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\def\lapproxeq{\lower .7ex\hbox{$\;\stackrel{\textstyle                  
<}{\sim}\;$}}                                         
\def\gapproxeq{\lower .7ex\hbox{$\;\stackrel{\textstyle                                         
>}{\sim}\;$}} 
\def\btop{\mathchar"1339}                                                                                       
\def\bmid{\mathchar"133D}                                                                                       
\def\bbot{\mathchar"133B}                                                                                       
\def\bs{\mathchar"1343}                                                                                      
\def\msb{\overline{\rm MS}}                                                                                       
\def\GeV{{\rm GeV}}                                                                                       
\def\TeV{{\rm TeV}}                                                                                       
\def\funp{{I\!\!P}}                                                                                       
\def\gup{{g\uparrow}}  
\def\gdown{{g\downarrow}}  
\def\asup{{\alpha_S\uparrow\uparrow}}                                                    
\def\asdown{{\alpha_S\downarrow\downarrow}} 

%\begin{document}
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\title*{Diagonal and Skewed Parton Distributions}
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\toctitle{Diagonal and Skewed Parton Distributions}
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\titlerunning{Diagonal and Skewed Parton Distributions}
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\author{Alan Martin}
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\authorrunning{A.D. Martin}
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\institute{University of Durham, Durham, DH1 3LE, UK}

\maketitle              % typesets the title of the contribution

\begin{abstract}
We briefly review the global analyses that are used to determine parton distributions from data, and discuss 
recent topical issues that have arisen.  We outline the properties of skewed parton distributions, and show 
that in the small $x$ domain they are completely determined by the conventional distributions.  The relevance 
to the description of vector meson production at HERA is emphasized.
\end{abstract}


\section{Introduction}
%
The cross section of hard scattering processes involving incoming protons (such as deep 
inelastic electron-proton scattering or the hadroproduction of jets of large transverse 
momentum) can be written as the sum of parton distributions multiplied by the cross 
sections of hard subprocesses calculated at the parton level using perturbative QCD.  That 
is we can factor off the long distance (non-perturbative) effects into universal, process 
independent, parton distributions, $f_i (x, \mu^2)$ with $i = q, \bar{q}, g$, which represent 
the probability of finding parton $i$ in the proton carrying a longitudinal fraction $x$ of the 
proton's momentum.  The parameter $\mu$ is a scale typical of the hard partonic subprocesses.  
Perturbative QCD determines the evolution of the parton distributions as a function of $\mu$, 
provided they are known as some starting scale $\mu_0$.  Calculating the parton distributions 
from first principles at some low scale $\mu_0$ is one of the most challenging problems of 
non-perturbative QCD.  The most promising approach is lattice QCD, but much remains to be done.  
On the other hand, from a practical point of view, the parton distributions of the proton are 
determined with good precision from global analyses of deep inelastic and related hard scattering 
data.  We will describe these analyses in Section 2 and focus on some of the recent issues that 
result.

Formally parton distributions can be expressed as matrix elements $\langle p | \hat{O} | p \rangle$ 
where $\hat{O}$ is a twist-2 quark or gluon operator, and $p$ is the four momentum of the proton.  
Recently there has been much renewed interest in off-diagonal or skewed parton distributions which 
are given by matrix elements $\langle p^\prime | \hat{O} | p \rangle$ in which the momentum $p^\prime$ of the 
outgoing proton is not the same as that of the incoming proton.  For example, the {\it amplitudes} 
for processes such as deeply virtual Compton scattering $(\gamma^* p \rightarrow \gamma p)$ or vector 
particle electro- or photo-production $(\gamma^* p \rightarrow Zp$ or $J/\psi p$) depend on 
off-diagonal distributions.  Since $p \neq p^\prime$ the parton returning to the proton has a 
different momentum to the one which is outgoing, and so we need two momentum variables to specify 
the skewed distributions.  In Section 3 we will discuss the phenomenological developments involving 
skewed distributions (concentrating entirely on the small $x$ domain) and emphasize their potential 
role in the global parton analyses.


\section{Global parton analyses}

As mentioned above, the parton densities $f_i (x, Q^2)$ are determined from a global analysis of a wide 
range of data for deep inelastic and related `hard' scattering processes, which are listed in the first 
column of Table 1.  The procedure is to parametrize the $x$ dependence of the `starting' distributions in 
physically motivated forms, such as
\begin{equation}
xf_i (x, Q_0^2) \; = \; A_i x^{-\alpha_i} (1 - x)^{\beta_i} (1 + \gamma_i x^{\frac{1}{2}} + \delta_i x)
\label{eq:a1}
\end{equation}
at a low scale $Q_0^2$, but which is nevertheless sufficiently high to be in the perturbative region.  
Next-to-leading order DGLAP equations are then used to evolve up in $Q^2$ to determine $f_i (x, Q^2)$ at all 
the $x, Q^2$ at which data are to be fitted, and the parameters $A_i, \alpha_i, \ldots \delta_i$ varied until an
optimum description of the data is achieved, subject to satisfying the momentum and flavour sum rules.  The last 
column of Table 1 gives an indication of the parton properties that are constrained by particular data sets.  Two 
independent groups have carried out global analyses:  MRST and CTEQ.  Their latest sets of partons are known as 
MRST \cite{MRST1}, MRST99 \cite{MRST2} and CTEQ5 \cite{CTEQ5}.  Semi-global analyses are also being undertaken by the 
experimental collaborations (see, for example, \cite{ZOMER}).  In this section we review topical issues concerning 
these global parton analyses.  We adopt the conventional notation $u \equiv f_u, d \equiv f_d$ etc. 

