\documentclass[fleqn,12pt,twoside]{article}
\usepackage{espcrc1}
% Use the option 'headings' if you want running headings
%\usepackage[headings]{espcrc1}

% identification
%\readRCS
%$Id: espcrc1.tex,v 1.2 2004/02/24 11:22:11 spepping Exp $
%\ProvidesFile{espcrc1.tex}[\filedate \space v\fileversion
%     \space Elsevier 1-column CRC Author Instructions]

% change this to the following line for use with LaTeX2.09
%\documentstyle[12pt,twoside,fleqn,espcrc1]{article}

% if you want to include PostScript figures
\usepackage[dvips]{graphicx}
% if you have landscape tables
%\usepackage[figuresright]{rotating}

% put your own definitions here:
%   \newcommand{\cZ}{\cal{Z}}
%   \newtheorem{def}{Definition}[section]
%   ...
\newcommand{\ttbs}{\char'134}
\newcommand{\AmS}{{\protect\the\textfont2
  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}

\newcommand{\nn}{\nonumber}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\ci}[1]{\cite{#1}}
\def\vk{{\bf k}_{\perp}}
\def\vl{{\bf l}_{\perp}}
\def\vd{{\bf \Delta}_\perp}
\def\vbs{{\bf b}}
\def\vb0{{\bf b}_0}
\newcommand{\LQCD}{\Lambda_{\rm{QCD}}}
\def\als{\alpha_s}
\def\ale{\alpha_{\rm elm}}
\def\mev{\,{\rm MeV}}
\def\gev{\,{\rm GeV}}
\def\kev{\,{\rm keV}}
\def\xbj{x_{\rm Bj}}
\newcommand{\sla}{\hspace*{-0.20cm}/}
\newcommand{\Sla}{\hspace*{-0.22cm}/}
\newcommand{\da}{{distribution amplitude}}
\newcommand{\wf}{wavefunction}
\newcommand{\lsim}{\raisebox{-4pt}{$\,\stackrel{\textstyle
                                                         <}{\sim}\,$}}
\newcommand{\gsim}{\raisebox{-4pt}{$\,\stackrel{\textstyle
                                                         >}{\sim}\,$}}
\newcommand{\tr}[1]{{\bf #1}_\perp}
\newcommand{\ov}[1]{\overline#1}
                                                         
\newcommand{\req}[1]{(\ref{#1})}
\def\xb{\bar{x}}
\def\sh{\hat{s}}
\def\uh{\hat{u}}
\def\={\,=\,}
\def\eps{\epsilon}
\def\veps{\varepsilon}


% add words to TeX's hyphenation exception list
%\hyphenation{author another created financial paper re-commend-ed Post-Script}

% set the starting page if not 1
% \setcounter{page}{17}
\begin{document}
% declarations for front matter
\title{Wide-angle exclusive scattering  - an update} 
       
\author{P.\ Kroll 
        Fachbereich Physik, Universit\"at Wuppertal, \\ 
        D-42097 Wuppertal, Germany}%
       % \thanks{Footnotes should appear on the first page only to
               
\runtitle{1-column format camera-ready paper in \LaTeX}
\runauthor{P.\ Kroll}

%\begin{document}

% typeset front matter
\maketitle

\begin{abstract}
The handbag mechanism for wide-angle exclusive scattering is 
discussed and compared with other theoretical approaches. Its application 
to  Compton scattering, meson photoproduction and two-photon annihilations 
into pairs of hadrons is reviewed in some detail.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recently a new approach to wide-angle Compton scattering off protons
has been proposed \cite{rad98,DFJK1} where, for Mandelstam variables 
$s,-t,-u$ that are large as compared to a typical hadronic scale, $\Lambda^2$ 
of  the order of $1\gev^2$, the process amplitudes factorize into a hard 
parton-level subprocess, Compton scattering off quarks, and in soft form 
factors which represent $1/x$ moments of generalized parton distributions 
(GPDs) and encode the soft physics (see Fig.\ \ref{fig:handbag}a ). 
Subsequently it has been realized that this so-called handbag mechanism 
also applies to a number of other wide-angle reactions such as virtual 
Compton scattering \cite{DFJK2} (provided the photon virtuality, $Q^2$ is 
smaller than $-t$), meson photo- and electroproduction \cite{hanwen} 
or two-photon annihilations into pairs of mesons \cite{DKV2} or baryons 
\cite{DKV3,weiss} (see Fig.\ \ref{fig:handbag}b). Last not least the 
photon-pseudoscalar meson transition form factor is to be mentioned 
(see Fig.\ \ref{fig:handbag}c) which is controlled by the
same subprocess, $\gamma^* \gamma^{(*)}\to q\bar{q}$ as the other
two-photon annihilation processes. It should be noted that the handbag
mechanism bears resemblance to the treatment of inelastic Compton 
scattering advocated for by Bjorken and Paschos \cite{bjo} long time ago.  

