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TUM/T39-98-21
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  {\Large\bf
    \setcounter{footnote}{1}
    Exclusive $J/\psi$ photoproduction
    and gluon polarization\footnote{Work supported in part by BMBF}
  }\\ 

\vspace{2cm}

\setcounter{footnote}{0}
M. V\"anttinen\footnote{Alexander von Humboldt fellow}$^{,1}$
and L. Mankiewicz$^{1,2}$

\vspace{1 cm}

{\em $^1$ Physik Department, Technische Universit\"{a}t M\"{u}nchen, \\
D-85747 Garching, Germany}

\vspace{0.5 cm}

{\em $^2$ N. Copernicus Astronomical Center, Polish Academy of Science, \\
ul.\ Bartycka 18, PL--00-716 Warsaw}

\vspace{1cm}

{\em \today}

\vspace{1cm}

{\bf Abstract:\\[5pt]} \parbox[t]{\textwidth}{In exclusive
$J/\psi$ production by polarized photons incident on polarized
protons, a finite polarization asymmetry arises because of 
$c\bar c$ Fermi-motion and binding-energy effects. The asymmetry
depends on the polarized nonforward gluon distribution of the
proton and thus gives information on gluon polarization in the
proton. The analyzing power, however, is rather small.}

%\vspace{1cm}
%{\em Submitted to Physics Letters B}
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Gluon polarization in the proton is a topic of current interest
in hadronic physics (see e.g.\ \cite{COMPASS,RHIC}).
A useful probe of gluon distributions is the production of open or
hidden charm, because charm quarks dominantly couple to gluons and
not to light quarks in the proton. In this paper we discuss what the
exclusive photoproduction of $J/\psi$ mesons could tell about
the polarized gluon content of the proton.

In general, hard exclusive processes probe nonforward matrix elements
of the target nucleon, i.e.\ matrix elements of QCD operators between
nucleon states of different momenta \cite{Ji,Radyushkin}.
These matrix elements can be
expressed in terms of nonforward parton distributions, which are
functions of two momentum-fraction variables and generalize the usual
(forward) parton distributions. Nonforward distributions reduce to
forward distributions in the limit of equal proton momenta in the
initial and final states. This limit cannot be realized in exclusive
reactions because of simple kinematical reasons, but may be a good
approximation at high energy \cite{MPW97,Frankfurt,Martin97}.
Thus exclusive $J/\psi$
production in electron-proton collider experiments, e.g.\ at HERA,
could yield information on the small $x$ behaviour of the usual
gluon distribution of the proton, whereas fixed-target experiments
will probe the nonforward parton distributions in a more general case.

The dependence of unpolarized exclusive $J/\psi$ cross sections on
unpolarized gluon distributions has been discussed in Refs.\ 
\cite{Ryskin-unpolarized,Brodsky}. In principle, $J/\psi$ production
by polarized beams on polarized targets should depend on polarized
gluon distributions $\Delta G$. Unfortunately, contrary to earlier
belief, the production amplitude does not depend on $\Delta G$
in the first approximation (i.e.\ assuming a nonrelativistic $J/\psi$
bound state), as we showed in a recent paper \cite{MV-LM}.

However, as will be shown in detail in this paper, relativistic
corrections to $J/\psi$ production will depend on $\Delta G$.
Gluon polarization can therefore be accessed in $J/\psi$ production,
although the analyzing power may be rather small.

The expansion of $J/\psi$ amplitudes around the nonrelativistic
limit was first discussed by Keung and Muzinich \cite{KeungMuzinich}.
Their approach is based on expanding the perturbative amplitudes for
$c\bar c$ production in powers of the heavy quark relative momentum.
Higher Fock states of the $J/\psi$ are neglected, and the calculation
is therefore not gauge invariant. The issue of gauge invariance
was discussed more recently by Khan and Hoodbhoy \cite{KhanHoodbhoy}.
They pointed out that the gluonic contributions necessary to restore
gauge inariance are proportional to $v^3$, where $v$ is the relative
velocity of the charm quark and antiquark. Gluonic contributions are
thus subleading as compared to the first relativistic corrections in the
Keung--Muzinich approach, which are proportional to $v^2$.

We shall now calculate the $O(v^2)$ corrections to exclusive $J/\psi$
production following \cite{KeungMuzinich,KhanHoodbhoy}. We work in the
photoproduction limit (photon virtuality $q^2=-Q^2=0$) and consider
only the case of collinear scattering.