\subsection{Light quark distributions:  $u, d, s$}

The $u$ and $d$ distributions are pinned down by the DIS data, with the slope of $d/u$ being constrained 
for $x \lapproxeq 0.3$ by the CDF $W^\pm$ rapidity asymmetry data.  In fact $d/u$ is determined up to 
$x \sim 0.7$ by the NMC $F_2^n/F_2^p$ data, {\it provided} that we assume that there are no deuterium binding 
corrections for $x > 0.3$.  However, Yang and Bodek \cite{YB} have recently stressed a point made by Melnitchouk 
and Thomas \cite{MT} that the $F_2^n$ data may have a nuclear binding correction which would lead to a larger $d/u$ 
ratio at large $x$.  This leads to considerable uncertainty in the $d$ distribution at large $x$.  Descriptions with $d/u 
\rightarrow 0$ or 0.2 as $x \rightarrow 1$ are equally possible.  In principle measurements of charged current 
DIS $e^+ p \rightarrow \bar{\nu} X$ (and $e^- p \rightarrow \nu X$) at HERA, or $W^\pm$ asymmetry at large 
rapidity at the LHC, can resolve the issue \cite{MRST2}.

The ratio $\bar{d}/\bar{u}$ of sea quark distributions is determined for $0.05 \lapproxeq x \lapproxeq 0.3$ by 
the E866 data for $pp, pn \rightarrow \mu^+ \mu^- X$ \cite{E866}.  Information is also coming from the HERMES data 
for $ep, en \rightarrow e \pi^\pm X$.  It is found that $\bar{d}/\bar{u}$ rises from unity at $x = 0$ to about 
1.5 at $x = 0.15$ and then falls sharply back through unity around $x \sim 0.3$.

\newpage

\noindent Table 1. Processes studied in global analyses.  The essential features of the partons that are 
probed by the particular data sets are indicated in the last column.

