of
the Wandzura--Wilczek integral~\cite{WW}. It emerges seeking the
representation of vector--valued distributions~(\ref{vecval}) in terms
of scalar distributions, cf.~\cite{BGR,BR}. In most of the applications
these integrals remain. An exception is the Callan--Gross relation,
see Refs.~\cite{BR,BRDIFF}, where all these integrals cancel and only
scalar distribution functions remain.
Here
the distribution function $\widehat{f}_5(z,\eta)$ is related to
$f_5(z_+,z_-)$ by
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqFF1}
\widehat{f}_5(z,\eta) = \int^{\eta(1-z)}_{\eta(1+z)} d\rho~
\theta(1-\rho) \theta(\rho+1) f_5(\rho,z-\rho/\eta)~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $\rho = z_+/\eta$.

The Compton amplitude takes the following form:
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqCOMP1}
T_{\mu\nu}^{\rm pol}(p_-,S,q) &=&
4i  \varepsilon_{\mu\nu\lambda\sigma} \int_{+1/\eta}^{-1/\eta}
d\vartheta \Biggl\{\frac{q^\lambda S^\sigma}{Q_z^2+i\varepsilon}
- q.S
\frac{\vartheta q^\lambda p_-^\sigma}
{(Q_z^2+i\varepsilon)^2}\Biggr\} \nonumber\\
& &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\times \int_\vartheta^{-{\rm sign}(\vartheta)/\eta}\frac{dz}{z}
\widehat{f}_5(z,\eta)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The $\vartheta$--integral in Eq.~(\ref{eqCOMP1}) can be simplified
using the identities
%-----------------------------------------------------------------------
\begin{eqnarray}
\int_{+1/\eta}^{-1/\eta} d\vartheta \frac{\vartheta^k}
{(\vartheta-2\beta+i\varepsilon)^2}\int_\vartheta^{{\rm sign}(\vartheta)
/\eta} \frac{dz}{z} \widehat{f}_5(z,\eta)
&=&
\int_{+1/\eta}^{-1/\eta} d\vartheta \frac{k \vartheta^{k-1}}
{(\vartheta-2\beta+i\varepsilon)}\int_\vartheta^{{\rm sign}(\vartheta)
/\eta} \frac{dz}{z} \widehat{f}_5(z,\eta) \nonumber\\    & &
~~~~~~~-\int_{+1/\eta}^{-1/\eta} d\vartheta \frac{\vartheta^{k-1}
\widehat{f}_5(\vartheta,\eta)}
{(\vartheta-2\beta+i\varepsilon)}~.
\end{eqnarray}
%-----------------------------------------------------------------------
As we work in the approximation of $M^2, t << |q^2|$ 
the vector $p_-$ obeys the representation
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{vect}
p_- &=& - x_\PP p_1~.
\end{eqnarray}
%-----------------------------------------------------------------------
Using these variables the Compton amplitude reads
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqCOMP2}
T_{\mu\nu}^{\rm pol}(p_1,S,q) &=&
-4i  \varepsilon_{\mu\nu\lambda\sigma} \int_{+1/\eta}^{-1/\eta}
\frac{d\vartheta}{\vartheta - 2 \beta + i \varepsilon} \Biggl\{
\left[\frac{q^\lambda S^\sigma}{q.p_1}
 + \frac{q.S}{(q.p_1)^2} q^\lambda p_1^\sigma
\right]
\int_{\vartheta}^{-{\rm sign}(\vartheta)/\eta} \frac{dz}{z}
\Hat{\Hat{f}}_5(z,\eta) \nonumber\\
& & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- \frac{q.S}{(q.p_1)^2}
q^\lambda p_1^\sigma
\Hat{\Hat{f}}_5(\vartheta,\eta) \Biggr\}~.
\end{eqnarray}
%-----------------------------------------------------------------------
Here,
%-----------------------------------------------------------------------
\begin{eqnarray}
\Hat{\Hat{f}}_5(z,\eta) =  \frac{1}{x_\PP} \widehat{f}_5(z,\eta)~.
\end{eqnarray}
%-----------------------------------------------------------------------
Taking the absorptive part one obtains\footnote{The `imaginary part'
concerns that of the Schwartz-distribution Eq.~(\ref{eqDENO}).
Because of the relations, Eqs.~(\ref{eqP2C},\ref{eqP2B}) an overall $i$
emerges in the hadronic tensor.}
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqIMPAR}
W_{\mu\nu}^{\rm pol} &=& 
\frac{1}{2\pi}~{\sf Im}~T_{\mu\nu}^{\rm pol}(p_1,p_2,q) \nonumber\\
           &=&       
i \varepsilon_{\mu\nu\lambda\sigma} \frac{q^\lambda S_1^\sigma}{q.p_1}
{\sf G}_1(\beta,\eta,Q^2)
+ i  \varepsilon_{\mu\nu\lambda\sigma} \frac{q^\lambda (p_1.q S^\sigma
- S.q p_1^\sigma)}{(p_1.q)^2}
{\sf G}_2(\beta,\eta,Q^2)~,
\end{eqnarray}
%-----------------------------------------------------------------------
where
%-----------------------------------------------------------------------
\begin{alignat}{4}
\label{eqG1}
{\sf G}_1(\beta,\eta,Q^2) &=& \sum_{q=1}^{N_f}
e_q^2\left[\Delta f_q^D(\beta,Q^2,x_{\PP})+ \Delta
\overline{f}_q^D(\beta,Q^2,x_{\PP})
\right]
&\equiv&  g_1^{D(3)}(x,Q^2,x_{\PP})~, \\
\label{eqG2}
{\sf G}_2(\beta,\eta,Q^2) &=& 
-{\sf G}_1(\beta,\eta,Q^2) + \int_\beta^1 \frac{d\beta'}{\beta'}
 {\sf G}_1(\beta',\eta,Q^2)
&\equiv&  g_2^{D(3)}(x,Q^2,x_{\PP})~,
\end{alignat}
%-----------------------------------------------------------------------
with $N_f$ the number of flavors, choosing the factorization scale
$\mu^2 = Q^2$. As we were working in the twist--2 approximation, the
Wandzura--Wilczek relation (\ref{eqG2}) describes
${\sf G}_2(\beta,\eta,Q^2)$.

To derive the representation for the diffractive parton densities
$\Delta f_q^D$, Eq.~(\ref{eqG1}), we consider the symmetry relation
for the polarized distribution functions $F^A(z_1,z_2)$, Ref.~\cite{BR},
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSYM1}
F^A_5(z_1,z_2) =  F^A_5(-z_1,-z_2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
It translates into
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSYM2}
\widehat{F}^A_5(\vartheta,\eta) =  \widehat{F}^A_5(-\vartheta,\eta)~,
\end{eqnarray}
%-----------------------------------------------------------------------
and, cf. Eq.~(\ref{eqFF}),
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSYM3}
\Hat{\Hat{f}}^A_5(\vartheta,\eta) =  \Hat{\Hat{f}}^A_5(-\vartheta,\eta)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The polarized diffractive quark and anti--quark densities are given by
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqPART}
\sum_{q=1}^{N_f} e_q^2 \Delta
f_q^D(\beta,Q^2,x_{\PP}) &=&
\Hat{\Hat{f}}_5(2\beta,\eta,Q^2) \nonumber\\
\sum_{q=1}^{N_f} e_q^2 \Delta
\overline{f}_q^D(\beta,Q^2,x_{\PP}) &=&
  \Hat{\Hat{f}}_5(-2\beta,\eta,Q^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
Unlike in the deep--inelastic case, where the scaling variable
$x~\epsilon~[0,1]$, the support of the distributions
$\Delta f_q^D(\beta, Q^2, x_\PP)$ is $x~\epsilon~[0, x_\PP]$.

We express the diffractive parton densities in terms of
the distribution function $f_5(z_+,z_-)$ directly
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqREL}
\Delta
\Hat{\Hat{f}}_5(\pm 2\beta,\eta,Q^2)
 =  \frac{1}{x_\PP}
\int_{-\frac{x_{\PP} \pm 2x}{2-x_{\PP}}}
                              ^{-\frac{x_{\PP} \mp 2x}{2-x_{\PP}}}
d \rho f_5(\rho,\pm 2\beta + \rho(2-x_{\PP})/x_{\PP};Q^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The latter relations are needed to compare experimental quantities with
those which might be obtained measuring the corresponding operators
on the lattice.

Finally, we would like to make a remark on the evolution of the
diffractive parton densities being derived above. In a previous paper
\cite{BRDIFF} the corresponding evolution equations for unpolarized 
diffractive scattering have been derived in detail. Also here one
may start with the general formalism for non-forward scattering,
see e.g.~\cite{BGR}, and discuss the evolution of the scalar operators.
The evolution equations are independent of the parameter $\kappa_+$
emerging in the anomalous dimensions
$\gamma^{AB}_5(\kappa_+,\kappa_-,\kappa_+',\kappa_-';\mu^2)$ which 
therefore may be set to zero. Moreover, the all-order rescaling relation
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqresc}
\gamma^{AB}(\kappa_+,\kappa_-,\kappa_+',\kappa_-';\mu^2) = \sigma^{d_{AB}}
\gamma^{AB}(\sigma\kappa_+, \sigma\kappa_-,\sigma\kappa_+',
\sigma\kappa_-')~,
\end{eqnarray}
%-----------------------------------------------------------------------
holds, with $d_{AB} = 2 + d_A - d_B$, $d_q=1, d_G=2$. A straightforward
calculation leads to the evolution equation for the polarized (singlet)
diffractive parton densities $f^A_5(\vartheta,\eta;\mu^2)$ in the
momentum fraction $\vartheta$
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqEV2}
\mu^2 \frac{d}{d\mu^2} f^A(\vartheta,\eta;\mu^2)
=  \int_\vartheta^{-{\rm sign}(\vartheta)/\eta}  \frac{d\vartheta'}
{\vartheta'} P^{AB}_5\left(\frac{\vartheta}{\vartheta'},\mu^2\right)
f_B(\vartheta',\eta;\mu^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The splitting functions $P_5^{AB}$ are the {\it forward} splitting 
functions~\cite{SPLIT}\footnote{For the non--forward anomalous dimensions
see \cite{SPNF}.}, which are independent of $\eta$ resp. 
$x_{\PP}$. Taking the 
absorptive part the usual evolution equations are obtained, with the
difference that the evolution takes place in the variable $\beta$.
The non-forwardness $\eta$ or $x_\PP$ behave as plain parameters.
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqEV4}
\mu^2 \frac{d}{d\mu^2} f^{D}_A(\beta,x_{\PP};\mu^2)
= \int_\beta^1 \frac{d\beta'}{\beta'} P_{5,A}^B
\left(\frac{\beta}{\beta'}; \mu^2\right)
f_B^D(\beta',x_{\PP};\mu^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------

We expressed the Compton amplitude with the help of the light--cone 
expansion at short distances and applied this representation to the
process of deep--inelastic diffractive scattering using Mueller's
generalized optical theorem. This representation is {\it not} limited to
leading twist operators but can be extended to all higher twist
operators. The corresponding evolution equations for
the higher twist hadronic matrix elements depend on more than one
momentum fraction $\vartheta_i$, which have a less trivial connection to
the outer kinematical variables similar to the case of deep--inelastic
scattering~\cite{BRRZ}. The construction is similar to the
above  and applies as well the generalized optical theorem. The evolution
of the associated parton correlation functions is for the same reason
forward.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\setcounter{equation}{0}
\label{sec-5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1mm}
\noindent
The differential cross section of polarized deep--inelastic 
$ep$--diffractive scattering for pure photon exchange is described by 
eight structure functions. They depend on the four kinematic variables, 
$x, Q^2, x_{\PP}$ and $t$. In the limit of small values of $t$ and%       target
neglecting target
masses two structure functions contribute. In the generalized Bjorken
range and the presence of a sufficiently large rapidity gap the scaling 
violations of hard diffractive scattering can be described within 
perturbative QCD. In this range processes, which are dominated by
light--cone contributions, are described. The scattering amplitude can be
rewritten using
Mueller's generalized optical theorem moving the outgoing diffractive
proton into an incoming anti-proton. In this kinematical domain
diffractive scattering {\it is} deep--inelastic scattering off a state
$\langle p_1,S_1,-p_2|$. Non-forward techniques may be used to describe
this process. In this way the two--variable polarized amplitudes
turn into the polarized diffractive parton densities, which depend
on one momentum fraction and a parameter $\eta$, which describes
the non--forwardness, and is directly related to the variable $x_\PP$.
