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%---------------------------------------------------------------------- 
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\def\aem{\alpha_{\rm em}} 
\def\aems{\alpha^2_{\rm em}} 
\def\dd{\partial} 
\def\pT{p_{\rm T}} 
\def\cO{{\cal O}} 
\def\T{{\cal T}} 
\def\pom{{I\!\!P}} 
\def\alphapom{{\alpha_{\pom}}} 
\def\regg{{I\!\!R}} 
\def\alpharegg{{\alpha_{\regg}}} 
\def\n{\!\!} 
\def\percent{{\%\ }} 
%----------------------------------------------------------------------% 
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\def\lsim{\;\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}\;} 
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\newcommand{\tr}{{\rm tr}} 
\newcommand{\Tr}{{\rm Tr}} 
\def\pT{p_{\rm T}} 
\def\mev{{\rm\, MeV}} 
\def\gev{{\,\mbox{GeV}  }} 
\def\gevs{{\,\mbox{GeV}^2}} 
\def\gevms{{\,\mbox{GeV}^{-2}}} 
\def\to{\rightarrow} 
\def\n{\!\!} 
\def\qb{\bar q} 
\def\({\left(} 
\def\){\right)} 
\def \los{\ln\frac{s}{\mu^2}} 
\def \rvt{\vec{r_{\perp}}} 
\def\kvt{\vec{k_{\perp}}} 
\def\bvt{\vec{b_{\perp}}} 
\def\lvt{\vec{\ell_{\perp}}} 
\def\qvt{\vec{q_{\perp}}} 
\def\rt{r_{\perp}} 
\def\rhot{\rho_{\perp}} 
\def\bt{b_{\perp}} 
\def\kt{k_{\perp}} 
\def\rts{$r^2_{\perp}$} 
\def\bkt{{\mathbf k_t}} 
\def\bkts{{\mathbf k^2_t}} 
\def\aa{$z(1 - z)\, Q^2\,\,+\,\,m^2_Q$} 
 
%%%%%%%%%%%%%%5  M A T E X   %%%%%%%%%%%%%% 
%        written by orlando alvarez 
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%% This defines et al., i.e., e.g., cf., etc. 
\def\ie{\hbox{\it i.e.}}         
\def\etc{\hbox{\it etc.\ }} 
\def\eg{\hbox{\it e.g.}}         
\def\cf{\hbox{\it cf.\ }} 
\def\etal{\hbox{\it et al.}} 
\def\dash{\hbox{---}} 
 
%% COMMON PHYSICS SYMBOLS 
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\def\cok{\mathop{\rm cok}} 
\def\tr{\mathop{\rm tr}} 
\def\Tr{\mathop{\rm Tr}} 
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\def\Re#1{\mathop{\rm Re}\{#1\}} 
\def\bR{\mathop{\bf R}} 
\def\bC{\mathop{\bf C}} 
\def\lie{\hbox{\it \$}}             % fancy L for the Lie derivative 
\def\pder#1#2{{\partial #1\over\partial #2}} 
\def\secder#1#2#3{{\partial^2 #1\over\partial #2 \partial #3}} 
\def\bra#1{\left\langle #1\right|} 
\def\ket#1{\left| #1\right\rangle} 
\def\VEV#1{\left\langle #1\right\rangle} 
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% double-headed superior arrow added 9.2.86 
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    \raisebox{1.3ex}{\rlap{$\leftrightarrow$}} \raisebox{ 0ex}{$#1$}} 
\def\com#1#2{ 
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% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\def\beq{\begin{equation}} 
\def\eeq{\end{equation}} 
\def\bea{\begin{eqnarray}} 
\def\eea{\end{eqnarray}} 
\def\eq#1{{\mbox{Eq.\hspace{1mm}(\ref{#1})}}} 
\def\eqs#1#2{{\mbox{Eqs.\hspace{1mm}(\ref{#1})--(\ref{#2})}}} 
\def\fig#1{{\mbox{Fig.\hspace{1mm}\ref{#1}}}} 
\def\figs#1#2{{Figs.~\ref{#1}--\ref{#2}}} 
% 
\def\ds{\displaystyle} 
\def\scrbox#1{\mbox{\scriptsize #1}} 
\def\eqfill{\hspace*{\fill}} 
\def\hfill{\hspace*{\fill}} 
 