\vspace{0.5cm}                         
                                                                                                                                              
\begin{center}                                                                                      
\begin{tabular}{|l|l|l|}    \hline                                                                                         
%                   &                           &                           \\                                                                                        
{\bf Process/}     &     {\bf Leading order}   & {\bf Parton behaviour probed}\\                                                                                        
{\bf Experiment}   &  {\bf subprocess}         &                           \\                                                                                        
%                   &                           &                  \\                        
\hline                                                                                        
&\hfill \raisebox{-0.5ex}[0.5ex][0.5ex]{$\btop$}&                      \\                                                                                        
{\bf DIS} $\mbox{\boldmath $(\mu N \rightarrow \mu X)$}$ &  $\gamma^*q                                                                                        
\rightarrow q$ \hfill {\arrayrulewidth=1pt\vline}\hspace*{4pt}&  \\                                                                                        
$F^{\mu p}_2,F^{\mu d}_2,F^{\mu n}_2/F^{\mu p}_2$                                                                                        
& \hfill {\arrayrulewidth=1pt\vline}\hspace*{4pt}&                                                                                          
Four structure                                                                                        
functions $\rightarrow$  \\                                                                                        
(SLAC, BCDMS,& \hfill {\arrayrulewidth=1pt\vline}\hspace*{4pt}& \hspace*{1cm}                                                                                         
$u + \bar{u}$  \\                                                                                        
NMC, E665) & \hfill {\arrayrulewidth=1pt\vline}\hspace*{4pt}&                                                                                        
\hspace*{1cm} $d + \bar{d}$   \\                                                                                        
      &\hfill $\bmid$ & \hspace*{1cm}  $\bar{u} + \bar{d}$  \\                                                                                       
{\bf DIS} $\mbox{\boldmath $(\nu N \rightarrow \mu X)$}$ & $W^*q \rightarrow                                                                                        
q^{\prime}$  \hfill {\arrayrulewidth=1pt\vline}\hspace*{4pt}& \hspace*{1cm}                                                                                          
$s$ (assumed = $\bar{s}$),\\                                                                                        
$F^{\nu N}_2,xF^{\nu N}_3$    &\hfill {\arrayrulewidth=1pt\vline}\hspace*{4pt}&                                                                                       
but only                                                                                        
$\int xg(x,Q_0^2)dx \simeq 0.35$ \\                                                                                        
(CCFR)     &\hfill {\arrayrulewidth=1pt\vline}\hspace*{4pt}&                                                                                        
and $\int(\bar{d}-\bar{u})dx\simeq 0.1$  \\                                                                                       
                &\hfill \raisebox{0.5ex}[1.5ex][1.5ex]{$\bbot$} &         \\                                                                                        
{\bf DIS (small $x$)}    &   $\gamma^* (Z^*)q \rightarrow q$   &   $\lambda$     \\                                                                                        
$F^{ep}_2$ (H1, ZEUS)   &                  & $(x\bar{q} \sim x^{-\lambda_S},                                                                                       
\ xg \sim  x^{-\lambda_g})$    \\[2mm]                                                                                       
%& & \\                        
{\bf DIS ($\mbox{\boldmath $F_L$}$)}    &   $\gamma^* g \rightarrow q\bar{q}$                      
&   $g$     \\                                                                                        
NMC, HERA   &                  &     \\[2mm]                                  
%                      &                         &                     \\                                                                                         
%{\bf DIS (GSR)}    &   $\gamma^*q \rightarrow q$                                                                                         
%& $\int_{0.004}^{0.8} (\bar u -\bar d)dx$ \\                                                                                        
%$\int (F^{\mu p}_2 - F^{\mu n}_2 )dx/x$    &    &   \\                                                                                        
%  (NMC)              &                            &                           \\                                                                                        
%               &                            &                           \\                                                                                        
$\mbox{\boldmath $\ell N \rightarrow c \bar{c} X$}$ &                                                                                       
  $\gamma^*  c\rightarrow  c$    & $c$ \\                                                                                        
 $F_2^{c}$  (EMC; H1, ZEUS)            &     &     $  \quad (x \gapproxeq 0.01;\                                                                                        
x \lapproxeq 0.01 )$    \\[2mm]                                                                                       
%               &                            &                           \\                                                                                        
$\mbox{\boldmath $\nu N \rightarrow \mu^+\mu^-X$}$ &  $W^* s \rightarrow                                                                                         
c$    & $s \approx \frac{1}{4} (\bar{u} + \bar{d}) $ \\                                                                                        
(CCFR)            &   $\;\;\;\;\;\;\;\;\;\;\;\;\;\hookrightarrow \mu^+$     &         \\[2mm]                                                                                       
%                      &                         &                     \\                                                                                         
$\mbox{\boldmath $p N \rightarrow \gamma X$}$  &  $qg \rightarrow \gamma q$  &                                                                                       
$g$ at $x \simeq 2 p_T^\gamma/\sqrt{s} \rightarrow$ \\                                                                                        
(WA70, UA6, E706, \ldots)    &      &  \hfill   $x \approx 0.2 - 0.6  $             
\\[2mm]                                                                                       
 %          &                   &                          \\                                                                                        
$\mbox{\boldmath $pN \rightarrow \mu^+\mu^- X$}$   &  $q\bar{q} \rightarrow                                                                                        
\gamma^*$  &  $\bar{q} = ...(1-x)^{\eta_S}$ \\                                                                                        
(E605, E772)          &                     &                             \\[2mm]                                                                                       
%               &                     &                                 \\                                                                                        
$\mbox{\boldmath $pp, pn \rightarrow \mu^+\mu^- X$}$ & $u\bar{u},d\bar{d}                                                                                        
\rightarrow \gamma^*$   & $\bar{u} - \bar{d}~~(0.04 \lapproxeq x \lapproxeq 0.3)$                     
\\                                                                                        
(E866, NA51)             &  $u\bar{d},d\bar{u} \rightarrow \gamma^*$   &     \\[2mm]                                                                                      
%                 &                   &                               \\                                                                                        
$\mbox{\boldmath $ep, en \rightarrow e \pi X$}$    &  $\gamma^* q \rightarrow q$          
with   &   $\bar{u} - \bar{d}~~(0.04 \lapproxeq x  \lapproxeq 0.2)$     \\                                                                                        
(HERMES)   & $q = u, d, \bar{u}, \bar{d}$ &     \\[2mm]                                 
%     &     &    \\                         
$\mbox{\boldmath $p\bar{p} \rightarrow WX(ZX)$}$    &  $ud \rightarrow W$                                                                                        
&  $u,d$ at $x \simeq M_W/\sqrt{s} \rightarrow$  \\                                                                                        
(UA1, UA2; CDF, D0)          &   & \hfill  $x \approx 0.13;~0.05$ \\[2mm]                               
%&    &   \\                         
\quad $\;\;\mbox{\boldmath $\rightarrow \ell^{\pm}$}$ {\bf asym} (CDF)   &         &                                                                                        
slope of $u/d$ at $x \approx 0.05 - 0.1$  \\[2mm]                                                                                       
%                 &                   &                               \\                                                                                       
$\mbox{\boldmath $p\bar{p} \rightarrow t\bar{t} X$}$ & $q\bar{q}, gg \rightarrow           
t\bar{t}$ & $q, g$ at $x \gapproxeq 2m_t/\sqrt{s} \simeq 0.2$ \\          
(CDF, D0) &                       & \\[2mm]          
% & &            
$\mbox{\boldmath $p\bar{p} \rightarrow ${\bf jet}$\, + \, X$}$    &                                                                                        
 $gg,qg,qq\rightarrow 2j$  &  $q,g$                                                                                        
at $x \simeq 2 E_T/\sqrt{s} \rightarrow$  \\                                                                                        
(CDF, D0)         &   & \hfill  $x \approx 0.05 -  0.5$ \\ \hline                                                                                       
%                         &                         &          \\                                                                                       
%\hline                                                                                          
\end{tabular}
\end{center}