For the absorptive part the scaling variable can be expressed by
the variable $\beta$, which also is the variable on which the
evolution kernels act in the twist--2 contributions, whereas $x_\PP$ 
remains as a simple parameter of the process. In the limit 
$t, M^2 \rightarrow 0$ the twist--2 contributions to the two structure 
functions $g_{1,2}^{D(3)}(x,Q^2,x_\PP)$ are related by a 
Wandzura--Wilczek relation in the variable $\beta = x/x_\PP$.
The approach followed in the present paper for twist--2 operators
can be synonymously extended to higher twist--operators in the kinematic
domain of the general Bjorken limit.

\vspace{2mm}
\noindent
{\bf Acknowledgement.}~For discussions we would like to thank J.~Eilers 
B.~Geyer, and X.~Ji. We thank J.~Dainton, M.~Erdmann, and D.~Wegener  for their
interest in the present work.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{999}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%[1]
\bibitem{EXP1}
M. Derrick et al., ZEUS collaboration, Phys. Lett. {\bf B315} (1993) 481;
\\
T. Ahmed et. al., H1 collaboration, Nucl. Phys. {\bf B429} (1994) 477;\\
J. Breitweg et al., ZEUS collaboration, Eur. Phys. J. {\bf C6} (1999) 43;
\\
C. Adloff et al., H1 collaboration, Z. Phys. {\bf C76} (1997) 613.
%------------------------------------------------------------------------
%
%[2]
\bibitem{SXPOL}
R. Holt et al., EICC, {\sf A Whitepaper on the Electron Ion Collider},
March 2001;\\
see e.g. {\sf Workshop on the Prospects of Spin Physics at HERA},
DESY 95--200, eds.~J. Bl\"umlein and W.-D. Nowak;\\
{\sf Deep Inelastic Scattering off Polarized Targets: Theory Meets
Experiment}, DESY 97--200, eds.~J. Bl\"umlein et al.
%------------------------------------------------------------------------
%
%[3]
\bibitem{BRDIFF}
J. Bl\"umlein and D. Robaschik, Phys. Lett. {\bf B517} (2001) 222.
%------------------------------------------------------------------------
%
%[4]
\bibitem{FACT}
A. Berera and D.E. Soper, Phys. Rev. {\bf D50} (1994) 4328; {\bf D53}
(1996) 6162;\\
J. Collins, Phys. Rev. {\bf D57} (1998) 3051; Erratum: {\bf D61} (2000)
019902;\\
F. Hautmann, Z. Kunszt, and D.E. Soper, Nucl. Phys. {\bf B563} (1999) 
153.
%------------------------------------------------------------------------
%
%[5]
\bibitem{BK}       
J. Bl\"umlein and N. Kochelev, Phys. Lett. {\bf B381} (1996) 296;
Nucl. Phys. {\bf B498} (1997) 285;\\
J. Bl\"umlein and A. Tkabladze, Nucl. Phys. {\bf B553} (1999) 427;
{\tt hep-ph/9905524}.
%------------------------------------------------------------------------
%
%[6]
\bibitem{PRAD}
D. Bardin, J. Bl\"umlein, P. Christova, and L. Kalinovskaya,
Nucl. Phys. {\bf B506} (1997) 295;\\
A. Arbuzov, D. Bardin, J. Bl\"umlein,  L. Kalinovskaya, and T. Riemann,
{\sf Code HECTOR}, Comp. Phys. Commun. {\bf 94} (1996) 128,
{\sf hep-ph/9511434}.
%------------------------------------------------------------------------
%[7]
\bibitem{WW}
S. Wandzura and F. Wilczek, Phys. Lett. {\bf B72} (1977) 95.
%------------------------------------------------------------------------
%
%[8]
\bibitem{COL}
J.C. Collins, {\tt hep-ph/9705393}.
%------------------------------------------------------------------------
%
%[9]
\bibitem{POM}
For recent surveys see:\\
M. W\"usthoff and A.D. Martin, J. Phys. {\bf G24} (1999) R309;\\
W. Buchm\"uller, {\tt hep-ph/9906546};\\
G. Ingelman, {\tt hep-ph/9912534};\\
J. Bartels and M. Kowalski, Eur. Phys. J. {\bf C19} (2001) 693.
%------------------------------------------------------------------------
%
%[10]
\bibitem{THOOFT}
G. t'Hooft, Phys. Rev. Lett. {\bf 37} (1976) 8.
%------------------------------------------------------------------------
%
%[11]
\bibitem{THETA}
see e.g. S. Aoki and T. Hatsuda, Phys. Rev. {\bf D45} (1992) 2427.
%------------------------------------------------------------------------
%
%[12]
\bibitem{TM}
R.D. Tangerman and P.J. Mulders, Phys. Rev. {\bf D51} (1995) 3357.
%------------------------------------------------------------------------
%
%[13]
\bibitem{DJT}
D.A. Dicus, R. Jackiw, and V.L. Teplitz, Phys. Rev. {\bf D4} (1971) 1733.
%------------------------------------------------------------------------
%
%[14]
\bibitem{SCHOUT}
M. Veltman, Nucl. Phys. {\bf B319} (1989) 253;\\
B. De Wit and J. Smith, {\sf Field Theory in Particle Physics},
(North Holland, Amsterdam, 1986), p.~414.
%------------------------------------------------------------------------
%
%[15]
\bibitem{BARY}
J. Bartels, T. Gehrmann, and M. Ryskin, Eur. Phys. J. {\bf C11} (1999)
325.
%------------------------------------------------------------------------
%
%[16]
\bibitem{BGR}
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, Nucl. Phys. {\bf B560} (1999) 
283.
%------------------------------------------------------------------------
%
%[17]
\bibitem{BR}
J. Bl\"umlein and D. Robaschik, Nucl. Phys. {\bf B581} (2000) 449.
%------------------------------------------------------------------------
%
%[18]
\bibitem{SPLIT}
H. Ito, Progr. Theor. Phys. {\bf 54} (1975) 555;\\
K. Sasaki, Progr. Thepr. Phys. {\bf 54} (1975) 1816;\\
M. A. Ahmed and G.G. Ross, Nucl. Phys. {\bf B111} (1976) 441;\\
G. Altarelli and G. Parisi, Nucl. Phys. {\bf B126} (1977) 298;\\
R. Mertig  and W.L. van Neerven, Z. Phys. {\bf C70} (1996) 637;\\
W. Vogelsang, Nucl. Phys. {\bf B475} (1996) 47.
%------------------------------------------------------------------------
%
%[19]
\bibitem{SPNF}
J. Bl\"umlein, B. Geyer and D. Robaschik, Phys. Lett. {\bf B406} (1997) 
161;\\
I. Balitsky and A. Radyushkin, Phys. Lett. {\bf B413} (1997) 114;\\
For the near-forward representation: X. Ji, Phys. Rev. {\bf D55}
(1997) 7114.
%------------------------------------------------------------------------
%
%[20]
\bibitem{AHM}
A.H. Mueller, Phys. Rev. {\bf D2} (1970) 2963; Phys. Rev. {\bf D4} (1971)
150;\\
P.D.P. Collins, {\sf An Introduction to Regge Theory and High Energy
Physics}, (Cambridge University Press, Cambridge, 1977), pp.~331.
%------------------------------------------------------------------------
%
%[21]
\bibitem{PARA}
J. Bl\"umlein, J. Eilers, B. Geyer, and D. Robaschik, 
{\tt hep-ph/0108095}.
%------------------------------------------------------------------------
%
%[22]
\bibitem{TRAC}
B. Geyer and M. Lazar, Nucl. Phys. {\bf B559} (1999) 339;  {\bf B581}
(2000) 341;\\
B. Geyer, M. Lazar, and D. Robaschik, Nucl. Phys. {\bf B618} (2001) 99.
%------------------------------------------------------------------------
%
%[23]
\bibitem{BRRZ}
R.K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. {\bf B212}
(1983) 29;\\
J. Bl\"umlein, V. Ravindran, J. Ruan, and W. Zhu, Phys. Lett. {\bf B594}
(2001) 235.
%------------------------------------------------------------------------
\end{thebibliography}
\end{document}
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%-----------------------------------------------------------------------
\begin{eqnarray}
S_-^\mu = - \overline{u}(p_-) \gamma_5 \gamma^\mu u(p_-)~.
\end{eqnarray}
%-----------------------------------------------------------------------








%
%[4]
\bibitem{MD}
M. Diehl, Nucl. Phys. {\bf B} (Proc. Suppl. {\bf 79} (1999) 723, Proc.
DIS 99, eds. J. Bl\"umlein  and T. Riemann.
%------------------------------------------------------------------------
%
%[8]
\bibitem{SCHOUT}
M. Veltman, Nucl. Phys. {\bf B319} (1989) 253;\\
B. De Wit and J. Smith, {\sf Field Theory in Particle Physics},
(North Holland, Amsterdam, 1986), p.~414.
%------------------------------------------------------------------------
%
%[9]
\bibitem{DJT}
D.A. Dicus, R. Jackiw, and V.L. Teplitz, Phys. Rev. {\bf D4} (1971) 1733.
%------------------------------------------------------------------------
%
%[10]
\bibitem{RAPGA}
J.C. Collins, {\tt hep-ph/9705393};\\
H. Navelet and R. Peschanski, {\tt hep-ph/0105030}.
%------------------------------------------------------------------------
%
%[11]
\bibitem{BT}
J. Bl\"umlein and A. Tkabladze, Nucl. Phys. {\bf B553} (1999) 427.
%------------------------------------------------------------------------










%
%[9]
\bibitem{BGRY}
J. Bartels, T. Gehrmann, and M. Ryskin, Eur. Phys. J. {\bf C11} (1999)
325.
%------------------------------------------------------------------------
%
%[1]
\bibitem{THOOFT}
G. t'Hooft, Phys. Rev. Lett. {\bf 37} (1976) 8.
%------------------------------------------------------------------------
%
%[2]
\bibitem{THETA}

%------------------------------------------------------------------------
%
%[3]
\bibitem{DJT}
D.A. Dicus, R. Jackiw, and V.L. Teplitz, Phys. Rev. {\bf D4} (1971) 1733.
%------------------------------------------------------------------------
%
%[1]
\bibitem{EXP1}
M. Derrick et al., ZEUS collaboration, Phys. Lett. {\bf B315} (1993) 481;
\\
T. Ahmed et. al., H1 collaboration, Nucl. Phys. {\bf B429} (1994) 477;\\
J. Breitweg et al., ZEUS collabortaion, Eur. Phys. J. {\bf C6} (1999) 43;
\\
C. Adloff et al., H1 collaboration, Z. Phys. {\bf C76} (1997) 613.
%------------------------------------------------------------------------
%
%[2]
\bibitem{SXPOL}
small x polarized facilities.
%------------------------------------------------------------------------
%
%[3]
\bibitem{BR2}
J. Bl\"umlein and D. Robaschik, DESY 01-072, {\tt hep-ph/0106037}.
%------------------------------------------------------------------------












%




%[3]
\bibitem{FL}
C. Adloff et al., H1 collaboration, Phys. Lett. {\bf B393} (1997) 452.
%------------------------------------------------------------------------
%
%[4]
\bibitem{EXP3}
H. Abramowicz and J. Dainton, J. Phys. {\bf G22} (1996) 911.
%------------------------------------------------------------------------
%
%[5]
\bibitem{BAKO}
%------------------------------------------------------------------------
%
%[6]
\bibitem{DL}
A. Donnachie and P.V. Landshoff, Phys. Lett. {\bf B191} (1987) 309;
Nucl. Phys. {\bf B286} (1987) 704;  Phys. Lett. {\bf B437} (1998) 408;
{\tt hep-ph/0105088} and references  quoted therein.