\newcommand{\grv}{xG^{\scrbox{GRV}}} 
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% ABBREVIATED JOURNAL NAMES 
% 
\def\ap#1#2#3{     {\it Ann.\ Phys.\ (NY) }{\bf #1} (#2) #3} 
\def\arnps#1#2#3{  {\it Ann.\ Rev.\ Nucl.\ Part.\ Sci.\ }{\bf #1} (#2) #3} 
\def\npb#1#2#3{    {\it Nucl.\ Phys.\ }{\bf B#1} (#2) #3} 
\def\npa#1#2#3{    {\it Nucl.\ Phys.\ }{\bf A#1} (#2) #3} 
\def\plb#1#2#3{    {\it Phys.\ Lett.\ }{\bf B#1} (#2) #3} 
\def\prd#1#2#3{    {\it Phys.\ Rev.\ }{\bf D#1} (#2) #3} 
\def\prep#1#2#3{   {\it Phys.\ Rep.\ }{\bf #1} (#2) #3} 
\def\prl#1#2#3{    {\it Phys.\ Rev.\ Lett.\ }{\bf #1} (#2) #3} 
\def\ptp#1#2#3{    {\it Prog.\ Theor.\ Phys.\ }{\bf #1} (#2) #3} 
\def\rmp#1#2#3{    {\it Rev.\ Mod.\ Phys.\ }{\bf #1} (#2) #3} 
\def\zpc#1#2#3{    {\it Z.\ Phys.\ }{\bf C#1} (#2) #3} 
\def\mpla#1#2#3{   {\it Mod.\ Phys.\ Lett.\ }{\bf A#1} (#2) #3} 
\def\nc#1#2#3{     {\it Nuovo Cim.\ }{\bf #1} (#2) #3} 
\def\yf#1#2#3{     {\it Yad.\ Fiz.\ }{\bf #1} (#2) #3} 
\def\sjnp#1#2#3{   {\it Sov.\ J.\ Nucl.\ Phys.\ }{\bf #1} (#2) #3} 
\def\jetp#1#2#3{   {\it Sov.\ Phys.\ }{JETP }{\bf #1} (#2) #3} 
\def\jetpl#1#2#3{  {\it JETP Lett.\ }{\bf #1} (#2) #3} 
\def\epj#1#2#3{    {\it Eur.\ Phys.\ J.\ }       {\bf #1} (#2) #3} 
\def\ppsjnp#1#2#3{{\it  Sov.\ J.\ Nucl.\ Phys.\ }{\bf #1} (#2) #3} 
\def\ppjetp#1#2#3{ {\it Sov.\ Phys.\ JETP }{\bf #1} (#2) #3} 
\def\ppjetpl#1#2#3{{\it JETP Lett.\ }{\bf #1} (#2) #3} 
\def\zetf#1#2#3{   {\it Zh.\ ETF }{\bf #1}(#2) #3} 
\def\cmp#1#2#3{    {\it Comm.\ Math.\ Phys.\ }{\bf #1} (#2) #3} 
\def\cpc#1#2#3{    {\it Comp.\ Phys.\ Commun.\ }{\bf #1} (#2) #3} 
\def\dis#1#2{      {\it Dissertation, }{\sf #1 } #2} 
\def\dip#1#2#3{    {\it Diplomarbeit, }{\sf #1 #2} #3 } 
\def\ib#1#2#3{     {\it ibid.\ }{\bf #1} (#2) #3} 
\def\jpg#1#2#3{    {\it J.\ Phys}.\ {\bf G#1}#2#3} 
% 
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\newcommand{\email}[1]{${\!}^{\scrbox{#1)}}$} 
\newcommand{\bm}[1]{\mbox{\boldmath{$#1$}}} 
\newcommand{\zf}[1]{\sum Z_{\scrbox{f}}^{#1}} 
\newcommand{\Qbar}{\overline{Q}} 
\newcommand{\Qbars}{\overline{Q}^{\:2}} 
\newcommand{\stot}{\sigma_{\scrbox{tot}}} 
\newcommand{\sighat}{\hat\sigma} 
\newcommand{\sigin}{\sigma_{\scrbox{input}}} 
\newcommand{\xgdglap}{xG^{\scrbox{DGLAP}}} 
\newcommand{\rsat}{r_{\perp,\scrbox{sat}}} 
\newcommand{\rta}[1]{r_{#1_\perp}} 
\newcommand{\qsat}{Q_{\scrbox{sat}}} 
\newcommand{\chisquare}{\chi^2/\mbox{n.d.f.}} 
\newcommand{\jpsi}{J/\psi} 
\newcommand{\mpsi}{M_{\psi}} 
\newcommand{\sd}{\sigma_{\mbox{\scriptsize dipole}}} 
\newcommand{\dsize}[1]{\mathbf{x}_{#1}} 
\newcommand{\sdsize}[1]{x^2_{#1}} 
\newcommand{\tn}{\tilde N} 
\def\df2dlnq2{\dd{F_2}/\dd\log{Q^2}} 
\def\dlnxg{\dd\log{xG}/\dd\log(1/x)} 
% 
 
\relax 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{document} 
%%% 
\begin{titlepage} 
\noindent 
\begin{flushright} 
\parbox[t]{14em}{
\begin{tabular}{ll}
TAUP &   2721/2003\\ 
DESY &   03-011\\
hep-ph/ & 0302010
\end{tabular}
%To appear in {\it XXX }{\bf X} 
} 
\end{flushright} 
\vspace{1cm} 
\begin{center} 
  {\Large \bf  
    ${\mathbf{J}/\mathbf{\psi}}$ Photo- and DIS Production via Nonlinear Evolution} 
  \\[4ex] 
\begin{center}\large{ 
        E.~Gotsman  $^{a}$ \email{1},  
        E.~Levin    $^{a,\,b}$ \email{2},  
        M.~Lublinsky $^{b}$  \email{3},  
        U.~Maor     $^{a}$ \email{4} and  
        E.~Naftali  $^{a}$ \email{5}} 
\end{center} 
 
\footnotetext{\email{1} gotsman@post.tau.ac.il } 
\footnotetext{\email{2} leving@post.tau.ac.il } 
\footnotetext{\email{3} lublinm@mail.desy.de } 
\footnotetext{\email{4} maor@post.tau.ac.il } 
\footnotetext{\email{5} erann@post.tau.ac.il } 
\vfill 
{\it  $^{a)}$ School of Physics and Astronomy}\\ 
{\it  Raymond and Beverly Sackler Faculty of Exact Science}\\ 
{\it  Tel Aviv University, Tel Aviv, 69978, ISRAEL}\\[4.5ex] 
\vfill 
 {\it $^{b)}$ DESY Theory Group}\\  
{\it 22603, Hamburg, GERMANY}\\[4.5ex]

\end{center} 
~\,\, 
\vspace{1cm} 
 
{\samepage {\large \bf Abstract:}}  
\vfill  
% 
 We extend our recent investigations of the  solution of the 
 nonlinear evolution equation to the case of $J/\psi$ photo- and DIS 
 production.  The uncertainty in the $\jpsi$ wavefunction normalization due 
 to Fermi motion is treated as a free parameter. We obtain good reproduction 
 of the experimental data.  Calculations of the production of $\jpsi$ mesons 
 on nuclei targets are presented and discussed. 
 
