text:
\section{Introduction} \renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}

\vspace{1mm}
\noindent
The process of deep--inelastic diffractive lepton--nucleon scattering
can be measured at high energy
colliders and constitutes a large fraction of the inclusive statistics,
although being a
semi--inclusive process. It was first observed at the electron--proton collider HERA some
years ago~\cite{Derrick:1993xh} and is now  measured in detail~\cite{Derrick:1996ma}.
The structure function $F_2^D(x,Q^2)$ was extracted. In the same manner it is desirable to compare
the longitudinal
diffractive structure function $F_L^D(x,Q^2)$  with the
longitudinal structure function in the inclusive case~\cite{Adloff:1996yz,FLH1}. The
measurement
of the polarized diffractive structure functions $g_{1,2}^D(x,Q^2)$ will be possible at
future facilities like EIC \cite{EIC}, which are currently planned.
The experimental measurements clearly showed that the scaling violations of the
deep-inelastic and the diffractive structure functions in the deep-inelastic regime,
after an appropriate change of kinematic variables, are the same. Furthermore, the ratio
of the two quantities, did not vary strongly, cf. \cite{Abramowicz:1996ha}. While the
former property is clearly of perturbative nature, the latter is of non--perturbative
origin. For diffractive scattering, however, another mass scale is of importance, which is
given by the invariant mass $t = (p_2 - p_1)^2$. Here $p_{1(2)}$ denote the 4--momenta of
the incoming and outgoing proton, where for the latter a sufficiently large rapidity gap
between this particle and the remainder final state hadrons is demanded as process
signature. A similar class of processes are the so-called meson--exchange processes, cf.
e.g. \cite{Kopeliovich:1996iw}, where
the finite rapidity gap is not required, but the r\^{o}le of the formerly diffractive final
state proton is taken by a leading hadron, which distinguishes itself due to its high
momentum from the remaining hadrons. Also in this case one
may try a leading twist description, although the signature for this process is
less clear than in the diffractive case.

The process of deep--inelastic diffractive scattering was first described
phenomenologically \cite{PHEN}. A consistent field-theoretic description of the process
requires factorization for the twist--2 contributions \cite{Berera:1995fj}.
It is due to this description that reference to specific pomeron models are
thoroughly avoided. In Refs.~\cite{Blumlein:2001xf,Blumlein:2002fw,Blumlein:2002ax}
two of the present authors gave a corresponding field--theoretic description of the
process in the limit $t, M^2 \rightarrow 0$. In \cite{Blumlein:2001xf}
we proved that under these conditions the scaling violations
for diffractive scattering and inclusive deeply inelastic scattering are the same, up to
a change in the momentum-fraction variable in the former case.

At low 4--momentum transfer $Q^2$ both target mass $(M^2)$ and finite
momentum transfer
$(t)$ corrections have to be considered for the diffractive and leading hadron processes
with meson exchange. In the deep-inelastic case the target mass corrections
were studied in Refs.~\cite{Nachtmann:1973mr, Georgi:1976ve,Blumlein:1998nv,Piccione:1997zh},
see also \cite{TM2}. The kinematics of the diffractive and leading hadron
processes
is similar to that in deeply-virtual non--forward scattering. Considering this general
class of processes,
one finds that the treatment of target mass effects and finite $t$--effects can
only be performed by combining both, see~\cite{gey2,belm}. If compared with the
deep--inelastic case the number of hadronic structure functions enlarges in the
diffractive case from two to four for  unpolarized scattering and to eight for  polarized
scattering, as shown in \cite{Blumlein:2001xf,Blumlein:2002fw}, if the
general kinematics is considered. In Ref.~\cite{BGR2006} we worked out these corrections
for the hadronic tensor in general, yet without quantifying the result. If one departs
form the limit $t, M^2 \rightarrow 0$ the corresponding representations require to carry
out a one-dimensional {\sf definite} integral which kinematically relates the two proton
momenta $p_1$ and $p_2$. As the integration is to be performed over  unknown
non-perturbative functions there is no a priori experimental way to unfold the
non-perturbative  distributions, which also would invalidate the partonic description
in case of diffractive scattering. Moreover, the $M^2$ and $t$ effects dealt with in this
case are not yet complete, since there emerge other contributions more in the scattering
cross section. One may expand the complete solution in two variables
$t/Q^2, M^2/Q^2$. It is found that these terms multiply at least with a factor $x = x_{{\rm BJ}(\PP)}$,
which is bounded in the diffractive case to values below 0.01 and in the meson--exchange case
$\lessapprox 0.3$. Thus the leading terms beyond $t, M^2 = 0$ give a good first estimate for
the corrections. The further corrections turn out to be widely suppressed in the
diffractive case, while they are larger for leading particle cross sections in the
meson--exchange case.

In the present paper we will discuss both the unpolarized and polarized case.
The paper is organized as follows. In Section~2
we derive the differential scattering cross section for inclusive diffractive scattering
at the Lorentz level. Main aspects of the relation of this
process to the Compton
amplitude within the light--cone expansion including finite $M^2$ and $t$ effects are
summarized in Section~3. The hadronic tensors for the unpolarized and polarized case
are expanded in terms of the variables $t/Q^2,~~M^2/Q^2$ in Section~4 to show the size of
the correction terms. Section~5 contains
the conclusions. In Appendix~A we summarize some kinematic relations. The present formalism
is specified to the case of deep--inelastic forward scattering (DIS) in
Appendix~B, where we obtain the target mass corrections given in
\cite{Georgi:1976ve,Blumlein:1998nv,Piccione:1997zh}  before.
\section{The Lorentz Structure}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}

\vspace{1mm}
\noindent
The process of deep--inelastic diffractive scattering belongs to
the class of semi--inclusive processes. It is described by an effective $2 \rightarrow 3$
diagram, cf.~Figure~1 Ref.~\cite{Blumlein:2001xf}, with incoming and outgoing
charged lepton and nucleon lines and an effective 4-vector for all the other hadron lines
in the final state, which are well separated in rapidity from the outgoing diffractive nucleon
line.

The differential scattering cross section for single--photon exchange is given by
\begin{equation}
\label{eqD1}
 \text{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 $s=\kln{p_1 + l_1}^2$ is the cms energy squared of the process  and $M$
denotes
the nucleon mass. The phase space $d P S^{\kln{3}}$ depends on five variables
since the mass $M_X$ of the diffractively produced inclusive set of hadrons varies. We
choose as  basic variables
\begin{eqnarray}
 x_{\rm BJ} &=& \frac{Q^2}{Q^2 + W^2 - M^2} = - \frac{q^2}{2 \, q p_1} \; , \\
 y          &=& \frac{Q^2}{x_{\rm BJ}(s-M^2)}~,
\end{eqnarray}
$t=\kln{p_2 - p_1}^2$ the 4--momentum difference squared between incoming and outgoing
nucleon, a variable describing the non-forwardness w.r.t.~the incoming
proton direction,
\begin{equation}
\label{eqV1}
x_{\PP} = \frac{Q^2 + M_X^2 - t}{Q^2 + W^2 - M^2} = - \frac{q p_-}{q p_1}
\geq x_{\rm BJ}~,
\end{equation}
and the angle $\phi_b $ between the lepton plane $\pvec_1 \times \lvec_1 $ and
the hadron plane  $\pvec_1 \times \pvec_2$,
\begin{equation}
\label{eqV3}
\cos(\phi_b) = \frac{(\pvec_1 \times \lvec_1).(\pvec_1 \times  \pvec_2)}
                 {|\pvec_1 \times \lvec_1 ||\pvec_1 \times  \pvec_2|}~.
\end{equation}
Here $Q^2 = - q^2$ denotes
the photon virtuality  and
$W$ is the hadronic mass with $W^2 = \kln{p_1 + q}^2$. We also refer to $x = Q^2/qp_+$.
It is useful to introduce the 4--vectors
\begin{equation}
p_\pm = p_2 \pm p_1~.
\end{equation}
The diffractive mass squared is given by
$M_X^2 = \kln{q - p_-}^2$.
The momenta $p_\pm$ obey
\begin{equation}
(p_+~p_-) = 0, \qquad \frac{p_+^2}{p_-^2} = \frac{4 M^2}{t} - 1\,.
\end{equation}
For later use we refer to the non-forwardness $\eta$ and the variable $\beta$ defined by
\begin{equation}
 \eta = \frac{q p_-}{q p_+} = \frac{-x_{\PP}}{2-x_{\PP}}
 \in \kle{-1 \, , \, \frac{-x}{2-x} } \; ,
\qquad
  \beta  =   \frac{q^2}{2\,qp_-} =\frac{x_{\rm BJ}}{x_{\PP}} \leq 1\,.
\end{equation}
The variable $x_{\PP}$ is directly related to $\eta$ but is more commonly used
in
experimental analyzes,
\begin{equation}
x_{\PP} = \frac{2 \eta}{\eta -1}~.
\end{equation}
More kinematic invariants are given in Appendix~A.

The transverse momentum variable, introduced as $\hat\pi_-$, \cite{BGR2006}, or $\pi_- =
-\eta \hat\pi_- $   is of special importance,
\begin{equation}
\label{tmv}
\pi_- = p_- - {p_+}{\eta}, \qquad (q \pi_-) = 0\,.
\end{equation}
Later on it plays the role of an expansion parameter.
The variables $x_{\rm BJ}, x_\PP, \beta$ and $\eta$ obey the inequalities
\begin{align}
&0 \leq x_{\rm BJ} \leq x_\PP \leq 1, \qquad 0 \leq x_{\rm BJ} \leq \beta \leq 1,
\\
-\infty \leq 1& - \frac{2}{x_{\rm BJ}} \leq 1 - \frac{2 \beta}{x_{\rm BJ}} =
\frac{1}{\eta} \leq -1 \leq \eta
\leq \frac{-x_{\rm BJ}}{2-x_{\rm BJ}} \leq 0~.
\end{align}

For the spin averaged cross section, the leptonic tensor is symmetric.
Taking into account conservation of the electromagnetic current one obtains
\cite{Blumlein:2001xf}
\begin{eqnarray}
\label{eqD2}
 W_{\mu\nu}^s
&=&
  -g_{\mu\nu}^T  W_1^s +  p_{1\mu}^T p_{1\nu}^T \frac{W_2^s}{M^2}
   + p_{2\mu}^T  p_{2\nu}^T \frac{W_4^s}{M^2}
   + \kle{ p_{1\mu}^T  p_{2\nu}^T +  p_{2\mu}^T  p_{1\nu}^T } \frac{W_5^s}{M^2} \; .
\end{eqnarray}
Here and in the following we do not assume that azimuthal
integrals are performed as sometimes is done in experiment.
In the latter case the number of contributing structure function reduces.

In the case of polarized nucleons we consider the initial state spin--vector
$S_1\equiv S,~S^2 = -M^2$,  only and sum over the spin of the outgoing
hadrons. One usually refers to the longitudinal $(||)$ and transverse
($\perp$) spin projections choosing
\begin{eqnarray}
S_{||}    &=& (\sqrt{E^2-M^2};0,0,0,E)~, \\
S_{\perp} &=& (0;\cos\gamma,\sin\gamma,0) M~,
\end{eqnarray}
in the laboratory frame with $p_1 = (E;0,0,\sqrt{E^2-M^2})$, with $S.p_1 = 0$.
Here $\gamma$ denotes the azimuthal angle. In the case of
longitudinal polarization the contraction of $S_{||}$ with $l_1$ and $p_2$ being nearly
collinear to $p_1$ are of ${\cal O}(\mu^2/Q^2),~~\mu^2 = |t|, M^2$, see Appendix A.

The antisymmetric part of the hadronic tensor was derived in
\cite{Blumlein:2002fw} and is given by
\begin{alignat}{10}
\label{eqH2}
W_{\mu\nu}^a &=&~~
i \left[ {p}_{1\mu}^T {p}_{2\nu}^T-
{p}_{1\nu}^T {p}_{2\mu}^T \right] \varepsilon_{p_1 p_2 q S}
&\frac{W_1^a}{M^6}&~~
                &+&~~
i \left[ {p}_{1\mu}^T \varepsilon_{\nu S p_1 q}
     - {p}_{1\nu}^T \varepsilon_{\mu S p_1 q} \right]
&\frac{W_2^a}{M^4}&
\nonumber \\ &+&~~
i \left[ {p}_{2\mu}^T \varepsilon_{\nu S p_1 q}
     - {p}_{2\nu}^T \varepsilon_{\mu S p_1 q} \right]
&\frac{W_3^a}{M^4}&~~
                &+&~~
i \left[ {p}_{1\mu}^T \varepsilon_{\nu S p_2 q}
     - {p}_{1\nu}^T \varepsilon_{\mu S p_2 q} \right]
&\frac{W_4^a}{M^4}&
\nonumber \\ &+&~~
i \left[ {p}_{2\mu}^T \varepsilon_{\nu S p_2 q}
     - {p}_{2\nu}^T \varepsilon_{\mu S p_2 q} \right]
&\frac{W_5^a}{M^4}&~~
                &+&~~
i \left[ {p}_{1\mu}^T {\varepsilon}_{\nu p_1 p_2 S}^T
     - {p}_{1\nu}^T {\varepsilon}_{\mu p_1 p_2 S}^T \right]
&\frac{W_6^a}{M^4}&
\nonumber\\ &+&~~
i \left[ {p}_{2\mu}^T {\varepsilon}_{\nu p_1 p_2 S}^T
     - {p}_{2\nu}^T {\varepsilon}_{\mu p_1 p_2 S}^T \right]
&\frac{W_7^a}{M^4}&~~
                &+&~~   i\; \varepsilon_{\mu \nu q S}
&\frac{W_8^a}{M^2}&~,
\end{alignat}
where $\varepsilon_{\mu\nu\alpha\beta}$ denotes the Levi--Civita symbol.
The kinematic factors above are constructed out of the
four--vectors $q,p_1,p_2$ and $S$ as well as $g_{\mu\nu}$
and $\varepsilon_{v_0 v_1 v_2 v_3}$ using
\begin{eqnarray}
\label{trans}
p_\mu^T &=& p_\mu - q_\mu \frac{q.p}{q^2},\,\,\,
  g_{\mu\nu}^T=  g_{\mu\nu} - \frac{q_\mu q_\nu}{q^2},
\\
{\varepsilon}_{\mu v_1 v_2 v_3}^T            &=&
    {\varepsilon}_{\mu v_1 v_2 v_3}            -
    {\varepsilon}_{q v_1 v_2 v_3} \frac{q_\mu}{q^2}~,  \\
{\varepsilon}_{\mu \nu v_1 v_2}^{TT}            &=&
    {\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}
One may rewrite (\ref{eqH2}) into an equivalent form using the
Schouten-identities~\cite{SCHOUT}.

Target mass and finite $t$ corrections to the differential scattering cross
section (\ref{eqD1}) in the leading twist approximation emerge from three sources:~~$(i)$ from
kinematic terms at the Lorentz level after contracting the leptonic and hadronic tensor;
~~$(ii)$ from the expectation value of the Compton operator; ~~$(iii)$
the $t$-behaviour of the non--perturbative distribution functions.

We will first consider the contributions $(i)$ and discuss the terms $(ii)$ in Section~4.
The non-perturbative effects cannot be calculated by rigorous methods within Quantum
Chromodynamics at present, but are left to phenomenological models or are determined
through fits to data, cf.~\cite{PHEN}.

For pure photon exchange the leptonic tensor is given by
\begin{eqnarray}
L_{\mu\nu} = 2(l_{1\mu} l_{2\nu} + l_{2\mu} l_{1\nu} - g_{\mu\nu} l_1.l_2 - i
\varepsilon_{\mu\nu\alpha\beta} l_1^\alpha q^\beta),
\end{eqnarray}
cf.~\cite{LR}, in case of longitudinal lepton polarization.