%------------------------------------------------------------------------
%
%[7]
\bibitem{SUR}
A. Hebecker, Phys. Rep. {\bf 331} (2000) 1 and references therein.
%------------------------------------------------------------------------
%
%[8]
\bibitem{NP}
H. Navelet and R. Peschanski, {\tt hep-ph/0105030} and references 
therein.
%------------------------------------------------------------------------
%
%[10]
\bibitem{BH}
W. Buchm\"uller and A. Hebecker, Nucl. Phys. {\bf B476} (1996) 293;\\
W. Buchm\"uller, T. Gehrmann, and A. Hebecker, Nucl. Phys. {\bf B537} 
(1999) 477.
%------------------------------------------------------------------------
%
%[11]
\bibitem{EBKW}
J. Bartels, J. Ellis, M. Kowalski, and M. W\"usthoff,  
Eur. Phys. J. {\bf C7} (1999) 443.
%------------------------------------------------------------------------
%
%[12]
\bibitem{FS}
D.~M\"uller, D. Robaschik, B. Geyer, F.~Dittes, and J.~Ho\v{r}ej\v{s}i,
Fortschr. Phys.  {\bf 42} (1994) 2; \\
X. Ji,  J. Phys. {\bf G24} (1998) 1181.
%------------------------------------------------------------------------
%
%[13]
\bibitem{RAD}
A.V.~Radyushkin, Phys. Rev. {\bf D56} (1997) 5524; {\tt hep-ph/0101225}.
%------------------------------------------------------------------------
%
%[14]
\bibitem{BGR}
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, Nucl. Phys. {\bf B560} (1999) 
283.
%------------------------------------------------------------------------
%
%[15]
\bibitem{KROLL}                           
M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Phys. Lett. {\bf B460}
(1999) 204; Eur. Phys. J. {\bf C8} (1999) 409;\\
P. Kroll, Nucl. Phys. {\bf A666} (2000) 3.
%------------------------------------------------------------------------
%
%[16]
\bibitem{LAT}
G. Martinelli and C.T. Sachrajda, Phys. Lett. {\bf B190} (1987) 151;
{\bf 196} (1987) 184;\\
M. Guagnelli, K. Jansen, and R. Petronzio, Phys. Lett. {\bf B459} (1999)
594; {\bf B493} (2000) 77.
%------------------------------------------------------------------------
%
%[17]
\bibitem{MD}
M. Diehl, Nucl. Phys. {\bf B} (Proc. Suppl. {\bf 79} (1999) 723, Proc.
DIS 99, eds. J. Bl\"umlein  and T. Riemann.
%------------------------------------------------------------------------
%
%[18]
\bibitem{TARR}
B. Tarrach, Nuov. Cim. {\bf 28A} (1975) 409.
%------------------------------------------------------------------------
%
%[19]
\bibitem{COL}
J.C. Collins, {\tt hep-ph/9705393}.
%------------------------------------------------------------------------
%
%[20]
\bibitem{BR}
J. Bl\"umlein and D. Robaschik, Nucl. Phys. {\bf B581} (2000) 449.
%------------------------------------------------------------------------
%
%[21]
\bibitem{AHM}
A.H. Mueller, Phys. Rev. {\bf D2} (1970) 2963; Phys. Rev. {\bf D4} (1971)
150;\\
P.D.P. Collins, {\sf An Introduction to Regge Theory and High Energy
Physics}, (Cambridge University Press, Cambridge, 1977), pp.~331.
%------------------------------------------------------------------------
%
%[22]
\bibitem{PARA}
S. Weinberg, {\sf The Quantum Theory of Fields}, {\bf Vol.~1}, (Cambridge
University Press, Cambridge, 1995), pp.~452;\\
J. Bl\"umlein, J. Eilers, B. Geyer, and D. Robaschik, to appear.
%------------------------------------------------------------------------
%
%[23]
\bibitem{NFLO}
T.~Braunschweig, B.~Geyer, and D.~Robaschik, Ann. Phys. (Leipzig) 
{\bf 44} (1987) 407;\\
I.I.~Balitsky and V.M.~Braun, Nucl. Phys. {\bf B311} (1988/89) 541;\\
A.V.~Radyushkin,
Phys. Lett. {\bf B385} (1996) 333; Phys. Rev. {\bf D56} (1997) 5524;\\
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, Phys. Lett. {\bf B406} (1997) 
161 and Erratum;\\
I.I. Balitsky and A.V. Radyushkin, Phys. Lett. {\bf B413} (1997) 114;\\
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, {\tt hep-ph/9711405}, in~:
Proc. of the Workshop~{\sf Deep Inelastic Scattering Off Polarized
Targets: Theory Meets Experiment}, eds.~J.~Bl\"umlein et al., (DESY,
Hamburg, 1997) DESY 97--200, p.~196.\\
X.~Ji, Phys. Rev. Lett. {\bf 78} (1997) 610; Phys. Rev. {\bf D55} (1997)
7114;\\
L. Mankiewicz, G. Piller, and T. Weigl, Eur. J. Phys. {\bf C5} (1998) 
119.
%------------------------------------------------------------------------
%
%[24]
\bibitem{NFNLO}
F.M. Dittes and A.V. Radyushkin, Phys. Lett. {\bf B134} (1984) 359;\\
S.V. Mikhailov and A.V. Radyushkin, Nucl. Phys. {\bf B254} (1985) 89;\\
F.M.~Dittes, D.~M\"uller, D. Robaschik, B. Geyer, and J.~Ho\v{r}ej\v{s}i,
Phys. Lett. {\bf B209} (1988) 325;\\
A.V. Belitsky, D. M\"uller, L. Niedermaier, and A. Sch\"afer,
Nucl. Phys. {\bf B546} (1999) 279.
%------------------------------------------------------------------------
%
%[26]
\bibitem{RSBJP}
C. Royon, L. Schoeffel, J. Bartels, H. Jung, and R. Peschanski, 
Phys. Rev. {\bf D63} (2001) 074004.
%------------------------------------------------------------------------
\end{thebibliography}
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\begin{document}
\noindent
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\begin{flushleft}
DESY 01--XXX \hfill
{\tt hep-ph/0202077}\\
June  2001
\end{flushleft}
%
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% 1
%\mbox{}
\vspace*{\fill}
\begin{center}
{\LARGE\bf Polarized Deep Inelastic Diffractive} 

\vspace{2mm}
{\LARGE\bf {\boldmath $ep$} Scattering: Operator Approach}
%

\vspace{2cm}
\large
Johannes Bl\"umlein$^a$  and
Dieter Robaschik$^{a,b}$
\\
\vspace{2em}
\normalsize
{\it $^a$~Deutsches Elektronen--Synchrotron, DESY,\\
Platanenallee 6, D--15738 Zeuthen, Germany}
\\
\vspace{2em}
{\it $^b$~Brandenburgische Technische Universit\"at Cottbus, 
Fakult\"at 1,}\\
{\it  PF 101344, D--03013  Cottbus, Germany} \\
%%\today
\end{center}
\vspace*{\fill}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
\noindent
A quantum field theoretic treatment of inclusive deep--inelastic 
diffractive scattering is given. The process can be described in the
general framework of non--forward scattering processes using the 
light--cone expansion in the generalized Bjorken region. Evolution 
equations of the diffractive hadronic matrix elements are derived at the 
level of the twist--2 contributions and are compared to those of inclusive 
deep--inelastic forward scattering (DIS). The diffractive parton densities 
are obtained as projections of two--variable parton distributions. We 
also comment on the higher twist contributions in the light--cone 
expansion.
%
\end{abstract}
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\section{Introduction}
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\label{sec-1}
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\vspace{1mm}
\noindent
Unpolarized deep inelastic diffractive lepton--nucleon scattering was 
observed at the electron--proton collider HERA some years ago~\cite{EXP1}.
In the region of hard diffractive scattering this process is described by structure functions which
by diffractive parton distributions, which depend on two scaling variables
$x$ and $x_{\PP}$ and are different from the parton densities of deep 
inelastic scattering. New diffractive parton densities are expected
to occur in polarized deep inelastic diffractive lepton nucleon 
scattering. They can be measured at potential future polarized $ep$
factilities capeable to probe the kinematic range of small $x$~, 
c.f.~\cite{SXPOL}. For virtualities in the range $Q^2 \gsim 10 \GeV^2$
values of $x \gsim 10^{-3} \ldots 10^{-4}$ can be probed.
Dedicated future experimental studies of this process
can reveal the helicity structure of the non--perurbative color--neutral 
exchange of diffractive scattering with respect to the quark and
gluon structure and how the nucleon spin is viewed under a diffractive
exchange. At short distances the problem can be clearly separated into
a part, which can be described within perturbative QCD, and another
part which is thoroughly non--perturbative. In this paper we use the
light--cone expansion to describe the process of polarized diffractive
deep--inelastic scattering similar to a recent study for the unpolarized
case~\cite{BR2}. While the scaling violations of the process can be
calculated within perturbative QCD, the polarized difractive two--variable
parton densities are non--perturbative and can be related to expectation
values of (non--)local operators. Their Mellin-moments with respect to
the variable $\beta = x/x_{\PP}$ may, in principle, be calculated
on the lattice and one may try to understand the ratios of these
moments and those for the related deep--inelastic process w.r.t. their
scaling violations.



The diffractive events are
characterized by a rapidity gap between the final state nucleon and the 
set of the diffractively produced hadrons due to a color--neutral 
exchange. The (semi-inclusive) structure functions emerging in the 
diffractive scattering cross section show the same scaling violations as 
the structure functions in deep inelastic scattering within the current 
experimental resolution~\cite{EXP3}. This is a remarkable fact, which 
should also be understood with the help of field--theoretic methods within 
Quantum Chromodynamics (QCD). In the description of diffractive $ep$ 
scattering one has to clearly distinguish~\cite{BAKO} the case of hard 
scattering, i.e. large $Q^2$, from that of softer hadronic interactions, 
cf. also~\cite{DL}. A perturbative description of the scaling violations 
can only by hoped for in the former case, to which we limit our 
considerations in this paper.

There is a vast amount of varying descriptions of the underlying dynamics
of the diffractive scattering process\footnote{For surveys and a recent
account see~\cite{SUR}.}, ranging from intuitive phenomenological 
models~\cite{INGEL} to non-perturbative semi-classical 
descriptions~\cite{BH}, being able to describe and to parameterize 
\cite{EBKW} the existing data differentially.

In this paper we describe the process of inclusive deep--inelastic
diffractive scattering, being a non--forward process, at large space--like 
momentum transfer using the general light--cone expansion for the 
non--forward case, cf.~[12--14]. As will be shown, the scaling
violations of 
the diffractive structure functions can be described perturbatively in 
this region. We show that the diffractive parton densities can be derived
as a special projection of  two--variable distribution functions 
$f^A(z_+,z_-)$, where $z_+$ and $z_-$ are light--cone momentum fractions;
for other projections in similar cases related to different observables 
see e.g.~[13--15]. In particular we have {\it no need} to
refer to the specific mechanism of the non--perturbative color--singlet 
exchange between the proton and the set of diffractively produced hadrons
and consider instead the expectation values of operators of a given twist
and their anomalous dimensions. In this way we can generalize the analysis 
to operators of higher twist. This formulation is very appropriate for 
potential later studies of the corresponding operator matrix elements with 
the help of lattice techniques similar to the case of deep--inelastic 
scattering \cite{LAT} calculating their Mellin--moments.

The paper is organized as follows. In section~2 we consider the Lorentz
structure of the differential cross section for inclusive deep--inelastic 
diffractive scattering. The short--distance structure of the matrix 
element is derived in section~3 using the (non--local) light--cone 
expansion in the generalized Bjorken region. In section~4 we derive the 
anomalous  dimensions for the case of twist--2, discuss the implications 
for higher twist operators and compare the case of diffractive scattering
to that of deep--inelastic scattering, and section~5 contains the
conclusions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lorentz Structure}
\vspace*{-5mm}\noindent
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\setcounter{equation}{0}
\label{sec-2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1mm}
\noindent
The process of deep--inelastic diffractive scattering is described by the
diagram Figure~1.