In terms of nonforward parton distributions $G$ and $\Delta G$, the
helicity amplitude for
\beq
  \gamma(q,\lambda) + p(p_1, S=\pm 1/2)
  \rightarrow J/\psi(K, \lambda'=\lambda) + p(p_2, S'=\pm 1/2)
\eeq
reads \cite{MV-LM}
\beqa
  {\cal A}_{\lambda\lambda \pm\pm}
  & = & \frac{1}{2} \int_{-1}^1 du 
        \frac{1}{(u-\xi+i\epsilon)(u+\xi-i\epsilon)} 
        \nonumber \\
  &   & \times \sum_{\lambda_1}
        \; A_{\lambda\lambda\lambda_1\lambda_1}
        \; \left[
        G(u,\xi;\mu^2) \pm \lambda_1 \, \Delta G(u,\xi;\mu^2)
        \right] \, ,
  \label{hadronic-amplitude-I}
\eeqa  
where $\xi = M^2/(2s-M^2)$ with $s=(q+p_1)^2$.        
The helicity amplitude $A_{\lambda\lambda'\lambda_1\lambda_2}$ for
\beq
 \gamma(q,\lambda) + g(k_1,\lambda_1)
 \rightarrow J/\psi(K,\lambda') + g(k_2,\lambda_2) \, ,
\eeq
where
\beq
  k_1 = \frac{u+\xi}{1+\xi} \; p_1 \, ,
\eeq
is the convolution of a perturbative $c\bar c$ production
amplitude\footnote{$H_{\lambda \lambda_1 \lambda_2}(q,K,k_1;\ell)$
denotes the $\gamma g \rightarrow c\bar c g$ amplitude with heavy-quark
spinors truncated, whereas in Ref.\ \protect\cite{MV-LM} we denoted
by $H^{\mu\nu}_{\lambda\lambda'}(q,K,k_1)$ the $\gamma g \rightarrow
J/\psi g$ amplitude with gluon polarization vectors truncated.} 
$H_{\lambda \lambda_1 \lambda_2}$ and a $J/\psi$ matrix element
(both are Dirac matrices):
\beqa
  \lefteqn{A_{\lambda\lambda'\lambda_1\lambda_2}(q,K,k_1)} \nonumber \\
  & = & {\rm Tr} \; \int \frac{d^4 \ell}{(2\pi)^4}
        H_{\lambda \lambda_1 \lambda_2}(q,K,k_1;\ell) 
        \int d^4x e^{i\ell\cdot x}
        \left\langle K,\lambda' \left| \psi(x/2)
        \bar\psi(-x/2) \right| 0 \right\rangle
        \, .
  \label{convolution}
\eeqa
There are six terms in $H_{\lambda \lambda_1 \lambda_2}$, corresponding
to permutations of the photon and the two gluons on the charm quark line.
A representative one, corresponding to the diagram in
Fig.~\ref{fig:diagram}, is
\beqa
  \lefteqn{H_{\lambda \lambda_1 \lambda_2}(q,K,k_1;\ell)} \nonumber \\
  & \sim & \slas{\epsilon}(q,\lambda) S_F(K/2-q+\ell)
           \slas{\epsilon}(k_1,\lambda_1) S_F(-K/2-k_2+\ell)
           \slas{\epsilon}^*(k_2,\lambda_2) \, ,
   \label{partonic-amplitude1}
\eeqa
where $S_F(p) \equiv (p\!\!\!/ - m_c)^{-1}$.
We have omitted normalization factors which cancel in a polarization
asymmetry calculation. Expanding the hard amplitude to second order in
$\ell$, we obtain
\beqa
  \lefteqn{A_{\lambda\lambda'\lambda_1\lambda_2}(q,K,k_1)} \nonumber \\
  & = & {\rm Tr} \, \Biggl[
        \left. H_{\lambda \lambda_1 \lambda_2}(q,K,k_1;\ell) \right|_{\ell=0}
        \left. \left\langle K,\lambda' \left| \psi(x/2) 
               \bar\psi(-x/2) \right| 0 \right\rangle  \right|_{x=0}
        \nonumber \\
  &   & \mbox{} + \left. \frac{\partial}{\partial\ell^\alpha}
        H_{\lambda \lambda_1 \lambda_2}(q,K,k_1;\ell) \right|_{\ell=0}
        \left. \left\langle K,\lambda' \left| \psi(x/2) 
               \left(
