%Title:  Testing J/psi Production and Decay Properties  in e+ e- Annihilation
%Authors:   V.M. Driesen, J.H. Kuehn and  E. Mirkes
%Published:  <EM>Phys. Rev.</EM><B> D49 </B><EM>(1994) 3197--3208.</EM>
%
%  TTP93-30  Oct. 93
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\begin{document}
%--------------------------------------
\newcommand{\slh}{\hspace{-.5em}/}
\newcommand{\Jpsi}{J\!/\!\psi}
\newcommand{\Tr}{\mbox{Tr}}
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\noindent
\hfill TTP93--$30$\\
\mbox{}
\hfill  Oct.  1993   \\   %                 \hfill \today \\
\vspace{0.5cm}
\begin{center}
  \begin{Large}
  \begin{bf}
Testing \boldmath{$\Jpsi$} Production and Decay Properties \\
in \boldmath{$e^{+}e^{-}$} Annihilation
\footnote{\normalsize Supported by
 BMFT Contract 055KA94P} \\
  \end{bf}
  \end{Large}
  \vspace{0.8cm}
  \begin{large}
 V.M. Driesen, J.H. K\"uhn and  E. Mirkes\footnote{
Address after Sept. 1, 1993:
Physics Dept., University of Wisconsin, Madison WI 53706, USA
} \\[5mm]
    Institut f\"ur Theoretische Teilchenphysik\\
    Universit\"at Karlsruhe\\
    Kaiserstr. 12,    Postfach 6980\\[2mm]
    76128 Karlsruhe, Germany\\
  \end{large}
  \vspace{2.5cm}
  {\bf Abstract}
\end{center}
\begin{quotation}
\begin{center}
\parbox{12.cm}{
Within the framework of pertubative QCD we calculate inclusive $\Jpsi$ 
production using the ``color singlet model'' (CSM). 
Predictions for the total cross section including the leading corrections 
from initial state radiation and the $\psi'$ contribution are given. 
We analyze the production and decay properties by considering
suitable lepton-hadron correlations.
It is shown that the production and the decay process
of an $\Jpsi$  are described by four structure functions, respectively.
We give analytical expressions 
for these
 structure functions
and  suggest energy dependent coefficients of angular 
distributions as a test for the model. Finally we give detailed predictions
for an energy of 10.5 GeV that could be tested at the CLEO-II experiment.
 The effect of arbitrarily polarized electron/positron beams is included.
}
\end{center}
\end{quotation}
\noindent
\newpage
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%------------------------------------------
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\section{Introduction}

The production of heavy quarks has become an important tool to test the 
validity of pertubative QCD predictions. Tagging of heavy quarks allows to 
separate quark and gluon jets and to examine a variety of characteristic
angular distributions. In the case of hadronic collisions heavy flavour 
productions serves as an important tool to extract parton --- in particular
gluon --- distributions. This holds true for the production of charm and bottom
mesons and similarly of the nonrelativistic bound states $\Jpsi$ or
$\Upsilon$. The latter have been observed abundantly in hadronic collisions.
In electron positron annihilation, however, inclusive production of $\Jpsi$ 
(not to speak of $\Upsilon$) has been limited to a few data points at a cm
energy of about 4.5 GeV \cite{PLUTO} 
and preliminary results at the $\Upsilon(4S)$
resonance and slightly below \cite{CLEO}. These results demonstrate that the 
predictions for continuum production of $\Jpsi$ are well compatible with
experiment. In particular no resonant contribution from the $\Upsilon(4S)$ 
decay is required, consistent with theoretical expectations. The large data
sample collected at CLEO II in the meantime will improve this situation 
considerably \cite{Honsch} in the near future.

In the energy region $\sqrt{s}=4-5$ GeV $\Jpsi$ is produced together with a 
few mesons $\pi\pi,\eta,\eta'$ in an $I=0$ and  $C=+$ configuration.
The rates of these exclusive reactions can be predicted 
\cite{jhk1,jhk2} individually on the basis of our
knowledge of $\psi' \rightarrow \Jpsi + X$ transitions. For higher energies,
relevant e.g. for the CLEO experiment, two QCD models have been proposed to 
predict the inclusive $\Jpsi$ cross section:

i) 
The color evaporation model (CEM) where a $c\overline{c}$ pair with invariant
mass in the charmonium region is produced in conjuction with a gluon and
hence in a color octet configuration \cite{FriKu}. Spin and color, 
appropriate for one of the charmonium states are arranged by soft gluon
emission with probability one.

