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\title{
Theory of diffractive meson production
%\footnote{\uppercase{T}his work is supported by etc, etc.}
}

\author{ 
L.\ Motyka}
%\footnote{e-mail: leszekm@th.if.uj.edu.pl }
\address{Institute of Physics, Jagellonian University, \\
Reymonta 4, 30-059 Krak\'{o}w, Poland\\
E-mail: leszekm@th.if.uj.edu.pl}


\maketitle

\abstracts{
In this talk some recent developments in the theory of diffractive meson production 
are reviewed:
(1) the issue of end point divergencies and diffractive factorisation; 
(2) elastic vector meson production in the context of the gluon saturation 
    in the impact parameter plane; 
(3) diffractive vector meson photoproduction at large momentum transfer;   
(4) $C$-even meson photoproduction and the Odderon.
}




\section{Introduction}

Diffractive meson production offers unique opportunities 
to probe the colour singlet exchange in high energy scattering.
In HERA experiments one measures properties of the scattering amplitude 
in a wide range of energy $W$, momentum transfer $t$, and the incoming 
photon virtuality $Q^2$, for various mesons.\cite{paul}
The multidimensional set of data is, with no doubts, a challenge 
for the theory. Standard methods of QCD, like the fixed order perturbative  
calculus and the hard factorisation, are not always sufficient to get a 
correct picture, and the use of resummation schemes, like the BFKL 
formalism\cite{BFKL} or phenomenological models as the saturation model,\cite{gbw} 
is necessary. Therefore a lot of effort is directed towards improving
the phenomenological models and rooting them in the fundamental theory.
The main goal is to obtain a robust global description of the 
diffractive meson production data, in a simple conceptual framework 
grounded in QCD.  

On the other hand, new measurements may be performed that probe some interesting 
mechanisms (e.g.\ the Odderon exchange) which are not fully understood yet.
Furthermore, some quantities which are not calculable from first 
principles may be determined or constrained. The hadron structure is a good
example here. 

In this talk I am presenting short notes on four selected 
topics in which substantially new results have recently appeared
and further progress is expected in the near future.



\section{Diffractive factorisation}

The theory of exclusive meson electroproduction is based on the diffractive 
factorisation theorem.\cite{Collins} The proof holds for a longitudinally 
polarized incoming photon that carries a large virtuality. 
In this case the scattering amplitude is related to parton distributions 
in the target. To be specific, for the diffraction at high energies the 
dominant contribution comes from the gluon evaluated at the scale of the order of 
the photon virtuality or of the meson mass. It should be added, however,
that the photon -- meson transition requires a non-zero energy-momentum transfer 
from the proton to the photon. Hence, the operators defining the gluon 
distribution in the proton are evaluated in non-forward kinematics. 
and the conventional gluon distribution has to be replaced by its off-diagonal 
counterpart.\cite{offwd} The evolution of the off-diagonal partons is  
known in QCD,\cite{offwd} and it was applied to construct models of these 
quantities.


In the case of transversely polarized photons, though, one finds some 
troublesome pieces in the amplitude which seem to spoil the diffractive 
factorisation. The hard part of the amplitude at the leading twist 
gives $1/[z(1-z)]^2$ dependence on the longitudinal momentum fraction $z$ 
of the quark. The singular behaviour of this part at the end-points, 
$z \to 0 $ and $z \to 1$, is not  compensated by the asymptotic form of 
the meson distribution amplitude $\phi(z) = 6 z (1-z)$, and infrared 
logarithmic divergencies appear in the integrals over $z$, breaking down 
the hard factorisation. Speaking in terms of the colour 
dipole model the photon wave function does not suppress large colour 
dipoles with an asymmetric sharing of the longitudinal momentum and those 
dipoles may interact with the target exchanging soft gluons. 


An interesting concept was put forward\cite{hoyer} to use the singular
behaviour of the hard amplitude to explain some peculiarities of the
high-$t$ diffractive meson photoproduction. Namely, the hard 
physics dictates a  $1/|t|^4$ dependence of the amplitude without a helicity 
flip between the photon and the meson and a $1/|t|^3$ dependence of the 
helicity flip amplitude. Indeed, the data for $\rho$ and $\phi$ exhibit  
approximately $1/|t|^3$ behaviour. Surprisingly, the analysis of angular 
distribution of the vector meson decay products, points to a strong dominance of 
the no-flip amplitude for all the range of momentum transfer.\cite{zeus} 