There are other mechanisms which may also contribute to wide-angle scattering 
besides the handbag which is characterized by one active parton, 
i.e.\ one parton from each hadron participates in the hard subprocess
(e.g.\ $\gamma q\to \gamma q$ in Compton scattering) while all others are
spectators. On the one hand, there are the so-called cat's ears graphs 
(see Fig.\ \ref{fig:handbag}d) with two active partons participating in the 
subprocess (e.g.\ $\gamma qq \to \gamma qq$). It can be shown however that 
in these graphs either a large parton virtuality or a large parton 
transverse momentum occurs. This forces the exchange of at least one hard 
gluon. Hence, the cat's ears contribution is expected to be suppressed 
as compared to the handbag one. The next class of graphs are characterized 
by three active quarks (e.g.\ $\gamma qqq \to \gamma qqq$) which 
require the exchange of at least two hard gluons (see Fig.\
\ref{fig:handbag}e). One can go on and 
consider four active partons and so forth. The series generated that way,
bears resemblance to an expansion in terms of n-body operators used 
in many-body theory. In principle, all the different contributions have 
to be added coherently. In practice, however, this is a difficult, 
currently almost impossible task since each contribution has its own 
associated soft hadronic matrix element which, as yet, cannot be 
calculated from QCD and is in general phenomenologically unknown.
%have to learn from experiment, 
%presently characterized by momentum transfers of the order of 
%$10 \gev^2$, whether one of the mentioned mechanisms is dominant  
%or whether the coherent sum of some or all topologies is 
%actually needed. 

Let me return to the case of three active partons. In valence quark
approximation there is no spectator left. Hence, the GPD blob decays into 
two hadron distribution amplitudes and one arrives at the ERBL
factorization scheme \ci{ERBL} (see Fig.\ \ref{fig:handbag}f). This contribution 
is expected to dominate at asymptotically large momentum transfer; the 
handbag contribution is formally a power correction to it. However,
for momentum transfer of the order of $10\,\gev^2$, characteristic of 
current wide-angle experiments, the ERBL contribution is about a
factor of $10^{-2} - 10^{-3}$  below experiments for reactions
involving baryons. The onset of the ERBL region is expected to occur 
above $-t (-u) > 100\,\gev^2$. 
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.28\textwidth,bb=123 238 344 404,clip=true]{handbag-c.ps} 
\includegraphics[width=0.65\textwidth,bb=123 520 590 647,clip=true]{handbag-c.ps}
\includegraphics[width=0.28\textwidth,bb=338 582 485 692,clip=true]{hand-graph.ps}
\includegraphics[width=0.28\textwidth,bb=286 123 452 242,clip=true]{hand-graph.ps}
\includegraphics[width=0.4\textwidth,bb=108 480 383 610,clip=true]{comp-pqcd-c.ps} 
\caption{Upper panel from left to right: Handbag diagram for Compton
 scattering (a), two-photon annihilation into a pair of hadrons (b) and the
 $\gamma P$ transition form factor (c). Lower panel: Cat's ears (d),
 three-particle correlation (e) and the ERBL contribution (f).}
\label{fig:handbag}
\end{center}
\vspace*{-0.08\textwidth}
\end{figure}

A property of the ERBL factorization scheme are the dimensional
counting rules which say that for asymptotically  large scales where
all soft scales can be ignored, exclusive observables behave
proportional to certain powers of the hard scale. The powers are 
determined by the number of point-like particles participating in the
hard scattering. Perturbative QCD modifies these power behaviour by
logs arising from the running coupling constant and the evolution of
the soft hadronic matrix elements. As yet no evidence for these logs
has been observed in experiment. In view of this and the ununderstood
normalization it is premature to claim dominance of the ERBL
contribution from the occasionally observed success of dimensional
counting. Moreover, recent precise experiments reveal clear deviations
from dimensional counting. Thus, for instance the Compton cross
section, measured recently at JLab \ci{bogdan}, should scale as
$s^{-6}$ at fixed c.m.s.\ scattering angle but it rather behaves as
$s^{-7...8}$, see Fig.\ \ref{fig:compton-scaling}. The Pauli form
factor should fall down as $1/t^3$ but, as a matter of fact, it
behaves as $1/t^2$. The integrated cross section for $\gamma\gamma\to
p\bar{p} (\pi^+\pi^-)$ drops with the power $\simeq 7.2 (4)$ instead
of $5 (3)$ \ci{belle1,belle2}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.35\textwidth,bb=118 392 448 724,clip=true]{comp-scal.ps} 
\hspace*{2em}
\includegraphics[width=0.37\textwidth,bb=46 105 385 431,clip=true]{fig-comptonff.ps}
\caption{Left: The Compton cross section, scaled by $s^6$ versus $s$ at a
 c.m.s.\ scattering angle of $90^\circ$. Data are taken from
 \ci{bogdan}. Right: The Compton form factors $R_V$, $R_A$ and $R_T$
 scaled by $t^2$.}
\label{fig:compton-scaling}
\end{center}
\vspace*{-0.07\textwidth}
\end{figure}