We consider the Bjorken limit,
\begin{eqnarray}
2 p_1.q = 2 M \nu
\rightarrow \infty,~~~~p_2.q \rightarrow \infty,~~~~Q^2
\rightarrow \infty,~~{\rm with}~~~~x_{\rm BJ}~~~{\rm and}~~~x_\PP = {\sf
fixed.}
\end{eqnarray}
Here,
\begin{eqnarray}
M W_1^s   &\rightarrow& F_1 \\
\nu W_k^s &\rightarrow& F_k,~~~k=2,4,5~,
\end{eqnarray}
with $\nu = y (s-M^2)/(2M)$.

In the unpolarized case we obtain in the limit
$M^2, t \rightarrow 0$ w.r.t. the kinematics of the momenta $p_1$ and $p_2$,
keeping the target mass dependence
\begin{eqnarray}
\label{xs1}
\frac{d^s \sigma^{\rm unpol}}{d x_{\rm BJ}\, d Q^2} = \frac{2 \pi \alpha^2}{Q^4 x_{\rm BJ}}
\left[2xF_1 \cdot y^2
                          + \left[F_2 + (1-x_\PP) F_4
                          + (1-x_\PP)^2 F_5\right]
\cdot 2\left(1-y - \frac{x^2_{\rm BJ} y^2 M^2}{Q^4}\right)\right],
\end{eqnarray}
where $F_k = F_k(x_{\rm BJ},x_\PP,Q^2;t)$ are the diffractive
structure functions, cf.~\cite{Blumlein:2001xf}. The correction terms are
of ${\cal O}(M^2/Q^2,~t/Q^2)$. In the limit $M^2, t \rightarrow 0$ the azimuthal
dependence on $\phi_b$ vanishes.

Likewise we obtain in the polarized case for longitudinal nucleon
polarization,
\begin{eqnarray}
\label{xs2}
\frac{d^3 \sigma^{\rm pol}(\lambda, \pm S_{||})}{d x_{\rm BJ} d Q^2 d x_\PP} &=& \mp 4 \pi s
\lambda \frac{\alpha^2}{Q^4} \left[y \left(2-y - \frac{2x_{\rm BJ}yM^2}{s}\right) xg_1
- 4x_{\rm BJ}y \frac{M^2}{s} g_2\right]~,\\
\label{xs3}
\frac{d^4 \sigma^{\rm pol}(\lambda, \pm S_{\perp})}{d x_{\rm BJ} d Q^2 d x_\PP. d \Phi} &=& \mp 4
\pi s
\lambda \sqrt{\frac{M^2}{s}} \frac{\alpha^2}{Q^2} \sqrt{x_{\rm BJ}y \Bigl[1-y- \frac{x_{\rm BJ}y
M^2}{s}\Bigr]}
\cos(\gamma - \Phi) \left[yx_{\rm BJ} g_1 + 2 x_{\rm BJ} g_2\right]~.
\nonumber\\
\end{eqnarray}
Here $\phi$ denotes the angle between the $\vec{l}_1-\vec{S}$ and the $\vec{l}_1-\vec{l}_2$
plane and $\alpha$ is the anle between $\vec{l}_1$ and $\vec{S}$.
The structure functions $g_{1,2}(x_{\rm BJ},x_\PP,Q^2;t)$ are obtained from $W_2^a, W_3^a, W_4^a,
W_5^a$ and $W_8^a$ by
\begin{eqnarray}
g_1 &=& \frac{p.q_1}{M^2} W_8^a~,
\\
g_2 &=& \frac{(p.q_1)^3}{q^2 M^4} \left[W_2^a + (1-x_\PP)[W_3^a + W_4^a] + (1-x_\PP)^2
W_5^a \right]
\end{eqnarray}
and the different structure functions $F_i$ and $g_i$ depend on the variables
$x_{\rm BJ},x_\PP,Q^2$ and $t$.

\section{The Compton Amplitude}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}

\vspace{1mm}
\noindent
The hadronic tensor for deep--inelastic diffractive scattering can be obtained from a
Compton amplitude as has been outlined in \cite{Blumlein:2001xf,Blumlein:2002fw,BGR2006}
before. We limit the description to the level of the twist--2 contributions, where
factorization holds for the semi--inclusive diffractive process \cite{Berera:1995fj}.
Furthermore, A.~Mueller's generalized optical theorem \cite{Mueller:1970fa}
allows to move the final state proton into an initial state anti-proton, where
both particle momenta are separated by $t$ and form a formal `quasi
two--particle' state $|p_1,-p_2,S; t\rangle$. These states are used to form the operator
matrix elements.
The correctness of this procedure within the
light--cone expansion relies, first, on the rapidity gap between
the outgoing proton and the remaining hadronic part with invariant
mass $M_X$ and, second, on the special property of matrix elements
of the contributing light-cone operators to contain no absorptive part.
Independently, one could argue that the corresponding matrix element is a
pure phenomenological quantity satisfying restrictions imposed by quantum field theory.
The general structure of the scattering amplitude is completely determined by the
off-cone structure of the twist-2 Compton operator (\ref{FTampl}), cf. \cite{gey1}.

The structure functions for the diffractive process
can thus be obtained by analyzing the absorptive part
\begin{equation}
W_{\mu\nu} ={\sf Im} T_{\mu\nu}
\end{equation}
of the expectation value
\begin{equation}
\label{matr}
 T_{\mu\nu}\kln{x} = \Matel{p_1,-p_2, S; t}{\widehat T_{\mu\nu}(x)}{p_1,-p_2,S; t}\,,
\end{equation}
with the well-known operator $\widehat T_{\mu\nu}$ of (virtual) Compton scattering
 defined as
\begin{equation}
\label{int_input}
 \widehat T_{\mu\nu}(x) \equiv i R \, T\kle{J_\mu\kln{\frac{x}{2}} \,
                                    J_\nu\kln{-\frac{x}{2}} {\cal S}} \, .
\end{equation}
In \cite{BGR2006}, based on a general quantum field theoretic consideration of virtual
Compton scattering at twist 2\cite{Geyer:2004by,gey2,Geyer:1999uq}, we specified
the various terms which contribute to the general structure  of the hadronic tensor
$W_{\mu\nu}= {\sf Im} T_{\mu\nu}$ in case of deep--inelastic diffractive scattering.
As shown in \cite{Blumlein:1999sc,Blumlein:2001sb} the operator $\widehat T_{\mu\nu}$
in lowest order of the non--local light--cone expansion \cite{LCE} contains the
vector or axial vector operators only.
The scattering amplitude is obtained by the Fourier transform of the operator
$\widehat T_{\kls{\mu\nu}}\kln{x}$ and forming the matrix element (\ref{matr}).
Here, we want to study its twist--2  contributions including  target mass
and finite momentum transfer corrections. This is obtained by {\sf harmonic extension}
\cite{Geyer:1999uq,BT,Eilers:2004mp} of the twist--2 light-cone operators
to twist--2 off-cone operators
\cite{Geyer:2001qf}, leading to
\begin{align}
\label{FTampl}
    \widehat T_{\mu\nu}^{ {\text{tw2}} }\kln{q}
         &=
   -\,e^2 \int \frac{\text{d}^4 \! x}{2\im\pi^2} \;
      \frac{\e^{\im qx}\,x_{\lambda} }{\kln{x^2- \im\epsilon}^2}
   \left\{ {S_{\mu\nu |}}^{ \alpha\lambda }\,
    O_\alpha^ {\text{tw2}} (\kappa x ,- \kappa x )
     {+} {\epsilon_{\mu\nu}}^{\alpha\lambda}\,
    O_{5\,\alpha}^{ {\text{tw2}} }(\kappa x ,-\kappa x )
    \right\} \,,

\end{align}
{
with
\begin{eqnarray}
   O_\alpha^ {\text{tw2}} (\kappa x ,- \kappa x )&=&
     i[\overline \psi (\kappa x)\gamma_\alpha \psi(-\kappa x)
     -[\overline \psi (-\kappa x)\gamma_\alpha \psi(\kappa x)]^ {\text{tw2}} , \nonumber \\
   O_{5\,\alpha}^ {\text{tw2}} (\kappa x ,- \kappa x )&=&
     [\overline \psi (\kappa x)\gamma_5 \gamma_\alpha \psi(-\kappa x)
     +[\overline \psi (-\kappa x)\gamma_5 \gamma_\alpha \psi(\kappa x)]^ {\text{tw2}} , \nonumber
\end{eqnarray}
and }
$\kappa =1/2$. The matrix elements can be written in terms of vectors
$ {\mathcal{K}} ^a_{\mu,(5)}$ and 2--dimensional Fourier-integrals over partonic distributions
$f_{a(5)}(z_+,z_-,t)$ summing over $a$,
\begin{align}
\label{non1}
\langle p_1,-p_2;t|\,e^2\,O_{\mu}^{ {\text{tw2}} }(\kappa  x,-\kappa   x)\,|p_1,-p_2;t\rangle
& =  {\mathcal{K}} ^a_{\mu}(p_\pm) \!\int \!\frac{D{\mathbb Z}}{(2\pi)^4}\,
    e^{i\kappa  x( p_-z_- + p_+ z_+)}\,
    f_{\,a}(z_+ ,z_-,t)\,,\\
\label{non15}
\langle p_1,-p_2,S;t|\,e^2 \,O_{5\mu}^{ {\text{tw2}} }(\kappa x, -\kappa  x)\,|p_1,-p_2,S;t\rangle
& =   {\mathcal{K}} ^a_{5\,\mu}(p_\pm,S) \!\int\! \frac{D{\mathbb Z}}{(2\pi)^4}\,
    e^{i\kappa  x( p_-z_- + p_+ z_+)}\,
    f_{5\,a}(z_+ ,z_-,t)\,, \nonumber  \\
\end{align}
which is defined as asymptotic expression on the light-cone at $x^2=0 $.

We choose as kinematic factors for the representation of the
matrix element of the non-local operator  for the symmetric part (\ref{non1})
\begin{eqnarray}
\label{kinsym+}
 {\mathcal{K}} ^{1 \, \mu} = p_+^\mu\,, \qquad \qquad
 {\mathcal{K}} ^{2 \, \mu} = \pi_-^\mu \equiv p^\mu_- - {\eta}{p^\mu_+}\,,
\qquad
\end{eqnarray}
and for its antisymmetric part (\ref{non15})
\begin{eqnarray}
\label{kinasym+}
 {\mathcal{K}} ^{1\, \mu}_{5} = S^\mu\,, \qquad
 {\mathcal{K}} ^{2\, \mu}_{5} = p_+^\mu \,{(p_2 S)}/{M^2}\,, \qquad
 {\mathcal{K}} ^{3\, \mu}_{5} = \pi_-^\mu\,{(p_2 S)}/{M^2}\,.
\end{eqnarray}
The normalization to $M^2$ in  (\ref{kinasym+})
is arbitrary and has to be arranged with the definition of the corresponding
distribution
functions $ f_{a,(5\,a)}(z_+,z_-)$, respectively. The corresponding
Lorentz-invariant has to be formed out of the hadronic momenta, except the
spin vector, since the polarization--symmetries are assumed to be linear in
the spin.

The momentum fractions $z_{\pm}$ in (\ref{non1}, \ref{non15}) corresponding to the
momenta $p_\pm$ are
\begin{equation}
\mathbb P = (p_+, p_-)=(p_2+p_1,p_2-p_1), \,\quad
\mathbb Z = (z_+, z_-)=((z_2+z_1)/2,(z_2-z_1)/2)\,,
\end{equation}
with the measure $ D{\mathbb Z}$
\begin{eqnarray}
 D{\mathbb Z} = 2\, dz_+ dz_-\,
 \theta(1-z_++z_-)\,\theta(1+z_+-z_-)\,
 \theta(1-z_+-z_-)\,\theta(1+z_++z_-)\,.
 \end{eqnarray}

We refer to $f_{a(5)}(z_+,z_-,t)$ as {\sf diffractive generalized parton distribution
functions, (dGPD),} in distinction to the GPDs emerging in deeply virtual Compton
scattering~\cite{MRGDJ}. These amplitudes are directly connected to the total cross sections
and polarization asymmetries, respectively. Both kinds of GPDs are expectation values of
the same light--cone operator, however, between different states.
Interesting limiting cases can be derived from them.
For the dGPDs these are the quasi collinear limit: $\pi_-
\rightarrow 0, M^2 \rightarrow 0$, \cite{Blumlein:2001xf,Blumlein:2002fw}, and the limit
of deep--inelastic scattering, see Appendix~B. Furthermore, for both types of GPDs the
evolution equations are derived from the renormalization group equation for the {\sf same}
light--cone operators. It is remarkable, that the evolution equations for the dGPDs are
two--variable equations which reduce to the simple evolution equation for forward
scattering in the quasi collinear limit, cf.~\cite{Blumlein:2001xf}.

The (dimensionless) amplitudes $f_{(5)\,a}(z_+ ,z_-,t)$  depend
on $t$ and $\eta$ explicitly.
In addition, there appears a $t$-- and $M^2$--dependence
of the amplitude (\ref{matr}) in momentum space, which finally, on the one hand,
results from the Fourier transform in (\ref{FTampl}) where the operator
$ O_{(5)\,\alpha}^{ {\text{tw2}} }(\kappa x ,-\kappa x )$ is off the light-cone, i.e.~with all
trace subtractions. On the other hand, the dependence
results from the kinematic
pre--factors
$ {\mathcal{K}} ^a_{(5)\,\mu}(p_\pm,S)$.\footnote{
In the following the explicit $t$--dependence of the distribution
functions is always understood and we drop this variable to lighten the
notation.}

Concerning the independent kinematic factors one has
two possibilities, which are  mathematically equivalent, depending on whether one chooses
$p_-$ or
$p_+$ as essential variable as we did in our previous papers \cite{BGR2006} and
\cite{gey2}, respectively. The corresponding choices lead to different dGPDs.
\smallskip

\noindent
(1) In the first case, which we considered in \cite{BGR2006}, cf. also
\cite{Blumlein:2001xf} and   \cite{Blumlein:2002fw},
 $p_-$ was chosen as essential variable,  by starting from
the physical picture using the generalized optical theorem,
and the parameterization~\footnote{
For later convenience the notation $(\vartheta, \zeta)$ of Ref.~\cite{BGR2006} has been
changed into $(\hat\lambda,\hat\zeta)$.}
\begin{eqnarray}
 p_- z_- + p_+ z_+ &= \hat\lambda\,[p_- + \hat\zeta(p_+ - p_-/{\eta}) ]
                   = \hat\lambda\,[p_- + \hat\zeta \hat\pi_-]
                   \equiv \hat\lambda \,\hat{\cal P},
\end{eqnarray}
{ with}
\begin{eqnarray}
  \hat\lambda &=& z_- + z_+/{\eta}~, \nonumber\\
   z_+ &=& \hat\lambda \,\hat\zeta~,  \nonumber\\
   z_- &=& \hat\lambda\, (1 - \hat\zeta /{\eta})~.
\end{eqnarray}
\smallskip

\noindent
(2) A mathematically equivalent description is obtained starting from $p_+$
as the essential variable \cite{gey2}.
In this approach we introduce the new variables $\lambda$ and $\zeta$
instead of $z_+$ and $z_-$,
\begin{align}
\label{z+}
 p_- z_- + p_+ z_+ &= \lambda\,[\, p_+ + \zeta\,(p_- - {\eta} \,p_+)] =
                   \lambda\, (p_+ + \zeta\, \pi_-)
                   \equiv \lambda\, {\cal P} { = 2\,\Pi\,},
\end{align}
{with}
\begin{eqnarray}
   \lambda  &=&  z_+ + {\eta} z_-~, \nonumber\\
   z_- &=& \lambda \zeta~, \nonumber\\
   z_+ &=& \lambda (1 - \zeta {\eta})~.
\end{eqnarray}
Here the variable $\lambda $  plays the role
of a {\sf common} scale for $z_\pm $.
Compared to Ref.~\cite{gey2} we list the essential kinematic variables using the
above parameterization
\begin{eqnarray}
\label{cP2}
    {\cal P}(\eta , \zeta) &=& p_+ (1 - \eta\,\zeta) + p_- \zeta~, \\
    {\cal P}^2 &=& p_+^2 - 2\,\zeta\,\eta p_+^2  + \zeta^2 (p_-^2 +p_+^2 \eta^2),
\\
     q{\cal P} &=& \, qp_+ ,\nonumber\\
    {\cal P}^2/ ({\cal P}^{\mathrm T})^2 &=& x^2 ({\cal P}^2/Q^2)\,/\left[1 + x^2 ({\cal
P}^2/Q^2)\right],
\end{eqnarray}
and
\begin{eqnarray}
\label{xicd}
 \xi_\pm
 =  \frac{2x}{1\pm \sqrt{1 + x^2 \mathcal{P}^2/Q^2}}\,,
 \quad x=\frac{Q^2}{qp_+} = \frac{Q^2}{2qp_1}(1-\eta) = x_{\rm BJ}(1-\eta) = - 2\,
\beta\,\eta.
\end{eqnarray}
Obviously, $\xi_+ \equiv \xi $ is the appropriate generalization of the Nachtmann variable.
With these definitions the measure of the $\mathbb Z$--integration  is
\begin{align}
\label{varlz}
 D{\mathbb Z}
 = 2|\lambda|\, d\lambda\, d\zeta \,
 &\theta\big(1-\lambda+(1+\eta)\lambda\,\zeta \big)\,
 \theta\big(1+\lambda- (1+\eta)\lambda\,\zeta\big) \nonumber \\
 \times\,
 &\theta\big(1-\lambda-(1-\eta)\lambda\,\zeta \big)\,
 \theta\big(1+\lambda +(1-\eta)\lambda\,\zeta\big)\,~.
 \end{align}
In the present treatment we choose $p_+$ as the essential variable.