In principle $\nu N \rightarrow \mu^- X$ and $\bar{\nu} N \rightarrow \mu^+ X$ DIS data can determine $s$ and 
$\bar{s}$ distributions separately.  However so far the global analyses have used the studies of the CCFR dimuon 
production data, which yields 
\begin{equation}
\label{eq:b1}
s + \bar{s} \simeq (\bar{u} + \bar{d})/2
\end{equation}
independent of $x$ at $Q^2 = 1$~GeV$^2$, with typically a $\pm 10\%$ error \cite{MRST2}.

\subsection{Heavy quark distributions:  $c$ and $b$}

The improved theoretical treatments have recently enabled these distributions to be put on a sound basis in the global 
analyses (see, for example, \cite{HARRIS}).  For a heavy quark distribution (say $c$) we should match, at $Q^2 = m_c^2$, the fixed flavour number scheme, 
describing $\gamma g \rightarrow c\bar{c}$ with $n_f = 3$ to the variable flavour number scheme with $n_f = 4$ 
flavours for $Q^2 > m_c^2$.  MRST \cite{MRST1} use the Thorne-Roberts prescription \cite{TR} in which $F_2^c$ and 
$\partial F_2^c/\partial \ln Q^2$ are required to be continuous at $Q^2 = m_c^2$, whereas CTEQ \cite{CTEQ5} use an 
alternative \lq ACOT\rq~prescription \cite{ACOT}.  In this way the charm distribution is generated perturbatively 
with the mass of the charm quark $m_c$ as the only free parameter.  Not surprisingly, due to the $g \rightarrow 
c\bar{c}$ transition, the charm distribution mirrors the form of the gluon distribution.  At some level we may 
expect an \lq intrinsic\rq~component of the charm distribution.

\subsection{The gluon distribution $g$}

In principle many processes are sensitive to the gluon distribution, but it is still difficult to determine 
for $x \gapproxeq 0.2$ where it becomes increasingly small.  In this region MRST \cite{MRST1} used the WA70 
prompt photon data to determine the gluon.  However this process suffers from scale dependence (reflecting the 
higher order corrections), effects of intrinsic $k_T$ (needed to describe the E706 prompt photon data) and 
uncertainties due to fragmentation effects and isolation criteria.  In an attempt to provide an acceptable 
spread of gluon distributions at large $x$, MRST presented three sets of partons ($g\uparrow$, the default MRST set, 
$g\downarrow$) corresponding, respectively, to an average intrinsic $k_T = 0, 0.4$ and 0.64~GeV for the WA70 data.  The 
arrows indicate the relative sizes of the gluons in the large $x$ region.  The corresponding $\langle k_T \rangle$ 
needed to obtain a good description of the E706 prompt photon data is worryingly large at about 1~GeV.

Due to the uncertainties associated with prompt photon production, the recent CTEQ analysis \cite{CTEQ5} omits these 
data and instead determines the large $x$ behaviour of the gluon using the single jet inclusive $E_T$ distributions 
measured at the Tevatron.  The above three MRST sets of partons \lq predict\rq~the shape of the jet $E_T$ 
distributions remarkably well, but the predicted normalization is 7\%, 13\%, 17\% below the data respectively with 
$g\uparrow$ being closest to CTEQ5.

Turning now to the small $x$ domain, the gluon is well constrained by the observed behaviour of 
$\partial F_2/\partial \log Q^2$.  To be precise the H1 and ZEUS data determine the gluon for $x \lapproxeq 0.01$, 
and the NMC data for $x \lapproxeq 0.1$.  Inspection of $F_2$ as a function of $\ln Q^2$ at low fixed $x$ 
\cite{MON,ZOMER} shows that the data flatten out at low $Q^2$, indicating a smaller value of 
\begin{equation}
\label{eq:b2}
\partial F_2/\partial \ln Q^2 \simeq \alpha_S (Q^2) P_{gg} \otimes g.  
\end{equation}
Also $\alpha_S$ increases with decreasing $Q^2$, and as a consequence $g (x)$ becomes valence-like for $Q^2 \simeq 1$~GeV$^2$, which 
is about the lowest value of $Q^2$ for a DGLAP description.  In fact $Q^2 \simeq 2$~GeV$^2$, where the gluon 
becomes flat (that is $g (x) \rightarrow$ constant as $x \rightarrow 0$), is a more reasonable lower limit for DGLAP 
evolution.