ii)
The model adequate for a more rigorous treatment is the color singlet model 
(CSM): The $c\overline{c}$ is produced in the ${}^{3}S_{1}\,\,(J^{PC}=1^{--})$ configuration
in a color singlet state and hence in conjunction with two gluons
\cite{KuSch,KuSchT,Keung}.
\begin{equation}
e^+e^-\rightarrow \gamma^{\ast}
\rightarrow J/\Psi\,gg
\rightarrow l_{1}l_{2}\,gg
\label{lh1}
\end{equation}
This reaction would allow particularly clean studies of purely
gluon induced final states: Events with energetic $\Jpsi$ and correspondingly
low invariant mass $M_{gg}$ of the two gluon system might allow to search
for glueballs. Events with $\Jpsi$s of low momentum and hence large $M_{gg}$
will involve two gluon jets which could be compared to $q\overline{q}$
events at the equivalent cm energy. This could provide important clues on
potential difference between quark and gluon fragmentation.

In a first step the validity of the model has to be explored. Predictions
for the total cross section are rather sensitive to the input assumptions, 
that is the wave function at the origin, the choice of $\alpha_{s}$ and the
estimate of (uncalculated) higher order corrections, and we estimate these
uncertainties to be about 50\%. Predictions for the energy distribution, the 
angular distribution and the $\Jpsi$ polarization are far less sensitive
to these uncertainties and will, therefore, be described in some detail.
We will show that an analysis of the angular distribution of the
intitial $(e^+e^-)$ and final $(l_1l_2)$ state lepton  pairs with respect to the
($\Jpsi gg$) system allows to test the underlying dynamics of the production and
decay models of (polarized) $\Jpsi$'s in a much more detailed way than by rate measurements
alone. Technically the physics of these lepton-hadron correlations
is
described by contraction of the lepton tensor $L_{\mu\nu}$ with the hadron tensor
$H^{\mu\nu}$. The contraction may be written in a rather suggestive
manner depending on whether the lepton pair is in the initial or final state.
For our process one would write
\begin{equation}
L_{\mu\nu}^{e^+e^-}\,\,
H^{\mu\nu;\,\,\delta\delta'}\,\,
L_{\delta\delta'}^{l_1l_2}
\end{equation}
where $L_{\mu\nu}^{e^+e^-}$ acts as a {\it polarizer} of the
$\gamma^{\ast}$ (and therefore for the ($\Jpsi\,gg$)-system)),
whereas $L_{\delta\delta'}^{l_1l_2}$
acts as an {\it analyzer}
of the polarization of the $\Jpsi$.
We will 
present compact analytical results for the relevant components of
the hadron tensor and derive formulae which can easily be 
compared to fourthcoming experimental results. 

This paper expands on results that can be found in the literature already. The
total cross section as well as the $\Jpsi$ energy distribution has been
derived in \cite{KuSch,KuSchT,Keung}, angular distributions can be found in 
\cite{KuSchT}. Here we shall present a more comprehensive treatment which 
includes a description of the angular distribution of the production plane
and a complete description of the angular distribution of the leptons 
in the decay.
We take into account the contribution from $\psi'$ and incorporate the
effect of initial state radiation. Finally we give angular distributions for 
polarized electron and positron beams.

Our paper is organized as
follows: The formalism for the calculation of bound states will
be introduced in section 2 and applied to the reaction under study. 
Angular distributions for $\Jpsi$ production are presented, including
the predictions with polarized beams.
Section 3
is concerned with the information that can be derived from the analysis of 
$\Jpsi$ polarization through its decay to lepton pairs. In section 4 we
shall present numerical results and the comparison with the scarce data.
Section 5 contains our conclusions. 