In Ref.~\cite{BGR2006}  deep--inelastic diffractive scattering has been worked out
within the first approach. The resulting expressions contain an internal
$\hat\zeta$--integral which is not well suited for the direct comparison of
experimental data with
the
diffractive GPDs. One way out is to introduce new `integrated distributions'.
Furthermore, we can perform a systematic $1/Q^2$ expansion which leads to an
expansion in terms of ${\cal P}^2/Q^2$ directly. Since ${\cal P}^2$ is a
polynomial of
second order
in the variable $\zeta $ we are led  to a $\zeta$-- expansion,
\begin{eqnarray}
 \hat{\cal P}^2 &=& t -2\,\hat\zeta\,t/\eta  + (4M^2 - t + t/\eta^2)\,
 \hat{\zeta}^2|_{\hat\zeta \rightarrow 0} = t  \,,   \\
 {\cal P }^2 &=& \hat{\cal P}^2/\,\eta^2 =
                  ( 4\,M^2 -t)(1 -2\,\eta\,\zeta) + \left[\,t + (4\,M^2 -t)\,\eta^2 \right]
   \zeta^2|_{\zeta \rightarrow 0} = (4M^2 - t).
\label{psquared}
\end{eqnarray}
We prefer the second parameterization
which leads to expressions which contain as lowest approximation the mass corrections
known from deep--inelastic scattering, without requiring any further redefinition of the
dGPDs.
We use the original expression for the
Compton scattering amplitude \cite{gey2} with the $\lambda$--parameterization and
apply the matrix elements (\ref{non1}, \ref{non15}).

\section{The Hadronic Tensor}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
\label{sec-unp}
In the following we discuss the symmetric and antisymmetric contributions
to the hadronic tensor, which correspond to the unpolarized and
polarized case, separately.
\subsection{The Symmetric Part}

\vspace{1mm}
\noindent
The symmetric part of the hadronic tensor for diffractive
scattering, cf.~\cite{BGR2006,gey2} is given by
\begin{eqnarray}
W^ {\text{tw2}} _{\{\mu\nu\}}\kln{q} &=& {\sf Im}~\frac{q^2}{2}\int D{\mathbb Z} \,
      \frac{{\cal A}_{\{\mu\nu\}}(q,{\cal P })}
      {\lambda \sqrt{(q{\cal P })^2- q^2 {\cal P }^2 }}
       \Bigg( \frac{1}{1 - \xi_+/\lambda + i \varepsilon}
         -  \frac{1}{1 - \xi_-/\lambda - i \varepsilon} \Bigg) \nonumber\\
&=&
- 2\pi\! \int\! d\zeta
 \frac{q^2}{\sqrt{(q{\cal P })^2- q^2 {\cal P }^2 }}
     \Bigg\{
 \frac{q{\cal K}^a}{q{\cal P}}\; \bigg[
 g_{\mu\nu}^{\mathrm T} F_{a\, 1}(\xi,\zeta)
 -
 \frac{{\cal P}_\mu^{\mathrm T} {\cal P}_\nu^{\mathrm T}}
      {({\cal P}^{\mathrm T})^2}\,
 F_{a\, 2}(\xi,\zeta) \bigg]
 \nonumber \\
 && \hspace*{4.5cm}
 +\Big( \frac{q{\cal K}^a}{q{\cal P}}
     - \frac{{\cal P}{\cal K}^a}{{\cal P}^2} \Big)
 \bigg[
 g_{\mu\nu}^{\mathrm T} F_{a\, 3}(\xi,\zeta)
 - \frac{{\cal P}_\mu^{\mathrm T} {\cal P}_\nu^{\mathrm T}}
      {({\cal P}^{\mathrm T})^2}\,
 F_{a\, 4}(\xi,\zeta) \bigg]
\nonumber\\
 && \hspace*{4.5cm}
 -\bigg(\frac{{\cal K}_{\mu}^{a\,\mathrm T} {\cal P}_\nu^{\mathrm T}
 + {\cal P}_\mu^{\mathrm T} {\cal K}_{\nu}^{a\,\mathrm T}}
      {({\cal P}^{\mathrm T})^2}\,
 - 2\, \frac{q{\cal K}^a}{q{\cal P}}\,
      \frac{{\cal P}_\mu^{\mathrm T} {\cal P}_\nu^{\mathrm T}}
      {({\cal P}^{\mathrm T})^2} \bigg)
 F_{a\, 5}(\xi,\zeta)
\Bigg\}.\! \nonumber\\
\label{Ts_nonfim}
\end{eqnarray}
The relevant imaginary part belongs to the $\delta$--distribution $
\delta(1 - \xi_+/\lambda)$ in terms of variables $(\xi_+ \equiv \xi, \zeta)$,
with the $\lambda$--integration, (\ref{varlz}), being carried out.
It implies the pole condition, cf.\cite{gey2} Eqs.~(6.6--6.10) and
\cite{BGR2006},
\begin{align}
1 + \hbox{\large$\frac{1}{2}$} \xi\,x\,{\cal P}^2/Q^2 = \sqrt{1+x^2{\cal P}^2/Q^2} = - (1-2\,x/\,\xi)~.
\label{pc}
\end{align}
which we use below. The structure functions $F_{a i},i=1, \ldots, 5$ are given by
\begin{align}
\label{F1x}
F_{a\,1}(\xi,\,\zeta )
  &=
  \Phi^{(0)}_a(\xi,\zeta)
  + \frac{1}{2} \frac{x\,{\cal P}^2/Q^2}{\sqrt{1 + x^2{\cal P}^2/Q^2}}\;
  \Phi^{(1)}_a(\xi, \zeta)
  + \frac{1}{4} \frac{(x\,{\cal P}^2/Q^2)^2}{1 + x^2{\cal P }^2/Q^2}\;
  \Phi^{(2)}_a(\xi, \zeta) \,,
\\
F_{a\,2}(\xi,\, \zeta)
  &=
  \Phi^{(0)}_a(\xi,\zeta)
  + \frac{3}{2}
  \frac{x\,{\cal P}^2/Q^2}{\sqrt{1 + x^2{\cal P}^2/Q^2}}\;
  \Phi^{(1)}_a(\xi, \zeta)
  + \frac{3}{4}
  \frac{(x\,{\cal P}^2/Q^2)^2}{1 + x^2{\cal P}^2/Q^2}\;
  \Phi^{(2)}_a(\xi, \zeta) \,.
\label{F2x}
\end{align}
\begin{eqnarray}
\label{F3x}
F_{a\,3}(\xi,\zeta) &=&
    -\frac{1}{2} \frac{\xi\,x\,{\cal P}^2/Q^2} {\sqrt{1+x^2{\cal P}^2/Q^2}}\,
     \Phi^{(0)}_a(\xi,\zeta) \\
  && +\frac{1}{2\xi}
   		\bigg( \frac{\xi\,x\,{\cal P}^2/Q^2}{\sqrt{1+x^2{\cal P}^2/Q^2}}
    	- \frac{(\xi\,x\,{\cal P}^2/Q^2)^2}{1+x^2{\cal P}^2/Q^2} \bigg)
     	\Phi_a^{(1)}(\xi,\zeta)
\nonumber\\
 && -\frac{1}{\xi}
     \bigg( \frac{\xi\,x\,{\cal P}^2/Q^2} {\sqrt{1+x^2{\cal P}^2/Q^2}}
     -\frac{(\xi\,x{\cal P}^2/Q^2)^2}
      {1+x^2{\cal P}^2/Q^2}
     +\frac{3}{8}\frac{(\xi\,x\,{\cal P}^2/Q^2)^3}
       {\sqrt{1+x^2{\cal P}^2/Q^2}^{\;3}}\bigg)
      \int_\xi^1 \frac{dy}{y}\, \Phi_a^{(1)}(y,\zeta)\nonumber  \\
  && - \frac{1}{\xi}
    \bigg(\frac{(\xi\,x\,{\cal P}^2/Q^2)^2} {1+x^2{\cal P}^2/Q^2}
    -\frac{3}{4} \frac{(\xi\,x\,{\cal P}^2/Q^2)^3}
     {\sqrt{1+x^2{\cal P}^2/Q^2}^{\;3}}
     +\frac{3}{16} \frac{(\xi\,x\,{\cal P}^2/Q^2)^4}
     {[1+x^2{\cal P}^2/Q^2]^2}\bigg)
      \int_\xi^1 \frac{dy}{y^2}\, \Phi_a^{(2)}(y,\zeta)\,,
      \nonumber\\
\label{F4x}
F_{a\,4}(\xi,\zeta) &=&
    -\frac{1}{2}  \frac{\xi\,x{\cal P}^2/Q^2} {\sqrt{1+x^2{\cal P}^2/Q^2}} \,
     \Phi^{(0)}_a(\xi,\zeta) \\
  && +\frac{1}{\xi}\bigg( \frac{5}{2} \frac{\xi\,x{\cal P}^2/Q^2}{\sqrt{1+x^2{\cal P}^2/Q^2}}
    - \frac{3}{2} \frac{(\xi\,x{\cal P}^2/Q^2)^2}{1+x^2{\cal P}^2/Q^2} \bigg)
     \Phi_a^{(1)}(\xi,\zeta) \nonumber \\
  && -\frac{3}{\xi} \bigg( \frac{\xi\,x{\cal P}^2/Q^2} {\sqrt{1+x^2{\cal P}^2/Q^2}}
     -2\frac{(\xi\,x{\cal P}^2/Q^2)^2}{1+x^2{\cal P}^2/Q^2}
     +\frac{5}{8}\frac{(\xi\,x{\cal P}^2/Q^2)^3}
       {\sqrt{1+x^2{\cal P}^2/Q^2}^{\;3}}\bigg)
      \int_\xi^1 \frac{dy}{y}\, \Phi_a^{(1)}(y,\zeta)\nonumber  \\
   &&  - \frac{3}{\xi}\bigg(
     \frac{(\xi\,x{\cal P}^2/Q^2)^2} {1+x^2{\cal P}^2/Q^2}
     -\frac{5}{4} \frac{(\xi\,x{\cal P}^2/Q^2)^3}
       {\sqrt{1+x^2{\cal P}^2/Q^2}^{\;3}}
     +\frac{5}{16} \frac{(\xi\,x{\cal P}^2/Q^2)^4}
       {[1+x^2{\cal P}^2/Q^2]^2}\bigg)
      \int_\xi^1 \frac{dy}{y^2}\, \Phi_a^{(2)}(y,\zeta)\, ,
      \nonumber\\
F_{a\, 5}(\xi,\zeta) &=&  \frac{1}{\xi}\bigg[\Phi_a^{(1)}(\xi,\zeta)
  + \frac{3}{2}\frac{\xi\,x{\cal P}^2/Q^2}{\sqrt{1 + x{\cal P}^2/Q^2}}
        \int_\xi^1 \frac{dy}{y}\,\Phi_a^{(1)}(y,\zeta)
        \label{F5x}\nonumber\\
  && +  \frac{3}{4}\frac{(\xi\,x{\cal P}^2/Q^2)^2}{1 + x{\cal P}^2/Q^2}
        \int_\xi^1 \frac{dy}{y^2}\,\Phi_a^{(2)}(y,\zeta)\bigg] \,.
\end{eqnarray}
Whereas $F_{a\, 1(2)}(\xi,\zeta)$  are direct generalizations of the well-known
deep--inelastic structure functions. $F_{a\, k}(\xi,\zeta)|_{k=3,4,5}$ are
new structure functions, which vanish in the forward limit, cf.~Appendix B.
The typical square roots ${\sqrt{1 + x^2{\cal P}^2/Q^2}} $ for the mass corrections
depend on the generalized momentum ${\cal P} = {\cal P}(\zeta).$
After substituting
$\lambda \rightarrow \xi $
in (\ref{F1x}--\ref{F5x}),
we introduce the following iterated representations for the basic
dGPDs $f_a (\lambda,\zeta)$, cf.~(\ref{non1}):
\begin{align}
\label{phia0}
\Phi_a^{(0)}(\xi,\zeta) &\equiv f_a (\xi,\zeta)\,,
\\
\label{phia1}
\Phi_a^{(1)}(\xi,\zeta)
&\equiv
      \int_\xi^{1} dy_1 \;f_a (y_1,\zeta)
 = \xi \int_0^1 \frac{d\tau}{\tau^2}\;
   f_a \Big(\frac{\xi}{\tau},\zeta\Big)\,,
\\
\label{phia2}
\Phi_a^{(2)}(\xi,\zeta)
&\equiv
\int_\xi^{1} dy_2
\int_{y_2}^{1} dy_1 \, f_a (y_1,\zeta)
= \xi^2 \int_0^1 \frac{d\tau_1}{\tau_1^3}
  \int_0^1 \frac{d\tau_2}{\tau_2^2}\;
   f_a \Big(\frac{\xi}{\tau_1\tau_2},\zeta\Big)\,,
\\
\label{eqphiaa}
\Phi_a^{(i)}(\xi,\zeta)
&\equiv
\int_\xi^{1} dy\; \Phi_a^{(i-1)}(y,\zeta)\, , \qquad \mbox{for}\quad i \geq 1 \,, \\
\label{eqphiaa1}
\int_\xi^1 \frac{dy}{y}\,\Phi_a^{(1)}(y,\zeta)
 &\equiv \int_\xi^1 \frac{dy_1}{y_1}\int_{y_1}^1\!\! dy\, \Phi_a^{(0)}(y,\zeta)
 = \xi\!\int_0^1 \frac{d\tau_1}{\tau_1^2}\int_0^1 \frac{d\tau_2}{\tau_2^2}\;
   f_a \Big(\frac{\xi}{\tau_1\tau_2},\zeta\Big)\,,
\\
\label{eqphiaa2}
\int_\xi^1 \frac{dy}{y^2}\,\Phi_a^{(2)}(y,\zeta)
 &\equiv \int_\xi^1 \frac{dy_1}{y_1^2}\int_{y_1}^
 \!\! dy\, \Phi_a^{(1)}(y,\zeta)
   =    \xi\! \int_0^1 \frac{d\tau_1}{\tau_1^3}
  \int_0^1 \frac{d\tau_2}{\tau_2^2}\int_0^1 \frac{d\tau_3}{\tau_3^2}    \;
   f_a \Big(\frac{\xi}{\tau_1\tau_2\tau_3 },\zeta\Big)
\end{align}

\vspace{2mm}\noindent
Let us now investigate the effect of target masses and finite terms in $t$ in more
detail. It turns out that both the $M^2$-- and $t$--contributions in the diffractive
structure
functions emerge due to the parameter $\rho$
\begin{eqnarray}
\label{rho}
\rho =  \epsilon \, x^2 \frac{p_+^2}{Q^2} \frac{1}{1+x^2 p_+^2/Q^2}~,
\end{eqnarray}
with  $\epsilon$ given by ${\cal P}^2 = p_+^2(1+\epsilon)$,
\begin{eqnarray}
\label{epsi}
\epsilon = \frac{1}{p_+^2} \left[2 \zeta p_+ \pi_- + \zeta^2 \pi_-^2 \right] = - 2 \eta \zeta
           +\left(\eta^2 + \frac{t}{p_+^2}\right) \zeta^2~.
\end{eqnarray}
Since
\begin{eqnarray}
-\eta \simeq x_\PP~ \ll 1~,
\end{eqnarray}
$\rho$ effectively takes values $\rho \lesssim 10^{-3}$ for $x_\PP \lesssim 10^{-2}, |t|
\approx
(0.1 ... 1 ) M^2, Q^2 \approx (1 ... 5 ) M^2.$ The range of $\zeta$ is
determined both
by the support condition (\ref{varlz}) and the condition $ {\cal P}^2 =
p_+^2(1+ \epsilon) > 0 $ in the diffractive case.