               -i\stackrel{\leftrightarrow}{\partial}_\alpha
               \right)
               \bar\psi(-x/2) \right| 0 \right\rangle  \right|_{x=0}
        \nonumber \\
  &   & \mbox{} + \left. \frac{1}{2}
        \frac{\partial^2}{\partial\ell^\alpha\partial\ell^\beta}
        H_{\lambda \lambda_1 \lambda_2}(q,K,k_1;\ell) \right|_{\ell=0}
        \nonumber \\
  &   & \times
        \left. \left\langle K,\lambda' \left| \psi(x/2) 
               \left(
               -i\stackrel{\leftrightarrow}{\partial}_\alpha
               \right)
               \left(
               -i\stackrel{\leftrightarrow}{\partial}_\beta
               \right)
               \bar\psi(-x/2) \right| 0 \right\rangle
        \right|_{x=0} \Biggr] \, .
 \label{expansion}
\eeqa
The $J/\psi$ matrix elements can be expressed as \cite{KhanHoodbhoy,Hoodbhoy}
\beqa
   \lefteqn{\left.\left\langle K,\lambda' \left| \psi(x/2) 
         \bar\psi(-x/2) \right| 0 \right\rangle  \right|_{x=0}}
         \nonumber \\
   & = & \frac{1}{2} M^{1/2} \left( \phi + \frac{\nabla^2\phi}{M^2} \right)
         \slas{\epsilon}^* \left( 1 + \frac{\slashK}{M} \right)
         - \frac{1}{6} M^{1/2} \frac{\nabla^2\phi}{M^2}
         \slas{\epsilon}^* \left( 1 - \frac{\slashK}{M} \right) \, ,
         \label{projector0} \\
%%%%%%%%%%%%%%%%%%%%%
   \lefteqn{\left.\left\langle K,\lambda' \left| \psi(x/2) 
         \left( -i \stackrel{\leftrightarrow}{\partial}_\alpha \right)
         \bar\psi(-x/2) \right| 0 \right\rangle  \right|_{x=0}}
         \nonumber \\
   & = & -\frac{1}{3} M^{3/2} \frac{\nabla^2\phi}{M^2}
         \epsilon^{*\beta} \left( g_{\alpha\beta}
         + i\epsilon_{\alpha\beta\mu\nu} \gamma^\mu \gamma_5 \frac{K^\nu}{M}
         \right) \, ,
         \label{projector1} \\
%%%%%%%%%%%%%%%%%%%%%
   \lefteqn{\left.\left\langle K,\lambda' \left| \psi(x/2) 
         \left( -i \stackrel{\leftrightarrow}{\partial}_\alpha \right)
         \left( -i \stackrel{\leftrightarrow}{\partial}_\beta  \right)
         \bar\psi(-x/2) \right| 0 \right\rangle  \right|_{x=0}}  
         \nonumber \\
   & = & \frac{1}{6} M^{5/2} \frac{\nabla^2\phi}{M^2}
         \left( g_{\alpha\beta} -  \frac{K_\alpha K_\beta}{M^2} \right)
         \slas{\epsilon}^* \left( 1 + \frac{\slashK}{M} \right) \, .
         \label{projector2}
\eeqa
The amplitude given by eqs.\ (\ref{expansion}-\ref{projector2}) is
gauge invariant at $O(v^2)$. In eqs.\ (\ref{projector0}-\ref{projector2}),
$\phi$ and $\nabla^2\phi$ are the $J/\psi$ meson's Coulomb-gauge wavefunction
and its Laplacian at the origin. For a wavefunction $\phi \sim e^{-r/a}$,
corresponding to a $1/r$ heavy-quark potential appropriate for
a heavy quarkonium state, $\nabla^2\phi$ is actually infinite at the origin.
The quantity $\nabla^2\phi$ in eqs.\ (\ref{projector0}-\ref{projector2})
has to be understood as representing the average of the Laplacian over a
region of volume $\sim 1/m_c^3$ and thus regarded as a free parameter.