\subsection{$W$ production at the Tevatron and the LHC}

The total cross sections for $W$ and $Z$ hadroproduction are known to NNLO \cite{WLN} and the input electroweak 
parameters are known to high accuracy.  The main uncertainty in predicting the size of the cross sections comes from 
the parton distributions and, to a lesser extent, from the value of $\alpha_S$.  The MRST predictions are \cite{MRST2}
\begin{equation}
\label{eq:a2}
B (W \rightarrow \ell \nu) \sigma_W \; = \; \left \{ \begin{array}{lcl}
2.45~{\rm nb} & (\pm 3\%) & {\rm at~the~Tevatron} \\
20.3~{\rm nb} & (\pm 5\%) & {\rm at~the~LHC}. \end{array} \right .
\end{equation}
The errors arise from considering the spread of predictions arising from taking the full range of parton sets, 
including a $\pm 0.005$ uncertainty in $\alpha_S (M_Z^2)$.  In principle the predictions allow a check on the 
luminosity of the collider.  Of course, there are uncertainties in the parton distributions due to the normalization errors 
on present data.  To make an attempt to allow for this, the range of parton sets used to specify the errors in (\ref{eq:a2}) 
includes sets obtained from global fits with the HERA deep inelastic data renormalized up by 2.5\%, and then down by 2.5\%.

\subsection{Comparison of MRST and CTEQ partons}

The predictions for $\sigma_W$ serve another purpose.  The long evolution length to $Q^2 = M_W^2$, together with 
the expected precision of the predictions, can expose small defects in the evolution codes used for the global analyses.  
An unexpectedly large 
difference was found between the cross sections predicted using MRST98 \cite{MRST1} and CTEQ5 \cite{CTEQ5} partons.  
The value for $\sigma_W$ obtained from CTEQ5 was 10\% greater than from MRST98 partons at the LHC energy.  It means that the 
evolution codes should be carefully checked.  Recall that neither CTEQ, nor MRS, agreed completely with the 
\lq\lq standard HERA workshop code \cite{ST}\rq\rq~for a test evolution from $Q^2 = 4$ to 100~GeV$^2$.  In the 
interval $10^{-4} \lapproxeq x \lapproxeq 10^{-1}$, the CTEQ sea was some 4\% high, and the MRS sea was 1\% low and 
the gluon about 1.5\% low.  MRST checked their code and found that it had to be corrected by a factor $C_F 
\rightarrow C_A$ in one of the terms appearing in the NLO $P_{gg}$ splitting function.  The effect is to make the 
gluon, and hence the sea, evolve a little more rapidly at small $x$.  With the correction the MRST99 code 
\cite{MRST2} is found to agree precisely with the \lq\lq standard\rq\rq~evolution code and, moreover, is consistent with exact 
momentum conservation as a function of $Q^2$, as it should be.  However the discrepancy for $\sigma_W$ remains:  the 
CTEQ5 HQ prediction is about 6\% (10\%) higher at Tevatron (LHC) energies \cite{MRST2}.  Differences between MRST and 
CTEQ partons can arise from the choice of the data sets to be included in the global fit, the $Q^2, W^2$ cuts used, the 
different renormalization of some data sets, the different treatment of heavy flavours, etc.  However none of these 
differences appears sufficient to explain the discrepancy in the predictions for $\sigma_W$, which sample partons at 
$Q^2 = M_W^2$ with $x \simeq M_W/\sqrt{s}$.

\subsection{Outstanding problems of global analyses}

There is a need to determine realistic errors on the parton distributions extracted in a global analysis.  This is far from 
easy.  In particular, among other things, we need to quantify the uncertainties due to (i) heavy target and other 
nuclear corrections, (ii) the possible effects of higher twist contributions, (iii) the choice of the form of 
parametrization of the initial distributions, and (iv) the possible effects of $k_T$ smearing on the analysis of exclusive 
processes.

A second, and related, problem is the need for a next-to-next-leading order (NNLO) analysis.  The $P_{gg}$ splitting 
function is not yet known to this order.  Also, as data at smaller and smaller $x$ are included, there is a need 
to resum $\log (1/x)$ effects, and to allow for parton shadowing corrections.