The handbag mechanism in real Compton scattering is reviewed in some 
detail in Sect.\ 2. The large $-t$ behaviour of the GPDs and their 
associated form factors is discussed in Sect.\ 3 and predictions for 
Compton scattering are given. A few results for wide-angle meson 
photoproduction and two-photon annihilations into pairs of hadrons  
are presented in Sect.\ 4 and 5, respectively. The paper ends with a 
summary (Sect.\ 6).   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Wide-angle Compton scattering}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For Mandelstam variables $s$, $-t$ and $-u$ that are large as compared
to a typical hadronic scale $\Lambda^2$ where $\Lambda$ being of order
$1\, \gev$, it can be shown that the handbag diagram shown in
Fig.\ \ref{fig:handbag}a, is of relevance in wide-angle Compton scattering. 
To see this it is of advantage to work in a symmetrical frame which is 
a c.m.s rotated in such a way that the momenta of the incoming ($p$) 
and outgoing ($p'$) proton momenta have the same light-cone plus 
components. In this frame skewness, defined as 
\be 
\xi \= \frac{(p - p')^+}{(p + p')^+}\,,
\ee
is zero. The bubble in the handbag is viewed as a sum over all possible 
parton configurations as in deep ineleastic lepton-proton scattering. 
The crucial assumptions in the handbag approach are that of restricted
parton virtualities, $k_i^2<\Lambda^2$, and of intrinsic transverse
parton momenta, ${\bf k_{\perp i}}$, defined with respect to their
parent hadron's momentum, which satisfy $k_{\perp i}^2/x_i
<\Lambda^2$, where $x_i$ is the momentum  fraction parton $i$ carries.   
 One can then show \cite{DFJK1} that the subprocess Mandelstam variables
$\hat{s}$ and $\hat{u}$ are the same as the ones for the full process,
Compton scattering off protons, up to corrections of order
$\Lambda^2/t$:
\ba
\hat{s}\=(k_j+q)^2 \simeq (p+q)^2 =s\,, \quad 
\hat{u}\=(k_j-q')^2 \simeq (p-q')^2 =u\,.
\ea
The active partons, i.e.\ the ones to which the photons couple, are
approximately on-shell, move collinear with their parent hadrons and
carry a momentum fraction close to unity, $x_j, x_j' \simeq 1$.
Thus, like in deep virtual Compton scattering, the physical 
situation is that of a hard parton-level subprocess, 
$\gamma q\to \gamma q$, and a soft emission and reabsorption of quarks 
from the proton. 

The handbag approach leads to the following result for the
Compton cross section  
\ba
\frac{d\sigma}{dt} &=& \frac{d\hat{\sigma}}{dt} \left\{ \frac12\, \big[
R_V^2(t)\,+\frac{-t}{4m^2}\,R_T^2(t) + R_A^2(t)\big] \right.\nn\\
&&\left. \qquad  - \frac{u s}{s^2+u^2}\, \big[R_V^2(t)\, 
                 +\frac{-t}{4m^2}\,R_T^2(t)
                - R_A^2(t)\big]\right\} + O(\alpha_s)\,,
\label{dsdt}
\ea
where $d\hat{\sigma}/dt$ is the Klein-Nishina cross section for
Compton scattering off massless, point-like spin-1/2 particles of
charge unity. The hard scattering has been calculated to
next-to-leading order (NLO) perturbative QCD \cite{HKM}. It turned out 
that the NLO amplitudes are ultraviolet regular but those amplitudes 
which are non-zero to LO, are infrared divergent. As usual the infrared
divergent pieces are interpreted as non-perturbative physics and absorbed
into the soft form factors, $R_i$. Thus, factorization of the wide-angle
Comton amplitudes within the handbag approach is justified to (at least)
NLO. To this order the gluonic subprocess, $\gamma g\to \gamma g$, has 
to be taken into account as well which goes along with corresponding 
gluonic GPDs and their associated form factors. In the numerical
results on Compton scattering the NLO corrections are included.

The form factors in Eq.\ \req{dsdt} are specific to Compton
scattering. They represent $1/\xb$ moments of the GPDs $H$,
$\widetilde{H}$ and $E$ 
\be
R_{V(T)}(t)\=\sum_a e_a^2 \int_{-1}^1 \frac{d\xb}{\xb} H^a(\xb,0;t)(E^a)\,, \quad
R_A(t)\=\sum_a e_a^2 \int_{-1}^1 \frac{d\xb}{\xb} {\rm sign}(\xb)\, 
\widetilde{H}^a(\xb,0;t)\,.
\label{Compton-formfactors}
\ee
Determinations of the zero-skewness GPDs from nucleon form
factor data have been attempted recently \ci{DFJK4,rad-weiss}. The
nucleon form factors only provide the first moment of the GPDs, e.g.\
\be
F_1(t)\=\sum_a e_a \int_{-1}^1 d\xb H^a(\xb,0;t)\,.
\label{formfactors}
\ee
Hence, a determination of the GPDs is only possible with the help of a
parameterization. The following ansatz is used in Ref.\ \ci{DFJK4}:
\be
H^a(\xb,0,t) \= q^a(\xb) \exp{[f_a(\xb)t]}\,,
\ee
where $q^a$ is the usual parton distribution for quarks of flavor $a$
and the profile function $f_a$ combines properties of the Regge pole
model at low $\xb$ with light-cone wavefunction overlaps at large
$\xb$. The parton distributions are taken from CTEQ \ci{cteq}. Fits 
to the data on the Dirac and Pauli form factors for proton
and neutron as well on the axialvector form factor yield the GPDs for
valence quarks. The Compton form factors evaluated from them, are
shown in Fig.\ \ref{fig:compton-scaling}.   