 
% 
\vfill 
 
\end{titlepage} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 \section{Introduction} \label{sec:introduction} 
% 
 It has been recognized long ago \cite{GLR} that the interactions of virtual
 photons at high-energies and low virtualities are governed by nonlinear
 evolution. Over the years, different theoretical methods of calculating
 processes in the kinematical region of high-density QCD (hdQCD) have been
 proposed \cite{MU86,MV,SAT,ELTHEORY,BA,KO}, leading to a construction of an
 equation, which incorporates both linear evolution due to parton splitting
 and nonlinear evolution due to recombination of partons at high-density.
 The equation is given in \eq{eq:BK} below, using the notations of \cite{KO},
 for $\tn(\rt,x; b)$, the imaginary part of the amplitude of a dipole of size
 $\rt$ which scatters elastically at impact parameter $b$.
 
In this paper we extend our recent investigations \cite{NLE01,NLE02} of 
 the nonlinear evolution to vector mesons, focusing on $\jpsi$ photo- 
 \cite{ZEUSpsiPhoto2002} and DIS \cite{ZEUSpsiDIS2002} production.  As this 
 process is characterized by a scale of the order of the charmed quark mass, 
 $m_c$, it is considered a primary candidate for investigating the 
 kinematical region of QCD on the boundary between perturbative and hdQCD. 
 In our previous publications \cite{GLMNpsi,GLMNpsi2} we demonstrated that 
 the experimental data of the cross section for $\jpsi$ production is well 
 reproduced if one introduces two damping factors, calculated using 
 a Glauber like shadowing-correction (SC) formalism \cite{SC}. 
 
Broadly speaking, each damping factor may be considered as one iteration in 
 an iterative procedure directed at obtaining $\tn(\rt,x; b)$.  These 
 iterations correspond to Glauber rescatterings where the first iteration is 
 attributed to SC due to the passage of a dipole through the target, and the 
 second iteration is attributed to one small $x$ gluon emitted by the dipole 
 prior to the interaction.  Hence, we believe that the SC formalism was, at 
 the time, an important preliminary step towards the more rigorous treatment 
 of \cite{NLE02}. 
 
A particular outcome of \cite{GLMNpsi2}, is the fact that the experimental 
 data for the forward differential cross section slope, $B$, can be well 
 reproduced by taking the profile function in the impact parameter space to 
 be a Fourier transform of the dipole electromagnetic form factor.  Our 
 numerical calculations of $B$ showed a good fit to the experimental data 
 with a hadron radius $R^2=10\,\gevms$.  On the other hand, in \cite{NLE02}, 
 a good fit to the $F_2$ data was obtained with $R^2=3.1\,\gevms$ for an 
 exponential form factor, and $R^2=4.5\,\gevms$, for a dipole form factor. 
 We consider the radii in \cite{NLE02} as effective radii which are reduced 
 due to ({\it i}) the presence of inelastic diffractive processes; and ({\it 
 ii}) the observation that in the simple expression typically used for $\sd$, 
 the dipole-proton cross section, a small anomalous dimension has been 
 assumed.  As we shall see, when calculating the cross section for $\jpsi$ 
 production, it is necessary to include an effective $b$-dependence, which we 
 choose to extract from the experimental data. 
 
This paper is organized as follows: in section \ref{sec:N} we briefly review 
 the nonlinear equation and its numerical approximate solution, as obtained in 
 \cite{NLE02}; in section \ref{sec:cross} we calculate the integrated cross 
 section for $\jpsi$ photo- and DIS- production, and compare to the 
 experimental data; and in section \ref{sec:nuc} we present our prediction 
 for  the production of $\jpsi$ from scattering off heavy nuclei. Our 
 summary and conclusions are given in section \ref{sec:summary}. 
 
 
\section{The Dipole Scattering Amplitude} \label{sec:N} 
 
The imaginary part of the dipole scattering amplitude, $\tn$, is a solution 
 of a nonlinear evolution equation, which characterizes the low $x$ behavior 
 of the parton densities, while taking into account hdQCD effects, thereby 
 obeying the unitarity constraints.  The equation describes the interaction 
 with a target of a parent dipole, of size $\dsize{01}$, and of two dipoles, 
 of sizes $\dsize{12}$ and $\dsize{02}$, which were produced by the dipole of 
 size $\dsize{01}$.  The probability for the decay of the dipole of size 
 $\dsize{01}$ is given by the square of its wavefunction, which, in a 
 simplified form, can be written as $\sdsize{01}/\sdsize{02}\sdsize{12}$. 