\section{Skewed (or off-diagonal) parton distributions}

Data of higher and higher precision are becoming available for processes which are described by off-diagonal 
(or skewed) parton distributions in the small $x$ domain.  A relevant example is diffractive vector meson 
production at HERA, $\gamma^* p \rightarrow Vp$ with $V = \rho, J/\psi$ or $\Upsilon$ (see, for example, 
\cite{T}).  At high $\gamma^* p$ c.m.~energy, $W$, the cross section is dominated by the two-gluon exchange 
amplitude
\begin{equation}
\label{eq:a3}
\left . \frac{d\sigma}{dt} (\gamma^* p \rightarrow Vp) \right |_{t = 0} \; = \; \ldots [x_2 g (x_1, x_2, \mu^2)]^2
\end{equation}
where $g$ is the off-diagonal $(x_1 \neq x_2)$ gluon distribution with
\begin{eqnarray}
\label{eq:a4}
x_1 & = & (Q^2 + M_{q\bar{q}}^2)/W^2, \nonumber \\
x_2 & = & (M_{q\bar{q}}^2 - M_V^2)/W^2 \: \ll \: x_1, 
\end{eqnarray}
see \cite{MR}.  The process is shown schematically in Fig.~1.  $M_{q\bar{q}}$ is the mass of the $q\bar{q}$ system 
produced by a photon of virtuality $Q^2$.  The 
relevant scale is $\mu^2 = z (1 - z) Q^2 + k_T^2 + m_q^2$ where $z, 1 - z$ and $\pm \mbox{\boldmath $k$}_T$ specify 
the momenta of the $q$ and $\bar{q}$ which form the vector meson of mass $M_V$.  The quadratic dependence of $g$ in 
(\ref{eq:a3}) shows that these data may offer a sensitive constraint on the gluon.  Indeed we will see that the 
off-diagonal distributions are fixed by the conventional (diagonal) parton distributions of Section 2, so that 
the data can, in principle, be included in a global parton analysis.

\begin{figure}
\begin{center}
\leavevmode
\includegraphics[width=.8\textwidth]{martin1.eps}
\end{center}
\caption{Schematic diagram for diffractive vector meson production at HERA, $\gamma^* p \rightarrow Vp$.  
The longitudinal fractions $x_1$ and $x_2$ of the in-going and out-going proton momentum are given by 
(\ref{eq:a4}).  $z$ and $1-z$ are the longitudinal fractions of the $q$ and $\bar{q}$ momenta.  There are 
four possible couplings of the two gluons to the $q$ and $\bar{q}$, represented by the upper circle.}
\label{2ndfig.eps}
\end{figure}

\subsection{Ji's symmetrized distributions}

It is convenient to use the skewed parton distributions $H (x, \xi) \equiv H (x, \xi, t, \mu^2)$, with 
support $-1 \leq x \leq 1$, introduced by Ji \cite{JI}, with the minor difference that the gluon $H_g = x H_g^{\rm Ji}$
\cite{GM}.  The distributions depend on the momentum fractions
\begin{equation}
\label{eq:a5}
x_{1.2} \; = \; x \pm \xi
\end{equation}
carried by the emitted and absorbed partons at each scale $\mu^2$, and on the momentum transfer variable $t = 
(p - p^\prime)^2$.  The variables do not change as we evolve the distributions up in the scale $\mu^2$.  In the limit 
$\xi \rightarrow 0$ they reduce to the conventional parton distributions
\begin{eqnarray}
\label{eq:a6}
H_q (x, 0) & = & \left \{ \begin{array}{lcr}
q (x) & {\rm for} & x > 0 \\
- \bar{q} (-x) & {\rm for} & x < 0 \end{array} \right . \nonumber \\
H_g (x, 0) & = & xg (x), 
\end{eqnarray}
and satisfy DGLAP evolution.  In the limit $\xi \rightarrow 1$ they obey ERBL evolution \cite{ER,BL}.  If we consider 
$H_q$ at arbitrary values of $\xi$, then for $x > \xi$ and $x < - \xi$ we have DGLAP-like evolution for quarks and 
antiquarks respectively, while for $-\xi < x < \xi$ we have ERBL-like evolution for the emitted $q\bar{q}$ pair.

On account of the $x_1 \leftrightarrow x_2$ symmetry the distributions $H_q$ and $H_g$ are symmetric in $\xi$
\begin{equation}
\label{eq:a7}
H_i (x, \xi) \; = \; H_i (x, -\xi).
\end{equation}
We also have symmetry relations in terms of the $x$ variable
\begin{eqnarray}
\label{eq:a8}
H_q^{NS} (x, \xi) & = & H_q^{NS} (-x, \xi) \nonumber \\
H_q^S (x, \xi) & = & -H_q^S (-x, \xi) \nonumber \\
H_g (x, \xi) & = & H_g (-x, \xi) 
\end{eqnarray}
where the superscripts $S$ and $NS$ denote singlet and non-singlet quarks respectively.