With the Compton form factors at hand various Compton observables can
be calculated in a parameter-free way. The predictions for the 
differential cross section are shown in Fig.\ \ref{fig:cross} and
compared to the Jlab data \ci{bogdan}. For sufficiently large $-t$ and
$-u$, actually for $-t, -u >2.5\,\gev^2$, fair agreement with
experiment is to be seen. Another interesting observable is the helicity
correlation, $A_{LL}$,  between the initial state photon and proton
or, equivalently, the helicity transfer, $K_{LL}$, from the incoming
photon to the outgoing proton. In the handbag approach one obtains
\cite{DFJK2,HKM} 
\be
A_{LL}\=K_{LL}\simeq \hat{A}_{LL}\, \frac{R_A}{R_V} + O(\kappa_T,\alpha_s)\,,
\label{all}
\ee  
where $\hat{A}_{LL}=(s^2 - u^2)/(s^2 + u^2)$ is the corresponding
observable for $\gamma q\to \gamma q$. The result \req{all} is a
robust prediction of the handbag mechanism, the magnitude of the
subprocess helicity correlation is only slightly diluted by the 
ratio of the form factors $R_A$ and $R_V$. Predictions for
$A_{LL}=K_{LL}$ are shown in Fig.\ \ref{fig:cross}. It is to be
emphasized that the handbag prediction for the helicity correlation is
only mildly energy dependent (the dashed line in Fig.\ \ref{fig:cross}
represents $A_{LL}$ at $s=20\,\gev^2$). The JLab E99-114 collaboration
\cite{nathan} has measured $K_{LL}$ at a c.m.s.\ scattering angle of
$120^\circ$ and a photon energy of $3.23 \gev$. Although the
kinematical requirement of large Mandelstam variables is not satisfied
for this measurement - and in so far one has to reckon with large
dynamical and kinematical corrections - fair agreement with the
handbag predicition is to be observed.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.43\textwidth,bb=147 290 570 715,clip=true]{fig-comptonsi.ps} 
\hspace*{2em}
\includegraphics[width=0.39\textwidth,bb=119 340 434 685,clip=true]{fig-all.ps}
\caption{Handbag predicitions for the Compton cross section at
$s=6.79, 8.9, 10.92$ and $20\,\gev^2$ (left)  and for the helicity 
correlations $A_{LL}=K_{LL}$ at $s=11\,\gev^2$ (right). Data are taken
form Refs.\ \ci{bogdan,nathan}. The green bands represent the
uncertainties of the handbag predictions due to the errors of the
Compton form factors. The yellow bands are estimates of the target
mass corrections \ci{DFHK}.}
\label{fig:cross}
\end{center}
\vspace*{-0.07\textwidth}
\end{figure}
Calculations on the basis of the ERBL factorization scheme, e.g.\ Ref.\ 
\cite{dixon}, reveal severe difficulties in getting the normalization 
of the Compton cross section correctly, the numerical results are way 
below experiment. There is growing evidence~\cite{bolz}~\footnote{
A perturbatively calculated $J/\Psi\to p\bar{p}$ decay width only agrees
with experiment if a proton distribution amplitude close to the
asymptotic form is employed~\cite{bolz96}} that the proton's distribution 
amplitude is close to the asymtotic form $\propto x_1 x_2 x_3$. 
Evaluating the Compton cross section from such a distribution amplitude  
one obtains results that are too small by a factor of about $10^{-3}$. 
Moreover, the ERBL approach~\cite{dixon} leads to a negative value for 
$K_{LL}$ at angles larger than $90^\circ$ in conflict with the JLab 
result~\cite{nathan}. Thus, one is forced to conclude that wide-angle
Compton scattering at energies available at JLab is not dominated by the 
ERBL contribution. Fairly good agreement with the Jlab results is, on
the other hand, obtained with a constituent quark model \ci{miller}.      