To prepare the expansion in $\rho$ we rewrite the hadronic tensor as
\begin{align}
W^ {\text{tw2}} _{\{\mu\nu\}}\kln{q}
 =
  2 \pi \!&\int\! d\zeta\;
   \frac{q{\cal K}^a}{q\cal P}\,
  \bigg[
 - g_{\mu\nu}^{\mathrm T}\, {W}^{\rm diff}_{a\,1}
     \Big(x,\frac{{\cal P}^2}{Q^2};\zeta\Big)
 + \frac{{\cal P}_\mu^{\mathrm T} {\cal P}_\nu^{\mathrm T}}{M^2}
 \,{W}^{\rm diff}_{a\,2}\Big(x,\frac{{\cal P}^2}{Q^2};\zeta\Big) \bigg]\,,
 \nonumber
 \\
& +
 \Bigg\{\!
 \bigg( \frac{q{\cal K}^a}{q{\cal P}}
     - \frac{{\cal P}{\cal K}^a}{{\cal P}^2} \bigg)\frac{{\cal P}^2}{Q^2}
 \bigg[
 - g_{\mu\nu}^{\mathrm T}{W}^{\rm diff}_{a\,3} \Big(x,\frac{{\cal P}^2}{Q^2};\zeta\Big)
 + \frac{{\cal P}_\mu^{\mathrm T}{\cal P}_\nu^{\mathrm T}}{M^2}
  {W}^{\rm diff}_{a\,4} \Big(x,\frac{{\cal P}^2}{Q^2};\zeta\Big)
  \bigg]
  \nonumber\\
&  +
 \bigg({\cal P}_\mu^{\mathrm T}{\cal K }_\nu^{a\,\mathrm T}
       + {\cal P}_\nu^{\mathrm T}{\cal K }_\mu^{a\,\mathrm T}
       - 2\,
 {\cal P}_\mu^{\mathrm T}{\cal P}_\nu^{\mathrm T} \frac{q {\cal K}^a}{q{\cal P}}
  \bigg)
 \frac{1}{M^2}
   {W}^{\rm diff }_{a\,5}\Big(x,\frac{{\cal P}^2}{Q^2};\zeta\Big)
  \Bigg\}\,~.
\label{TSext}
\end{align}
The integral over $\zeta$ cannot be  performed easily. Here, the un-integrated
structure functions ${W}^{\rm diff}_{a\,k }(x,{{\cal P}^2(\zeta)}/{Q^2};\zeta)$ are given by
\begin{align}
     {W}^{\rm diff}_{a\,1 }\Big(x,\frac{{\cal P}^2(\zeta)}{Q^2};\zeta\Big)
  & \equiv
   - \frac{x}{\sqrt{1+x^2\mathcal{P}^2/Q^2}}
    \; F_{a\,1}(\xi,\zeta)~,
    & &
\label{WF1}
    \\
     {W}^{\rm diff}_{a\,3 }\Big(x,\frac{{\cal P}^2(\zeta)}{Q^2};\zeta\Big)
  & \equiv
   - \frac{x}{\sqrt{1+x^2\mathcal{P}^2/Q^2}} \frac{Q^2}{{\cal P}^2}
    \; F_{a\,3}(\xi,\zeta)~,
    & &
\label{WF3}
    \\
     {W}^{\rm diff}_{a\,k }\Big(x,\frac{{\cal P}^2(\zeta)}{Q^2};\zeta\Big)
  & \equiv
 - \frac{M^2}{Q^2}\,
   \bigg(\frac{x}{\sqrt{1+x^2\mathcal{P}^2/Q^2}} \bigg)^{\!\!3}
    F_{a\,k}(\xi,\zeta)
    &  &\mbox{for~} k= 2,5\;,
\label{WFj}
     \\
     {W}^{\rm diff}_{a\,4 }\Big(x,\frac{{\cal P}^2(\zeta)}{Q^2};\zeta\Big)
  & \equiv
 - \frac{M^2}{{\cal P }^2}\,
   \bigg(\frac{x}{\sqrt{1+x^2\mathcal{P}^2/Q^2}} \bigg)^{\!\!3}
    F_{a\,4}(\xi,\zeta)~.
    &  &
\label{WF4}
\end{align}
As noted in \cite{BGR2006}
a generalized Callan--Gross \cite{CG} relation between ${W}_{a\,1}^{\rm diff}$ and
${W}_{a\,2}^{\rm diff}$, which holds for diffractive scattering in the limit
$M^2,~t~\rightarrow~0$, \cite{Blumlein:2001xf}, is broken as in the case of
deep--inelastic scattering \cite{Georgi:1976ve}.
Correspondingly, the distribution
functions
${W}_{a\,(1,2)}^{\rm diff}$ are related to ${W}_{a\, \rm L}^{\rm diff}$, the diffractive
analogue of the longitudinal structure function of deep--inelastic scattering, by
\begin{align}
{W}^{\rm diff }_{a\,\rm L}(x,{\cal P}^2/Q^2;\zeta)
&=
- \,{W}^{\rm diff }_{a\,1}(x,{\cal P}^2/Q^2;\zeta)
+  \left(1+\frac{x^2{\cal P}^2}{Q^2}\right)\frac{qp_+}{x\, M^2}\,
 {W}^{\rm diff }_{a\,2}(x,{\cal P}^2/Q^2;\zeta)\,.
 \label{w1}
\intertext{
To see this in detail, we insert (\ref{F1x}), (\ref{F2x})  and  (\ref{WFj}), so that}
{W}^{\rm diff }_{a\,\rm L}(x,{\cal P}^2/Q^2;\zeta)
&= \frac{x}{\sqrt{1 + x^2{\cal P}^2/Q^2 }}
     \big( F_{a\,1}(\xi,\zeta)  -  F_{a\,2}(\xi,\zeta) \big) \approx
{\cal O}\left(\frac{x^2{\cal P}^2}{Q^2}\right).
\label{CGk}
\intertext{
The last relation follows by direct inspection of $ F_{a\,i}$ and is explicit in}
  {W}^{\rm diff }_{a\,\rm L}(x,{\cal P}^2/Q^2;\zeta)& =
      -\frac{x^2 {\cal P}^2}{2Q^2}\frac{\partial}{\partial {x}}
       \Big(\frac{x}{\xi \sqrt{1 + x^2{\cal P}^2/Q^2 }}
        \Phi_a^{(2)}(\xi,\zeta)\Big)
\end{align}
derived in \cite{gey2}, cf. also \cite{Georgi:1976ve,Blumlein:1998nv} for the case of
forward scattering.

Most of the above quantities depend on ${\cal P}^2$ (\ref{cP2})  which we write now as
\begin{eqnarray}
\label{psquared1}
{\cal P}^2 &=& p_+^2 + 2\zeta p_+ \pi_- +\zeta^2 \pi_-^2 =  p_+^2 (1+ \epsilon)\,~.
\end{eqnarray}
Let us simplify the contraction of the kinematic coefficients in (\ref{TSext}).
For ${\cal K}_1 = p_+ $, observing $q{\cal P} = qp_+$ and Eq.~(\ref{psquared1})
for ${\cal P}^2$, we obtain
\begin{align}
\label{kin11}
 \frac{q{\cal K}_1}{q{\cal P}} &= 1\,, \\
\label{kin21}
 \Big(\frac{q{\cal K}_1}{q{\cal P}} - \frac{{\cal P}{\cal K}_1}{{\cal P}^2}\Big)
  \frac{{\cal P}^2}{Q^2}
 & = \frac{p_+^2}{Q^2}\bigg(-\eta \zeta  + \zeta^2\left(\eta^2 + \frac{t}{p_+^2}\right) \bigg)
   \\
\label{kin31}
{\cal K}_{1\mu}^{\mathrm T} {\cal P}_\nu^{\mathrm T}
 + {\cal P}_\mu^{\mathrm T} {\cal K}_{1\nu}^{\mathrm T}
 -2\frac{q{\cal K}_1}{q{\cal P}}
 {\cal P}_\mu^{\mathrm T} {\cal P}_\nu^{\mathrm T}
 &=
 -\,\zeta\left(p_{+\mu}^{\mathrm T}\pi_{-\nu} + p_{+\nu}^{\mathrm T}\pi_{-\mu}\right)
 - 2\,\zeta^2 \pi_{-\mu}\pi_{-\nu}\,,
 \\
\intertext{and for ${\cal K}_2 = \pi_- $, due to the transversality of $\pi_-$,
one finds}
\label{kin12}
 \frac{q{\cal K}_2}{q{\cal P}} &= 0\,, \\
\label{kin22}
\Big( \frac{q{\cal K}_2}{q{\cal P}} - \frac{{\cal P}{\cal K}_2}{{\cal P}^2}\Big)
    \frac{{\cal P}^2}{Q^2}
& = \frac{p_+^2}{Q^2}\bigg( \eta -\zeta\left(\eta^2 +
\frac{t}{p_+^2}\right)\bigg) , \\
\label{kin32}
{\cal K}_{2\mu}^{\mathrm T} {\cal P}_\nu^{\mathrm T}
 + {\cal P}_\mu^{\mathrm T} {\cal K}_{2\nu}^{\mathrm T}
 -2\frac{q{\cal K}_2}{q{\cal P}}
 {\cal P}_\mu^{\mathrm T} {\cal P}_\nu^{\mathrm T}
 &=
 \left(p_{+\mu}^{\mathrm T}\pi_{-\nu} + p_{+\nu}^{\mathrm T}\pi_{-\mu}\right)
 + 2\,\zeta \,\pi_{-\mu}\pi_{-\nu}\,.
 \end{align}
It is remarkable that only for ${\cal K}_1 = p_+ $  the first invariant
${q{\cal K}_1}/{q{\cal P}}$  contributes to the zeroth power in $\zeta$,
whereas the other ones start at most with the {first} power.
The contributions of  invariants belonging to kinematic coefficients
containing  $\pi_-$  are less important because this variable is transverse
to $q$ with $\pi_- q =0$. The corresponding invariants
\begin{equation}
\pi_-^2 = t +\eta^2\, p_+^2,\; \pi_-p_+ = -\eta \,p_+^2,\; \pi_-p_- = t~,
\end{equation}
are small compared to $Q^2$.

Having now expressed the $\zeta$--dependence in all kinematic factors explicitly,
we may perform the $\zeta$--integral introducing $n$th moments~:
\begin{eqnarray}
\label{mg}
 {W}^{(n)\,\rm diff}_{a\,k}(x,\eta,t,p_+^2/Q^2)= \int\! d\zeta\, \zeta^n
           \,{W}^{\,\rm diff}_{a\,k}(x,{\cal P}^2/Q^2;\zeta)\,~.
\end{eqnarray}
The hadronic tensor reads
\begin{align}
\frac{1}{2\pi}{\sf Im}\,&T^ {\text{tw2}} _{\{\mu\nu\}}\kln{q}
  =\nonumber  \\
 &  - g_{\mu\nu}^{\mathrm T}\,
    \left\{\, {W}^{(0)\,\rm diff}_{1\,1} + \frac{p_+^2}{Q^2}\left[
             \eta \, ({W}^{(0)\,\rm diff}_{2\,3}
           - {W}^{(1)\rm diff}_{1\,3}) +\left(\eta^2
+\frac{t}{p_+^2}\right)\,({W}^{(2)\rm diff}_{1\,3}
           - {W}^{(1)\rm diff}_{2\,3}) \right]
         \right\}\nonumber \\
 & + \frac{p_{+\,\mu}^{\mathrm T} p_{+\,\nu}^{\mathrm T}}{M^2} \left\{
      {W}^{(0)\,\rm diff}_{1\,2} +
      \frac{p_+^2}{Q^2} \left[\eta \, \left({W}^{(0)\,\rm diff}_{2\,4}
      -{ {W}^{(1)\,\rm diff}_{1\,4}}\right) +
      \left(\eta^2 + \frac{t}{p_+^2}\right)\,( {W}^{(2)\,\rm diff}_{1\,4} -
      {W}^{(1)\,\rm diff}_{2\,4})
         \right]
          \right\} \nonumber
\end{align}\begin{align}
 &  +\frac{p_{+\,\mu}^{\mathrm T} \pi_{-\,\nu}+ p_{+\,\nu}^{\mathrm T} \pi_{-\,\mu}}
     {M^2}\left\{{W}^{(1)\,\rm diff}_{1\,2} +
        \frac{p_+^2}{Q^2} \Bigl[\eta ({W}^{(1)\,\rm diff}_{2\,4}
         - {W}^{(2)\,\rm diff}_{1\,4})  \right.
\nonumber  \\
 & \qquad \qquad \qquad \qquad \left.
   +\left(\frac{t}{p_+^2} +\eta^2\right)({W}^{(3)\,\rm diff}_{1\,4}
-{W}^{(2)\,\rm diff}_{2\,4})\Bigr]
     +{{W}^{(0)\,\rm diff}_{2\,5}} -  {W}^{(1)\,\rm diff}_{1\,5}
\right\}
\nonumber \\
 &  +  \frac{\pi_{-\,\mu} \pi_{+\,\nu}}{M^2}\,
           \left\{{W}^{(2)\,\rm diff}_{1\,2} + 2{W}^{(1)\,\rm diff}_{2\,5}
            -2 {W}^{(2)\,\rm diff}_{1\,5} \right.\nonumber \\
& \quad \qquad  \left.
  + \frac{p_+^2}{Q^2}
\left[
     \eta \, \left(
             {W}^{(2)\,\rm diff}_{2\,4}- {W}^{(3)\,\rm diff}_{1\,4}\right)
           + \left(\eta^2 + \frac{t}{p_+^2} \right)
             \left( {W}^{(4)\,\rm diff}_{1\,4}- {W}^{(3)\,\rm diff}_{2\,4}
             \right)
\right]
\right\}~.
\label{WPhi0}
\end{align}
Here the momentum fraction argument of the structure functions
 $W_{ak}^{\rm diff}$  is the original Nachtmann variable (\ref{xicd}),
whereas for the functions ${W}^{(n)\,\rm diff}_{a\,k}(x,\eta,t,p_+^2/Q^2)$ it is $x$.
These structure functions are in principle accessible experimentally,
varying the
external kinematic parameters $x_{\rm BJ}, Q^2, t$ and $x_\PP, (\eta =
\eta(x_\PP))$.