\subsection{$H (x, \xi)$ in terms of conformal moments}

In order to relate the skewed distributions to the conventional parton densities it is convenient to work in terms 
of the so-called conformal moments\footnote{Conformal moments were introduced in \cite{ER,C} for $\xi = 1$, and in 
\cite{O} for $\xi \neq 1$; see also \cite{BFKL1}.} of the skewed distributions
\begin{equation}
\label{eq:a9}
\fbox{$O_N (\xi, \mu^2) \; = \; \displaystyle{\int_{-1}^1} dx R_N (x_1, x_2) H (x, \xi, \mu^2).$}
\end{equation}   
At leading order, these moments are not mixed by evolution.  In fact
\begin{equation}   
O_N (\xi, \mu^2) \; = \; O_N (\xi, \mu_0^2) \left ( \frac{\mu^2}{\mu_0^2} \right    
)^{\gamma_N},   
\label{eq:a10}   
\end{equation}   
where $\gamma_N$ are the same anomalous dimensions as for diagonal partons.  The $R_N$    
are known polynomials of degree $N$   
\begin{equation}   
R_N \; = \; \sum_{k = 0}^N \left ( \begin{array}{c} N \\ k \end{array} \right ) \left (    
\begin{array}{c} N + 2p \\ k + p \end{array} \right ) x_1^k x_2^{N - k}   
\label{eq:a11}   
\end{equation}   
with $p = 1,2$ for quarks and gluons respectively. The $O_N$ reduce to the usual moments    
in the limit $\xi \rightarrow 0$.  For example for quarks   
\begin{equation}   
O_N \rightarrow M_N \; = \; \int_0^1 x^N q(x) dx,   
\label{eq:a12}   
\end{equation}   
up to a normalizing factor $R_N (1, 1)$.   
   
The crucial step is to find the inverse relation to (\ref{eq:a9}).  That is to reconstruct $H (x,    
\xi)$ from a knowledge of its conformal moments.  The result, due to Shuvaev \cite{SHUV},    
is   
\begin{equation}   
\fbox{$H (x, \xi) \; = \; \displaystyle{\int_{-1}^1} dx^\prime K (x, \xi; x^\prime) f (x^\prime)    
$}   
\label{eq:a13}   
\end{equation}   
where the kernel $K$ is a known integral \cite{SHUV,SGMR} and $f$ is the Mellin    
transform   
\begin{equation}   
f (x^\prime) \; = \; \int \frac{dN}{2 \pi i} (x^\prime)^{-N} O_N (\xi) / R_N (1, 1).   
\label{eq:a14}   
\end{equation}   
The function $f$ reduces to the conventional diagonal parton distribution for $\xi^2 \ll 1$.  This follows since \cite{JI}   
\begin{eqnarray}   
O_N (\xi)  & = & \sum_{k = 0}^{[(N + 1)/2]} O_{Nk} \xi^{2k} \nonumber    
\\   
& \simeq & O_{N0} \; = \; O_N (0) = M_N R_N (1, 1)   
\label{eq:a15}   
\end{eqnarray}   
for small $\xi^2$.  So the skewed distribution $H$ is completely determined in terms of    
the diagonal distribution $f$ via (\ref{eq:a13}).  This identification can be made at any scale, and so there is no 
need to solve the off-diagonal evolution equations to obtain the $\mu^2$ dependence of $H (x, \xi, \mu^2)$.  That is 
the normal DGLAP evolution of $f (x, \mu^2)$ is sufficient to generate the $x, \xi$ and $\mu^2$ dependence of 
$H (x, \xi, \mu^2)$ at small $x$ and $\xi$.
   
\subsection{A good small $x, \xi$ approximation}   
   
We can simplify (\ref{eq:a13}) further if we assume that the diagonal partons have the forms   
\begin{equation}   
xq (x) \; = \; N_q x^{- \lambda_q}, \quad xg (x) \; = \; N_g x^{- \lambda_g},   
\label{eq:a16}   
\end{equation}   
which should be valid for very small $x$.  Then the $x^\prime$ integration in (\ref{eq:a13}) can be performed 
analytically, giving
\begin{equation}   
H_i (x, \xi) \; = \; \xi^{- \lambda_i - p} F_i \left ( \frac{x}{\xi} \right ) \nonumber   
\label{eq:a17}   
\end{equation}   
with $p = 1, 0$ for $i = q, g$ respectively, where the $F_i$ are known in terms of $N_i$ and $\lambda_i$.  
A full set of results can be found in \cite{SGMR} for the off-diagonal/diagonal    
ratios,    
\begin{equation}   
R_i (x, \xi) \; = \; H_i (x, \xi) / H_i (x + \xi, 0),   
\label{eq:a18}   
\end{equation}   
for which the only free parameter is $\lambda_i$.  In \cite{SGMR} the ratios $R_q^{NS,S}$ and $R_g$ are plotted as    
functions of $x/\xi$ for different values of $\lambda_i$. The scale dependence of the    
off-diagonal distributions, $H_i (x, \xi)$ of (\ref{eq:a17}), and hence of the $R_i$, is hidden    
in the $\mu^2$ dependence of the $\lambda_i$.  Both $\lambda_g$ and $\lambda_q$ increase    
with increasing $\mu^2$, see Fig.~13 of \cite{MRST1}.   
   