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two-photon annihilations into pairs of hadrons} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The arguments for handbag factorization hold as well for two-photon 
annihilations into pairs of hadrons as has recently been shown in Ref.\ 
\cite{DKV2} (see also Ref.~\cite{weiss}). 
The cross section for $\gamma\gamma \leftrightarrow p\ov{p}$
reads~\footnote{
Note that in the time-like region the pseudoscalar form factor
contributes while it decouples in the space-like region.} 
\ba
\frac{d\sigma}{dt}\,(\,\gamma\gamma\,\leftrightarrow\,\, p\ov{p}\,) &=& 
      \frac{4\pi\alpha^2_{\rm elm}} 
        {s^2 \sin^2 \theta} \Big\{ \big|R_{\rm eff}^\gamma(s)|^2
                  + \cos^2\theta\,\big|R_V^\gamma(s)\big|^2\, \Big\}\,, \nn\\
R^\gamma_{\rm eff} &=& \sqrt{|R_A^\gamma+R_P^p|^2+\frac{s}{4m^2}|R_\gamma^p|}\,.
\label{ppbar-cross-section}
\ea
In analogy to Eq.\ \req{Compton-formfactors} the form factors represent
moments of two-hadron distribution amplitudes (TDAs), $\Phi_{2h}$, which are 
time-like versions of GPDs. Since the TDAs are unknown at present one
has to extract the form factors from experiment. A fit \ci{schaefer}
to the recent BELLE data \ci{belle1} is shown in Fig.\
\ref{fig:pipi}. In combination with the data on the integrated cross
section for $s \geq 8\,\gev^2$ one finds for the form factors the
parameterizations
\ba   
s^2 R_V^\gamma&=& (8.20\pm 0.77)\,\gev^4\,(s/s_0)^{(-1.10\pm 0.15)}\,, \nn\\
s^2 R_{\rm eff}^\gamma &=&
              (2.90\pm 0.31)\,\gev^4\,(s/s_0)^{(-1.10\pm 0.15)}\,,
\label{ppbar-ff}
\ea
where $s_0=10.4\,\gev^2$. The energy dependence of the scaled form
factors manifests the violation of dimensional counting. The time
reversed process $p\ov{p}\to \gamma\gamma$ may be measured at the future
FAIR facility. One may measure the cross section and the form factors 
at higher energies, test factorization and, in combination with data on
the electromagnetic form factors $G_M$, $F_2$ for protons and neutrons
in the time-like region, one may attempt an analysis of the TDAs. 
Measurement of the correlation between the proton and antiproton
helicities will
allow to disentangle the axial vector and the pseudoscalar form
factors. The cross section \req{ppbar-cross-section} can be
straightforwardedly extended to other baryons. Relying on flavor 
symmetry one can evaluate the cross sections for two-photon
annihilations into $\Lambda\ov{\Lambda}$ and $\Sigma^0\ov{\Sigma}^0$
from the form factors \req{ppbar-ff}. Reasonable agreement with
experiment is found \ci{DKV3}. It is to be stressed that the ERBL
mechanism has again difficulties to account for the size of the cross 
sections \cite{farrar} while the diquark model \cite{schweiger} which 
is a variant of the ERBL approach in which diquarks are considered as 
quasi-elementary constituents of baryons, is infair agreement with
experiment for $\gamma\gamma\to B\bar{B}$.

Two-photon annihilations into meson pairs can also be investigated
within the handbag approach. For instance for pseudoscalar mesons the
cross section reads  
\be
\frac{d\sigma}{dt}(\gamma\gamma\to M\ov{M}) = 
     \frac{8\pi\alpha^2_{\rm elm}} 
              {s^2 \sin^4 \theta} \big|R_{M\ov{M}}(s)\big|^2\,,
\label{MM-cross-section}
\ee
The characteristic $1/\sin^4{\theta}$ dependence is in good agreement
with the BELLE data \ci{belle2} on the cross sections for pion and
Kaon pairs for $s\gsim 9\,\gev^2$, see Fig.\ \ref{fig:pipi}. 
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4\textwidth,bb=108 363 463 724,clip=true]{fig-annsigma-ppbar.ps}
\hspace*{2em}
\includegraphics[width=0.42\textwidth,bb=99 310 435 640,clip=true]{fig-dsdcos-meson.ps}
\caption{Handbag predictions for the differential cross sections
for $\gamma\gamma\to p\ov{p}$ (left) and $\pi^+\pi^- (\bullet), K^=K^-
(\circ)$ (right). Data are taken form Refs.\ \ci{belle1,belle2}.}
\label{fig:pipi}
\end{center}
\vspace*{-0.08\textwidth}
\end{figure}
A feature of the handbag mechanism in the time-like
region is the intermediate $q\ov{q}$ state implying the absence of 
isospin-two components in the final state. A consequence of this property is 
\be
\frac{d\sigma}{dt}(\gamma\gamma\to \pi^0\pi^0) = 
                 \frac{d\sigma}{dt}(\gamma\gamma\to \pi^+\pi^-)\,,
\ee
which is independent of the soft physics input and is, in so
far, a robust prediction of the handbag approach. This result also
holds for $\rho$ mesons. The ERBL approach leads to substantial
differences between the cross sections for charged and uncharged meson
pairs. Measurements of these cross sections are highly welcome.     
Neglecting non-valence quark form factors one obtains for Kaon pairs 
\be
\frac{d\sigma}{dt}(\gamma\gamma\to K_S K_S) \= \frac{2}{25}\,
                 \frac{d\sigma}{dt}(\gamma\gamma\to K^+K^-)\,,
\ee
in the strict flavor symmetry limit. Preliminary BELLE results
\ci{chen} on the $K_SK_S$ cross section seem to indicate that the
experimental ratio of these two cross sections is below that
limit.