Each of the produced dipoles can interact with the target independently, with
 respective amplitudes of $\tn(\dsize{12},y;{\mathbf{b}-\half\dsize{02}})$
 and $\tn(\dsize{02},y;{\mathbf{b}-\half\dsize{12}})$, where $y$ is the
 rapidity variable and $b$ is the impact parameter.  However, adding these
 contributions clearly overestimates the dipole-nucleon interaction, since
 one must consider the probability that during the interaction, one dipole is
 in the shadow of the other. This negative correction factor is given by
 $-\tn(\dsize{12},y;\mathbf{b}-\half\dsize{02})\tilde
 N(\dsize{02},y;\mathbf{b}-\half\dsize{12})$.
 
Thus, $\tn$ can be written in the form  
% 
\begin{eqnarray} 
\lefteqn{\tn({\mathbf{x_{01}}},Y;b) = \tilde 
N({\mathbf{x_{01}}},Y_0;b)\, {\rm exp}\left[-\frac{2 \,C_F\,\as}{\pi} 
\,\ln\left( \frac{{\mathbf{x^2_{01}}}}{\rho^2}\right)(Y-Y_0)\right ]\, + 
}\nonumber \\ & & \frac{C_F\,\as}{\pi^2}\,\int_{Y_0}^Y dy \, {\rm 
exp}\left[-\frac{2 \,C_F\,\as}{\pi} \,\ln\left( 
\frac{{\mathbf{x^2_{01}}}}{\rho^2}\right)(Y-y)\right ]\,\times 
\label{eq:BK} 
\\  
& & \int_{\rho} d^2 {\mathbf{x_{2}}}  
\frac{{\mathbf{x^2_{01}}}}{{\mathbf{x^2_{02}}}\, 
{\mathbf{x^2_{12}}}} \nonumber 
\left(\,2\,\tn({\mathbf{x_{02}}},y;{ \mathbf{ b- 
\frac{1}{2} x_{12}}}) 
-\tn({\mathbf{x_{02}}},y;{ \mathbf{ b - 
\frac{1}{2} 
x_{12}}})\tn({\mathbf{x_{12}}},y;{ \mathbf{ b- \frac{1}{2} 
x_{02}}})\right)\,, 
\end{eqnarray} 
% 
 where, $Y=-\ln x$, $Y_0=-\ln x_0$ and $\rho$ is an ultraviolet cutoff, which
 does not appear in the physical quantities.   
 
The linear part of (\ref{eq:BK}) is the LO BFKL equation \cite{BFKL}, which 
 describes the evolution of the multiplicity of the fixed size color dipoles 
 with respect to the energy $Y$. The equation sums the high twist 
 contributions. Note, that the linear part of (\ref{eq:BK}) also has higher 
 twist contributions and the main contribution of the nonlinear 
 part is to the leading twist (see \cite{MU86} for general arguments and 
 \cite{HTM} for explicit calculations). 
 
For completeness we provide a brief description of the steps taken in 
 \cite{NLE02} for obtaining an approximate solution for $\tn({\mathbf{x_{01}}},Y;b)$. 
 
The initial conditions for (\ref{eq:BK}) were taken at $x_0=10^{-2}$ in the 
 eikonal approximation, accounting for multiple dipole-target interactions: 
% 
\begin{equation}\label{initialcondition} 
\tn(\dsize{01},x_0;b)\,=\,1\,-\,e^{-\frac{1}{2}\sigin(\dsize{01},x_0)\,S(b)}, 
\end{equation} 
% 
where 
% 
\begin{equation}\label{SBP} 
\sigin(\dsize{01},x_0) = 
\frac{\alpha_s\,\pi^2}{N_c}\sdsize{01}\xgdglap(x_0,4/\sdsize{01}), 
\end{equation} 
% 
and $S(b)$ is the profile function in impact parameter space.  As stated, 
we found that the experimental data of the differential cross section slope, 
$B$, is well described by the following profile function, which is the 
Fourier transform of the dipole electromagnetic form factor: 
% 
\begin{equation}\label{dipS} 
S(b)=\frac{2}{\pi  
R^2}\frac{\sqrt{8}b}{R}K_1(\frac{\sqrt{8}b}{R}). 
\end{equation} 
 
As a first step, the $b$-dependence of (\ref{eq:BK}) was neglected. Thus, all 
 twist contributions for the evolution were summed, using the initial 
 condition (\ref{initialcondition}) at $b=0$.  Then, once an approximate 
 solution was obtained, the $b$ dependence is restored, assuming 
 similar $b$-dependences for both the solution and the initial 
 conditions. Specifically, the following ansatz was used for the 
 $b$-dependence of $\tn$: 
% 
\begin{equation} 
\label{Nb}  
\tn(r_\perp,x; b)\,=\, 
(1\,-\,e^{-\kappa(x,r_\perp)\, S(b)/S(0)})\,,  
\end{equation} 
% 
where  
% 
\begin{equation} \label{kappa} 
\kappa(x,r_\perp)\,=\,-\,\ln(1\,-\,\tn(r_\perp,x,b=0)).  
\end{equation} 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Cross section for $\jpsi$ production} \label{sec:cross} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
The cross section for $\jpsi$ production is given by: 
% 
\begin{equation}\label{eq:sigtot} 
\sigma(\gamma^*p\longrightarrow Vp) = \int d^2b 
\left|\int dz\, d^2\rt \Psi_{\gamma^*}(\rt,z,Q^2)A(\rt,x;b)\Psi_V(\rt,z)\right|^2, 
\end{equation} 
% 
 where $A(\rt,x;b)$ is the imaginary part of the production amplitude in the
 impact parameter space, and $\Psi_V$ and $\Psi_{\gamma^*}$ are, respectively
 the wavefunctions of the $\jpsi$ and the virtual photon.  Formally speaking
 $\Psi_V$ should depend on $b$. However, as the dominant contribition of the
 $\jpsi$ wavefunction comes from short distances, this dependence is
 neglected.  The evaluation of (\ref{eq:sigtot}) is done by first performing
 the polarization summation of the $\jpsi$-photon overlap
 function,$\Psi_V\times\Psi_{\gamma^*}$.  This overlap function can be
 derived in the $\rt$ representation using the spin structures of the vector
 meson wavefunction \cite{BM} and the well-known photon wavefunction
 \cite{WF}.
 