Up to this point no approximations have been made. We would now like to
discuss the above structure.
Note that  disregarding of $\pi_-$ as transversal degree of freedom corresponds
to the limit  $ \epsilon \rightarrow 0 $, (\ref{psquared1}). However, $\epsilon$,
(\ref{epsi}), is not necessarily a small quantity.
The Taylor expansion in $\epsilon$ would retain the DIS-like  target mass corrections
and lead to the physically relevant  power series in $p_+^2/Q^2$ of the denominators.
Because of the smallness of $\rho $ (\ref{rho}) and also
$x^2 p_+^2/Q^2 $ we use the latter as expansion parameter.
Thereby the Nachtmann variable is substituted by $x$ in lowest order,
whereas by setting $\pi_-=0$  we would retain an approximate Nachtmann variable,
\begin{eqnarray}
\label{zeta0}
\xi_0= 2x/(1+\sqrt{1 +x^2p_+^2/Q^2}).
\end{eqnarray}
For simplicity we proceed as follows:
\begin{itemize}
\item effective expansion w.r.t. the parameter $p_+^2/Q^2$,
 \begin{eqnarray}
 \label{epsilon2}
 \bigg(1+ x^2 \,\frac{{\cal P}^2}{Q^2}\bigg)^{-n} =
 \bigg(1+ x^2 \,\frac{p_+^2(1+\epsilon)}{Q^2}\bigg)^{-n} =
\left(1- n x^2\,\frac{ p_+^2}{Q^2}(1+\epsilon) + ...  \right),
\end{eqnarray}
\item expansion of the Nachtmann variable (\ref{xicd}),
\begin{eqnarray}
\label{xi_00}
  \xi- x = -\frac{1}{4}x \frac{x^2p_+^2}{Q^2}(1+\epsilon) + ...\, ,
\end{eqnarray}
\item use of $x$ instead of the Nachtmann variable $\xi$.
\item For the treatment of the denominators we shift the integration variable
      $ \lambda = \lambda' + \xi - x$,
 \begin{eqnarray}
  \frac{1}{\lambda -\xi + i\varepsilon \lambda} =\frac{1}{\lambda' -x + i\varepsilon \lambda} .
 \end{eqnarray}
\end{itemize}
Through this procedure we avoid the expansion of the denominator
 in favor of an expansion of the  dGPDs. In principle problems could arise because
 of possible differences in $ \varepsilon(\lambda - \lambda')$.
Therefore we have to expand the basic  dGPD
\begin{eqnarray}
\Phi^{(i)}_a(\lambda, \zeta)& =&
\Phi^{(i)}_a(\lambda' +\xi-x , \zeta)
   = \Phi^{(i)}_a(\lambda' , \zeta)
	+ \partial_{\lambda'}  \Phi^{(i)}_a(\lambda', \zeta)
          (\xi - x) + \ldots \nonumber \\
 &=&\Phi^{(i)}_a(\lambda' , \zeta)
    -\frac{1}{4}x \frac{x^2 p_+^2}{Q^2}(1+\epsilon) \partial_{\lambda'}
     \Phi^{(i)}_a(\lambda', \zeta) + \ldots\quad.
\nonumber \end{eqnarray}
As a test we can study the limit of
deep--inelastic scattering, whereby we reproduce the standard result. For
diffractive DIS it is sufficient to consider the lowest approximation which
extends our results \cite{Blumlein:2001xf,Blumlein:2002fw}.
In the following we define moments of the dGPDs by
\begin{eqnarray}
\label{mph}
   \Phi_a^{(i\,{n})}(x) = \int d\zeta  \zeta^n \Phi_a^{(i)}(x,\zeta).
\end{eqnarray}
This corresponds to a change from a GPD to a parton density.

Now we apply our approximation procedure directly to Eq.~(\ref{TSext}) using
(\ref{WF1}--\ref{WF4}) and (\ref{F1x}--\ref{F5x}). We write the result separately for the
invariants ${\cal K}^1 = p_+$,
\begin{align}
\frac{1}{2\pi}{\sf Im}\, T^ {\text{tw2}} _{\{\mu\nu\}}\kln{q}|_1
 & =    g_{\mu\nu}^{\mathrm T}\,\Bigg[
         x \Phi_1^{(0\,0)}(x) + \frac{x^2 p_+^2}{Q^2}
         \Big(t_{1\,1}^0 + t_{1\,3}^0 - \tilde t_{1\,3}^0 + \eta \tilde t_{1\,3}^1
         \Big) \Bigg]
 \nonumber \\
 & - \frac{p_{+\,\mu}^{\mathrm T} p_{+\,\nu}^{\mathrm T}}{Q^2}
          \Bigg[x^3 \Phi_1^{(0\,0)} + \frac{x^2 p_+^2}{Q^2}
          x^2\Big(t_{1\,2}^0 + t_{1\,4}^0 - \tilde t_{1\,4}^0 + \eta \tilde t_{1\,4}^{\,1}
           \Big) \Bigg]
 \nonumber \\
 &  -\frac{p_{+\,\mu}^{\mathrm T} \pi_{-\,\nu}+ p_{+\,\nu}^{\mathrm T} \pi_{-\,\mu}}
     {Q^2}\Bigg[ x^3 \Phi_1^{(0\,1)} - x^2 \Phi_1^{(1\,1)}
\nonumber\\ & \hspace*{4cm}
+ \frac{x^2 p_+^2}{Q^2}x^2
     \Big(t_{1\,2}^1 + t_{1\,4}^1 - \tilde t_{1\,4}^{\,1} - t_{1\,5}^1   +
      \eta \tilde t_{1\,4}^{\,2}
           \Big) \Bigg]
 \nonumber \\
 &  -  \frac{\pi_{-\,\mu} \pi_{-\,\nu}}{Q^2}\,
           \bigg[ x^3 \Phi_1^{(0\,2)} - 2x^2 \Phi_1^{(1\,2)}+ \frac{x^2 p_+^2}{Q^2}x^2
     \Big(t_{1\,2}^2 + t_{1\,4}^2 - \tilde t_{1\,4}^{\,2} -2 t_{1\,5}^2   +
      \eta \tilde t_{1\,4}^{\,3}
           \Big) \Bigg]~,
\label{Wa1}
\intertext{and for ${\cal K}^2 = \pi_- $,  }
\frac{1}{2\pi}{\sf Im}\, T^ {\text{tw2}} _{\{\mu\nu\}}\kln{q}|_2
 & =   g_{\mu\nu}^{\mathrm T}\,
         \frac{x^2 p_+^2}{Q^2}
         \Big[\eta \tilde t_{2\,3}^{\,0} - \left(\eta^2
+\frac{t}{p_+^2}\right)\tilde t_{2\,3}^{\,1}
          \Bigg]
 \nonumber \\
 &           - \frac{p_{+\,\mu}^{\mathrm T} p_{+\,\nu}^{\mathrm T}}{Q^2}
          \frac{x^2 p_+^2}{Q^2}
          x^2\Big[\eta \tilde t_{2\,4}^{\,0}
          -\left(\eta^2 +\frac{t}{p_+^2}\right)\tilde t_{2\,4}^{\,1}
           \Big) \Bigg]
 \nonumber \\
 &  -\frac{p_{+\,\mu}^{\mathrm T} \pi_{-\,\nu}+ p_{+\,\nu}^{\mathrm T} \pi_{-\,\mu}}
     {Q^2}\Bigg[ x^2 \Phi_2^{(1\,0)}  + \frac{x^2 p_+^2}{Q^2}x^2
     \Big(t_{2\,5}^0 +{\eta t_{2\,4}^1 -\left(\eta^2 +\frac{t}{p_+^2}\right)
      \tilde t_{2\,4}^{\,2}}
           \Big) \Bigg]
 \nonumber \\
 &    -  \frac{\pi_{-\,\mu} \pi_{-\,\nu}}{Q^2}\,
           \bigg[2 x^2 \Phi_2^{(1\,1)} + \frac{x^2 p_+^2}{Q^2}x^2
     \Big( 2t_{2\,5}^1 + \eta t_{2\,4}^2 -\left(\eta^2 +\frac{t}{p_+^2}\right)
         \tilde t_{2\,4}^{\,3}            \Big) \Bigg] .
\label{Wa2}
\end{align}
Here $t_{ai}^n$ and $\tilde t_{ai}^n$ are given by
\begin{align}
t_{a1}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(-\hbox{\large$\frac{1}{2}$} x \Phi^{(0)}_a(x, \zeta) + \hbox{\large$\frac{1}{2}$} \Phi^{(1)}_a(x, \zeta)
          -\frac{1}{4}x^2 \partial_{x}\Phi^{(0)}_a(x, \zeta)\right),
\nonumber \\
t_{a2}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(-\frac{3}{2} x\Phi^{(0)}_a(x, \zeta) +
          \frac{3}{2}  \Phi^{(1)}_a(\xi_{00}, \zeta)
          -\frac{1}{4} x^2 \partial_{x}\Phi^{(0}_a(x, \zeta)\right),
\\
t_{a3}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(-\hbox{\large$\frac{1}{2}$} x \Phi^{(0)}_a(x, \zeta) +
          \frac{1}{2} \Phi^{(1)}_a(\xi_{00}, \zeta)
          -\int_x^1\frac{dy}{y}\Phi^{(1)}_a(y, \zeta)\right),
\nonumber
\end{align}\begin{align}
\nonumber \\
t_{a4}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(-\hbox{\large$\frac{1}{2}$} x\Phi^{(0)}_a(x, \zeta) +
          \frac{5}{2} \Phi^{(1)}_a(x, \zeta)
          -3\int_x^1\frac{dy}{y}\Phi^{(1)}_a(y, \zeta)\right),
\nonumber \\
t_{a5}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(-\frac{5}{4} \Phi^{(0)}_a(x, \zeta) +
          \frac{3}{2} \int_x^1\frac{dy}{y}\Phi^{(1)}_a(y, \zeta)
          -\frac{1}{4} x \partial_{x}\Phi^{(1)}_a(x, \zeta)\right),
\end{align}
and
\begin{align}
\tilde t_{a1}^{\,n} & = \int d\zeta  \zeta^n
          \left(-\hbox{\large$\frac{1}{2}$} x \Phi^{(0)}_a(x, \zeta) + \hbox{\large$\frac{1}{2}$} \Phi^{(1)}_a(x, \zeta)
          -\frac{1}{4}x^2 \partial_{x}\Phi^{(0)}_a(x, \zeta)\right)~.
\end{align}
Similar
for all other terms $\tilde t_{ak}^n$, the factor $(1+\epsilon(\zeta)) $
           is absent compared to  $t_{ak}^n$.

It is remarkable that each kinematic coefficient ${\cal K }^a$ contributes to all possible
kinematic  structures. Because of the transversal behaviour of $\pi_-$
we expect that the last two structures
${p_{+\,\mu}^{\mathrm T} \pi_{-\,\nu}+ p_{+\,\nu}^{\mathrm T} \pi_{-\,\mu}}$ and
${\pi_{-\,\mu} \pi_{-\,\nu}}$ as well as the complete contributions of the second
invariant (\ref{Wa2})   are less important in comparison with the structures
$ g_{\mu\nu}^{\mathrm T}\,$ and  ${p_{+\,\mu}^{\mathrm T} p_{+\,\nu}^{\mathrm T}}$
and the first invariant in (\ref{Wa1}). Moreover the leading contributions to the first two
structures in (\ref{Wa2}) contain the small coefficient {$\eta$}.
In Eqs.~(\ref{Wa1}, \ref{Wa2}) the
contributions $\propto  M^2, t$ emerge as
\begin{eqnarray}
\frac{x^2 p_+^2}{Q^2} &=& \frac{x^2\left(4 M^2 - t\right)}{Q^2}~,\\
\frac{x^2 p_-^2}{Q^2} &=& \frac{x^2 t}{Q^2}~,
\end{eqnarray}
respectively. Noting that $|\eta| \simeq x_\PP$, and $x_\PP \sim {\cal O}(x_{\rm BJ})$ for
diffractive
scattering the target mass and finite $t$ corrections are suppressed by
${\cal O}(x^2 M^2/Q^2)$, with $ x \lesssim 10^{-2}$. In the meson-exchange case,
$x$--values of around $x \simeq 0.3$
may be reached and ${\cal O}(10 \% \times (M^2/Q^2))$
effects may be obtained.

Let us consider the complete zeroth order term
\begin{align}
\frac{1}{2\pi}{\sf Im}\, T^ {\text{tw2}} _{\{\mu\nu\}}\kln{q}|_0
  = & g_{\mu\nu}^{\mathrm T}\, \,x \Phi_1^{(0\,0)}(x)
       - \frac{p_{+\,\mu}^{\mathrm T} p_{+\,\nu}^{\mathrm T}}{Q^2}
               x^3\,   \Phi_1^{(0\,0)}
        \nonumber \\
 &  -\frac{p_{+\,\mu}^{\mathrm T} \pi_{-\,\nu}+ p_{+\,\nu}^{\mathrm T} \pi_{-\,\mu}}
     {Q^2} x^2\bigg[  x\,  \Phi_1^{(0\,1)} +
            \Phi_2^{(1\,0)}- \Phi_1^{(1\,1)} \bigg]
 \nonumber \\
 &  -  \frac{\pi_{-\,\mu} \pi_{-\,\nu}}{Q^2}\,x^2
           \bigg[ x\,\Phi_1^{(0\,2)} - 2\Phi_1^{(1\,2)} + 2 \Phi_2^{(1\,1)}
            \bigg]~.
\label{T_01}
\end{align}
Also here we can see that the contributions to the first two  kinematic structures
result from the distribution functions  $\Phi_1^{(0\,0)}(x) $   of the first kinematic
structure only. This reproduces our result \cite{Blumlein:2001xf} obtained for
vanishing $t$, target mass,
and negligible transversal momenta $\pi_-$,
\begin{align}
\frac{1}{2\pi}{\sf Im}\, T^ {\text{tw2}} _{\{\mu\nu\}}\kln{q}|_0
  =  g_{\mu\nu}^{\mathrm T}\, \,x \Phi_1^{(0\,0)}(x)
       - \frac{p_{+\,\mu}^{\mathrm T} p_{+\,\nu}^{\mathrm T}}{Q^2}
               x^3\,   \Phi_1^{(0\,0)}.
        \nonumber
\end{align}
The leading $t$-dependence is contained in the first structure
$ g_{\mu\nu}^{\mathrm T}\,$ of (\ref{Wa1})
\begin{align}
\frac{1}{2\pi}{\sf Im}\, T^ {\text{tw2}} _{\{\mu\nu\}}\kln{q}|_t = &  g_{\mu\nu}^{\mathrm T}\,
 \frac{x^2 t}{Q^2}\, \chi(x),\\
 \chi(x)\approx &\bigg\{ \frac{1}{2}x \big(  \Phi_1^{(0\,0)}-  \Phi_1^{(0\,2)}
                 -\eta(3 \Phi_1^{(0\,1)}
                 + \Phi_1^{(0\,3)})\big) -\frac{1}{2}\big( \Phi_1^{(1\,0)} -
\Phi_1^{(1\,2)}
                 -\eta(3 \Phi_1^{(1\,1)}+ \Phi_1^{(1\,3)})\big)\nonumber \\
         & +\frac{1}{4}x^2\partial_x( \Phi_1^{(0\,0)}- \Phi_1^{(0\,2)}-2\eta  \Phi_1^{(0\,1)})
    -\int_x^1\frac{dy}{y}(\eta \Phi_1^{(1\,1)}
+ \eta \Phi_1^{(1\,3)}+\Phi_1^{(1\,2)})\bigg\}~.
\label{tgsym}
\end{align}
Terms $\propto \eta^2 \simeq x_\PP^2$ are dropped.