The differential scattering cross section for single--photon 
exchange is given by
%-----------------------------------------------------------------------
\begin{equation}
\label{eqD1}
d^5 \sigma_{\rm diffr}
= \frac{1}{2(s-M^2)} \frac{1}{4} dPS^{(3)} \sum_{\rm spins}
\frac{e^4}{Q^2} L_{\mu\nu} W^{\mu\nu}~.
\end{equation}
%-----------------------------------------------------------------------
Here $s=(p_1+l)^2$ is the cms energy of the process squared and $M$ 
denotes the nucleon mass.

\vspace*{0.8cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{picture}(-50,100)(0,0)
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\SetWidth{1.5}
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\setlength{\unitlength}{1pt}
\Text(-160,110)[]{$l$}
\Text(-40,130)[]{$l'$}
\Text(-90,75)[]{$q$}
\Text(-110,-10)[]{$p_1$}
\Text(10,-10)[]{$p_2$}
\Text(10,110)[]{$M_X$}
\end{picture}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{1mm}
\noindent
\begin{center}
{\sf Figure~1:~The virtual photon-hadron amplitude for 
diffractive $ep$ scattering} 
\end{center}
The phase space $dPS^{(3)}$ depends on five variables\footnote{For the
commonly used notation of four variables see e.g.~\cite{MD}.} since one 
final state mass varies. They can be chosen as 
Bjorken~$x= Q^2/(W^2+Q^2-M^2)$, the photon virtuality $Q^2=-q^2$, 
$t = (p_1 - p_2)^2$, a variable describing the non--forwardness w.r.t.
the incoming proton direction,
%-----------------------------------------------------------------------
\begin{equation}
\label{eqV1}
x_{\PP} = - \frac{2\eta}{1-\eta} = 
\frac{Q^2 + M_X^2 - t}{Q^2 + W^2 -M^2} \geq  x~,
\end{equation}
%-----------------------------------------------------------------------
demanding $M_X^2 > t$ and where
%-----------------------------------------------------------------------
\begin{equation}
\label{eqV2}
\eta = \frac{q.(p_2 - p_1)}{q.(p_2+p_1)}~\epsilon~\left[-1,\frac{-x}{2-x}
\right]~,
\end{equation}
%-----------------------------------------------------------------------
and $\Phi$ the angle between the lepton plane $\pvec_1 \times \lvec$ and
the hadron plane $\pvec_1 \times \pvec_2$,
%-----------------------------------------------------------------------
\begin{equation}
\label{eqV3}
\cos \Phi = \frac{(\pvec_1 \times \lvec).(\pvec_1 \times  \pvec_2)}
                 {|\pvec_1 \times \lvec ||\pvec_1 \times  \pvec_2|}~.
\end{equation}
%-----------------------------------------------------------------------
$W^2 = (p_1+q)^2$ and  $M_X^2 = (p_1+q-p_2)^2$ denote the hadronic
mass squared and the square of the diffractive mass, respectively.

Unpolarized deep inelastic diffractive scattering was considered in
previous paper~\cite{BR2} in detail. Here we focus on the polarized part 
only, which can be measured in terms of a polarization asymmetry
%-----------------------------------------------------------------------
\begin{equation}
\label{eqP1}
A(x,Q^2,x_{\PP},S_\mu) = \frac{d^5 \sigma(S_{\mu}) - d^5 \sigma(-S_\mu)}
                              {d^5 \sigma(S_{\mu}) + d^5 \sigma(-S_\mu)}~,
\end{equation}
%-----------------------------------------------------------------------
where $S_\mu$ is a hadronic spin vector, with $S_1.p_1 = S_2.p_2 = 0$.
Since the cross sections are linear functions in the initial-- and final 
state proton spin--vectors, the denominator projects on the even and the 
numerator on the odd part in $S_\mu$. If the final--state spin remains 
unmeasured, the denominator refers to the unpolarized cross 
section~\cite{BR2}.

We consider the case of single photon exchange, which is projected by the
polarized contribution
%-----------------------------------------------------------------------
\begin{equation}
\label{eqPLE}
L_{\mu\nu}^{\rm pol} = 2i \varepsilon_{\mu\nu\rho\sigma} l^\rho q^\sigma
\end{equation}
%-----------------------------------------------------------------------
to the leptonic tensor. Since the electromagnetic current is conserved,
the strong interactions conserve parity and are even under 
time--reversal,\footnote{Here we disregard potential contributions due to
strong CP--violation~\cite{THOOFT}, because of the smallness of the 
$\theta$--parameter, $|\theta| < 3 \cdot 10^{-9}$~\cite{THETA}.} 
and the hadronic tensor has to be hermitic due to Eq.~(\ref{eqPLE}), the
following relations hold~\cite{TM}~:
%-----------------------------------------------------------------------
\begin{equation}
\begin{alignat}{2}
\label{eqP2}
{\sf Current~conservation:}&~~q^{\mu}~W_{\mu\nu}(q,p_1,S_1,p_2,S_2)
 & &=
W_{\mu\nu}(q,p_1,S_1,p_2,S_2)~q^{\nu} = 0~,\\
{\sf P~~invariance:}&~~W_{\mu\nu}(\overline{q},%
\overline{p}_1,-\overline{S}_1,\overline{p}_2,%
-\overline{S}_2) & &=
W^{\mu\nu}(q,p_1,S_1,p_2,S_2)~, \\
{\sf T~~invariance:}&~~W_{\mu\nu}(\overline{q},%
\overline{p}_1,-\overline{S}_1,\overline{p}_2,-\overline{S}_2) & &=
\left[W^{\mu\nu}(q,p_1,S_1,p_2,S_2)\right]^*, \\
{\sf Hermiticity:}&~~W_{\mu\nu}(q,p_1,S_1,p_2,S_2)
& &=  
\left[W_{\nu\mu}(q,p_1,S_1,p_2,S_2)\right]^*,
\end{alignat}
\end{equation}
%-----------------------------------------------------------------------
with $\overline{a}_\mu = a^{\mu}$. Constructing the hadronic tensor we 
seek a structure which is linear in the initial and final proton spin.
Upon noting that
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP3}
\varepsilon^{\mu\nu\alpha\beta} = - \varepsilon_{\mu\nu\alpha\beta}
\end{eqnarray}
%-----------------------------------------------------------------------
the terms which depend either on $S_{1\mu}$ or on $S_{2\mu}$ have to 
occur together with the Levi--Civita pseudo--tensor, whereas separate 
terms $\propto S_{1\mu} S_{2\nu}$ are possible.
The most general asymmetric parity--conserving hadronic tensor, which
obeys current conservation, is\footnote{A sub-set of this structure based
on $p,q$ and $S$ was considered in Ref.~\cite{DJT}.}
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP4}
W_{\mu\nu}^{\rm pol} &=&
~~i \varepsilon_{\mu\nu\alpha\beta} S_1^\alpha q^\beta H_1
+ i \varepsilon_{\mu\nu\alpha\beta} S_2^\alpha q^\beta H_2 \nonumber\\
& & + i \varepsilon_{\mu\nu\alpha\beta} p_1^\alpha q_\beta \left[
S_1.q H_3 + S_1.p_2 H_4 + S_2.q H_5 + S_2.p_1 H_6 \right] \nonumber\\
& & + i \varepsilon_{\mu\nu\alpha\beta} p_2^\alpha q^\beta \left[
S_1.q H_7 + S_1.p_2 H_8 + S_2.q H_9 + S_2.p_1 H_{10} \right] 
\end{eqnarray}
%-----------------------------------------------------------------------
Other terms as
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP5}
V_{\mu\nu} &=&~~
 i \left[
q^2 \varepsilon_{\mu\nu\alpha\beta} S_k^\alpha v_1^\beta
+ q_\mu \varepsilon_{\nu\rho\alpha\beta} q^\rho S_k^\alpha v_1^\beta
- q_\nu \varepsilon_{\mu\rho\alpha\beta} q^\rho S_k^\alpha v_1^\beta
\right] \widetilde{H}_{1}^k \nonumber\\  & &
+ i \left[
q^2 \varepsilon_{\mu\nu\alpha\beta} v_1^\alpha v_2^\beta
+ q_\mu \varepsilon_{\nu\rho\alpha\beta} q^\rho v_1^\alpha v_2^\beta
- q_\nu \varepsilon_{\mu\rho\alpha\beta} q^\rho v_1^\alpha v_2^\beta
\right] S_k.v_3 \widetilde{H}_{2}^k~ \nonumber\\ & &
+ i \left[\tilde{u}_{3\mu} \varepsilon_{\nu\rho\alpha\beta} q^\rho
-         \tilde{u}_{3\nu} \varepsilon_{\mu\rho\alpha\beta} \right]
q^\rho u_1^\alpha u_2^\beta u_4.u_5 \widetilde{H}_3(u_k|_{k=1}^5)~,
\end{eqnarray}
%-----------------------------------------------------------------------
with
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqPTE}
\tilde{u}_\mu = u_k - \frac{q.u_k}{q^2} q_\mu~,
\end{eqnarray}
%-----------------------------------------------------------------------
and $v_1 \neq q$, $v_k|_{k=1}^3~\epsilon~{p_1,p_2}$ leaving out the 
combinations where $v_3 = p_k$, and $u_k~\epsilon~\{q,p_1,p_2,S_1,S_2\}$
such that only one spin vector occurs, reduce in 4--dimensions
to the structures contained already in Eq.~(\ref{eqP5}) using
the Schouten--idendity~\cite{SCHOUT}
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP6}
X_\mu \varepsilon_{\nu\rho\sigma\tau} =
X_\nu \varepsilon_{\mu\rho\sigma\tau} +
X_\rho \varepsilon_{\nu\mu\sigma\tau} +
X_\sigma \varepsilon_{\nu\rho\mu\tau} +
X_\tau \varepsilon_{\nu\rho\sigma\mu}~.
\end{eqnarray}
%-----------------------------------------------------------------------
The structure functions $H_i|_{i=1}^{10}$ are chosen to be real.
Therefore one finally may 
parameterize the hadronic tensor in terms of the two structure functions 
$G_1$ and $G_2$ only
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP8}
W_{\mu\nu}^{\rm pol} &=& i \varepsilon^{\mu\nu\alpha\beta} S_\alpha
q_\beta~G_1 
+ i \varepsilon^{\mu\nu\alpha\beta} p_\alpha q_\beta~S.q~G_2~.
\end{eqnarray}
%-----------------------------------------------------------------------
with
%-----------------------------------------------------------------------
\begin{equation}
\label{eqD3}
G_i = G_i(x,Q^2,x_{\PP},t)~.
\end{equation}
%-----------------------------------------------------------------------
The size of the rapidity--gap \cite{RAPGA} is of the order
$\Delta y \sim \ln(1/x_{\PP})$ and large in the kinematic domain of the 
facilities envisaged for the future~\cite{SXPOL}.

If target masses can be neglected, $M^2 \sim 0$, and $t$ is considered
to be very small, the in-- and outgoing proton four--momenta become 
proportional,~$p_2 = z p_1$, where
%-----------------------------------------------------------------------
\begin{equation}
\label{eqD4}
z = 1 - x_{\PP} =  \frac{1 + \eta}{1 - \eta}~.
\end{equation}
%-----------------------------------------------------------------------
In this limit the polarized
hadronic tensor is described by two structure functions
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqP9}
W_{\mu\nu}^{\rm pol} &=& i \varepsilon_{\mu\nu\alpha\beta} q^\alpha
\left\{S_1^\beta \left[G_{11}+G_{21}\right]
+ S_2^\beta \left[G_{12}+G_{22}\right] \right\}
- i \varepsilon_{\mu\nu\alpha\beta} q^\alpha
\left\{\left(\frac{S_1.q}{p_1.q}\right)p_1^\beta G_{21}
+ \left(\frac{S_2.q}{p_.q}\right)p_2^\beta G_{22} \right\}~,
\end{eqnarray}
%-----------------------------------------------------------------------
where
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqD6}
G_{11} &=& H_1-G_{21}\\
G_{12} &=& H_2-G_{22}\\
G_{21} &=& H_3+(1-x_{\PP} H_7\\
G_{22} &=& H_5+(1-x_{\PP} H_9
\end{eqnarray}
%-----------------------------------------------------------------------
Although in this limit the hadronic tensor is structurally the same as
for the deep inelastic process the dependence of $\eta$, resp. $x_{\PP}$,
points to the non--forwardness on the side of the initial and final
state proton.