\begin{figure}
\centerline{\psfig{figure=fig1.ps,width=10cm}}
\caption{\sf One of 6 Feynman diagrams which contribute to the amplitude
(\protect\ref{partonic-amplitude1}). From Ref.\ \protect\cite{MV-LM}.} 
\label{fig:diagram}
\end{figure}

Further corrections of the same magnitude arise because the quark mass
$m_c$ which appears in perturbative propagators is not exactly one half
of the physical charmonium mass which appears in the matrix elements
(\ref{projector0}-\ref{projector1}). These binding-energy corrections
are obtained by expanding the amplitude to first order in
$\epsilon_B = 2m_c-M$.

The parton-level amplitude simplifies to
\beqa
  \lefteqn{A_{\lambda\lambda'\lambda_1\lambda_2}(q,K,k_1)} \nonumber \\
  & \sim & \left( 1 - \frac{\epsilon_B}{2M}
                    + \frac{\nabla^2\phi}{3M^2\phi} \right)
           (\epsilon_1 \cdot \epsilon_2^*)
           (\epsilon_\gamma \cdot \epsilon_\psi^*)
           \nonumber \\
  &      & \mbox{} + \left( \frac{\epsilon_B}{2M}
                            - \frac{\nabla^2\phi}{M^2\phi} \right)
           \frac{M^2}{\hat s (\hat s - M^2)}
           \left[ M^2 (\epsilon_1 \cdot \epsilon_2^*)
                      (\epsilon_\gamma \cdot \epsilon_\psi^*)
           \right.
           \nonumber \\
  &      & \left. \mbox{} - \hat s (\epsilon_1 \cdot \epsilon_\psi^*)
                                   (\epsilon_\gamma \cdot \epsilon_2^*)
                  + (\hat s - M^2) (\epsilon_1 \cdot \epsilon_\gamma)
                                   (\epsilon_\psi^* \cdot \epsilon_2^*)
           \right] \, ,
\eeqa
where $\hat s = (q+k_1)^2$ and $\epsilon_1, \epsilon_2, \epsilon_\gamma,
\epsilon_\psi$ are the usual transverse polarization vectors for the
gluons, the photon and the $J/\psi$. Inserting these into the
hadron-level amplitude (\ref{hadronic-amplitude-I}) gives
\beqa
  {\cal A}_{\lambda\lambda \pm\pm}
  & \sim & \int_{-1}^1 du \left\{
           \left( \frac{1}{u-\xi+i\epsilon} - \frac{1}{u+\xi-i\epsilon} \right)
           \, G(u,\xi;\mu^2) [1 + O(\eta)] \right.
           \nonumber \\
  &      & \left. \mbox{} \pm \lambda \xi \frac{\eta}{2} \,
           \left( \frac{1}{(u-\xi+i\epsilon)^2}
                  - \frac{1}{(u+\xi-i\epsilon)^2} \right)
           \, \Delta G(u,\xi;\mu^2) \right\} \, ,
  \label{hadronic-amplitude-II}
\eeqa  
where
\beq
  \eta = \frac{\epsilon_B}{M} - 2\frac{\nabla^2\phi}{M^2\phi} \, .
\eeq