A last remark concerns the generalized Callan-Gross relation (\ref{w1}).
This
relation can be written for $\zeta$--moments (\ref{mg}) as
\begin{eqnarray}
{W}^{(n)\rm diff }_{a\,\rm L}(x,p_+^2/Q^2)
&=&
- \,{W}^{(n)\rm diff }_{a\,1}(x,p_+^2/Q^2)
+ \frac{p_+^2 + (qp_+)^2/Q^2}{M^2}\,
 {W}^{(n)\rm diff }_{a\,2}(x,p_+^2/Q^2)\, \nonumber \\
&+&   \frac{p_+^2}{M^2}\int d\zeta \zeta^n \, \epsilon \,
   {W}^{\rm diff }_{a\,2}(x,{\cal P}^2/Q^2,\zeta ).
\label{cga}
\end{eqnarray}
Finally we remark that an equivalent kinematic parameterization can be obtained
using
\begin{eqnarray}
 p_\pm^{\mathrm T} = p_2^{\mathrm T} \pm p_1^{\mathrm T},
 \qquad \pi_- = p_- -\eta p_+ = p_2 (1-\eta) - p_1(1+\eta)
       =  p_2^{\mathrm T} (1-\eta) - p_1^{\mathrm T}(1+\eta) .
\end{eqnarray}
These relations allow to link different representations of the hadronic tensor,
which linearly relates various definitions of structure functions, cf.~(\ref{eqD2}).
All contributions due to $M^2$ and $t$--effects in the above are suppressed
like $\propto x^2_{(\PP)} \mu^2/Q^2$ with $\mu^2 = |t|, M^2$.

There are, however also other contributions emerging in the scattering cross
sections, which are of kinematic origin and stem from 4-vector products
contributing to the process contracting the leptonic and hadronic tensor, see
Appendix~A for details. Most of these
invariants are large, like $l_1.l_2$ and $l_1.p_1$. The invariant
$l_1.p_2$, (\ref{eqR1}),
leads to kinematic power corrections further to those considered in
(\ref{xs1}--\ref{xs3}). Here the leading contribution beyond the lowest
order term is of ${\cal O}(\cos(\phi_b) x_{\rm BJ} \sqrt{\mu^2/Q^2}), \mu^2 = |t|,
M^2$~.
The terms, which do not vary with the angle $\phi_b$ are of ${\cal O}(x_{\rm BJ}
\mu^2/Q^2)$. In conclusion, the largest dependences from the limiting case
$|t|, M^2 \rightarrow 0$ are obtained from the kinematic terms in the cross
section. Those resulting from the target-mass and $t$-corrections of the
hadronic matrix elements always occur with an extra power in $x_{\rm BJ}$ or
$x_\PP$.

\subsection{The Antisymmetric Part}

\vspace{1mm}\noindent
The contribution to the antisymmetric part of the hadronic tensor is given
by, cf. \cite{BGR2006,gey2},
\begin{align}
W^ {\text{tw2}} _{[\mu\nu]}\kln{q}  &=
 -\pi \,\epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta} \int d\zeta  \,
   \bigg\{
    \frac{q_{\alpha}\,{\cal K}^a_{5\beta}}{q{\cal P}}
     \Big[g_{a1}(x;\zeta) + g_{a2}(x;\zeta)\Big]
    - \frac{q_{\alpha}\,\mathcal{P}_\beta}{q{\cal P}}
      \frac{(q{\cal K}_5^a)}{q{\cal P}}\; g_{a2}(x;\zeta)
    \nonumber\\
   & \qquad\qquad\qquad\qquad
    + \frac{1}{2}
     \frac{q_{\alpha}\,\mathcal{P}_\beta}{q{\cal P}}
     \frac{(\mathcal{P}{\cal K}_5^a)}{Q^2}\; g_{a0}(x;\zeta)
    \bigg\}\,,
    \label{Tas7}
\end{align}
in terms of the $\zeta$-integral.
Here the coefficients ${\cal K}_{5\,\mu}^a$ are given by (\ref{kinasym+}) and
the functions $g_{a\,k}(x;\zeta)\equiv
g_{a\,k}(x,\xi,\mathcal{P}^2\!/Q^2,\zeta)|_{k=0,1,2}$
read
\begin{align}
   g_{a1}(x;\zeta)&
     =
    \frac{x}{\xi}\frac{1}{[1+x^2\,\mathcal{P}^2/Q^2]^{3/2}}\times
\label{ga1}
    \\
     \qquad\qquad &\quad
    \left[
    \Phi_{5a}^{(0)}(\xi,\zeta)
    +
    \frac{x(\xi +x)\, \mathcal{P}^2/Q^2}{[1+x^2\mathcal{P}^2/Q^2]^{1/2}}
    \Phi_{5a}^{(1)}(\xi,\zeta)
    -
    \frac{x\xi\, \mathcal{P}^2}{2\,Q^2}\,
    \frac{2-x^2\mathcal{P}^2/Q^2}{1+x^2\mathcal{P}^2/Q^2}
    \Phi_{5a}^{(2)}(\xi,\zeta)
    \right]\!,
    \nonumber\\
      g_{a2}(x;\zeta)
   & =
    - \frac{x}{\xi}\frac{1}{[1+x^2\mathcal{P}^2/Q^2]^{3/2}}\times
\label{ga2}
    \\
    \qquad\qquad&\quad
    \left[
    \Phi_{5a}^{(0)}(\xi,\zeta)
    -
    \frac{1 - x \xi\, \mathcal{P}^2/Q^2}{[1+x^2\mathcal{P}^2/Q^2]^{1/2}}
    \Phi_{5a}^{(1)}(\xi,\zeta)
    -
    \frac{3}{2}
    \frac{x \xi\, \mathcal{P}^2/Q^2}{1+x^2\mathcal{P}^2/Q^2}
    \Phi_{5a}^{(2)}(\xi,\zeta)
    \right]\!,
    \nonumber\\
\intertext{}
  g_{a1}(x;\zeta)& + g_{a2}(x;\zeta)
   \nonumber \\
    &
= \frac{x}{\xi}\frac{1}{[1+x^2\mathcal{P}^2/Q^2]^{3/2}}
    \left[\left(1 + \frac{x\xi \mathcal{P}^2}{2Q^2}\right)
    \Phi_{5a}^{(1)}(\xi,\zeta)\,
    +\frac{x\xi \mathcal{P}^2}{2Q^2}
    \Phi_{5a}^{(2)}(\xi,\zeta)\,
    \right]
\label{ga12}
    \\
 &= \frac{x}{\xi}
 		\frac{1}{1+x^2\mathcal{P}^2/Q^2}
 		\left[\Phi_{5a}^{(1)}(\xi,\zeta)\,
     + \frac{1}{2}\,
       \frac{x\xi\, \mathcal{P}^2/Q^2}{[1+x^2\mathcal{P}^2/Q^2]^{1/2}}
    \Phi_{5a}^{(2)}(\xi,\zeta)\right]\,,
    \nonumber\\
   g_{a0}(x;\zeta)
     &
    = \frac{x^2}{[1+x^2\mathcal{P}^2/Q^2]^{3/2}}
    \times
\label{ga0}
    \\
    &\quad
    \left[
    \Phi_{5a}^{(0)}(\xi,\zeta)
    -
    \frac{3}{[1+x^2\mathcal{P}^2/Q^2]^{1/2}}
    \Phi_{5a}^{(1)}(\xi,\zeta)
    +
    \frac{2- x^2 \mathcal{P}^2/Q^2}{1+x^2\mathcal{P}^2/Q^2}
    \Phi_{5a}^{(2)}(\xi,\zeta)
    \right]\!~.
    \nonumber
\end{align}
The dGPDs $\Phi^{(i)}_{5\,a}(\xi,\zeta)$  are based on (\ref{non15})
and, similar to the definitions (\ref{phia0}--\ref{eqphiaa})
of $\Phi^{(i)}_{a}(\xi,\zeta)$,
\begin{eqnarray}
    \Phi^{(0)}_{5\,a}(\xi,\zeta)
    &\equiv&  \xi\, f_{5\,a}(\xi,\zeta) \,,
\label{phia50}\\
\Phi^{(i)}_{5\,a}\left(\xi,\zeta\right)
    &=&
    \int^{1}_{\xi}\frac{dy}{y} \,
    \Phi^{(i-1)}_{5\,a}\left(y,\zeta\right)\,, i \geq 1~.
\label{NVF}
\end{eqnarray}
As shown before \cite{BGR2006,gey2} the Wandzura--Wilczek (WW) relation
\cite{WW}
holds for the un-integrated distribution functions
\begin{eqnarray}
\label{ww11}
    g^{ {\text{tw2}} }_{a2}(x;\zeta)
    &=&
    -\, g_{a1}^{\rm tw2}(x;\zeta) +
    \int_{x}^1 \frac{dy}{y}g_{a1}^{\rm tw2}(y;\zeta)\,,
\end{eqnarray}
between $g_{a2}^{\rm tw2}$ and $g_{a1}^{\rm tw2}$. All target mass and
$t$--corrections can uniquely be
absorbed into the structure functions. Note that this relation holds for
all invariants ${\cal K}_5^a $ independently. The validity of the Wandzura--Wilczek
relation for diffractive scattering at general hadronic scales
$M^2, t$ is a further example in a long list of cases. It was observed using
the covariant parton model and light-cone expansion
\cite{Roberts:1996ub,Blumlein:1996vs}. For forward scattering,
target- and quark-mass corrections could be completely absorbed into the
structure functions
maintaining the WW-relation \cite{Piccione:1997zh,Blumlein:1998nv}. It is valid
for gluon-induced heavy flavor production \cite{Blumlein:2003wk},
non--forward scattering \cite{Blumlein:2000cx}, and
diffractive scattering in the limit $M^2, t \rightarrow 0$
\cite{Blumlein:2002fw}. In the electro--weak case further sum--rules
exist~\cite{Blumlein:1996vs}. Considering the target mass corrections
there are
new twist--3 integral relations~\cite{Blumlein:1998nv}.
The distribution function $g_{a0}^{\rm tw2}$ is also related to $g_{a1}^{\rm tw2}$ but in a more complicated manner:
\begin{eqnarray}
    \label{g02}
    g_{a0}^{ {\text{tw2}} }(x;\zeta)
    &=&
    x\xi\,g_{a1}^{\rm tw2}(x;\zeta)
    - \frac{2 x^2 +x\xi}{[1+x^2\mathcal{P}^2/Q^2]^{1/2}}
    \int_{x}^1\!
    \frac{dy}{y}g_{a1}^{\rm tw2}(y;\zeta)
    \nonumber \\
    & &
    +\frac{2x^2}{[1+x^2\mathcal{P}^2/Q^2]^{3/2}}
    \int_{x}^1\! \frac{dy}{y} \!
    \int_y^1\! \frac{dy'}{y'}
    g_{a1}^{\rm tw2}(y';\zeta)\,.
\end{eqnarray}

From (\ref{Tas7}) -- (\ref{ga0}) we now extract the
$\zeta$-independent functions.
In the kinematic  factors $\zeta$ appears only up to second power.
As preliminary classification we therefore can perform the $\zeta$--integrals according
to the $\zeta$--powers of the kinematic factors, not counting the internal
$\zeta$-dependence of the GPDs $ g_{ak}(x;\zeta) $ itself.

For each invariant $a$ let us define
\begin{eqnarray}
\label{eqGG}
    G_{a\,k}^{(n)}(x,\eta,t,p_+^2/Q^2) =
                \int d\zeta \, \zeta^n \,g_{a\,k}(x;\zeta)\,,
                \qquad
    k = 0,1,2,
\end{eqnarray}
so that from (\ref{Tas7}) one obtains
\begin{align}
 {\mathrm{Im}}\, T^ {\text{tw2}} _{[\mu\nu]}\kln{q}
 &= -\pi  \,\epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\,
   \Bigg\{
    \frac{q_{\alpha}\,{\cal K}^a_{5\beta}}{qp_+}
          \Big(G_{a1}^{(0)} + G_{a2}^{(0)}\Big)
    \nonumber\\
   & \quad
    - \frac{q_{\alpha}\,p_{+\beta}}{qp_+}\left(
              \frac{q{\cal K}_5^a}{qp_+}\, G_{a2}^{(0)}
   - \frac{1}{2}\frac{p_+{\cal K}_5^a}{Q^2}\, G_{a0}^{(0)}
    - \frac{1}{2
        \frac{\pi_- {\cal K}_5^a}{Q^2}\, G_{a0}^{(1)}
   \right)
    \nonumber\\
    & \quad
    - \frac{q_{\alpha}\,\pi_{-\beta}}{qp_+}\left(
              \frac{q{\cal K}_5^a}{qp_+}\, G_{a2}^{(1)}
    - \frac{1}{2}\frac{p_+{\cal K}_5^a}{Q^2}\, G_{a0}^{(1)}
    - \frac{1}{2}\frac{\pi_-{\cal K}_5^a}{Q^2}\, G_{a0}^{(2)}\right)\!\!
     \Bigg\}\,.
      \label{as88}
      \\
  & \approx
     -\pi  \,\epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\,
   \Bigg\{
    \frac{q_{\alpha}\,{\cal K}^a_{5\beta}}{qp_+}
          \Big(G_{a1}^{(0)} + G_{a2}^{(0)}\Big)
- \frac{q_{\alpha}\,p_{+\beta}}{qp_+}
              \frac{q{\cal K}_5^a}{qp_+}\, G_{a2}^{(0)}
    - \frac{q_{\alpha}\,\pi_{-\beta}}{qp_+}
              \frac{q{\cal K}_5^a}{qp_+}\, G_{a2}^{(1)}
\Bigg\}\,.
\end{align}
In the last line the leading terms are written only.
Now,  inserting the three invariants ${\cal K}_{5}^a $ in (\ref{kinasym+})
explicitly, one obtains disregarding sub-leading terms in $1/Q^2$~:
\begin{align}
{\mathrm{Im}}\, &{ T}^{(0)\, {\text{tw2}} }_{[\mu\nu]}\kln{q}
    \approx
    -\pi \, \epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\,
    \Bigg\{
    \frac{q_{\alpha}S^{\mathrm T}_{\beta}}{qp_+}
          \big(G_{1\,1}^{(0)} + G_{1\,2}^{(0)}\big)
    - \frac{q_{\alpha} p^{\mathrm T}_{+\beta}}{qp_+}
     \bigg[ \frac{q S}{q p_+}\;  G_{1\,2}^{(0)}
    -
      \frac{p_2.S}{M^2}\;  G_{2\,1}^{(0)} \bigg]
      \nonumber    \\
&\quad
  	- \frac{q_{\alpha} \pi^{\mathrm T}_{-\beta}}{qp_+}
   \bigg[ \frac{q S}{q p_+}\;  G_{1\,2}^{(1)}
    + \frac{p_2.S}{M^2} \bigg( G_{2\,2}^{(1)} - G_{3\,1}^{(0)} -
G_{3\,2}^{(0)}
   \bigg) \bigg]
    \!\Bigg\}~.
     \label{ergas0}
\end{align}
Due to the presence of the Levi-Civita symbol, only transversal
components (\ref{trans})
contribute. Note that $p_1.S=0$.
The approximate expressions for $ G_{a\,k}^{(n)}$ are
\begin{align}
G_{a1}^{(n)}(x)
     &  \approx  \Phi_{5a}^{(0\,n)}(x) - \frac{x^2\,p_+^2}{Q^2}\gamma_{a1}^n
        \nonumber\\
G_{a2}^{(n)}(x)
    & \approx
    - \left[
    \Phi_{5a}^{(0\,n)}(x)   - \Phi_{5a}^{(1\,n)}(x) \right]  -
    \frac{x^2\,p_+^2}{Q^2}\gamma_{a2}^n ,
    \nonumber\\
G_{a1}^{(n)}(x) + G_{a2}^{(n)}(x)
   & \approx    	\Phi_{5a}^{(1\,n)}(x)\, - \frac{x^2\,p_+^2}{Q^2}\gamma_{a12}^n
    \nonumber \\
G_{a0}^{(n)}(x)
    &\approx x^2 \left[
    \Phi_{5a}^{(0\,n)}(x)- {3} \Phi_{5a}^{(1\,n)}(x)
    + {2} \Phi_{5a}^{(2\,n)}(x) \right] -
\frac{x^2\,p_+^2}{Q^2}\gamma_{a0}^n~,
    \label{G_rel}
\end{align}
with
\begin{align}
\gamma_{a1}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(\frac{5}{4}  \Phi^{(0)}_{a5}(x, \zeta) - 2 \Phi^{(1)}_{a5}(x, \zeta)
           + \Phi^{(2)}_{a5}(x, \zeta)
          +\frac{1}{4}x \partial_{x}\Phi^{(0)}_{a5}(x, \zeta)\right),
\nonumber \\
\gamma_{a2}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(-\frac{5}{4} \Phi^{(0)}_{a5}(x, \zeta) + \frac{11}{4} \Phi^{(1)}_{a5}(x, \zeta)
          -\frac{3}{2}  \Phi^{(2)}_{a5}(x, \zeta)
\right.
\nonumber\\ & \hspace*{6cm} \left.
          -\frac{1}{4} x \partial_{x}\big(\Phi^{(0}_{a5}(x, \zeta)- \Phi^{(1)}_{a5}(x, \zeta)\big)
          \right),
\nonumber \\
\gamma_{a12}^n & = \int d\zeta (1+\epsilon(\zeta)) \zeta^n
          \left(-\hbox{\large$\frac{1}{2}$} x \Phi^{(2)}_{a5}(x, \zeta) +
          \frac{3}{4} \Phi^{(1)}_{a5}(x, \zeta)
          +\frac{1}{4} x\partial_x\Phi^{(1)}_{a5}(x, \zeta)\right)
\nonumber
\intertext{and}
\widetilde\gamma_{a1}^{\,n} & = \int d\zeta  \zeta^n
          \left(\frac{5}{4}  \Phi^{(0)}_{a5}(x, \zeta)
- 2 \Phi^{(1)}_{a5}(x, \zeta)
           + \Phi^{(2)}_{a5}(x, \zeta)
          +\frac{1}{4}x \partial_{x}\Phi^{(0)}_{a5}(x, \zeta)\right)~,
\nonumber
\end{align}
with similar expressions for the other terms $\widetilde\gamma_{ak}^{\,n}$.
The functions $\Phi_{5a}^{(1\,n)}(x)$ are determined as in Eq. (\ref{mph}).
The last function $ G_{a0}^{(n)} $ does not contribute to leading order. As
final approximate result in leading order  we obtain
\begin{align}
{\mathrm{Im}}\, &{ T}^{(0)\, {\text{tw2}} }_{[\mu\nu]}\kln{q}|_0
    \approx
    -\pi \, \epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\,
    \Bigg\{
    \frac{q_{\alpha}S^{\mathrm T}_{\beta}}{qp_+} \Phi_{51}^{(1,\,0)}(x)
 \label{ergas00}
     \\
  &  + \frac{q_{\alpha} p^{\mathrm T}_{+\beta}}{qp_+}
     \bigg[ \frac{q S}{q p_+}\;
       \left(\Phi_{51}^{(0,\,0)}(x)-\Phi_{51}^{(1,\,0)}(x)\right)
    +\frac{p_2 S}{M^2}\Phi_{52}^{(0,\,0)}(x) \bigg]
     \nonumber \\
&  + \frac{q_{\alpha} \pi^{\mathrm T}_{-\beta}}{qp_+}
   \bigg[ \frac{q S}{q p_+}\;\left(\Phi_{51}^{(0,\,1)}(x)
                         - \Phi_{51}^{(1,\,1)}(x)\right)
   + \frac{p_2 S}{M^2}\left(\Phi_{53}^{(1,\,0)}(x)
   +\Phi_{52}^{(0,\,1)}(x)-  \Phi_{52}^{(1,\,1)}(x)
    \right) \bigg] \Bigg\}~.
  \nonumber
\end{align}
In Ref.~\cite{Blumlein:2002fw} the terms $ \propto p_2.S$ were neglected
treating $p_2 || p_1$ and vanishing contributions $\propto \pi_-$. While this
is correct for $t \rightarrow 0$, a finite
contribution remains for $M^2 \rightarrow 0$, Eq.~(\ref{eqSp2}),
\begin{eqnarray}
\lim_{t \rightarrow 0} \frac{p_2.S}{M^2} = x_\PP \frac{1-x_\PP/2}{1-x_\PP}
+ {\cal O}\left(x_{\rm BJ} x_\PP \frac{M^2}{Q^2}\right)~.
\end{eqnarray}
So the previous result \cite{Blumlein:2002fw} is to be modified by a third term
\begin{align}
\label{polNEW}
{\mathrm{Im}}\, { T}^{(0)\, {\text{tw2}} }_{[\mu\nu]}\kln{q}|_0
    \approx
    -\pi \, \epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\,
    \Bigg\{
    \frac{q_{\alpha}S^{\mathrm T}_{\beta}}{qp_+} \Phi_{51}^{(1,\,0)}(x)
  + \frac{q_{\alpha} p^{\mathrm T}_{+\beta}}{qp_+}
\Bigl[
\frac{q S}{q p_+}\;
       \left(\Phi_{51}^{(0,\,0)}(x)-\Phi_{51}^{(1,\,0)}(x)\right)
      + x_\PP \Phi_{52}^{(0,\,0)}(x) \Bigr]\Bigg\}~.
\end{align}