For the present process we will consider  a limit, similar to the
Bjorken limit,
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqBL}
p_1.q  \rightarrow \infty,~~~~~p_2.q \rightarrow \infty,~~~~~
Q^2 \rightarrow \infty~~~{\rm with}~~x~~{\rm and}~~x_{\PP} =~{\rm fixed}~,
\end{eqnarray}
%-----------------------------------------------------------------------
which leads to
$p_i.q G_{ki}(x,Q^2,x_{\PP}) \rightarrow g_{ki}^D(x,Q^2,x_{\PP})$.
Let us consider the case in which the spin of the final--state proton
is summed over. This removes all contributions $\propto S_{2\mu}$ from
$W_{\mu\nu}^{\rm pol}$.
The initial proton polarization vector may be choosen either
longitudinal or transversal to the momentum $p_1$,
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSPI}
S_L = (0,0,0,M),~~~~~~~~~~~~~~~~~~~~S_T=M(0,\cos\alpha,\sin\alpha,0)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The corresponding differential scattering cross sections read~\cite{BT}
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqPS1}
\frac{d^3 \sigma_{\rm diffr}^{\rm pol}(\lambda,\pm S_L)}
{dx dQ^2 d x_{\PP}}
&=& \pm \lambda \frac{4 \pi \alpha^2}
{x Q^4} \left[y \left(2 - y - \frac{2xyM^2}{S}\right) 
xg_1^D(x,Q^2,x_{\PP})
+ 4 \frac{x^2 y M^2}{S} g_2^D(x,Q^2,x_{\PP})\right]\nonumber\\ \\
%-------
\label{eqPS2}
\frac{d^4 \sigma_{\rm diffr}^{\rm pol}(\lambda,\pm S_T)}
{dx dQ^2 d \phi d x_{\PP}}
&=& \mp \lambda \frac{4 \alpha^2}
{x Q^4} \sqrt{\frac{M^2}{S}}\sqrt{xy\left[1 - y - \frac{xyM^2}{S}\right]
} \cos(\alpha - \phi) \nonumber\\ & &~~~~~~~~~\times
\left[yx g_1^D(x,Q^2,x_{\PP}) +
2 x g_2^D(x,Q^2,x_{\PP}) \right]~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $y = q.p_1/l.p_1$ and $\phi$ the azimuthal scattering angle.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Compton Amplitude}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\setcounter{equation}{0}
\label{sec-3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1mm}
\noindent
The renormalized and time--ordered product of two electromagnetic
currents is given by
%-----------------------------------------------------------------------
\begin{eqnarray}
\widehat{T}_{\mu\nu}(x) &=&
i RT \left[J_\mu\left(\frac{x}{2}\right)J_\nu\left(-\frac{x}{2}\right) S 
\right] \nonumber\\
&=&
 -e^2 \frac{\tilde x^\lambda}{2 \pi^2 (x^2-i\epsilon)^2}
 RT
 \left[
\overline{\psi}
\left(\frac{\tilde x}{2}\right)
\gamma^\mu \gamma^\lambda \gamma^\nu \psi
\left(-\frac{\tilde x}{2}\right)
- \overline{\psi}
\left(-\frac{\tilde x}{2}\right)
\gamma^\mu \gamma^\lambda \gamma^\nu \psi
\left(\frac{\tilde x}{2}\right)
\right] S
\end{eqnarray}
%-----------------------------------------------------------------------
$\tilde x$ denotes a light--like vector corresponding to $x$,
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{xtil}
\tilde x = x + \frac{\zeta}{\zeta^2}\left[ \sqrt{x.\zeta^2 - x^2 \zeta^2}
- x.\zeta\right]~,
\end{eqnarray}
%-----------------------------------------------------------------------
and $\zeta$ is a subsidiary vector. Following Refs.~\cite{BGR,BR} the
operator $\widehat{T}_{\mu\nu}$ can be expressed in terms of a vector
and an axial--vector operator by
%-----------------------------------------------------------------------
\begin{eqnarray}
 \widehat{T}_{\mu\nu}(x)  =
 - e^2 \frac{\tilde x^\lambda}{i \pi^2 (x^2-i\epsilon)^2}
 \left[S_{\alpha \mu\lambda \nu} 
 O^\alpha\left(\frac{\tilde x}{2}, -\frac{\tilde x}{2}\right)
+i \varepsilon_{\mu\lambda \nu \sigma} 
O_5^\alpha  \left(\frac{\tilde x}{2}, -\frac{\tilde x}{2}\right)
\right]~,
\end{eqnarray}
%-----------------------------------------------------------------------
where
%-----------------------------------------------------------------------
\begin{eqnarray}
S_{\alpha \mu\lambda \nu} = g_{\alpha\mu}g_{\lambda \nu}
                          + g_{\lambda\mu}g_{\alpha \nu}
                          - g_{\mu\nu}g_{\lambda \alpha}~.
\end{eqnarray}
%-----------------------------------------------------------------------
The  bilocal light--ray operators are
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{oo}
O^{\alpha}\left(\frac{\tilde x}{2},-\frac{\tilde x}{2}\right)
&=&
\frac{i}{2}RT
\left[\overline{\psi}\left(\frac{\tilde x}{2}\right)
\gamma^\alpha\psi\left(-\frac{\tilde x}{2}\right)
- \overline{\psi}\left(-\frac{\tilde x}{2}\right)
\gamma^\alpha\psi\left(\frac{\tilde x}{2}\right)\right]S~,
\\
\label{oo5}
O^{\alpha}_5\left(\frac{\tilde x}{2},-\frac{\tilde x}{2}\right)
&=&
\frac{i}{2} RT
\left[\overline{\psi}\left(\frac{\tilde x}{2}\right)
\gamma_5\gamma^\alpha\psi\left(-\frac{\tilde x}{2}\right)
+ \overline{\psi}\left(-\frac{\tilde x}{2}\right)
\gamma_5\gamma^\alpha\psi\left(\frac{\tilde x}{2}\right)\right]S~.
\end{eqnarray}
%----------------------------------------------------------------------
The general operator $\widehat{T}_{\mu\nu}$ is now to be related to the
diffractive scattering cross section. This is possible upon applying
Mueller's generalized optical theorem~\cite{AHM} (Figure~2), which
moves the final state proton into an initial state anti-proton.

\vspace*{7mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
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\Text(50,50)[]{\large $=~~{\rm Disc}$}
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\Text(5,-15)[]{$p_2$}
\Text(75,115)[]{$q$}
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\Text(95,-15)[]{$p_2$}
\Text(60,35)[]{$X$}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{10mm}\noindent
\begin{center}
{\sf Figure~2:~A. Mueller's optical theorem.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Consequently, the Compton amplitude is obtained as the expectation value
%-----------------------------------------------------------------------
\begin{eqnarray}
T_{\mu\nu}(x) = \langle p_1,-p_2|\widehat{T}_{\mu\nu}|p_1,-p_2\rangle~,
\end{eqnarray}
%-----------------------------------------------------------------------
which is forward w.r.t. to the direction defined by the state 
$\langle p_1,-p_2|$.
The twist--2 contributions to the expectation values of the operators 
(\ref{oo},\ref{oo5})  are obtained
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSCA}
\langle p_1,-p_2|O^{A,\mu}_{(5)}(\kappa_+ \xx,\kappa_-\xx)|
p_1,-p_2\rangle = \left.
\int_0^1 d\lambda
\partial^{\mu}_x
\langle p_1,-p_2|O_{(5)}^A(\lambda \kappa_+ x,\lambda \kappa_-x)
|p_1,-p_2\rangle \right|_{x=\xx}
\end{eqnarray}
%-----------------------------------------------------------------------
as partial derivatives of the expectation values of
%-----------------------------------------------------------------------
\begin{eqnarray}
O^A(\kappa_+x,\kappa_-x)
= x^\alpha O^A_\alpha(\kappa_+x,\kappa_-x)
\end{eqnarray}
%-----------------------------------------------------------------------
the corresponding scalar and pseudo-scalar operators. The index $A = q,G$
labels the quark-- or gluon operators, cf.~\cite{BGR}. In the unpolarized 
case only the vector operators contribute. The scalar twist--2 quark
operator
matrix element has the representation\footnote{For parameterizing hadronic
matrix elements see e.g.~\cite{PARA}.} due to the overall symmetry in $x$
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSCAM}
\langle p_1,-p_2|O^q(\kappa_+ x,\kappa_- x)|p_1,-p_2\rangle = xp_-
\!\!\!         \int \!\!\!
Dz ~\e^{-i \kappa_- x p_z} {f}^q(z_+,z_-)
+ x\pi_-\!\!\!         \int \!\!\!
Dz ~\e^{-i \kappa_- x p_z} {f}^q_\pi(z_+,z_-),
%\hspace*{11.2cm} (21)  \nonumber
\end{eqnarray}
%-----------------------------------------------------------------------
where we assume that all the trace--terms have been subtracted, see
\cite{BGR}.
${f}^A(z_+,z_-)$ and $f^A_\pi(z_+,z_-)$ denote the scalar 
two--variable distribution amplitudes and the measure $Dz$ is
%------------------------------------------------------------------------
\begin{eqnarray}
\label{Dz}
Dz = d z_+ d z_-
\theta(1 + z_+ + z_-) \theta(1 + z_+ - z_-)
\theta(1 - z_+ + z_-) \theta(1 - z_+ - z_-)~.
\end{eqnarray}
%------------------------------------------------------------------------
Here, we decomposed the vector $p_z$ as
%-----------------------------------------------------------------------
\begin{eqnarray}
p_z = p_- z_- + p_+ z_+ =  p_- \vartheta + \pi_- z_+~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $z_{1,2}$ momentum fractions along $p_{1,2}$ and $p_{\pm} = p_2 \pm
p_1$, $z_\pm = (z_2 \pm z_1)/2$ and
%-----------------------------------------------------------------------
\begin{eqnarray}
\vartheta = z_- + \frac{1}{\eta} z_+,~~~~~~~~~~~~~\pi_-
= p_+ - \frac{1}{\eta} p_-~.
\end{eqnarray}
%-----------------------------------------------------------------------
Note that, $q.\pi_- \equiv 0$. In the approximation $M^2, t \sim 0$, in 
which we work from now on, the vector $\pi_-$ vanishes.
In this limit only the first term contributes to the matrix element
Eq.~(\ref{eqSCAM}).

The Fourier--transform of the Compton amplitude is given by~\cite{BR}
%-----------------------------------------------------------------------
\begin{eqnarray}
T_{\mu\nu}(p_1,p_2,q) &=& i \int d^4x \e^{iqx} T_{\mu\nu}(x) \nonumber\\
&=& - 2 S_{\rho\mu\sigma\nu} \int Dz \left[
\frac{p_-^{\rho} Q_z^{\sigma}}{Q^2_z + i\varepsilon} - \frac{1}{2}
\frac{p_z^{\rho} p_-^{\sigma}}{Q^2_z + i\varepsilon}%
+ \frac{Q_z.p_-}{(Q^2_z+i\varepsilon)^2}
p_z^{\rho} Q_z^{\sigma}\right] F(z_+,z_-)~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $Q_z = q - p_z/2$. The function $F(z_+,z_-)$ is related to the
distribution function $f(z_+,z_-)$ by
%-----------------------------------------------------------------------
\begin{eqnarray}
F(z_+,z_-) = \int_0^1 \frac{d\lambda}{\lambda^2} 
f\left(\frac{z_+}{\lambda},\frac{z_-}{\lambda}\right)
\theta(\lambda-|z_+|)\theta(\lambda-|z_-|)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The denominators take the form
%-----------------------------------------------------------------------
\begin{eqnarray}
\frac{1}{Q^2_z+i\varepsilon} = - \frac{1}{qp_-}\frac{1}{(\vartheta
- 2\beta  + i \varepsilon)}~,
\end{eqnarray}
%-----------------------------------------------------------------------
where
%-----------------------------------------------------------------------
\begin{eqnarray}
\beta =  \frac{x}{x_{\PP}} = \frac{q^2}{2q.p_-}~.