An assumption\cite{hoyer} that the distribution amplitudes do not vanish 
at $z \to 0$ and $z\to 1$ leads to an appearance of a power divergence at the
end-points that replaces the usual logarithmic divergence.
The necessary physical infrared cut-off is naturally provided 
by some soft scale $\sim \Lambda_{QCD}$. Thus, the end-point contributions
would have a different scaling with $t$ than it follows from the 
perturbative analysis. This is how one could explain the observed $t$-dependence 
$\sim 1/|t|^3$ of the dominant no-flip amplitude. Another consequence of this 
scenario is that the helicity need not be conserved for the 
quark--anti-quark configurations which contain a very slow parton and 
which are sensitive to non-perturbative effects, e.g.\ the instantons.
Thus, this model could explain the qualitative features of the data for 
high-$t$ meson production but no quantitative predictions have been 
provided so far. Although the idea is interesting, the assumptions behind
it are controversial. For instance, the invariant mass $M_{q\bar q}$ 
of the end-point configurations at the quark level, given by  
$M^2 _{q\bar q} \simeq (m_q ^2 + k_T^2) / [z (1-z)]$,  
is much larger than the meson mass $M_V$
(with $m_q$ and $k_T$ denoting quark mass and transverse momentum respectively).
Thus, such configurations should be strongly suppressed. The reason is 
that in an exclusive process, the invariant mass at the parton level should
be close to the meson mass if the local parton-hadron duality is to hold. 


Recently, a thorough analysis of the factorisation at the end-points beyond the 
leading twist contribution has been carried out.\cite{IK} The momentum 
dependent meson wave function is modelled in the following way. 
The starting point is the perturbative photon wave function, next its point-like 
component is removed using the Borel transform.  The obtained meson wave 
function  
$\Psi_V(z,k_T) \sim \exp \left( - \frac{k_T^2+m_q^2}{ M_V^2\, z(1-z)} \right)$ 
contains contributions from all twists. The resulting helicity structure is the same as 
the helicity structure of the photon. Note that this wave function gives the
asymptotic form of the distribution amplitudes after the integration over 
the transverse momentum, and that it naturally suppresses the configurations 
with large invariant mass at the parton level. 

This model permits a detailed analysis of the end-point contributions at all
twists. An apparent singular behaviour of the integrals at the end-points 
is found to be only an effect of inappropriate extrapolations at the leading 
twist. In the complete model of the wave function the end-point divergences 
do not appear. Instead, at large photon virtuality $Q^2$ one finds 
logarithms of $Q^2$ which can be fully absorbed into the generalized 
evolution of $t$-channel $gg \to q\bar q$ exchange. Thus the end point 
contributions are separated into two factorised terms, 
containing the gluon distribution and the an effective $q\bar q$ distribution. 
The latter term may be neglected at small~$x$ and the gluon dominates.\cite{IK} 



\section{Elastic production of vector mesons}

It was shown a long time ago\cite{ryskin,brodsky} that elastic production 
of vector mesons may be used as a probe of the collinear parton densities 
in the proton, giving direct access to the off-diagonal gluon. In fact,
a detailed measurement of the forward peak provides even more 
information. The $t$-dependence of the scattering is related to the  profile of the 
scattering amplitude in the impact parameter plane by the Fourier transform. 
Furthermore, in collisions at high energies the transverse position 
of the dipole is conserved in the scattering, thus, in the $b$-representation
the $S$-matrix is diagonal and much easier to determine.
There is one difficulty left, however. Experiments measure cross sections, 
and the information about relative complex phases of the scattering matrix 
elements is partly lost. A procedure that was recently proposed for $S$-matrix 
determination\cite{msm} is based on a simplifying assumption that 
this matrix is real. In this scheme a closed formula was derived for
the $S$-matrix elements in a dipole scattering off the proton, 
at a given impact parameter. The necessary experimental input was the differential 
cross section $d\sigma /dt$ for the elastic electroproduction of longitudinally 
polarised $\rho$ meson.\cite{msm} 
The differential cross section of this process $d\sigma / dt$ is known for a 
few values of $Q^2$, in each case covering about a decade in~$x$ and 
the range of $-t<0.6$~GeV$^2$. The results obtained were parameterised in a
form following from the hypothesis of $b$-dependent saturation, and the saturation 
scale was determined as a function of the impact parameter.   

In the colour dipole representation the dependence of dipole scattering amplitude 
on the impact parameter is influenced by the matter (mostly gluons) 
distribution in the proton. Thus, in principle, the cross section for 
vector meson production could be used to provide some clues on the three 
dimensional structure of the proton. 