The $t$-dependent correction terms result from the $(\eta^2 + t/p_+^2)$--contributions
in $\epsilon$ and they are entirely
contained in correction terms $\gamma_{ak}^n $~:
\begin{eqnarray}
\label{ERG}
{\sf Im}\, { T}^{(0)\, {\text{tw2}} }_{[\mu\nu]}\kln{q}|_t
    &\approx&
    +\pi \, \epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\,\frac{x^2\,t}{Q^2}
    \Bigg\{
    \frac{q_{\alpha}S^{\mathrm T}_{\beta}}{qp_+}
          \big( (\widetilde\gamma_{1\,1}^{2} + \widetilde\gamma_{1\,2}^{2})
             +2\eta  (\widetilde\gamma_{1\,1}^{1} + \widetilde\gamma_{1\,2}^{1})
             - (\widetilde\gamma_{1\,1}^{0} + \widetilde\gamma_{1\,2}^{0})
                      \big)  \nonumber\\
&&\quad    - \frac{q_{\alpha} p^{\mathrm T}_{+\beta}}{qp_+}
     \bigg[ \frac{q S}{q p_+}\;
 \big( \widetilde\gamma_{1\,2}^{2} +2\eta \widetilde\gamma_{1\,2}^{1}
           - \widetilde\gamma_{1\,2}^{0}\big)

  - \frac{p_2 S}{M^2}\; \big( \widetilde\gamma_{2\,1}^{2}
       +2\eta \widetilde\gamma_{2\,1}^{1} -\widetilde\gamma_{2\,1}^{0}
      \big) \bigg]
      \nonumber    \\
&&\quad    	- \frac{q_{\alpha} \pi^{\mathrm T}_{-\beta}}{qp_+}
   \bigg[ \frac{q S}{q p_+}\;
    \big(  \widetilde\gamma_{1\,2}^{3}  +2\eta \widetilde\gamma_{1\,2}^{2}
           - \widetilde\gamma_{1\,2}^{0} \big) \nonumber \\
&&\quad \qquad \qquad    + \frac{p_2 S}{M^2} \bigg(( \widetilde\gamma_{2\,2}^{3} -
     \widetilde\gamma_{3\,1}^{2} - \widetilde\gamma_{3\,2}^{2})
     +2\eta (\widetilde\gamma_{2\,2}^{2} -
     \widetilde\gamma_{3\,1}^{1} - \widetilde\gamma_{3\,2}^{1})
\nonumber\\ &&\hspace*{3cm}
  -(\widetilde\gamma_{2\,2}^{1} -
     \widetilde\gamma_{3\,1}^{0} - \widetilde\gamma_{3\,2}^{0})
     \bigg) \bigg]
    \!\Bigg\}~.
\end{eqnarray}
The corresponding terms are of the same size as in the unpolarized case, Section~4.1,
and may have a quantitative effect only in the low $Q^2$--region in the meson--exchange case.

It is remarkable that the Wandzura-Wilczek relation \cite{WW} remains intact after
$\zeta$-integrations and is  valid for the experimentally observable moments,
\begin{eqnarray}
    G^{(n)}_{a\,2}(x,\eta, t, p_+^2/Q^2)  = -\, G_{a\,1}^{(n)}(x,\eta, t, p_+^2/Q^2)
               + \int_{x}^1\!\! \frac{dy}{y}\;G_{a\,1}^{(n)}(y,\eta, t, p_+^2/Q^2)\,.
    \label{G22}
\end{eqnarray}
The second integral relation (\ref{g02}) contains the $\zeta$--dependent denominator
$\sqrt{1 +x^2 {\cal P}^2/q^2}$ so that we obtain after $\zeta$-integration more
complicated expressions.
In the approximation  $\pi_- =0 $
one obtains
\begin{align}
   \label{G02}
    G_{a\,0}^{(n)}(x,\eta, t, p_+^2/Q^2)
    & \approx
     x\xi_0\;G_{a\,1}^{(n)}(x,\eta, t, p_+^2/Q^2)
    - \frac{2x^2+ x\xi_0}{[1+4x^2\,p_+^2/Q^2]^{1/2}}
    \int_{x}^1\!
    \frac{dy}{y}\;G_{a\,1}^{(n)}(y,\eta, t, p_+^2/Q^2)
    \nonumber \\
    & \quad
    +\frac{2x^2}{[1+4 x^2\, p_+^2/Q^2]^{3/2}}
    \int_{x}^1\!\! \frac{dy}{y} \!
    \int_y^1\! \frac{dy'}{y'}\;
    G_{a\,1}^{(n)}(y',x,\eta, t, p_+^2/Q^2)\,.
\end{align}
$\xi_0$ denotes the Nachtmann variable (\ref{zeta0}).
However the functions $ G_{a\,0}^{(n)}(x,\eta, t, p_+^2/Q^2) $ contribute  to sub-leading terms
only.

\section{Conclusions}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
\vspace{2mm}
\noindent
Deep--inelastic diffractive scattering, like other hard scattering
processes off nucleons, requires target mass corrections in the region of
lower $Q^2$-scales. In fact, the nucleon mass $M$ is not the only hadronic scale
relevant to that process where both the incoming and outgoing nucleon play a role.
The invariant $t = (p_2 - p_1)^2$ on average is of the same size as $M^2$.\footnote{
In case of related semi--exclusive processes in which more than one final--state
hadron is well separated in rapidity from the inclusively produced hadrons other
invariants more would emerge.} In the present paper we investigated in detail
the conditions under which terms like $M^2/Q^2$ or $|t|/Q^2$ contribute.

We considered the leading twist contributions for which factorization theorems
allow a partonic description. With the help of A. Mueller's generalized optical
theorem it was possible to reformulate diffractive scattering in terms of
deep--inelastic scattering off an effective two--nucleon pseudo-state
accounting for $t$. All essential expressions determining experimentally
relevant quantities are the diffractive generalized parton densities (dGPD)
defined as expectation values of non-local light-cone operators
(\ref{non1},\ref{non15}). The involved iterated diffractive dGPDs
(\ref{phia0}--\ref{eqphiaa2}), respectively (\ref{phia50}) and
(\ref{NVF}), $ \Phi_{(5)a}^{(i)}(\lambda , \zeta, t , \eta; \mu^2)$ depend on at
least three variables, $ \lambda , \zeta$ and $ t $. Hereby $t$ is
an external variable, whereas  $\lambda$ is defined as overall scale
multiplied with  a generalized momentum in  $(p_+z_+ + p_-z_-) =\lambda {\cal P}$.
In the hadronic tensor $ W_{\mu\nu} = \textrm{Im} \, T_{\mu \nu}$ it
is fixed by $\xi$, the  generalized Nachtmann variable (\ref{xicd}). Moreover
the generalized momentum ${\cal P}= p_+ + \zeta \pi_- $ splits into a
``longitudinal'' and a ``transversal'' part $\pi_-$ multiplied by a new
variable $\zeta$ and can be treated separately.
The problem in applying the results of our previous work \cite{BGR2006} is the
dependence of the dGPDs on the `internal' variable $\zeta $ which is not measurable in experiment
since it contributes through a definite integral in the final
expressions. We performed an expansion w.r.t. the external variable $p_+^2/Q^2$.
This leads to a set of integrated dGPDs which describe the process and the relevant mass
corrections in a well-defined approximation.

One of our results is a prescription of experimental data in terms of
experimentally accessible integrated diffractive GPD's,
\begin{eqnarray}
\Phi_{(5)a}^{(i\, n)} (\xi, t, \eta)
= \int d\zeta\, \zeta^n \,\Phi_{(5)a}^{(i)} (\xi, \zeta, t,\eta)\,,
\end{eqnarray}
or approximately by the functions (\ref{mph}),
which could be considered as diffractive parton densities, as it is the case for
vanishing masses \cite{Blumlein:2001xf}. For our approximation a similar relation
holds,
where $\xi$ is substituted by the variable $x=Q^2/qp_+ $.
Note that one and the same diffractive input GPD
$\Phi_{(5)a}^{(i)} (\xi, \zeta, t,\eta)$  determines several amplitudes with different
kinematic factors. This can be seen in  the lowest approximations (\ref{T_01}) or
(\ref{ergas00}) and  for the $t$-dependent corrections  (\ref{tgsym}) and
(\ref{ERG}).

The $t$-- and $M^2$--dependence due to the functions $\Phi_{(5)a}^{(in)}
(\xi, t,\eta)$, besides the non-perturbative $t$--behaviour, turns out to be of
${\cal O}(x^2_{{\rm BJ}(\PP)} \mu^2/Q^2),~\mu^2 = |t|, M^2$. Some of the kinematic factors emerging in
the
scattering cross section turn out to be less suppressed and are of ${\cal O}(x_{{\rm BJ}(\PP)} \mu^2/Q^2)$.
In the case of diffractive scattering the region of $x_{\rm BJ}$ and $x_\PP$ is effectively
limited
by $\lessapprox 10^{-2}$. The corresponding corrections cannot be resolved at the
experimental
accuracy. The effects are larger in the case of meson--exchange processes with a fast hadron
due to the range $x \lessapprox 0.3$.
Due to the smallness of these corrections the diffractive distribution
functions obey a partonic description, where $t$ plays the role of an additional variable
besides $\beta=x_{\rm BJ}/x_\PP$.

At the level of twist--2 the structure functions the scattering cross section can be built from
the corresponding operator expectation values (\ref{non1}--\ref{non15}) as in the case of
deep--inelastic scattering since the specifics of diffractive scattering is
moved into the corresponding two--particle wave functions. Consequently,
the logarithmic scaling violations, which can be completely associated
with that of the {\sf operators},
cf.~\cite{Blumlein:1999sc,Blumlein:2001xf}, are found to be the same as in
DIS or DVCS, if the complete diffractive GPDs are used.