A detailed analysis of the overlap function has already been made in 
 \cite{LMR}, where it was found that, in the momentum representaion, the 
 overlap function for transverse (T) and longitudinal (L) polarized photon is 
 given by \cite{LMR}: 
% 
\begin{equation}\label{overkt} 
\Psi_V(\kt,z)\,\times\,\Psi_{\gamma^*,T}(\kt,z)\propto 
\frac{2(z^2+(1-z)^2)\kt^2a^2+m_c^2(a^2-\kt^2)} 
{m_c\,(a^2+\kt^2)^3}\Psi_V(\kt,z)\,, 
\end{equation} 
\begin{equation}\label{overkl} 
\Psi_V(\kt,z)\,\times\,\Psi_{\gamma^*,L}(\kt,z)\propto 
\frac{2z(1-z)Q(a^2-\kt^2)}{(a^2+\kt^2)^3}\Psi_V(\kt,z)\,, 
\end{equation} 
% 
 with $a^2=z(1-z)Q^2+m_c^2$. Here $\times$ denotes the polarization and
 helicity summation.  For a given $\jpsi$ spatial distribution, the
 transformation to configuration space is straightforward and is given in
 \eqs{overlapT}{overlapL} below.  Although many models exists for the spatial
 distributions of vector mesons (see for example,
 \cite{BM,Fermi,Kulzinger,Nemchik,Ivanov,Caldwell}), we choose to approximate
 $\Psi_V(\rt,z)$ to be $\Psi_V(\rt=0,z=\half)$. As in our previous
 publications \cite{GLMNpsi,GLMNpsi2}, we consider below a deviation from
 this approximation due to relativistic effects, produced by the Fermi motion
 of the bound quarks within the vector meson.
 
The effect of this motion, however, strongly depends on the charmed quark 
 mass, $m_c$.  If one assumes, for example, that $m_c=M_{\psi}/2 \simeq 
 1.55\gev$, then, by definition, there is no correction due to Fermi motion. 
 On the other hand, in \cite{Fermi}, it has been assumed that $m_c \simeq 
 1.50\gev$, and a suppression factor of the cross section of about $0.25$ was 
 obtained, with almost no energy dependence.  Hence, as it stands, the 
 contribution of this effect lies within a substantial range of uncertainty 
 in which $m_c$ varies by no more than $0.05\gev$. 
 
We therefore consider the effect of Fermi motion as an uncertainty of the 
 wavefunction normalization, introducing it as an overall 
 (energy-independent) suppression factor, $K_F$, which we use as a fitting 
 parameter.  Using the above approximation we Fourier transform 
 on Eqs.\ (\ref{overkt}) and (\ref{overkl}) and obtain: 
% 
\begin{eqnarray} 
\lefteqn{ 
\Psi_V(\rt=0,z=\half)\,\times\,\Psi_{\gamma^*,T}(\rt;Q^2)=  
\frac{K_F}{48\aem}\sqrt{\frac{3\Gamma_{ee}M_{\psi}}{\pi}}\,\times 
} \nonumber\\ 
& &  
\hspace{3cm}\left\{ 
\frac{a^2}{m_c}\, 
\left(\,\zeta\,K_1(\zeta) - \frac{\zeta^2}{4}K_2(\zeta) \,\right) 
+m_c 
\left(\frac{\zeta^2}{2}\,K_2(\zeta) - \zeta\,K_1(\zeta) \,\right) 
\right\} 
\label{overlapT} 
\end{eqnarray} 
% 
% 
\begin{equation}\label{overlapL} 
\Psi_V(\rt=0,z=\half)\,\times\,\Psi_{\gamma^*,L}(\rt;Q^2)=  
\frac{K_F}{48\aem}\sqrt{\frac{3\Gamma_{ee}M_{\psi}}{\pi}} 
\frac{Q}{2}\left(\frac{\zeta^2}{2}\,K_2(\zeta) - \zeta\,K_1(\zeta) \,\right). 
\end{equation} 
% 
 where $\zeta=a\rt$, $K_i,i=1,2$ are the modified Bessel functions and
 $\Gamma_{ee}=5.26\,\mbox{KeV}$ is the leptonic width of the $\jpsi$.

Before presenting our results for the cross section (\ref{eq:sigtot}), we
 will discuss the impact parameter dependence of the production amplitude,
 $A$. At first sight, $A(\rt,x;b)$ equals $\tn(\rt,x;b)$ where its
 $b$-dependence is related to the solution of \eq{eq:BK} at $b=0$ through the
 ansatz of \eq{Nb}.  The parameter $R$ in $S(b)$ [see \eq{dipS}] is related
 to the hadron size and has already been determined in \cite{NLE02} by
 fitting $\tn$ to the experimental data of $F_2$.  However, as stated in the
 introduction, a good fit to $F_2$ was obtained using a relatively small
 value of $R^2=4.5\gevs$, whereas we know that the the measured $\jpsi$
 differential slope, $B$, is consistent with a radius which is more than
 twice larger.
 