Evaluating the $\gamma\gamma\to M\ov{M}$ cross section within the ERBL
approach from DAs that are close to the asymptotic form one obtains 
results that are nearly an order of magnitude below experiment even if
the large NLO corrections are taken into account \ci{niciz}. The 
differential cross section in this approach is approximately
similar to the handbag result \req{MM-cross-section} with the
ERBL result for the pion's electromagnetic form factor instead of the
annihilation one. Brodsky and Lepage \ci{bro82} recommended the use of
the experimental form factor instead of the ERBL result. A recent CLEO
measurement \ci{cleo} of $F_\pi$ however refute this idea. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pion photoproduction and $p\ov{p}\to \gamma\pi^0$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Photo- and electroproduction of mesons as well as the
$s\leftrightarrow t$ crossed reaction, proton-antiproton annihilation 
into photon and meson have also been analyzed within the handbag 
approach~\cite{hanwen,schaefer,HJKP}. Considering the case of the
$\pi^0$ as an example, one obtains the following cross sections
\ba
\frac{d\sigma}{dt}(\gamma p\to \pi^0p) &=& \ale a^2
                  \frac{t(s-u)^2}{32 s^4 u^2}\,
           \left[\Big(R_V^{\pi^0}(t)\Big)^2 + \frac{-t}{4m^2}\,\Big(R_T^{\pi^0}(t)\Big)^2
           + \frac{t^2}{(s- u)^2}\,
	   \Big(R_A^{\pi^0}(t)\Big)^2\right]\,, \nn\\
\frac{d\sigma}{dt}(p\ov{p}\to \gamma\pi^0) &=& 
               \ale \frac{|\ov{a}|^2}{s^3\sin^4{\theta}}\,
              \left[\Big|R_{\rm eff}^{\pi^0}(s)|^2 
               + \cos^2{\theta}\Big|R_V^{\pi^0}(s)\Big|^2\right]\,.
\ea
Due to the universality of the GPDs/TDAs the new form factors can be
either evaluated from the GPDs determined in Ref.\ \ci{DFJK4} or
estimated from the annihilation form factors \req{ppbar-ff} assuming
a constant ratio of the form factors for $d$ and $u$ quarks with
values lying in the range $0.25 - 0.75$. 

For asymptotically large Mandelstam variables the meson is generated
by one-gluon exchange leading to
\be
a\=\ov{a} \= \frac{16}{9}\, \als f_\pi \langle 1/\tau \rangle_\pi\,,
\ee
where $f_\pi$ is the pion's decay constant and $\langle 1/\tau
\rangle_\pi$ the $1/\tau$ moment of its DA. This result is too small
to account for the experimental values of the cross
sections. Dynamical effects as for instance higher orders of
perturbative QCD, transverse degrees of freedom or power corrections,
are required in order to built up the proper values of $a$ and
$\ov{a}$. A quantitative estimates of these constants are however
lacking at present. Taking them as free parameters a fair
description of the photoproduction \ci{anderson} and $p\ov{p}\to
\gamma\pi^0$ \ci{e760} data is obtained. Interestingly the helicity
correlation parameter $A_{LL}=K_{LL}$ for photoproduction is similar
to that for Compton scattering. Also the ratio of $\pi^+$ and $\pi^-$
photoproduction 
\be
\frac{d\sigma(\gamma n\to \pi^- p)}{d\sigma(\gamma p\to \pi^+ n)} \simeq
\left[\frac{e_d \uh + e_u \sh}{e_u \uh + e_d \sh}\right]^2\,,
\label{pi-ratio}
\ee
is in fair agreement with a JLab measurement \ci{zhu}.
The latter two results are independent on the cross section normalization.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I briefly reviewed the theoretical activities on applications of the
handbag mechanism to wide-angle scattering. There are many interesting
predictions, some are in fair agreement with experiment, others still
awaiting their experimental examination. It seems that the handbag
mechanism plays an important role in exclusive scattering
for momentum transfers of the order of $10\gev^2$. However, before we
can draw firm conclusions more experimental tests are needed.
The ERBL approach, on the other hand, typically provides
cross sections which are way below experiment. 
I finally emphasize that the structure of the handbag amplitude,
namely its representation as a product of perturbatively calculable
hard scattering amplitudes and $t$-dependent form factors is the
essential result. Refuting the handbag approach necessitates  
experimental evidence against this factorization.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{9}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{rad98} A.~V.~Radyushkin,
%``Nonforward parton densities and soft mechanism for form factors and
%wide-angle Compton scattering in {QCD},'' 
Phys.\ Rev.\ D {\bf 58}, 114008 (1998) [hep-ph/9803316].
%%CITATION = HEP-PH 9803316;%%

\bibitem{DFJK1} M.~Diehl, T.~Feldmann, R.~Jakob and P.~Kroll,
%``Linking parton distributions to form factors and Compton
%scattering,'' 
Eur.\ Phys.\ J.\ C {\bf 8}, 409 (1999)  [hep-ph/9811253].
%%CITATION = HEP-PH 9811253;%%

\bibitem{DFJK2} M.~Diehl, T.~Feldmann, R.~Jakob and P.~Kroll, 
%``Skewed parton distributions in real and virtual Compton
%scattering,'' 
Phys.\ Lett.\ B {\bf 460}, 204 (1999)
[hep-ph/9903268].
%%CITATION = HEP-PH 9903268;%%

\bibitem{hanwen} H.~W.~Huang and P.~Kroll,
%``Large momentum transfer electroproduction of mesons,''
Eur.\ Phys.\ J.\ C {\bf 17}, 423 (2000)
[hep-ph/0005318].
%%CITATION = HEP-PH 0005318;%%