The integral relations (\ref{w1}), (\ref{ww11}) and (\ref{g02}) can be transformed in part
to the integrated functions only.
The presence of target mass and $t$-effects enlarges the number of
structure functions determining the hadronic tensor if compared to the case of forward scattering.
As shown in the present paper, these corrections are suppressed by at least one power in $x_{\rm
BJ}$ or $x_\PP$ and therefore the picture derived in \cite{Blumlein:2001xf,Blumlein:2002fw} remains
valid quantitatively. In the polarized case, there is a new term, cf. (\ref{polNEW}), which
contains $x_\PP$ as prefactor. The Wandzura--Wilzcek relation remains unbroken
and holds even separately for the contributions of the three different invariants
${\cal K}_a^5$ (\ref{kinasym+}). We have also shown how the present formalism can be used
to derive the target mass corrections in the limit of forward scattering.

redpajama_set_name: RedPajamaArXiv


text:
MS Governor Tate Reeves Signs Life Equality Act into Law
July 2, 2020 CONTACT: Mallory Quigley
mquigley@sbalist.org, (202) 223-8073
Historic Legislation Protects Unborn Children from Abortion on the Basis of Sex, Race, or Genetic Makeup
Jackson, MS. – The national pro-life group Susan B. Anthony List (SBA List) today celebrated as Mississippi Governor Tate Reeves signed the Life Equality Act, legislation to stop the practice of discrimination abortions carried out because of a child's race, sex, disability, or genetic makeup, into law. The bill passed by overwhelming margins in the state legislature.
State Representative Carolyn Crawford authored the House bill and State Senator Jenifer Branning sponsored the Senate bill. Both are members of SBA List's National Pro-Life Women's Caucus. The legislation has also been championed by House Speaker Philip Gunn, State Representative Nick Bain, Senate Judiciary Chairman Brice Wiggins and Lt. Governor Delbert Hosemann.
"The enactment of the Life Equality Act is an historic victory for the pro-life citizens of Mississippi, for the voiceless victims of discrimination and all who advocate for them," said SBA List State Policy Director Sue Liebel. "Starting now, unborn babies in Mississippi cannot be targeted for abortion based on their sex, race, or potential disability, such as Down syndrome. Such lethal discrimination, whether inside or outside the womb, should be unacceptable anywhere in our society.
"We thank Governor Reeves for his strong pro-life leadership in signing this bill swiftly into law, as well as the bill's author Rep. Carolyn Crawford, House Speaker Philip Gunn, and Rep. Nick Bain, Senate Judiciary Committee Chairman Brice Wiggins, Sen. Jenifer Branning, Sen. Jeremy England, Sen. Joey Fillingane, and Lt. Gov. Delbert Hosemann for their tireless efforts to get it across the finish line."
Mika Hartman and her son Hudson, who has Down syndrome, recently met with Mississippi legislators to advocate for the Life Equality Act. Mika and her husband chose life for Hudson after being encouraged to abort her son due to his disability. Today Hudson is a happy, healthy toddler.
Polling conducted early this year found that 65% of Americans, including 50% of those who identify as "pro-choice," oppose discrimination abortion carried out because a baby has Down syndrome.
SBA List is a network of more than 837,000 pro-life Americans nationwide, dedicated to ending abortion by electing national leaders and advocating for laws that save lives, with a special calling to promote pro-life women leaders.
Posted in Home, Press Releases

redpajama_set_name: RedPajamaCommonCrawl


Considering the speed they use in their example I'm wondering if what they have written there is still true today.
Yes, nothing has changed - what they wrote is still true. Your drive writes CD-R media at 32x max, rewrites CD-RW media at 12x max, and reads at 40x max.


text:
Q: Pros and cons of load specific view to mobile devices I'm using Codeigniter, and I've noticed using the User agent library we can know if the user device to know if is a mobile or not. And based on this, we can load one or another view with the same url. Something like this:
$this->load->library('user_agent');
$data = array (
     //some data
);
if (!$this->agent->is_mobile()) {
     $this->load->view('test_view',$data);
}
else {
    $this->load->view('test_view_mobile',$data);
}

Well, it works perfectly, and using this:

*

*We dont have to put css media-queries. Just put the mobile css in the corespondient view.

*We don't have to track exceptions for mobile user such as touch events or similar.

*We can load small images than desktop devices. Becouse the screen size is smaller.

These are the pros that I realiced. But what about the cons? Is there any problematic or issue using this practice?

A: Here are reasons not to use it - even if the library works perfectly - new devices come out all the time. And if the device is not recognized - then they are getting the full screen version.
And even if you have a team of people and every device is recognized - you are creating two different content resources for the same page. This means that as your content and resources change - you have to change it in two different places. And there are search engine issues with this.
All of these issues are why front end responsive libraries are so popular - they take care of these issues AND make your site content work better on tablets.  Twitter Bootstrap is by far the most popular. http://getbootstrap.com https://github.com/twbs/bootstrap
note that if you go to learn more about twitter bootstrap there are breaking changes between bootstrap 2 and bootstrap 3 (latest version)

A: Cons are: The documentation sucks.

*

*For me as a CodeIgniter novice, it's unclear what $this refers to.

*What exactly is a mobile device?

*Which mobile devices are known?

*What argument must I pass to the function to test for a particular mobile device?

*I've seen 'iphone' has been passed to the function in the example. Does that include all iPhones or just the ones that where available at the time that the function was written?

*What characteristics can I assume about the client when the function returns true?

redpajama_set_name: RedPajamaStackExchange


text:
Fresno County Preterm Birth Initiative honored for its research
Date: October 20, 2017Author: fresnostatechhs 0 Comments
The Fresno County Preterm Birth Initiative received the 2017 Excellence in Partnership Award in the Research and Evaluation category, presented by the University of California, San Francisco's Center for Community Engagement and Council.
Representatives from the initiative received the award on Oct. 19 at the Annual Partnerships Celebration at UCSF Medical Center.
Fresno County PTBi Steering Committee members, Emilia Reyes (middle left), executive director of First 5 Fresno County, and Erica Martinez (middle right), member of the PTBi Parent's Council, accept the award in San Francisco. Oct. 19, 2017.
The Excellence in Partnership Award, given annually, recognizes exemplary partnerships between community initiatives and UCSF that build on each other's strengths to improve higher education, civic engagement and overall health of communities.
The awardee must work in an existing partnership to achieve the systems and policy changes needed to overcome the root causes of health inequities. The award highlights the power and potential of university-community partnerships as a strategy for social justice.
Through a partnership between the UCSF California Preterm Birth Initiative and the Central California Center for Health and Human Services at Fresno State, local leaders and organizations across Fresno County have joined the initiative and are working collaboratively to address the epidemic of preterm birth through the Collective Impact.
In April 2015, a steering committee comprised of leaders from health care, education, community-based organizations, businesses and other sectors that serve women and families, was formed to help guide the initiative.
Since then, a community-based approach has been used to help reduce preterm births in Fresno County from 11.1 percent to 7 percent by 2025, resulting in a 37 percent reduction of babies born prematurely in Fresno County.
The initiative recently launched GLOW! – a research initiative that explores a new way to deliver group prenatal care with the goal of reducing early delivery.
In total, 22 committee members, 129 partners, 77 organizations, systems and community members, including moms with lived experience, are working with the initiative. They are providing support and services to communities at highest risk for preterm birth, including low-income families, families living in southwest Fresno and African American, Hispanic, and Southeast Asian families.
"This award celebrates what the Fresno County Preterm Birth Initiative has been able to achieve, utilizing a commitment to partnerships that greatly influence our region and bring a sense of pride to our community," said Fresno State President Joseph I. Castro, a member of the steering committee.
The Fresno County Preterm Birth Initiative is part of a 10-year Preterm Birth Initiative, led by UCSF and funded by philanthropists Lynne and Marc Benioff. Fresno State serves as the backbone organization for the Fresno initiative.
For more information on the initiative, contact program director Sandra Flores at 559.278.2126 or shflores@csufresno.edu.
Previous Previous post: Public health students bring clean water initiative to Dominican Republic
Next Next post: Meet 2017 Health & Human Services Hero, Lynne Ashbeck


text:
William Peter Blatty
1928-01-07 New York City
Novelist screenwriter film director
William Peter Blatty (born January 7, 1928) is an American writer and filmmaker. The novel The Exorcist, written in 1971, is his most well-known novel; he also penned the subsequent screenplay version of the film, for which he won an Academy Award. He also wrote and directed the sequel, The Exorcist III: Legion.His most recent works include the novels Elsewhere (2009), Dimiter (2010), and Crazy (2010). He is also featured in the upcoming 2014 Smoke And Mirrors anthology, featuring the teleplay "Hell Hospital" and the treatment "Faith".In 2013, Demons Five, Exorcist Nothing: A Fable (1996) and Dimiter (2010) were re-released as revised editions with new covers and interior artwork. Each were limited to 250 signed copies.
Items by William Peter Blatty


This is the 100th story at pfft.net. What a milestone, huh? I want to thank all the readers and writers who have contributed to this website over the.. months.. and I hope we get to 200 twice as fast as we got here. Keep up the good work. Also, we're trying to develop our "Movies" category a little bit, with either your PFFT! stories relating to going to the movies, or movie reviews. Any contributions in this area would be greatly appreciated.
Now, back to your regularly scheduled editorial.
There are some flaws in the US Postal Service. Anyone who lives in or ever deals with the US will know and understand that. This latest complaint comes from the 3 weeks I spent a short time ago where I received no mail because the mail carrier decided he shouldn't deliver stuff with my name on it to my mailbox. There were some important documents that were sent to me during that time, and although I did finally receive them, they were later than I had hoped and some stuff I was working on was delayed.
However, I want to thank the people at Copyleft for helping me out during this time. I ordered their DeCSS t-shirt (in support of 2600 and the programmers who wrote a DVD decrypting program), but because none of my mail was delivered, it got lost in the mail. The Post Office employees told me there was no way for them to locate the mail that was sent to me during that time, but Copyleft was kind enough to send another shirt. Many companies would have said, "Too bad. We shipped it, it's not our fault."
The post office could use some reorganization. I know it would be impossible for them to track every piece of mail that's sent like UPS or FedEx, but it would be nice. It would also be nice if they would just deliver mail to the address it's sent to, no matter what the name is on it!


Keep your online message on-point, with 'hands-free' web page updating.
Publishing content to the Web doesn't have to be difficult and it should have to require IT involvement. By simply placing content into a designated folder on a user's network, a Website is automatically updated with the content.
Group contributions to a single Website can now be easily managed through storage network submissions. Access rights can even be established to restrict or grant access to a single user or entire group.
By simply dropping content into a folder on the network, Pointspin updates and publishes the web page.
Eliminating IT involvement, users and department groups can manage and publish their own web content.
Individuals can govern access to their pages by requiring a password to keep unwanted viewers restricted.
An encapsulated webserver engine simplifies administration while minimizing configuration and maintenance.
Tired of reading? Check out our videos!
Will Pointspin™ provide the web publishing capabilities w/in Novell Open Enterprise Server that were available in mod_edir?
Absolutely! One of the principle reasons we developed Pointspin was to address the absence of a web publishing tool in Novell Open Enterprise Server. We're certain that former mod-edir users will be pleased with this product.
How does Pointspin assign website URLs?
When web content is added to a designated subfolder, the Pointspin Engine serves it up by assigning a URL based on the webserver name and user home folder or group home folder name. For example, the URL for web content located in a user home folder would be: http://www.domainname/~username.
All administration work takes place in a web browser based administration interface. Much of the administrative work pertains to initial setup and configuration and once established, requires no follow-up administration. The minimalistic administrative work combined with the ability to automatically publish websites make Pointspin a product that is very cost-effective to manage.
Does Pointspin have measures in place for preventing the publication or download of copyrighted materials from our websites?
Pointspin provides a Content Management utility through the web interface. Here, an administrator can block Pointspin from serving files with a provided file extension (.mp3, .swf, etc.). This is on a domain-wide basis, not an individual basis. Lastly, content types over a certain size (in megabytes) can also be blocked in the administrative web interface.
What is the websever software that Pointspin utilizes?
The Pointspin Engine includes a proprietary web server that Condrey Corporation first introduced in 2005 with its Kanaka for eDirectory product. Since its initial release, the web server has continued to be enhanced in its features and performance. Thousands of Condrey Corporation customers utilize this web server every day.
How does Pointspin utilize the identity information stored in Microsoft Active Directory or Novell eDirectory?
Pointspin looks for the requested username in the directory service index and serves the website out of the proper folder.
Can we install multiple Pointspin Engine servers within our directory tree?
Yes you can. In fact, we recommend that you do so to support high loads or to support the various differences in Pointspin publication settings among the users in an organization.
In addition to serving web content from a subfolder of a user home folder, from where else can Pointspin serve web content?
What types of security is built into Pointspin?
The client/server data is sent via industry-standard HTTPS, and the Pointspin server rejects HTTP POST requests except in the password-protected administrator page (meaning a client cannot send arbitrary data to the server; it can only receive data from the server).
Can some web pages be accessible only through a username and password, while allowing other to be fully accessible?
Of course. Through the administrative interface, a network administrator can establish settings that require that a valid username and password to access the web content. If these settings are not specified, the web site by default, is accessible to everyone.
If we already have an Apache web server or Microsoft IIS web server, can those work in conjunction with the Pointspin Engine?
Yes, if different ports are used for Pointspin and the other server software. However, we strongly recommend a dedicated server for Pointspin.
From a single class, is it possible to prevent some students from publishing web content while allowing others to publish?
Using the administrative interface, Pointspin allows you to disable specific users and groups from publishing.


text:
Press Release: ProQuest [April 30, 2014]
KERIS Selects ProQuest to Provide Unprecedented Collection of Scholarly Humanities and Social Science Historical Periodicals
ANN ARBOR, MI, April 30, 2014 � Korean Education and Research Information Service (KERIS), a consortium with 300 members, expands its holdings of scholarly periodicals from ProQuest. Periodicals Archive Online connects researchers to a rich digital archive of leading historical periodicals, spanning more than two centuries of scholarship in the humanities and social sciences. The ability to search across all of these journals simultaneously allows researchers to uncover valuable material and insights in subject areas and publications that they might not otherwise explore, thereby, helping to increase unique research output.
The purchase of Collection 2 provides access to 100 additional titles that comprise more than 500,000 articles, covering 30 key subject areas and 12 languages. Researchers can swiftly access relevant material from a variety of publication types, encompassing major scholarly titles, magazines/reviews, venerable trade publications and older general interest titles from the nineteenth century.
"This addition to our ProQuest collection illustrates our continued commitment to providing our universities with important research resources in humanities and social sciences, and represents a key academic asset for our country. This shared access sponsored by KERIS enables institutions of all sizes to benefit from this permanent digital archive and increases our return on investment," said D.H. Son, director, academic research information division, KERIS. "Our members highly value the marketing and training support they receive from ProQuest. Attention to end-user satisfaction and providing key user statistics further demonstrates the value delivered by ProQuest."
"Working with KERIS is a collaborative effort and ensures we provide the best solution for meeting the requirements of its members to improve researchers and librarians outcomes," said Boe Horton, vice president sales, Asia Pacific region, ProQuest. "We understand the unique attributes of large consortia and recognize the importance of promoting and improving library resource sharing among Korean universities."
With each journal in Periodicals Archive Online digitized from its first issue, the archive offers an overview of the history of scholarship in numerous fields, as well as thousands of articles that illuminate contemporary attitudes, events and interests at the time of their publication. Advanced search capabilities and post-search filters allow users to quickly and efficiently navigate this wealth of content and locate the most relevant items.
About KERIS
Established in 1999, the Korean Education and Research Information Service (KERIS) promotes public education through the use of Information and Communications Technology (ICT) in schools. For higher education and academic research, the organization aims to increase Korea's competitiveness by operating the participatory academic research content service for advancing the teaching/learning methods and sharing content. The mission of KERIS is to develop human resources through e-Learning, regain public trust in education, lay a foundation for a knowledge and information-based society by activation of e-Learning, and to enhance national education and research competitiveness through academy digitalization.
ProQuest (www.proquest.com) connects people with vetted, reliable information. Key to serious research, the company's products are a gateway to the world's knowledge including dissertations, governmental and cultural archives, news, historical collections and ebooks. ProQuest technologies serve users across the critical points in research, helping them discover, access, share, create and manage information.
The company's cloud-based technologies offer flexible solutions for librarians, students and researchers through the ProQuest, Bowker, Dialog, EBL, ebrary and Serials Solutions businesses � and notable research tools such as the Summon discovery service, Flow, Pivot and the Intota library services platform. The company is headquartered in Ann Arbor, Michigan, with offices around the world.
Korean Education and Research Information Service, a consortium with 300 members, expands its holdings of scholarly periodicals from ProQuest. Periodicals Archive Online connects researchers to a rich digital archive of leading historical periodicals, spanning more than two centuries of scholarship in the humanities and social sciences. The ability to search across all of these journals simultaneously allows researchers to uncover valuable material and insights in subject areas and publications that they might not otherwise explore, thereby, helping to increase unique research output.