\section{Structure Functions in $\Jpsi$ Production}

The differential production  cross section 
for the production process (\ref{lh1})
is given by
\begin{equation}
  {\rm d}\sigma= \frac{1}{2s}\; \frac{1}{2}\; \frac{1}{4}
  \sum_{pol} |M|^{2} \; {\rm d}P\!S^{(3)}
 \label{WinkelWQ}
\end{equation}
where $s=Q^{2}$ is the center of mass energy.
%
\begin{figure}[htb]
 \begin{center} \mbox{
% auf der DEC-maschine muss es leider in einer Zeile stehen!
 \epsfig{file=feyn.eps,width=7.cm,height=4.cm,bbllx=2.cm,bblly=3.cm,bburx=18.cm,bbury=12.cm} }   
 \end{center}
 \caption{ Notation: ``particle, momentum''}
\end{figure}
%
The amplitude which describes the effective coupling of the virtual photon to 
the $\Jpsi\,gg$-system can be calculated within the bound state formalism of
\cite{KuKaSa} to be
\begin{equation}
  A^{\gamma^{*}gg}= \frac{1}{2} \frac{1}{\sqrt{4\pi M_{\psi}}} \Phi(0)
   \Tr \left[ {\cal O}^{0} (P\slh-M_{\psi}) (-E\slh) \right]
\label{ampli}
\end{equation}
where $\Phi(0)$ denotes the radial wave function of the bound state, which 
can be calculated either from potential models or related to the leptonic
decay rate
\begin{equation}
|\Phi(0)|^{2} = \frac{M_{\psi}^{2}}{4\alpha^{2}(2/3)^{2}} \Gamma_{ee}
\end{equation}
$P$, $M_{\psi}$ and $E$ denote the $\Jpsi$ momentum, mass and polarization.
The amplitude ${\cal O}^{0}$ can be obtained from the amplitude
for the production of a free quark pair at threshold and two gluons:
\begin{eqnarray}
 {\cal O}^{0} & = & \frac{-1}{4}  \left[
 \frac{{\varepsilon\slh}^{*}_{2}(P\slh+2{K\slh}_{2}+M_{\psi}) {\varepsilon\slh}
 (-P\slh-2{K\slh}_{1}+M_{\psi}) {\varepsilon\slh}^{*}_{1}}{PK_{2} \;\; PK_{1}}
\nonumber \right.\\ & & \;\;\mbox{}
+\frac{{\varepsilon\slh}(-P\slh-2{K\slh}_{1}-2{K\slh}_{2}+M_{\psi}) 
 {\varepsilon\slh}^{*}_{1} (-P\slh-2{K\slh}_{2}+M_{\psi})
 {\varepsilon\slh}^{*}_{2}} {(PK_{1}+PK_{2}+2K_{1}K_{2}) \;\; PK_{2}}
\nonumber \\ & & \;\;\left.
 {}+\frac{{\varepsilon\slh}^{*}_{2}(P\slh+2{K\slh}_{2}+M_{\psi}) 
 {\varepsilon\slh}^{*}_{1} (P\slh+2{K\slh}_{1}+2{K\slh}_{2}+M_{\psi}) 
 {\varepsilon\slh}} {PK_{1} \;\; (PK_{1}+PK_{2}+2K_{1}K_{2})}
\right] 
  \nonumber \\ & &\;\; {}+[1\leftrightarrow 2]
\end{eqnarray}
Here $K_{1},K_{2},\varepsilon_{1},\varepsilon_{2}$ are the momenta and 
polarization vectors of the two gluons and $\varepsilon$ is the polarization 
vector of the virtual photon.
Coupling constants and color matrices contribute a factor
$ (4\pi\,\alpha_{s}\,e\,2/3)^{2}(2/3) $ to the rate.
%
The differential cross section can be expressed in terms of 
the lepton tensor
\begin{equation}
L_{\mu\nu}^{e^+e^-}=4({p_{1}}_{\mu}{p_{2}}_{\nu}
                  + {p_{1}}_{\nu}{p_{2}}_{\mu}
                  - g_{\mu\nu}p_{1}p_{2})
\end{equation}
and the hadron tensor
\begin{eqnarray}
 H^{\mu\nu}& = & 
   \left(-g_{\delta \delta'} + \frac{P_{\delta}P_{\delta'}}{M_{\psi}^{2}} 
   \right) \cdot 
   \Tr \left[ {\cal O}^{\alpha\mu\beta} (P\slh-M_{\psi})
              \gamma^{\delta} \right]  \nonumber \\
&& \mbox{}\hspace{3.4cm}
   \Tr \left[ {\cal O}^{\;\;\nu}_{\alpha\;\;\beta} (P\slh-M_{\psi}) 
              \gamma^{\delta'} \right]^{*}  
 \label{HJpsi}
\end{eqnarray}
as follows
\begin{equation}
 {\rm d}\sigma  =  \sigma_{\mu^{+}\mu^{-}} 
   \left( \frac{1}{\pi} \right)
   \left( \frac{\alpha_{s}}{\alpha} \right)^{2} 
   \frac{\Gamma_{ee}}{64} \frac{M_{\psi}}{16s} 
 \cdot L_{\mu\nu}^{e^+e^-}H^{\mu\nu}  \cdot 
   {\rm d}x_{1}\:{\rm d}x_{2}  \, \frac{{\rm d}\cos\theta}{2}\,
   \frac{{\rm d}\phi}{2\pi} \,  \frac{{\rm d}\alpha}{2\pi} 
\end{equation}
The scaled energies of the two gluons and the $\Jpsi$ are denoted by
$x_{1},x_{2}$ and $y$, respectively:
\begin{equation}
x_{1}=2E_{1}/\sqrt{s} \hspace{.7cm} x_{2}=2E_{2}/\sqrt{s}
\hspace{.7cm} y=2E_{\psi}/\sqrt{s} \hspace{.7cm} r=M_{\psi}^{2}/s
\end{equation}
The three Euler angles $\theta$, $\phi$ and $\alpha$ fix
the relative orientation of the $(\Jpsi\,gg)$ and the laboratory systems
and will be specified below. All cross sections are given in terms of
$\sigma_{\mu^{+}\mu^{-}}=4\pi\alpha^{2}/(3s)$.