In order to estimate the magnitude of the asymmetry by using model
distributions, we shall employ the formalism of nonforward double
distributions $G(x,y)$ introduced by Radyushkin \cite{Radyushkin}.
Using
\beq
  G(u,\xi)
  = \int_0^1 dx \int_0^{1-x} dy \, G(x,y)
    \delta(u - [x + (x+2y-1)\xi])
\eeq
and a similar relation for the polarized distribution, we can 
express the beam-target polarization asymmetry as
\beq
  \frac{d\sigma(\uparrow\uparrow) - d\sigma(\uparrow\downarrow)}{
        d\sigma(\uparrow\uparrow) + d\sigma(\uparrow\downarrow)}
          \nonumber \\
  = \eta \; \frac{{\rm Re}\, (I_G^* I_{\Delta G})}{|I_G|^2}
%   = \eta \cdot \frac{{\rm Re}\, I_G \cdot {\rm Re}\, I_{\Delta G}
%                + {\rm Im}\, I_G \cdot {\rm Im}\, I_{\Delta G}}{
%                ({\rm Re}\, I_G)^2 + ({\rm Im}\, I_G)^2} \, ,
\eeq
where
\beqa
  {\rm Re}\, I_G
    & = & 2 \int_0^1 dx \int_0^{1-x} dy
          \ln |(x+2y)^2 - (\bar\omega x)^2|
          \frac{\partial G(x,y)}{\partial y} \, ,
          \label{Re-G} \\
  {\rm Re}\, I_{\Delta G}
    & = & -\int_0^1 dx \int_0^{1-x} dy   
          \ln \left| \frac{x+2y-\bar\omega x}{x+2y+\bar\omega x} \right|
          \frac{\partial^2 \Delta G(x,y)}{\partial y^2} \, ,
          \label{Re-DeltaG} \\
  {\rm Im}\, I_G 
    & = & -2\pi \int_0^{2/(1+\bar\omega)} dx
          \left. G(x,y) \right|_{y = (\bar\omega-1)x/2} \, ,
          \label{Im-G} \\
  {\rm Im}\, I_{\Delta G}
    & = & \pi \int_0^{2/(1+\bar\omega)} dx
          \left. \frac{\partial \Delta G(x,y)}{\partial y}
          \right|_{y = (\bar\omega-1)x/2}
          \label{Im-DeltaG}
\eeqa
(we used the notation $\bar\omega = 1/\xi$). Thus the asymmetry is
proportional to an unknown parameter $\eta$ and depends on integrals
of both $G$ and $\Delta G$. Derivatives of the distributions
appear because we integrated by parts.

The values of the parameters $\epsilon_B/M$ and $\nabla^2\phi/(M^2\phi)$
were estimated in \cite{KhanHoodbhoy}, where it was reported that the
choice $\epsilon_B/M = -0.076$ and $\nabla^2\phi/(M^2\phi) = -0.073$
gives agreement with data on charmonium decays and inelastic $J/\psi$
photoproduction. In accordance with this, we shall use $\eta=0.07$ below.

We now evaluate the integrals (\ref{Re-G}-\ref{Im-DeltaG}) for simple
model distributions and plot the resulting polarization asymmetry.
Following the discussion of Ref.\ \cite{Radyushkin-model}, we choose 
\beqa
  G(x,y)        & = & \frac{30}{(1-x)^5} \, [y(1-x-y)]^2 \, xg(x) \, ,
  \label{model-unpolarized} \\
  \Delta G(x,y) & = & \frac{30}{(1-x)^5} \, [y(1-x-y)]^2 \, x\Delta g(x) \, .
  \label{model-polarized}
\eeqa
We use the GRV-LO unpolarized gluon distribution $g(x)$ \cite{GRV-LO}
and the Gehr\-mann--Stirling A(LO) polarized distribution $\Delta g(x)$
\cite{Gehrmann}, evaluated at $Q^2 = 4 \, ({\rm GeV})^2$.
 Fig.~\ref{fig:ratio} shows the asymmetry
$\eta \, {\rm Re}(I_G^* I_{\Delta G})/|I_G|^2$ for $\eta=0.07$
as a function of photon energy in the proton rest frame. 
The asymmetry is of $O(10^{-2})$ in the photon energy range relevant
for fixed-target energies. At photon energies relevant for e.g.\ the
HERA collider experiments, the asymmetry drops by an order of magnitude. 

\begin{figure}
\input{fig2.tex}
\caption{\sf Polarization asymmetry obtained by using the model
distributions of eqs.\ 
(\protect\ref{model-unpolarized}--\protect\ref{model-polarized})
and setting $\eta=0.07$, plotted as a function of photon energy in
the proton rest frame.}
\label{fig:ratio}
\end{figure}

In summary, we have shown that a small but finite polarization asymmetry
arises from charm quark Fermi-motion and binding-energy corrections 
to polarized exclusive $J/\psi$ photoproduction. The asymmetry
depends on particular integrals of the polarized and unpolarized
nonforward parton distributions of the proton. Using a simple model
for parton distributions and previous estimates of $J/\psi$ parameters,
we estimate that the asymmetry may be of $O(10^{-2})$ at fixed-target
energies and $O(10^{-3})$ at collider energies.

\bigskip
 
{\bf Acknowledgement.} We wish to thank P.~Hoodbhoy and W.-D.~Nowak
for discussions.

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\bibitem{Radyushkin}
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\bibitem{MPW97}
L.~Mankiewicz, G.~Piller and T.~Weigl,
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\bibitem{Frankfurt} 
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\end{thebibliography}

\end{document}

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