\subsection{Application to $\gamma^* p \rightarrow Vp$}   
   
The above results are exactly what we need to describe vector meson production at HERA at large $\gamma^* p$ c.m.~energies 
$W$.  From (\ref{eq:a4}) we see that $x_1$ is small and $x_2 \approx 0$, that is $x \approx \xi$.  The gluon  
is the relevant parton and the skewed distribution is larger than the conventional gluon by a factor
\begin{equation}   
R_g (x = \xi) \; = \; \frac{H_g (\xi, \xi)}{H_g (2 \xi, 0)} \; = \; \frac{2^{2 \lambda_g + 3}}{\sqrt{\pi}} 
\frac{\Gamma\!\left (\lambda_g + \frac{5}{2} \right )}{\Gamma (\lambda_g + 4)}.   
\label{eq:a19}   
\end{equation}   
The cross section formula (\ref{eq:a3}) may then be expressed in terms of the conventional    
gluon distribution $g$,   
\begin{equation}   
\left . \frac{d \sigma}{dt} (\gamma^* p \rightarrow Vp) \right |_{t = 0} \; = \; \ldots \left [R_g    
x_1 g (x_1, \mu^2) \right ]^2,   
\label{eq:a20}   
\end{equation}   
where all the off-diagonal effects are contained in the known (enhancement) factor $R_g^2$.     
Of course to calculate the cross section properly we must use the unintegrated gluon    
distribution and integrate over the transverse momenta of the exchanged gluons and of the    
$q$ and $\bar{q}$ forming the vector meson, see \cite{T}.   
   
To obtain the scale dependence of $R_g$, we first obtain the $\mu^2$ dependence of    
$\lambda_g$ of (\ref{eq:a16}) from the behaviour of the gluon found in the global parton    
analyses.  For example, the MRST partons \cite{MRST1} have $\lambda_g = 0.205$ and 0.38    
at $\mu^2 = 4$ and 100~GeV$^2$ respectively.  The appropriate scale for the diffractive    
process $\gamma^* (Q^2) p \rightarrow V (q\bar{q}) p$ is $\mu^2 \simeq m_q^2 + Q^2/4$.     
In this way, for diffractive $J/\psi$ and $\Upsilon$ photoproduction at HERA we find from (\ref{eq:a19}) that the    
off-diagonal enhancement, $R_g^2$, is $(1.15)^2$ and $(1.32)^2$ respectively.  However,    
for $\Upsilon$ photoproduction, $x$ is not sufficiently small $(\sim 0.01)$ and we have to    
improve the assumption made in (\ref{eq:a16}).  If we take $xg \sim x^{- \lambda_g} (1 -    
x)^6$ and perform the $x^\prime$ integration in (\ref{eq:a13}) numerically, then we find an    
enhancement of $(1.41)^2$ for $\Upsilon$ photoproduction \cite{TEUB}.  Moreover note that the skewedness, and 
hence the enhancement, increase with $Q^2$.  This is particularly noticeable in $\gamma^* p \rightarrow \rho p$, 
see \cite{T}.
   
\subsection{Validity of the skewed $\rightarrow$ diagonal parton connection}   
   
Before we can be sure that the skewed distribution $H (x, \xi)$, at any scale, is fully determined at small $x, \xi$ by    
knowledge of the diagonal parton distribution, at the same scale, we must consider the points below.  First, we have checked 
that the analytic continuation of the conformal moments $O_N$ in $N$ is allowed \cite{SGMR}.  A second consideration is that, from a    
formal point of view, we may add to the off-diagonal distribution any function which exists    
only in the ERBL-like region, $|x| < \xi$.  In \cite{SGMR} we show such a contribution is    
negligible $O (\xi^2)$ at small $\xi$.  So far our distributions allow the calculation of the    
imaginary part of the amplitude for the process.  At small $x$ and $\xi$ it turns out that the    
real part may be calculated easily using a dispersion relation in the c.m.~energy squared,    
$W^2$, and that the amplitude   
\begin{equation}   
A \; = \; i {\rm Im} A \: \frac{1 + e^{- i \pi \lambda}}{1 + \cos \pi \lambda},   
\label{eq:a21}   
\end{equation}   
where $A \propto (W^2)^\lambda$.  Finally we note that our result remains valid at NLO,    
since there is no conformal mixing for $\xi^2 \ll 1$.   
   
\subsection{Conclusions}

We conclude that, at small $x, \xi$, the skewed distributions $H (x, \xi; \mu^2)$ are    
completely known in terms of conventional partons.  Thus data for processes which are    
described by such distributions can, in principle, be included in a conventional global analysis    
to better constrain the low $x$ behaviour of the partons.  In particular, data for diffractive vector meson 
production, $\gamma^* p \rightarrow Vp$, which depend on the {\it square} of the gluon distribution, have the potential 
to put a tight constraint on the gluon at small $x$.  Before this can be done it is necessary to calculate the NLO 
contributions.  Finally here we have been concerned only with the skewed distributions in the small $x$ domain.  
Skewed distributions, in general, have become an area of intense activity with a large literature (see, for example, 
\cite{JI,RAD,GV} and references therein).
   
\section*{Acknowledgements}   
   
I thank Bernd Kniehl, G\"{u}nter Grindhammer and Gustav Kramer for their efficient organisation of the Ringberg 
Workshop.  I also thank Krzysztof Golec-Biernat, Dick Roberts, Misha Ryskin, Andrei Shuvaev, James Stirling, Thomas Teubner and 
Robert Thorne for enjoyable collaborations on the subject of this review.
   
     
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