\bibitem{DKV2} M.~Diehl, P.~Kroll and C.~Vogt,
%``The handbag contribution to gamma gamma $\to$ pi pi and K K,''
Phys.~Lett.~{\bf B} 532, 99 (2002) [hep-ph/0112274].
%%CITATION = HEP-PH 0112274;%%

\bibitem{DKV3} M.~Diehl, P.~Kroll and C.~Vogt,
%``Two-photon annihilation into baryon antibaryon pairs,''
Eur.\ Phys.\ J.\ C {\bf 26}, 567 (2003)
[hep-ph/0206288].
%%CITATION = HEP-PH 0206288;%% 

\bibitem{weiss} A.~Freund, A.~V.~Radyushkin, A.~Schafer and C.~Weiss,
%``Exclusive annihilation p anti-p $\to$ gamma gamma in a generalized 
%parton  picture,''
Phys.\ Rev.\ Lett.\  {\bf 90}, 092001 (2003) 
[hep-ph/0208061];
%%CITATION = HEP-PH 0208061;%%
%F.~E.~Close and Q.~Zhao,
%``On baryon antibaryon coupling to two photons,''
%Phys.\ Lett.\ B {\bf 553}, 211 (2003)
%[hep-ph/0210277].
%%CITATION = HEP-PH 0210277;%%

\bibitem{bjo} J.D.\ Bjorken and E.A.\ Paschos,
              Phys.\ Rev.\ {\bf 185}, 1975 (1969).
%              D.M.\ Scott,
%              Phys.\ Lett.\ {\bf 53B}, 185 (1974). 

\bibitem{ERBL}G.\ P.\ Lepage and S.\ J.\ Brodsky,
Phys.\ Rev.\ D {\bf 22}, 2157 (1980);
%%CITATION = PHRVA,D22,2157;%%
A.V.\ Efremov and A.V.\ Radyushkin,
             Phys.\ Lett.\ B {\bf 94}, 245 (1980). 

\bibitem{bogdan} A.~Danagulian, A.~M.~Nathan, M.~Roedelbronn,
  D.~J.~Hamilton, C.~E.~Hyde-Wright, V.~H.~Mamian and
  B.~Wojtsekhowski, 
%``Real Compton scattering on proton at high momentum transfers,''
Nucl.\ Phys.\ A {\bf 755}, 281 (2005).
%%CITATION = NUPHA,A755,281;%%

\bibitem{belle1}C.C.\ Kuo {\it et al.} [BELLE collaboration],
  %``Measurement of gamma gamma $\to$ p anti-p production at Belle,''
  Phys.\ Lett.\ B {\bf 621}, 41 (2005)
  [hep-ex/0503006].
  %%CITATION = HEP-EX 0503006;%%

\bibitem{belle2}H.~Nakazawa {\it et al.}  [BELLE Collaboration],
%``Measurement of the gamma gamma $\to$ pi+ pi- and gamma gamma $\to$ K+ K-
%processes at energies of 2.4-GeV - 4.1-GeV,''
Phys.\ Lett.\ B {\bf 615}, 39 (2005) [hep-ex/0412058].
%%CITATION = HEP-EX 0412058;%%

\bibitem{HKM} H.~W.~Huang, P.~Kroll and T.~Morii,
%``Perturbative and non-perturbative QCD corrections to wide-angle 
%Compton  scattering,''
Eur.\ Phys.\ J.\ C {\bf 23}, 301 (2002)
[hep-ph/0110208].
%%CITATION = HEP-PH 0110208;%%

\bibitem{DFJK4} M.~Diehl, T.~Feldmann, R.~Jakob and P.~Kroll,
  %``Generalized parton distributions from nucleon form factor data,''
  Eur.\ Phys.\ J.\ C {\bf 39}, 1 (2005)
  [hep-ph/0408173].
  %%CITATION = HEP-PH 0408173;%%

\bibitem{rad-weiss} M.~Guidal, M.~V.~Polyakov, A.~V.~Radyushkin and M.~Vanderhaeghen,
  %``Nucleon form factors from generalized parton distributions,''
  Phys.\ Rev.\ D {\bf 72}, 054013 (2005)
  [hep-ph/0410251].
  %%CITATION = HEP-PH 0410251;%%

\bibitem{cteq} J.~Pumplin, D.~R.~Stump, J.~Huston, H.~L.~Lai, P.~Nadolsky and W.~K.~Tung,
 %``New generation of parton distributions with uncertainties from global  QCD
 %analysis,''
  JHEP {\bf 0207}, 012 (2002)
  [hep-ph/0201195].
  %%CITATION = HEP-PH 0201195;%%

\bibitem{nathan} D.~J.~Hamilton {\it et al.}  [Jefferson Lab Hall A Collaboration],
 %``Polarization transfer in proton Compton scattering at high momentum
 %transfer,''
  Phys.\ Rev.\ Lett.\  {\bf 94}, 242001 (2005)
  [nucl-ex/0410001].
  %%CITATION = NUCL-EX 0410001;%%

\bibitem{DFHK} M.~Diehl, T.~Feldmann, H.~W.~Huang and P.~Kroll, 
%''Proton mass effects in wide-angle Compton scattering,''
Phys.\ Rev.\ D {\bf 67}, 037502 (2003)
[hep-ph/0212138].
%%CITATION = HEP-PH 0212138;%%