Hence, it turns out that $A$ differs from $\tn$.  The correct $b$-dependence of 
 $A$ is given by the following expression: 
 
\begin{equation} \label{AB} 
A(\rt,x;b) = \int d^2b' \tn(\rt,x; b')\, S'(b-b')\, . 
\end{equation} 
 
To understand \eq{AB} we need to recall the procedure of calculating the 
 total cross section for the deep inelastic process which is shown in 
 \fig{txsec}.  If one uses the specific model in which the proton consists of 
 two colour dipoles (see Refs. \cite{DFK,BGLLM} for details), the total deep 
 inelastic cross section is given solely by the scattering amplitude of one 
 particular dipole present in the proton target.  More specifically, as there 
 is no momentum transfer which is involved in the process, the total cross 
 section does not depend on the probability of finding a second (spectator) 
 dipole in the target. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=txsec.eps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
Rescatterings for total DIS  cross section. 
}} 
\label{txsec} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
On the other hand, in the process of $J/\psi$ production, the momentum 
 transfer, $q$, is not zero, see \fig{psisec}.  In this process, for fixed 
 momentum transfer, the $t=-q^2$ dependence of 
 the $J/\psi$ production relates both to $\tn(\rt,x;q)$ (the amplitude for 
 the photon to scatter off one dipole) and to the probability to find the 
 second dipole having a momentum $q$ inside of the recoiled proton. 
 Denoting this probability by $S'(q)$, the production amplitude is thus 
 proportional to $\tn(\rt,x;q)S'(q)$.  A product in momentum representation 
 is equivalent to a convolution in the impact parameter, hence the form of 
 \eq{AB}. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=psisec.eps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
Rescatterings for $J/\psi$ cross section at fixed $t = -q^2$ . 
}} 
\label{psisec} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
Actually, we can estimate $S'(q)$ using the Born approximation for the 
 scatttering amplitude shown in the diagram of \fig{bapsi}. Assuming a  
 simple factorized form of the proton wave function, 
% 
\begin{equation} 
\Psi_{\scrbox{proton}}(\rt,\rhot) 
\,=\,\phi(\rt)\,\phi(\rhot) 
\end{equation} 
% 
where $\rhot = r_{1,\perp} - r_{2,\perp}$ and  
$\rt = r_{3,\perp} - (r_{1,\perp} + r_{2,\perp})/2$, 
we can easily calculate $S'(q)$ 
% 
\begin{equation} \label{BAXSEC} 
S'(q)\,\,=\,\,\int d^2 \rt \,\left|\phi(\rt)\right|^2 \, 
e^{i\frac{1}{3}\vec{q}\cdot\vec{\rt}} 
\end{equation} 
% 
Comparing \eq{BAXSEC} with the expression for the electromagnetic form 
factor for proton 
% 
\begin{equation} \label{FFP} 
F(q)\,\,=\,\,\int d^2 \rt\,\left|\phi(\rt)\right|^2  
\,e^{i\frac{2}{3}\,\vec{q}\cdot\vec{\rt}} 
\end{equation} 
% 
 one can see that $S'(b)$ can be described by \eq{dipS} with
 $R=R_{\scrbox{proton}}/2$, where $R_{\scrbox{proton}}$ is the
 electromagnetic radius of proton.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=bapsi.eps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
The amplitude  for $J/\psi$ cross section at fixed $t = -q^2$ in the Born 
approximation. 
}} 
\label{bapsi} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
 Alternatively, we can choose a different strategy and extract the typical
 size for the profile function $S'(b)$ using the experimental data.
 Generally, the differential slope is related to the average of the square of
 impact parameter, weighted by the the dipole-proton cross section.  The
 correct averaging procedure would be to integrate both over the impact
 parameter and over the dipole size, thereby reconstructing the energy
 dependence of $B$ (the shrinkage of the diffraction peak):
% 
\begin{equation}\label{eq:2} 
B(x) = \half\langle b^2 \rangle =  
\frac 
{  \int d^2\rt b^2d^2b\, \tn(\rt,x,b)} 
{2 \int d^2\rt d^2b   \, \tn(\rt,x,b)} . 
\end{equation} 
% 
 The value of $B$ as obtained from (\ref{eq:2}) is, of course, not universal 
 for all processes, and it directly depends on the choice of $S(b)$.  For the 
 purpose of calculating the effective radius to be used in $S'(b)$, we define 
 the deviation of $\half\langle b^2\rangle$ from the measured slope to be: 
% 
\begin{equation}  
B' = B_{\scrbox{exp}} - \half\langle b^2\rangle, 
\end{equation} 
% 
 where, $B_{\scrbox{exp}}$ is taken from experimental data (see, \eg, 
 \cite{ZEUSpsiEPS2001}). $S'(b)$ is then calculated 
 from \eq{dipS} with the substitution $R\longrightarrow R'=2B'$. 
% 
 