\end{eqnarray}
%-----------------------------------------------------------------------
The conservation of the electromagnetic current is easily seen
%-----------------------------------------------------------------------
\begin{eqnarray}
q^{\mu}T_{\mu\nu}(p_1,p_2,q)  =
q^\nu T_{\mu\nu}(p_1,p_2,q)  =
- 2 p_{-,\nu(\mu)} \int Dz F(z_+,z_-) = 0~,
\end{eqnarray}
%-----------------------------------------------------------------------
since the symmetry relation, \cite{BR},
%-----------------------------------------------------------------------
\begin{eqnarray}
\int Dz F(z_+,z_-) = 0~,  
\end{eqnarray}
%-----------------------------------------------------------------------
holds.
Subsequently we will use the distribution
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqFF}
\widehat{F}(\vartheta,\eta) = \int Dz F(z_+,z_-) 
\delta(\vartheta - z_- -z_+/\eta) = \int_\vartheta^{-{\rm sign}(\vartheta)
/\eta} \frac{dz}{z} \widehat{f}(z,\eta)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The distribution function $\widehat{f}(z,\eta)$ is related to 
$f(z_+,z_-)$ by
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqFF1}
\widehat{f}(z,\eta) = \int^{\eta(1-z)}_{\eta(1+z)} d\rho~
\theta(1-\rho) \theta(\rho+1) f(\rho,z-\rho/\eta)~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $\rho = z_+/\eta$.

The Compton amplitude may be re-written as
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqCOMP1}
T_{\mu\nu}(p_1,p_2,q) &=& 
-2 \int_{+1/\eta}^{-1/\eta} d\vartheta \Biggl\{
\left[-q.p_- g_{\mu\nu}
-2\vartheta p_{-,\mu} p_{-,\nu} + (p_{-,\mu}q_\nu +p_{-,\nu} q_\mu)\right]
\frac{1}{Q_z^2+i\varepsilon} \nonumber\\
& & ~~~~+ \left[\vartheta (q_\mu p_{-,\nu} + q_\nu p_{-,\mu}) - \vartheta
q.p_- g_{\mu\nu} - \vartheta^2 p_{-,\mu} p_{-,\nu} \right]\frac{1}
{(Q_z^2+i\varepsilon)^2}\Biggr\} \nonumber\\
& &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\times \int_\vartheta^{-{\rm sign}(\vartheta)/\eta}\frac{dz}{z}
\widehat{f}(z,\eta)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The $\vartheta$--integrals in Eq.~(\ref{eqCOMP1}) can be simplified
using the identities
%-----------------------------------------------------------------------
\begin{eqnarray}
\int_{+1/\eta}^{-1/\eta} d\vartheta \frac{\vartheta^k}
{(\vartheta-2\beta+i\varepsilon)^2}\int_\vartheta^{{\rm sign}(\vartheta)
/\eta} \frac{dz}{z} \widehat{f}(z,\eta)
&=&
\int_{+1/\eta}^{-1/\eta} d\vartheta \frac{k \vartheta^{k-1}}
{(\vartheta-2\beta+i\varepsilon)}\int_\vartheta^{{\rm sign}(\vartheta)
/\eta} \frac{dz}{z} \widehat{f}(z,\eta) \nonumber\\    & &
~~~~~~~-\int_{+1/\eta}^{-1/\eta} d\vartheta \frac{\vartheta^{k-1}
\widehat{f}(\vartheta,\eta)}
{(\vartheta-2\beta+i\varepsilon)}~.
\end{eqnarray}
%-----------------------------------------------------------------------
One may further re-write Eq.~(\ref{eqCOMP1}) referring to $p_1$ instead
of $p_-$,
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqCOMP2}
T_{\mu\nu}(p_1,p_2,q) &=& 
2 \int_{+1/\eta}^{-1/\eta} d\vartheta
\left[-g_{\mu\nu} 
+ \frac{p_{1,\mu}q_\nu +p_{1\nu} q_\mu)}{q.p_1}
+\vartheta  x_{\PP} \frac{p_{1\mu} p_{1\nu}}{q.p_1}\right]
\frac{\widehat{f}(\vartheta,\eta)}{(\vartheta - 2\beta +i\varepsilon)}~.
\end{eqnarray}
%-----------------------------------------------------------------------
Before we consider the absorptive part of the Compton amplitude
we exploit the symmetry relation for the unpolarized distribution 
functions  $F^A(z_1,z_2)$, Ref.~\cite{BR},
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSYM1}
F^A(z_1,z_2) = - F^A(-z_1,-z_2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
It translates into
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSYM2}
\widehat{F}^A(\vartheta,\eta) = - \widehat{F}^A(-\vartheta,\eta)~,
\end{eqnarray}
%-----------------------------------------------------------------------
and, cf. Eq.~(\ref{eqFF}),
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqSYM3}
\widehat{f}^A(\vartheta,\eta) = - \widehat{f}^A(-\vartheta,\eta)~.
\end{eqnarray}
%-----------------------------------------------------------------------
Seeking the form of Eq.~(\ref{eqD5}), Eq.~(\ref{eqCOMP2}) transforms to
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqCOMP3}
T_{\mu\nu}(p_1,p_2,q) &=& \int_{1/\eta}^{-1/\eta} d\vartheta \left[
\left(-g_{\mu\nu} +\frac{q_\mu q_\nu}{q^2}\right) + \frac{2x}{q.p_1}
\left(p_{1\mu} - q_\mu \frac{p_1.q}{q^2}\right)
\left(p_{1\nu} - q_\nu \frac{p_1.q}{q^2}\right) \right] \nonumber\\
& &~~~~~~~~~~\times
\left[\frac{\widehat{f}(\vartheta,\eta)}
{\vartheta - 2 \beta + i\varepsilon}
-
\frac{\widehat{f}(-\vartheta,\eta)}
{\vartheta - 2 \beta + i\varepsilon}\right]\nonumber\\
& & ~~~~~~~~~~~+ 2\frac{p_{1\mu} p_{1\nu}}{q.p_1}
\int_{1/\eta}^{-1/\eta}
d\vartheta \left(\vartheta x_{\PP} - 2x
\right) \frac{\widehat{f}(\vartheta,\eta)}{\vartheta - 2\beta + 
i \varepsilon}~.
\end{eqnarray}
%-----------------------------------------------------------------------
Taking the absorptive part one obtains
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqIMPAR}
W_{\mu\nu} &=& \frac{1}{2\pi}~{\sf Im}~T_{\mu\nu}(p_1,p_2,q) \\
           &=&       
\left(-g_{\mu\nu} +\frac{q_\mu q_\nu}{q^2}\right) 
{\sf F}_1(\beta,\eta,Q^2)
+ \frac{1}{q.p_1}
\left(p_{1\mu} - q_\mu \frac{p_1.q}{q^2}\right)
\left(p_{1\nu} - q_\nu \frac{p_1.q}{q^2}\right) 
{\sf F}_2(\beta,\eta,Q^2) \nonumber
\end{eqnarray}
%-----------------------------------------------------------------------
where
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqF1}
{\sf F}_1(\beta,\eta,Q^2) = \sum_{q=1}^{N_f} 
e_q^2\left[f_q^D(\beta,Q^2,x_{\PP})+\overline{f}_q^D(\beta,Q^2,x_{\PP})
\right]
\equiv  F_1^{D(3)}(x,Q^2,x_{\PP})~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $N_f$ the number of flavors, choosing the factorization scale
$\mu^2 = Q^2$, and
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqCG}
{\sf F}_2(\beta,\eta,Q^2) = 2x {\sf F}_1(\beta,\eta,Q^2) \equiv
F_2^{D(3)}(x,Q^2,x_{\PP}).
\end{eqnarray}
%-----------------------------------------------------------------------
If we compare the distribution functions $f_A^{D}$ with the parton
distributions in the deep inelastic case, the role of $x$ in the latter
case is taken by the variable $\beta$. Both variables have the range
$[0,1]$, through which Mellin transforms w.r.t. these variables can
be evaluated. On the contrary, the support in $x$ of the diffractive
structure functions is limited to $[0,x_{\PP}]$. Although $\beta$ in the
diffractive case compares to $x$ in the deep inelastic case, the 
Callan--Gross relation, at lowest order in $\alpha_s$, Eq.~(\ref{eqCG}), 
is given by the factor $2x$ between the two structure functions. 

The diffractive quark and anti--quark densities are given by
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqPART}
\sum_{q=1}^{N_f} e_q^2 f_q^D(\beta,Q^2,x_{\PP}) &=&
-\widehat{f}(2\beta,\eta,Q^2) \nonumber\\
\sum_{q=1}^{N_f} e_q^2 \overline{f}_q^D(\beta,Q^2,x_{\PP}) &=&
~~\widehat{f}(-2\beta,\eta,Q^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
We finally can express the diffractive parton densities in terms of
the distribution function $f(z_+,z_-)$ directly
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqREL}
f^D(\beta,Q^2,x_{\PP}) &=& - \int_{-\frac{x_{\PP}+2x}{2-x_{\PP}}}
                              ^{-\frac{x_{\PP}-2x}{2-x_{\PP}}}
d \rho f(\rho,2\beta + \rho(2-x_{\PP})/x_{\PP},Q^2)\nonumber\\
\overline{f}^D(\beta,Q^2,x_{\PP}) &=& 
- \int_{-\frac{x_{\PP}+2x}{2-x_{\PP}}}
                              ^{-\frac{x_{\PP}-2x}{2-x_{\PP}}}
d \rho f(\rho,-2\beta + \rho(2-x_{\PP})/x_{\PP},Q^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Evolution Equations}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\setcounter{equation}{0}
\label{sec-4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1mm}
\noindent
The evolution equation of the scalar twist--2 quark and gluon operator 
Eq.~(\ref{eqSCA}) read, cf.~\cite{BGR},\footnote{The non--singlet
evolution equations are structurally the same as that with the
anomalous dimension $\gamma^{qq}$.}
%-----------------------------------------------------------------------
\begin{eqnarray}
\mu^2 \frac{d}{d\mu^2} O^A(\kappa_+\xx, \kappa_-\xx; \mu^2) =
\int D\kappa' \gamma^{AB}(\kappa_+,\kappa_-,\kappa_+',\kappa_-';\mu^2)
O_B(\kappa_+'\xx,\kappa_-'\xx;\mu^2)~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $\kappa_\pm =(\kappa_2 - \kappa_1)/2$ and the measure
%-----------------------------------------------------------------------
\begin{eqnarray}
D\kappa =   d\kappa_+ d\kappa_- \theta(1+\kappa_+ + \kappa_-)
                                \theta(1+\kappa_+ - \kappa_-)
                                \theta(1-\kappa_+ + \kappa_-)
                                \theta(1-\kappa_+ - \kappa_-)~.
\end{eqnarray}
%-----------------------------------------------------------------------
We consider the non--forward evolution equations to trace an eventual
$\eta$--dependence.
Here $\gamma^{AB}(\kappa_+,\kappa_-,\kappa_+',\kappa_-',\mu^2)$ is the
general non--forward twist--2 singlet evolution 
matrix.~\footnote{The leading order and next--to--leading
order non--singlet and singlet anomalous dimensions were calculated in
Refs.~\cite{NFLO,RAD} and \cite{NFNLO}.}
The same evolution
equation applies to the matrix elements
$\langle p_1,-p_2|O^A(\kappa_+,\kappa_-)|p_1,-p_2\rangle$, which may be
related to the distribution functions $f^A(\vartheta,\eta)$ 
by\footnote{Note that the we identified here $\tau = \xx.p_-/\xx.p_+$
with $\eta = q.p_-/q.p_+$ which is possible after the Bjorken limit is
taken.}
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqFOU}
f^A(\vartheta,\eta) = \int \frac{d\kappa_- \xx p_-}{2\pi}~\e^{i\kappa_-%
\xx p_- \vartheta}
\langle p_1,-p_2|O^A(\kappa_+,\kappa_-)|p_1,-p_2\rangle 
(\xx p_-)^{1-d_A}~,
\end{eqnarray}
%-----------------------------------------------------------------------
where $d_q=1$ and $d_G=2$. Eq.~(\ref{eqFOU}) and the inverse Fourier 
transform
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqFOUI}
\langle p_1,-p_2|O^A(\kappa_+,\kappa_-)|p_1,-p_2\rangle (\xx p_-)^{1-d_A}
= \int_{1/\eta}^{-1/\eta}
d\vartheta'~\e^{-i\kappa_-' \xx p_- \vartheta'} f^A(\vartheta',\eta)
\end{eqnarray}
%-----------------------------------------------------------------------
are used to obtain evolution equations for $f^A(\vartheta,\eta;\mu^2)$.