The $b$-dependent saturation of the elastic colour dipole -- proton cross 
section is a phenomenon which follows directly from the unitarity constraints.
A vector meson production process may probe dipole scattering in the 
kinematical domain where rescattering effects should be important.
It opens a possibility to analyse in detail how the amplitude saturates in the
impact parameter plane and for increasing collision energy.   
In the classical saturation model proposed by Golec-Biernat and W\"{u}sthoff 
(GBW)\cite{gbw} it is assumed that the nucleon-dipole interaction is 
confined to a disc in the transverse plane. In a more realistic picture 
the scattering matrix elements should be characterised by a smooth 
distribution $S(b)$ in the impact parameter. The large-$b$ behaviour 
is constrained by the fact that QCD is a theory with a mass gap, thus 
$S(b)$  should decrease faster than $\exp(-mb)$ for $b\to \infty$. 
It may be argued that the Gaussian tail $S(b) \sim \exp(-Ab^2)$ accounts
better for the confining properties of the QCD vacuum.\cite{pomb}

 
Two recent extensions of the GBW model\cite{kow,lev} use the Gaussian 
profile $S(b)$ of a single dipole scattering. The unitarity constraints 
are imposed by the Glauber-Muller prescription at each impact parameter\cite{kow}
or by inserting\cite{lev} the $b$-dependence into solutions of Balitsky-Kovchegov 
equation.\cite{BK} Both the models were confronted with the data for
elastic $J/\psi$ photo- and electroproduction, giving a good fit.
For the $J/\psi$ though, the typical dipole size is related to the inverse 
$c$-quark mass, and the unitarity corrections are not large.\cite{kow} 
An important goal to achieve in the near future is to provide a global fit 
of all the data for exclusive vector meson production in the 
saturation model with the impact parameter dependence. Such an analysis, 
based on the present and future HERA data, would provide stringent 
experimental constraints of the dynamics of $b$-dependent saturation.   


\section{Diffractive production at large t}

Recently, ZEUS\cite{zeus} and H1\cite{h1hight} measured
the differential cross section $d\sigma / dt$  for the $\rho^0$, $\phi$ and 
$J/\psi$ diffractive production over a wide range of~$t$, and also the corresponding 
spin density matrix elements. From those data estimates of the helicity amplitudes 
may be obtained. The amplitude without helicity flip dominates for all 
mesons and for the accessible~$t$ range. The double flip and single flip 
amplitudes have the magnitudes of one fifth and one tenth of the leading 
amplitude respectively. 
 
High energy diffractive meson production in non-forward direction 
tests colour singlet exchange at large momentum transfer $t$. Elastic 
scattering of a dipole with size $r$ is strongly suppressed for 
$r^2 |t| \gg 1$ because the dipole tends to break up. Also gluon 
radiation that spoils the rapidity gap is important in this regime. 
Therefore the momentum transfer provides the hard scale for the 
problem and the perturbative approach is justified for the photon-meson transition, 
even for light mesons and the photoproduction limit. 


In the colour dipole representation at high energies, the dipole scattering 
amplitude is diagonal in the quark helicity basis. In the leading twist 
approximation the quark momenta are collinear and the vector meson helicity 
is a simple sum of quark helicities. Beyond this approximation, the angular 
momentum originating from a motion of quarks inside the meson and a possible 
appearance of additional partons in higher Fock states has to be 
taken into account and the sum of quark helicities need not match the meson 
helicity. This explains how the double helicity flip occurs in the high-$t$ 
photoproduction of $\rho^0$ and $\phi$.\cite{IKSS,EFMP,PEFM}


The lowest order approximation to the $C$-even colour singlet in QCD is 
given by two gluons combined into a colour neutral state. This simple picture 
is not sufficient, though, when the rapidity gap $\hat y$ between the meson 
and the proton remnant is large. Perturbative corrections to the two-gluon 
exchange contain terms that behave like $(\hat y \alpha_s)^n$ at the $n$-th 
order of expansion. Since $\hat y \alpha_s \sim 1$, the fixed order 
perturbative expansion gives inaccurate results and a resummation of leading
logarithms $(\hat y \alpha_s)^n$  (with $\hat y \sim \log s$) is necessary. 
Those contributions are diagrammatically represented as gluon ladders.
The resummation is performed by the BFKL equation\cite{BFKL} providing a
consistent framework to analyse the non-forward diffractive amplitudes.