The resulting $b$-dependence of  
$A(x;b)\equiv\int d^2\rt A(\rt,x;b) \Psi_{\gamma}(\rt)\Psi_V$ are shown 
 in \fig{bdep}.  The convolution of $S'(b)$ with $\tn(\rt,x;b')$ is realized 
 by a shift in the maximum of the scattering amplitude in the impact 
 parameter space.  The decrease of the amplitude in the low $b$ region may be 
 understood as a signature of shadowing corrections to unitarity violating 
 effects. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=wfbdep.ps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
The impact parameter dependence of the amplitude for $J/\psi$ production. 
}} 
\label{bdep} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
 Our calculations of $\sigma(\gamma^*p\longrightarrow\jpsi)$ were basically
 in accordance with \eq{eq:sigtot}, with modifications due to the
 contribution from the real part of the production amplitude \cite{EDEN} and
 the skewed (off diagonal) gluon distribution \cite{OFFDIAGONAL}.
% 
The contribution of the real part is given by: 
\begin{equation} 
C_R^2 = (1\,+\rho^2)\,, 
\end{equation} 
% 
where, 
\begin{equation}\label{er:realpart1} 
\rho = ReA/ImA\,=\,tg(\frac{\pi \lambda}{2}) 
\end{equation} 
and, in our approach, 
% 
\begin{equation}\label{eq:realpart2} 
\lambda = \partial \ln (\tn)/\partial \ln (\frac{1}{x}). 
\end{equation} 
% 
 Note that with this definition of $\lambda$, its value obeys the unitarity 
 constraint according to which, at a fixed value of $Q^2$, $\lambda$ is a 
 decreasing function of the energy with $\lambda\rightarrow 0$ as 
 $x\rightarrow 0$ \cite{NLE02}. 
 
The off diagonal contribution is given by: 
% 
 \begin{equation}\label{eq:OD} C_G^2 = 
 \(\frac{2^{2\lambda+3}\,\Gamma(\lambda+\frac{5}{2})} 
 {\sqrt{\pi}\,\Gamma(\lambda+4)}\)^2. 
\end{equation} 
Formally, the definition (\ref{eq:realpart2}) of $\lambda$ is  
$\rt$-dependent.  In practice, the corrections due the $\rt$-dependence of 
$\lambda$ are rather small.  Thus, $\lambda$ 
is computed at fixed scale $\rt^2=4/(Q^2+M^2_{\jpsi})$. 
% 
 
 The results of our calculations of \eq{eq:sigtot}, given the above
 modifications, are shown in \fig{fig:photo} for $\jpsi$ photoproduction.
 For photoproduction we define $x=M_{\jpsi}^2/W^2$. We found that the H1 data
 \cite{H1psi2000} are consistent with $K_F=0.6$ ($\chisquare=0.5$) and the
 ZEUS data are consistent with $K_F=0.76$ ($\chisquare=0.6$).  As stated, the
 factor $K_F$ is the deviation of our approximation for the $\jpsi$ static
 potential wavefunction from a more realistic model, in which the spatial
 distribution of $\jpsi$ depends on $\rt$ and $z$.  Our results are
 consistent with \cite{Caldwell}, where a comparison between a realsitic and
 static $\jpsi$ wavefunctions has been made.  

The values of $K_F$ which were obtained for $Q^2=0$, were used to
 predict the cross section for $Q^2>0$.  In \fig{fig:DIS} we compare the
 $Q^2>0$ predictions to the preliminary data which were read off plots in
 \cite{ZEUSpsiDIS2002}.  Within the experimental errors, our reproduction of
 the data is good, except for data at $Q^2=16\gevs$ where we underestimate
 the H1 point ($W\simeq 80\gev$) and one of the ZEUS points ($W\simeq
 180\gev$).  Still, the overall $\chisquare$ for photoproduction and DIS data
 (including these two problematic points) is 0.73 for ZEUS and 0.84 for H1.
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=psiphoto.ps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
$J/\psi$ photoproduction ZEUS and H1 data together with our calculations
using NLE.  
}} 
\label{fig:photo} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=psidis.ps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
$J/\psi$ DIS and photoproduction, data and our calculations using NLE.  The 
upper (lower) curves at each $Q^2$ bin correspond to different values $K_F$, 
which were fitted to ZEUS (H1) photoproduction data. 
}} 
\label{fig:DIS} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\section{Predictions for Heavy Nuclei} \label{sec:nuc} 
 
In this section we wish to examine the possibility of observing unitarity 
 effects in processes with a nucleus target. In such processes, unitarity effects are 
 more pronounced than in $e-p$ collisions. Thus, with nuclear targets one 
 can access the region of hdQCD at values of $x$ which are larger than 
 those characterizing this region at HERA experiments. 
 
The production of vector mesons on a nuclear target can be calculated in a 
 straightforward manner using similar methods to the above.  It is recognized 
 that neither a dipole nor a Gaussian are adequate profile functions when 
 dealing with heavy nuclei interactions.  When the probing hadronic system 
 percolates through a heavy nucleus, it experiences multiple rescatterings 
 over a substantially long region of the impact parameter space. On the other 
 hand, once the system leaves the heavy nuclei, the number of strong 
 interactions drops rapidly. 
 