To perform the $\kappa_-$--integral the dependence on this variable has 
to be made explicit in the anomalous dimensions $\gamma^{AB}$ to all
orders. The scalar operator matrix elements, Eq.~(\ref{eqSCAM}), do not 
depend  on the light cone mark $\kappa_+$ due to translation invariance 
on the  light--cone, which therefore can be set to zero in 
Eq.~(\ref{eqEV1}). Furthermore, the anomalous dimension $\gamma^{AB}$ 
obeys the re-scaling relation, cf.~\cite{BGR},
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqresc}
\gamma^{AB}(\kappa_+,\kappa_-,\kappa_+',\kappa_-';\mu^2) = \sigma^{d_{AB}}
\gamma^{AB}(\sigma\kappa_+, \sigma\kappa_-,\sigma\kappa_+',
\sigma\kappa_-')~,
\end{eqnarray}
%-----------------------------------------------------------------------
with $d_{AB} = 2 + d_A - d_B$. The anomalous dimension transforms to
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqAN1}
\int D\kappa'~\kappa_-^{d_B-d_A}
\gamma^{AB}\left(0,1,\frac{\kappa_+'}{\kappa_-},
\frac{\kappa_-'}{\kappa_-};\mu^2\right) &=& 
\int D\alpha~\kappa_-^{d_B-d_A}
\widehat{K}^{AB}(\alpha_1,\alpha_2;\mu^2) \nonumber\\  &=&
\kappa_-^{d_B-d_A}
\int_0^1 du (1-u) \int_0^1 d\xi  \nonumber\\
& & ~~~~~~~~~~\times
\widehat{K}^{AB}(\xi(1-u),(1-\xi)(1-u);
\mu^2) \nonumber\\
&=& \kappa_-^{d_B-d_A} \int_0^1 du  \widehat{\widehat{K}}^{AB}(u;
\mu^2)~,
\end{eqnarray}
%-----------------------------------------------------------------------
with  $\alpha_1=[1-(\kappa_+'+\kappa_-')/\kappa_-]/2=\xi(1-u),
\alpha_2=[1-(\kappa_+'-\kappa_-')/\kappa_-]/2=(1-\xi)(1-u)$.
The $\kappa_-$--integral yields
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqKAPM}
\int \frac{d\kappa_- \xx p_-}{2\pi}~\e^{i\kappa_- \xx p_-(\vartheta-u
\vartheta')} (\xx p_- \kappa_-)^{d_B - d_A} 
= \widetilde{O}^{AB}(u\vartheta' -
\vartheta) =
\left\{\begin{array}{ll} \delta(u \vartheta' -  \vartheta) &~
{\rm for}~A=B=q,G \\
\partial_u \delta(u \vartheta'-\vartheta) &~
{\rm for}~A=q,~B=G \\
\theta(u \vartheta'-\vartheta)/\vartheta  &~
{\rm for}~A=G,~B=q
\end{array} \right. \nonumber
\end{eqnarray}
%-----------------------------------------------------------------------
The following evolution equations for the distribution functions 
$f^A(\vartheta,\eta)$ are obtained~:
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqEV1}
\mu^2 \frac{d}{d\mu^2} f^A(\vartheta,\eta;\mu^2)
= \int_0^1 du \int_\vartheta^{-{\rm sign}(\vartheta)/\eta}
d\vartheta'~\widetilde{O}^{AB}(u\vartheta'-\vartheta)~\widehat{\widehat
{K}}^{AB}(u;\mu^2) f_B(\vartheta',\eta;\mu^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
The functions
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqANOMF}
\vartheta' \int_0^1 du~\widetilde{O}^{AB}(u\vartheta'-\vartheta)
\widehat{\widehat{K}}^{AB}(u;\mu^2) \equiv P^{AB}\left(\frac{\vartheta}
{\vartheta'},\mu^2\right)
\end{eqnarray}
%-----------------------------------------------------------------------
are the {\it forward} splitting functions, which are independent of 
$\eta$ resp. $x_{\PP}$, leading to
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqEV2}
\mu^2 \frac{d}{d\mu^2} f^A(\vartheta,\eta;\mu^2)
=  \int_\vartheta^{-{\rm sign}(\vartheta)/\eta}  \frac{d\vartheta'}
{\vartheta'} P^{AB}\left(\frac{\vartheta}{\vartheta'},\mu^2\right)
f_B(\vartheta',\eta;\mu^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
They act on the momentum fraction $\vartheta$ of the diffractive parton
densities. Note that the range of $\vartheta'$ is yet different from that
of the corresponding quantity for deep inelastic scattering. After
identifying $\vartheta = 2\beta$ taking the absorptive part, however, one
arrives at the twist--2 evolution equation
%-----------------------------------------------------------------------
\begin{eqnarray}
\label{eqEV4}
\mu^2 \frac{d}{d\mu^2} f^{D}_A(\beta,x_{\PP};\mu^2)
= \int_\beta^1 \frac{d\beta'}{\beta'} P_{A}^B\left(\frac{\beta}{\beta'};
\mu^2\right)
f_B^D(\beta',x_{\PP};\mu^2)~.
\end{eqnarray}
%-----------------------------------------------------------------------
In the case of deep inelastic
scattering the corresponding identification for the momentum fraction is
$z = x$. The dependence of the diffractive parton densities w.r.t. $\eta$,
or $x_{\PP}$, is {\it entirely} parametric and not changed under the 
evolution, which affects $\beta$. In the case of the twist--2 
contributions factorization proofs were given~\cite{FACT}, leading
to the same evolution equations,~(\ref{eqEV4}), which are confirmed by
the present derivation. For phenomenological applications to 
next-to-leading order, see e.g.~\cite{RSBJP}.

We expressed the Compton amplitude with the help of the light--cone 
expansion at short distances and applied this representation to the
process of deep--inelastic diffractive scattering using Mueller's
generalized optical theorem. This representation is {\it not} limited to
leading twist operators but can be extended to all higher twist
operators synonymously. The corresponding evolution equations for
the higher twist hadronic matrix elements, which depend on more momentum
fractions $\vartheta_i$ than one, transform analogously to the case
of twist--2 being outlined above. A central parameter 
$\kappa_+ = \frac{1}{n} \sum_{i=1}^n \kappa_i$ may be set to zero, and 
analogous re--scaling relations apply. By virtue of this also here 
forward evolution equations are derived. However, the connection of the 
momentum fractions $\vartheta_i$ to the outer kinematic parameters is 
less trivial than in the case of twist--2 due to the structure of the 
corresponding Wilson coefficients. This is the case also for deep 
inelastic scattering, cf. e.g.~\cite{BRRZ}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\setcounter{equation}{0}
\label{sec-5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1mm}
\noindent
The differential cross section of unpolarized deep--inelastic 
$ep$--diffractive
scattering is described by four structure functions for pure photon
exchange, which depend on the four kinematic variables, $x, Q^2, x_{\PP}$
and $t$. In the limit of vanishing target masses and $t \rightarrow 0$
only two structure functions contribute. In the case of hard diffractive
scattering the scaling violations of these structure functions can be
described perturbatively. This is possible in transforming the 
non--forward amplitude of the process via Mueller's optical theorem, by
rotating the final--state proton into an initial--state anti--proton.
The Compton--amplitude is calculated for the hadronic two--particle state
$\langle p_1,-p_2|$ and w.r.t. this state in the forward direction.
The diffractive parton densities are associated to the general 
two--variable distribution functions, which describe the hadronic matrix
element. The scaling variable of the diffractive parton densities, which
directly compares to the Bjorken variable $x$ in the deep--inelastic case,
is $\beta = x/x_{\PP}$. However, the Callan--Gross relation takes the
usual form $F_2^D(x,Q^2,x_{\PP}) = 2x F_1^D(x,Q^2,x_{\PP})$. Due to
the transformation of the problem, which is made possible applying 
Mueller's optical theorem, the anomalous dimensions are the same as for
forward scattering. We demonstrated this by an explicit calculation
in the case of the twist--2 operators, however, the same mechanism
applies also for higher twist operators using the light cone expansion.
In the case of the twist--2 contributions the effective momentum fraction,
$\vartheta = z_- +z_+/\eta$ may be identified with the variable $2\beta$
for diffractive scattering. The dependence of the parton densities on
$x_{\PP}$ is not affected by QCD--evolution, which acts on the variable
$\beta$.

\vspace{2mm}
\noindent
{\bf Acknowledgment.}~For discussions we would like to thank
J. Bartels, W. Buchm\"uller, J.~Dainton, B.~Geyer, P. Kroll, and G. Wolf.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{999}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%












%




%[3]
\bibitem{FL}
C. Adloff et al., H1 collaboration, Phys. Lett. {\bf B393} (1997) 452.
%------------------------------------------------------------------------
%
%[4]
\bibitem{EXP3}
H. Abramowicz and J. Dainton, J. Phys. {\bf G22} (1996) 911.
%------------------------------------------------------------------------
%
%[5]
\bibitem{BAKO}
J. Bartels and M. Kowalski, Eur. Phys. J. {\bf C19} (2001) 693.
%------------------------------------------------------------------------
%
%[6]
\bibitem{DL}
A. Donnachie and P.V. Landshoff, Phys. Lett. {\bf B191} (1987) 309;
Nucl. Phys. {\bf B286} (1987) 704;  Phys. Lett. {\bf B437} (1998) 408;
{\tt hep-ph/0105088} and references  quoted therein.
%------------------------------------------------------------------------
%
%[7]
\bibitem{SUR}
M. W\"usthoff and A.D. Martin, J. Phys. {\bf G24} (1999) R309;\\
A. Hebecker, Phys. Rep. {\bf 331} (2000) 1 and references therein.
%------------------------------------------------------------------------
%------------------------------------------------------------------------
%
%[9]
\bibitem{INGEL}
G. Ingelman and P. Schlein, Phys. Lett. {\bf B152} (1985) 256;\\
J. Bartels and G. Ingelman, Phys. Lett. {\bf B235} (1990) 175;\\
G. Ingelman, {\tt hep-ph/9912534}.
%------------------------------------------------------------------------
%
%[10]
\bibitem{BH}
W. Buchm\"uller and A. Hebecker, Nucl. Phys. {\bf B476} (1996) 293;\\
W. Buchm\"uller, T. Gehrmann, and A. Hebecker, Nucl. Phys. {\bf B537} 
(1999) 477.
%------------------------------------------------------------------------
%
%[11]
\bibitem{EBKW}
J. Bartels, J. Ellis, M. Kowalski, and M. W\"usthoff,  
Eur. Phys. J. {\bf C7} (1999) 443.
%------------------------------------------------------------------------
%
%[12]
\bibitem{FS}
D.~M\"uller, D. Robaschik, B. Geyer, F.~Dittes, and J.~Ho\v{r}ej\v{s}i,
Fortschr. Phys.  {\bf 42} (1994) 2; \\
X. Ji,  J. Phys. {\bf G24} (1998) 1181.
%------------------------------------------------------------------------
%
%[13]
\bibitem{RAD}
A.V.~Radyushkin, Phys. Rev. {\bf D56} (1997) 5524; {\tt hep-ph/0101225}.
%------------------------------------------------------------------------
%
%[14]
\bibitem{BGR}
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, Nucl. Phys. {\bf B560} (1999) 
283.
%------------------------------------------------------------------------
%
%[15]
\bibitem{KROLL}                           
M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Phys. Lett. {\bf B460}
(1999) 204; Eur. Phys. J. {\bf C8} (1999) 409;\\
P. Kroll, Nucl. Phys. {\bf A666} (2000) 3.
%------------------------------------------------------------------------
%
%[16]
\bibitem{LAT}
G. Martinelli and C.T. Sachrajda, Phys. Lett. {\bf B190} (1987) 151;
{\bf 196} (1987) 184;\\
M. Guagnelli, K. Jansen, and R. Petronzio, Phys. Lett. {\bf B459} (1999)
594; {\bf B493} (2000) 77.