\bibitem{dixon}T.~C.~Brooks and L.~Dixon,
%``Recalculation of proton Compton scattering in perturbative QCD,''
{\em Phys.\ Rev.} D {\bf 62}, 114021 (2000)
[hep-ph/0004143].
%%CITATION = HEP-PH 0004143;%%

\bibitem{bolz} J.~Bolz and P.~Kroll,
%``Modelling the nucleon wave function from soft and hard processes,''
Z.\ Phys.\ A {\bf 356}, 327 (1996)
[hep-ph/9603289];
%%CITATION = HEP-PH 9603289;%%
V.~Braun {\it et al.},
%``Higher twist distribution amplitudes of the nucleon in QCD,''
Nucl.\ Phys.\ B {\bf 589}, 381 (2000)
[Erratum-ibid.\ B {\bf 607}, 433 (2001)]
[hep-ph/0007279];
%%CITATION = HEP-PH 0007279;%%
V.~Y.~Petrov and M.~V.~Polyakov,
%``Light cone nucleon wave function in the quark soliton model,''
hep-ph/0307077.
%%CITATION = HEP-PH 0307077;%%

\bibitem{bolz96} J.~Bolz and P.~Kroll,
%``Exclusive J/psi and psi' decays into baryon antibaryon pairs,''
Eur.\ Phys.\ J.\ C {\bf 2}, 545 (1998)
[hep-ph/9703252].
%%CITATION = HEP-PH 9703252;%%

\bibitem{miller} G.~A.~Miller,
  %``Handling the handbag diagram in Compton scattering on the proton,''
  Phys.\ Rev.\ C {\bf 69}, 052201 (2004)
  [nucl-th/0402092].
  %%CITATION = NUCL-TH 0402092;%%

\bibitem{schaefer} P.~Kroll and A.~Schafer,
  %``The Process P + Anti-P $\to$ Gamma + Pi0 Within The Handbag Approch,''
  Eur.\ Phys.\ J.\ A {\bf 26}, 89 (2005)
  [hep-ph/0505258].
  %%CITATION = HEP-PH 0505258;%%

\bibitem{farrar} G.~R.~Farrar {\it et al.},
%``QCD Predictions For Gamma Gamma Annihilation To Baryons,''
Nucl.\ Phys.\ B {\bf 259}, 702 (1985)
[Erratum-ibid.\ B {\bf 263}, 746 (1985)].
%%CITATION = NUPHA,B259,702;%%

\bibitem{schweiger} 
C.~F.~Berger and W.~Schweiger,
%``Hard exclusive baryon antibaryon production in two-photon collisions,''
Eur.\ Phys.\ J.\ C {\bf 28}, 249 (2003)
[hep-ph/0212066].
%%CITATION = HEP-PH 0212066;%%

\bibitem{chen} W.T.\ Chen [BELLE collaboration],
talk presented at the Intern. Conf. 'Photon 2005', Kazimierz, Poland (2005). 

\bibitem{niciz} G.~Duplancic and B.~Nizic,
  %``NLO perturbative QCD predictions for gamma gamma $\to$ M+ M- (M = pi,K),''
  hep-ph/0607069.
  %%CITATION = HEP-PH 0607069;%%

\bibitem{bro82} S.~J.~Brodsky and G.~P.~Lepage,
  %``Large Angle Two Photon Exclusive Channels In Quantum Chromodynamics,''
  Phys.\ Rev.\ D {\bf 24}, 1808 (1981).
  %%CITATION = PHRVA,D24,1808;%%

\bibitem{cleo} T.K.\ Pedlar {\em et al} [CLEO collaboration], Phys.\
  Rev.\ Lett.\ {\bf 95}, 261803 (2005).

\bibitem{HJKP} H.W.\ Huang, R.\ Jakob, P.\ Kroll and K.\ Passek,
%``Signatures of the handbag mechanism in wide-angle photoproduction
%of pseudoscalar mesons,'' 
hep-ph/0309071.
%%CITATION = HEP-PH 0309071;%%

\bibitem{anderson} R.~L.~Anderson {\it et al.},
%   ``Measurements Of Exclusive Photoproduction Processes At Large Values Of T
%And U From 4-Gev To 7.5-Gev,''
  Phys.\ Rev.\ D {\bf 14}, 679 (1976).
  %%CITATION = PHRVA,D14,679;%%

\bibitem{e760} T.~A.~Armstrong {\it et al.}  [Fermilab E760 Collaboration],
% ``Two-body neutral final states produced in anti-proton - proton
%annihilations at 2.911-GeV <= s**(1/2) <= 3.686-GeV,''
  Phys.\ Rev.\ D {\bf 56}, 2509 (1997).
  %%CITATION = PHRVA,D56,2509;%%

\bibitem{zhu} L.~Y.~Zhu {\it et al.}  [Jefferson Lab Hall A Collaboration],
%   ``Cross section measurement of charged pion photoproduction from hydrogen and
%deuterium,''
  Phys.\ Rev.\ Lett.\  {\bf 91}, 022003 (2003)
  [nucl-ex/0211009].
  %%CITATION = NUCL-EX 0211009;%%
\end{thebibliography}
\end{document}