To calculate the cross section for $J/\psi$ production off nuclei we have 
solved (\ref{eq:BK}), separately for each atomic number $A$. We used 
 the same Glauber-Mueller form of initial conditions (\ref{initialcondition})  
% 
\begin{equation} \label{GT} 
N_A(\rt,x_0;b;A) \,=\, 1\, -\, e^{ - \frac{1}{2}\sigma_{dipole}(\rt,x_0) 
\,S_A(b)}, 
\end{equation} 
where,  
\begin{equation}\label{dipole} 
\sigma_{dipole}(\rt,x_0)\,=\,2\,\int d^2b\,\tilde N(\rt,x_0;b) 
\end{equation} 
and $S_A(b)$ is the number of nucleons in a nucleus interacting with the 
 incoming dipole.  
Our approach is to use the Wood-Saxon parameterization \cite{Enge} for the 
impact parameter dependence ($S_A(b)$ in \eq{GT}): 
% 
\begin{equation}\label{eq:WS1} 
S_A(b) = \rho\int\,\frac{d r_{||}}{1+e^{\frac{r-R_A}{h}}} 
\end{equation} 
% 
 where $r_{||}$ is the longitudinal distance from the target, $h$ and $R_A$ are 
 parameters which are taken from experimental tables \cite{NUCTAB}, 
 $r=\sqrt{r_{||}^2+b^2}$ and the normalization factor $\rho$ is defined in the 
 following relation: 
% 
\begin{equation}\label{eq:WS2} 
\rho\int\,\frac{dr_{||}\,d^2b}{1+e^{\frac{r-R_A}{h}}} = A\,. 
\end{equation} 
% 
 Note that $R_A$ is associated with the radius of a nucleus with atomic
 number $A$. The $b$-dependence of the function $N_A$ is restored using
 \eq{eq:WS1} for $S_A(b)$ in our $b$-dependence ansatz \eq{Nb}.
 
To check the effect of nuclear targets as compared to a nucleon target, we 
 need to examine the $A$ dependence of the integrated cross section. 
 \fig{fig:psinuc} shows the cross section as a function of $A$ with an 
 arbitrary normalization, for $x=10^{-5}, 10^{-4}$ and $10^{-3}$.  To see the 
 saturation effect, we compared the curves to a power behavior of 
 $\sigma\propto A^\delta$, recalling that the maximal value for $\delta$ is 
 $4/3$, for a Gaussian profile, and larger than $4/3$ (depending on $A$) for 
 the Wood-Saxon parameterization.  A deviation from the maximal value of 
 $\delta$ would be a clear signature of saturation of the growth of the cross 
 section.   
 
Fitting the calculated $A$-dependence of the cross section to $A^{\delta}$, 
 we obtained the results shown in Table \ref{tab:1}.  Unitarity conserving 
 effects are appreciable already for $x=10^{-3}$, where we have found an 
 $A$-dependence of $A^{0.9}$. 

Our prediction for the energy dependence for $\jpsi$ production on a Gold
 target ($A\simeq 200$) is shown in \fig{fig:psinucdis} for several values of
 $Q^2$.  Due to the uncertainity in the wavefunction normalization the
 relative errors of the curves are of order 15\%.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{table} 
\begin{center} 
\begin{tabular}{||l|l||} 
\hline \hline \multicolumn{1}{||c|}{$x$} & \multicolumn{1}{c||}{$\delta$}\\ 
\hline $10^{-5}$ & 0.73 \\ $10^{-4}$ & 0.78 \\ $10^{-3}$ & 0.88 \\ \hline 
\hline 
\end{tabular} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
Comparison of our NLE predictions for $J/\psi$ photoproduction to $A^{\delta}$. 
}} 
\label{tab:1} 
\end{table} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=psitotA.ps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
$J/\psi$ photoproduction on a nuclear target, our NLE predictions for 
different values of $x$ as a function of the atomic number, $A$.  
Also shown the curve $A^{\frac{4}{3}}$ for comparison. 
The curves are normalized to unity at $A=50$. 
}} 
\label{fig:psinuc} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure} 
\begin{center} 
\epsfig{file=psinucdis.ps,width=12cm,angle=0} 
\end{center} 
  \caption[]{\parbox[t]{0.80\textwidth}{\small 
$J/\psi$ photo- and DIS production on a Gold target, our NLE predictions for 
different values of $Q^2$ as a function of $W$.  
}} 
\label{fig:psinucdis} 
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
 \section{Summary} \label{sec:summary} 
 
We have extend our recent investigations of the approximate solution of the
 nonlinear evolution to the case of $\jpsi$ photo- and DIS production. We
 have convoluted our previous ansatz for the $b$-dependence of the amplitude
 with an additional profile which is extracted from the electromagnetic form
 factor.  The resulting impact parameter dependence of the amplitude exhibits
 a decrease near $b=0$. We believe that this decrease is due to the deviation
 from the linear evolution equations, which is a signature of the onset of
 unitarity taming effects.
 
We have used the solution to the NLE and obtained a good 
 reproduction of both the photo- and the DIS production data.  Our free 
 parameter was the uncertainty in the $\jpsi$ wavefunction normalization due 
 to Fermi motion of the $c\bar{c}$ pair.  In addition, we took into account 
 modifications of the cross section due to the real part of the amplitude and 
 the existence of skewed gluons within the nucleon. 
 
Our predictions for the production of $\jpsi$ meson on a nucleus target 
 demonstrates that such processes are important for the understanding of 
 unitarity effects.  We have found that these effects start to dominate for 
 $x\le10^{-3}$. Hence, the predictions presented may be verified with the 
 future eRHIC experiments. 

It is important to note that, apart from the uncertainities due to vector
 meson wavefunction, the calculations on nuclei do not contain any fitting
 parameters. More specifically the uncertainity due to $b$ dependence of the
 solution for nuclei is quite small.  In other words, the initial conditions
 for the BK equation \cite{BA,KO} are determined solely by the proton DIS
 data.
 
\section*{Acknowledgments} 
 
Two of us (E.G. and E.L.) thank the DESY Theory Division for their 
 hospitality.  This research was supported in part by the BSF grant $\#$ 
 9800276, by the GIF grant $\#$ I-620-22.14/1999 and by Israeli Science 
 Foundation, founded by the Israeli Academy of Science and Humanities. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
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\end{document} 