%------------------------------------------------------------------------
%
%[18]
\bibitem{TARR}
B. Tarrach, Nuov. Cim. {\bf 28A} (1975) 409.
%------------------------------------------------------------------------
%
%[20]
\bibitem{BR}
J. Bl\"umlein and D. Robaschik, Nucl. Phys. {\bf B581} (2000) 449.
%------------------------------------------------------------------------
%
%[21]
\bibitem{AHM}
A.H. Mueller, Phys. Rev. {\bf D2} (1970) 2963; Phys. Rev. {\bf D4} (1971)
150;\\
P.D.P. Collins, {\sf An Introduction to Regge Theory and High Energy
Physics}, (Cambridge University Press, Cambridge, 1977), pp.~331.
%------------------------------------------------------------------------
%
%[22]
\bibitem{PARA}
S. Weinberg, {\sf The Quantum Theory of Fields}, {\bf Vol.~1}, (Cambridge
University Press, Cambridge, 1995), pp.~452;\\
J. Bl\"umlein, J. Eilers, B. Geyer, and D. Robaschik, to appear.
%------------------------------------------------------------------------
%
%[23]
\bibitem{NFLO}
T.~Braunschweig, B.~Geyer, and D.~Robaschik, Ann. Phys. (Leipzig) 
{\bf 44} (1987) 407;\\
I.I.~Balitsky and V.M.~Braun, Nucl. Phys. {\bf B311} (1988/89) 541;\\
A.V.~Radyushkin,
Phys. Lett. {\bf B385} (1996) 333; Phys. Rev. {\bf D56} (1997) 5524;\\
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, Phys. Lett. {\bf B406} (1997) 
161 and Erratum;\\
I.I. Balitsky and A.V. Radyushkin, Phys. Lett. {\bf B413} (1997) 114;\\
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, {\tt hep-ph/9711405}, in~:
Proc. of the Workshop~{\sf Deep Inelastic Scattering Off Polarized
Targets: Theory Meets Experiment}, eds.~J.~Bl\"umlein et al., (DESY,
Hamburg, 1997) DESY 97--200, p.~196.\\
X.~Ji, Phys. Rev. Lett. {\bf 78} (1997) 610; Phys. Rev. {\bf D55} (1997)
7114;\\
L. Mankiewicz, G. Piller, and T. Weigl, Eur. J. Phys. {\bf C5} (1998) 
119.
%------------------------------------------------------------------------
%
%[24]
\bibitem{NFNLO}
F.M. Dittes and A.V. Radyushkin, Phys. Lett. {\bf B134} (1984) 359;\\
S.V. Mikhailov and A.V. Radyushkin, Nucl. Phys. {\bf B254} (1985) 89;\\
F.M.~Dittes, D.~M\"uller, D. Robaschik, B. Geyer, and J.~Ho\v{r}ej\v{s}i,
Phys. Lett. {\bf B209} (1988) 325;\\
A.V. Belitsky, D. M\"uller, L. Niedermaier, and A. Sch\"afer,
Nucl. Phys. {\bf B546} (1999) 279.
%------------------------------------------------------------------------
%
%[25]
\bibitem{FACT}
A. Berera and D.E. Soper, Phys. Rev. {\bf D50} (1994) 4328; {\bf D53}
(1996) 6162;\\
J. Collins, Phys. Rev. {\bf D57} (1998) 3051; Erratum: {\bf D61} (2000)
019902;\\
F. Hautmann, Z. Kunszt, and D.E. Soper, Nucl. Phys. {\bf B563} (1999) 
153.
%------------------------------------------------------------------------
%
%[26]
\bibitem{RSBJP}
C. Royon, L. Schoeffel, J. Bartels, H. Jung, and R. Peschanski, 
Phys. Rev. {\bf D63} (2001) 074004.
%------------------------------------------------------------------------
%
%[27]
\bibitem{BRRZ}
R.K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. {\bf B212}
(1983) 29;\\
J. Bl\"umlein, V. Ravindran, J. Ruan, and W. Zhu, Phys. Lett. {\bf B594}
(2001) 235.
%------------------------------------------------------------------------
\end{thebibliography}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ i \left[(p^\mu q^2 - \nu q^\mu) \varepsilon^{\nu\rho\alpha\beta} 
q_\rho S_\alpha p_\beta
-(p^\nu q^2 - \nu q^\nu) \varepsilon^{\mu\rho\alpha\beta}
q_\rho S_\alpha p_\beta\right] H_4   \nonumber\\ & &
+~\left[(p^\mu q^2 - \nu q^\mu) \varepsilon^{\nu\rho\alpha\beta}
q_\rho S_\alpha p_\beta
+(p^\nu q^2 - \nu q^\nu) \varepsilon^{\mu\rho\alpha\beta}
q_\rho S_\alpha p_\beta\right] H_5~,

hep-ph0202077.extracted_bib
\begin{thebibliography}{999}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%[1]
\bibitem{EXP1}
M. Derrick et al., ZEUS collaboration, Phys. Lett. {\bf B315} (1993) 481;
\\
T. Ahmed et. al., H1 collaboration, Nucl. Phys. {\bf B429} (1994) 477;\\
J. Breitweg et al., ZEUS collaboration, Eur. Phys. J. {\bf C6} (1999) 43;
\\
C. Adloff et al., H1 collaboration, Z. Phys. {\bf C76} (1997) 613.
%------------------------------------------------------------------------
%
%[2]
\bibitem{SXPOL}
R. Holt et al., EICC, {\sf A Whitepaper on the Electron Ion Collider},
March 2001;\\
see e.g. {\sf Workshop on the Prospects of Spin Physics at HERA},
DESY 95--200, eds.~J. Bl\"umlein and W.-D. Nowak;\\
{\sf Deep Inelastic Scattering off Polarized Targets: Theory Meets
Experiment}, DESY 97--200, eds.~J. Bl\"umlein et al.
%------------------------------------------------------------------------
%
%[3]
\bibitem{BRDIFF}
J. Bl\"umlein and D. Robaschik, Phys. Lett. {\bf B517} (2001) 222.
%------------------------------------------------------------------------
%
%[4]
\bibitem{FACT}
A. Berera and D.E. Soper, Phys. Rev. {\bf D50} (1994) 4328; {\bf D53}
(1996) 6162;\\
J. Collins, Phys. Rev. {\bf D57} (1998) 3051; Erratum: {\bf D61} (2000)
019902;\\
F. Hautmann, Z. Kunszt, and D.E. Soper, Nucl. Phys. {\bf B563} (1999) 
153.
%------------------------------------------------------------------------
%
%[5]
\bibitem{BK}       
J. Bl\"umlein and N. Kochelev, Phys. Lett. {\bf B381} (1996) 296;
Nucl. Phys. {\bf B498} (1997) 285;\\
J. Bl\"umlein and A. Tkabladze, Nucl. Phys. {\bf B553} (1999) 427;
{\tt hep-ph/9905524}.
%------------------------------------------------------------------------
%
%[6]
\bibitem{PRAD}
D. Bardin, J. Bl\"umlein, P. Christova, and L. Kalinovskaya,
Nucl. Phys. {\bf B506} (1997) 295;\\
A. Arbuzov, D. Bardin, J. Bl\"umlein,  L. Kalinovskaya, and T. Riemann,
{\sf Code HECTOR}, Comp. Phys. Commun. {\bf 94} (1996) 128,
{\sf hep-ph/9511434}.
%------------------------------------------------------------------------
%[7]
\bibitem{WW}
S. Wandzura and F. Wilczek, Phys. Lett. {\bf B72} (1977) 95.
%------------------------------------------------------------------------
%
%[8]
\bibitem{COL}
J.C. Collins, {\tt hep-ph/9705393}.
%------------------------------------------------------------------------
%
%[9]
\bibitem{POM}
For recent surveys see:\\
M. W\"usthoff and A.D. Martin, J. Phys. {\bf G24} (1999) R309;\\
W. Buchm\"uller, {\tt hep-ph/9906546};\\
G. Ingelman, {\tt hep-ph/9912534};\\
J. Bartels and M. Kowalski, Eur. Phys. J. {\bf C19} (2001) 693.
%------------------------------------------------------------------------
%
%[10]
\bibitem{THOOFT}
G. t'Hooft, Phys. Rev. Lett. {\bf 37} (1976) 8.
%------------------------------------------------------------------------
%
%[11]
\bibitem{THETA}
see e.g. S. Aoki and T. Hatsuda, Phys. Rev. {\bf D45} (1992) 2427.
%------------------------------------------------------------------------
%
%[12]
\bibitem{TM}
R.D. Tangerman and P.J. Mulders, Phys. Rev. {\bf D51} (1995) 3357.
%------------------------------------------------------------------------
%
%[13]
\bibitem{DJT}
D.A. Dicus, R. Jackiw, and V.L. Teplitz, Phys. Rev. {\bf D4} (1971) 1733.
%------------------------------------------------------------------------
%
%[14]
\bibitem{SCHOUT}
M. Veltman, Nucl. Phys. {\bf B319} (1989) 253;\\
B. De Wit and J. Smith, {\sf Field Theory in Particle Physics},
(North Holland, Amsterdam, 1986), p.~414.
%------------------------------------------------------------------------
%
%[15]
\bibitem{BARY}
J. Bartels, T. Gehrmann, and M. Ryskin, Eur. Phys. J. {\bf C11} (1999)
325.
%------------------------------------------------------------------------
%
%[16]
\bibitem{BGR}
J.~Bl\"umlein, B.~Geyer, and D.~Robaschik, Nucl. Phys. {\bf B560} (1999) 
283.
%------------------------------------------------------------------------
%
%[17]
\bibitem{BR}
J. Bl\"umlein and D. Robaschik, Nucl. Phys. {\bf B581} (2000) 449.
%------------------------------------------------------------------------
%
%[18]
\bibitem{SPLIT}
H. Ito, Progr. Theor. Phys. {\bf 54} (1975) 555;\\
K. Sasaki, Progr. Thepr. Phys. {\bf 54} (1975) 1816;\\
M. A. Ahmed and G.G. Ross, Nucl. Phys. {\bf B111} (1976) 441;\\
G. Altarelli and G. Parisi, Nucl. Phys. {\bf B126} (1977) 298;\\
R. Mertig  and W.L. van Neerven, Z. Phys. {\bf C70} (1996) 637;\\
W. Vogelsang, Nucl. Phys. {\bf B475} (1996) 47.
%------------------------------------------------------------------------
%
%[19]
\bibitem{SPNF}
J. Bl\"umlein, B. Geyer and D. Robaschik, Phys. Lett. {\bf B406} (1997) 
161;\\
I. Balitsky and A. Radyushkin, Phys. Lett. {\bf B413} (1997) 114;\\
For the near-forward representation: X. Ji, Phys. Rev. {\bf D55}
(1997) 7114.
%------------------------------------------------------------------------
%
%[20]
\bibitem{AHM}
A.H. Mueller, Phys. Rev. {\bf D2} (1970) 2963; Phys. Rev. {\bf D4} (1971)
150;\\
P.D.P. Collins, {\sf An Introduction to Regge Theory and High Energy
Physics}, (Cambridge University Press, Cambridge, 1977), pp.~331.
%------------------------------------------------------------------------
%
%[21]
\bibitem{PARA}
J. Bl\"umlein, J. Eilers, B. Geyer, and D. Robaschik, 
{\tt hep-ph/0108095}.
%------------------------------------------------------------------------
%
%[22]
\bibitem{TRAC}
B. Geyer and M. Lazar, Nucl. Phys. {\bf B559} (1999) 339;  {\bf B581}
(2000) 341;\\
B. Geyer, M. Lazar, and D. Robaschik, Nucl. Phys. {\bf B618} (2001) 99.
%------------------------------------------------------------------------
%
%[23]
\bibitem{BRRZ}
R.K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. {\bf B212}
(1983) 29;\\
J. Bl\"umlein, V. Ravindran, J. Ruan, and W. Zhu, Phys. Lett. {\bf B594}
(2001) 235.
%------------------------------------------------------------------------
\end{thebibliography}