Vector meson production data at $|t| > 1$~GeV$^2$ are dominated by  
events with a dissociated proton. Thus, in most theoretical analyses
all the final states of the proton remnant are included, having a given
transverse momentum and the total colour charge equal to zero.
Then, the hard colour singlet couples predominantly to individual partons 
in the nucleon\cite{bart} resolving them  at the hard scale $\sim |t|$. 
This approximation improves with increasing $|t|$, and this allows to 
factorise the cross sections into parton densities in the proton and a 
partonic cross section for the elastic meson production. 

The scattering amplitude $A$ in the BFKL formalism is given by a 
convolution over the gluon transverse momenta, of the universal 
gluon $2 \to 2$ Green function~$G$ and of two specific impact 
factors  $\Phi_i$ describing the coupling of two gluons to the systems 
at the two sides of the rapidity gap, $A \sim \Phi_1 \otimes G \otimes \Phi_2$.
The analytic approach to the BFKL equation is 
based on its conformal invariance\cite{lipatov} which permits a decomposition
of the Green function in the basis of its conformal Eigenfunctions, indexed
by a discrete conformal spin~$n$ and a continuous variable, $\nu$.  
This method was successfully applied to the $J/\psi$ diffractive production. 
At first, only the leading conformal component with $n=0$~\cite{fr,FP} was taken
into account, and then the calculation was extended 
to all the conformal spins.\cite{EMP}
    

For the production of light vector mesons, the impact factor representing 
photon-meson transition depends substantially on the details of both the photon
and the meson structure. For the photon, one may use the perturbative wave 
function with a constituent quark mass.
% (see e.g.\ \cite{pirner}). 
An alternative approach is based on the Operator Product Expansion
in which only the leading dependence on the dipole vector $\vec{r}$ 
is retained at $r \to 0$. The dependence on the longitudinal momentum 
fraction is parameterised by the distribution amplitudes.\cite{BB} 
The approximation  may be systematically improved by including higher twist 
terms and the QCD evolution. The same method may be applied also for the vector 
mesons.\cite{BBKT} 


Recently, a few attempts have been made\cite{IKSS,FP,EMP,EFMP,PEFM,IK3} 
to describe the high-$t$ photoproduction data. In those papers important 
observations about the meson production dynamics are performed, but none 
of the approaches is completely successful. 
It was argued\cite{IKSS} that non-perturbative condensates in the QCD vacuum 
generate a sizeable chiral odd coupling of the photon to the quark anti-quark pair. 
This component significantly enhances the amplitude without 
helicity flip, and the discrepancy between the theory and the data decreases.  
%Still, the fit of the light vector meson data is not satisfactory. 
Furthermore, an analysis was carried out\cite{FP,EMP,EFMP,PEFM} that went 
beyond the two-gluon exchange approximation\cite{IKSS} and the complete BFKL 
amplitudes for all helicities were determined. The differential cross sections 
for all the mesons were fitted very well\cite{FP,EMP,PEFM} as well as the energy 
dependence and spin density matrix for $J/\psi$ production. 
% 
Although the inclusion of BFKL effects greatly improves the fit of 
the spin density matrix elements, this description is not satisfactory 
for light vector mesons. In particular, both the sign and the magnitude of the 
single helicity flip amplitude come out wrong when the most probable values of 
wave function parameters are used. The model is capable to provide a reasonable fit 
of the data with an unrealistically large value of the constituent quark 
mass in the photon wave function. This may indicate that the model 
lacks some physical mechanism that suppresses the scattering of large 
dipoles e.g.\ the Sudakov form-factor.\cite{PEFM}    

Another important idea is to use the meson wave function beyond the fixed twist
approximation.\cite{IK3} In the model, the BFKL evolution is included 
in the doubly logarithmic approximation but the chiral odd components in the photon
are not taken into account. As a consequence, the single helicity flip amplitude 
is found to dominate in severe contradiction with the data. 

The lesson learned from those three approaches may be summarised in
the following. The chiral odd components of the photon play an important
r\^{o}le in the vector meson photoproduction. Calculations based on the BFKL
equation\cite{FP,EMP,EFMP,PEFM} give a better picture of all the 
observables than the two gluon approximation. An improvement is needed 
in the treatment of the large dipole scattering, perhaps an inclusion
of the Sudakov form-factor. Another interesting task is to apply the 
BFKL formalism to compute the helicity amplitudes at a large photon 
virtuality and non-zero $|t|$.

\section{The Odderon}

The Odderon is the leading part of the colour singlet exchange at high energies 
with odd $C$-parity, a partner of the $C$-even Pomeron. The surprising 
feature of the Odderon is that only very little evidence of this exchange
has been found so far.\cite{ewerz} HERA experiments may be able to shed some 
light on the puzzle. 

A simple and robust signal of the Odderon could be provided by a 
diffractive photoproduction of a $C$-even meson.
In the perturbative QCD, three $t$-channel gluons is the simplest system 
which can take the quantum numbers of the Odderon and that contributes at 
asymptotic energies. The necessary inclusion of the leading logarithmic
corrections leads to the BKP evolution equation\cite{bkp} for the Odderon.
The equation yields amplitudes with the intercept equal to one,\cite{oddint} 
that is the diffractive cross sections constant with the energy.

Diffractive $\eta_c$ photoproduction was studied within the perturbative 
framework. The charmed quark mass sets the scale for the problem, and the 
$\gamma \to \eta_c$ impact factor may be calculated in perturbative QCD. 
For the coupling of the Odderon to the proton, one is forced to use a model, 
and thus, to introduce some uncertainties. The first 
estimates of the cross section based on the three gluon approximation
gave $\sigma (\gamma p \to \eta_c p) \simeq 10$~pb,\cite{ckms} a signal probably too 
weak to detect given the low acceptance for $\eta_c$. The QCD evolution
effects increased the prediction significantly\cite{vacca1} to yield 
$\sigma (\gamma p \to \eta_c p)\simeq 50$~pb. In addition, the 
cross section for a similar diffractive process with the proton 
dissociation was found to be even larger,\cite{vacca2} 
$\sigma (\gamma p \to \eta_c X)\simeq 60$~pb.
Thus, the combined cross section for the diffractive $\eta_c$ production
exceeds earlier estimates by one order of magnitude and may turn out 
to be large enough for HERA experiments to discover the perturbative 
Odderon exchange.

In search of the Odderon, other promising observables have been also proposed.
Estimates of the diffractive photoproduction cross-section of 
$\pi^0$, $f_2$, $a_2^0$ were obtained in the Stochastic Vacuum Model.\cite{heidodd} 
The H1 collaboration set the upper limits for the corresponding
cross sections\cite{h1odd} and, unfortunately, the results contradict 
the prediction for $\pi^0$ production. Besides that, an important idea was 
developed to use the Pomeron-Odderon interference effects in diffractive 
processes in the Odderon searches. The charge and spin 
asymmetries in $\pi^+ \pi^-$ diffractive electroproduction  
were shown to be sensitive to the Odderon amplitude.\cite{oddpipi} 
 

\section{Conclusions}
 
The theory of diffractive meson production has been developing rapidly 
since  about ten years. It touches some central problems
of QCD, transition between the soft and the hard regime of the scattering, 
the resummation schemes at large energies and the saturation phenomenon. 
In addition, the analysis of exclusive processes showed importance of the 
non-forward parton distributions and stimulated studies of their properties 
and their QCD evolution. Analysis of the high-$t$ photoproduction of vector mesons
clearly indicates that the photon and light vector mesons contain sizeable 
chiral odd components. Diffractive production of $C$-even 
mesons may shed some light on the Odderon puzzle. All of this shows how much 
inspiration originates from this field.
 

Recent advances of the theory narrow the gap between the first principles of QCD 
and some successful phenomenological models. The mechanism of diffractive factorisation 
which is the basis for the perturbative approach is getting better understood. 
The Balitsky-Kovchegov equation provides a QCD description of the 
perturbative parton saturation phenomenon within a well defined approximation 
scheme. Exchange amplitudes of the perturbative Pomeron and Odderon are 
computed for increasing number of processes at the leading logarithmic accuracy, 
and estimates of non-leading corrections are also known in some cases. 
The good news is, that together with the improving theoretical foundations of the
phenomenological analysis, the data become better described. This may indicate that
the crucial mechanisms that govern the diffractive meson production are correctly 
identified. Still, the quantitative description of the data is rather crude. 


Wealth of HERA data, to be supplemented by the new data from HERA2 will certainly 
keep stimulating efforts to obtain a more complete QCD description of the 
diffractive meson production. With improved statistics, the constraints
on the theoretical estimates will become more stringent. We will learn more on some 
important phenomena, like the unitarisation of QCD amplitudes in the impact 
parameter plane. Finally, there is a lot of room left for measurements of new 
observables, for surprising results and for new theoretical ideas.





 


\section*{Acknowledgments}
I would like to thank the Organizers for creating an excellent, 
inspiring atmosphere during the Workshop.
The author is partially supported by the Polish Committee for Scientific 
Research (KBN) grant no.\ 5P03B~14420.

\newpage

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\end{thebibliography}

\end{document}