The general structure of the hadronic tensor $H^{\mu\nu}$
can be readily exhibited by 
writing down the  general covariant expansion\footnote{
The most general covariant expansion is obtained by adding four additional
terms:
$H_{6}\;\epsilon(\mu,\nu,q,K_1),
H_{7}\;\epsilon(\mu,\nu,q,K_2),
H_{8}(K_1^{\mu}\epsilon(\nu,K_1,K_2,q) + (\mu\leftrightarrow\nu))$ and
$H_{9}(K_2^{\mu}\epsilon(\nu,K_1,K_2,q) + (\mu\leftrightarrow\nu))$.
However, parity conservation requires $H_6-H_9$=0.
}.
\begin{eqnarray}
H^{\mu\nu}&=& H_1 \left(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^2}\right)
            + H_2 \hat{K}_1^{\mu}\hat{K}_1^{\nu}
            + H_3 \hat{K}_2^{\mu}\hat{K}_2^{\nu}\nonumber\\
       & + & H_4 \left( \hat{K}_1^{\mu}\hat{K}_2^{\nu}
                        +\hat{K}_1^{\nu}\hat{K}_2^{\mu}\right)
            + H_5 \left( \hat{K}_1^{\mu}\hat{K}_2^{\nu}
                        -\hat{K}_1^{\nu}\hat{K}_2^{\mu}\right)\label{hidef}
\end{eqnarray}              
where we have introduced the four momenta
$\hat{K}_i^{\mu}=K_i^{\mu}-\frac{(K_iq)}{q^2}q^{\mu}$.
From the hermiti\-city of the hadron tensor
$H^{\mu\nu}=H^{\nu\mu \,\,\ast}$ we conclude that $H_1 - H_4$ are real
and $H_5$ is imaginary.
Therefore, in Born approximation, $H_5$=0.
An equivalent representation of the hadron tensor is obtained
in the helicity basis
\begin{equation}
h_{m
m^{\prime}}= \epsilon_{\mu}^{\ast}(m)H^{\mu\nu}\epsilon_{\nu}(m^{\prime})
\label{hmdef}
\end{equation}
($m,m^{\prime}=+,0,-$)
where
  \begin{equation}
    \begin{array}{ll}
\epsilon_{\mu}(\pm)&=(0;\pm1,-i,0)/\sqrt{2}\\[3mm]
\epsilon_{\mu}(0)  &=(0;0,0,1)
    \end{array}
\hspace{5mm}
\label{polvekdef}
  \end{equation}
 are the polarization vectors
for the $\gamma^{\ast}$ defined with respect to the  coordinate frame 
specified below.
The matrix $h_{m m^{\prime}}$
represents the {\it polarization density
matrix} of the vitual photon.
Its elements can be analyzed by considering angular distributions of
the initial state lepton pair with respect to the $(\Jpsi gg)$ system.
The general angular distributions in terms of two Euler angles $\theta$ and
$\phi$ is given by:
\begin{eqnarray}
\frac{{\rm d}\sigma}{{\rm d}x_1 {\rm d}x_2 {\rm d}\cos\theta {\rm d}\phi}
 & = &
    \frac{3}{16\pi}  
 \Big[ (1+\cos^{2}\theta) \sigma_{U+L}
          \; +\; (1-3\cos^{2}\theta) \sigma_{L}  \label{winkel} \\
&  & \mbox{}\hspace{4.mm}
 +\; (2\sin^{2}\theta\:\cos (2\phi))\sigma_{T}
 \;+\; (2\sqrt{2}\sin (2\theta)\:\cos\phi)\sigma_{I} \Big] \nonumber
\end{eqnarray}
%
Here $\theta$ and $\phi$ are the polar and azimuthal angles of the 
incoming electron in the $\gamma^{\ast}$ rest frame and will be specified 
below. The third Euler angle has been integrated out.
The angular coefficients $\sigma_X$ are linearly related to the 
density matrix elements in 
(\ref{hmdef}) (or alternatively to the structure functions
$H_i$ in (\ref{hidef})):
$$
\sigma_X = C\frac{M_{\psi}^{2}}{2048}
\sigma_{\mu^{+}\mu^{-}}
 H_X
\hspace{2cm}
X\in\{U+L,L,T,I\}
$$
where
\begin{eqnarray} 
 H_{U+L} & = & h_{00}+h_{++}+h_{--}\hspace{2.cm}  
   =\;  H^{11}+H^{22}+H^{33} \label{hul}\\   
 H_{L} & = & h_{00} \hspace{4.45cm} = \;H^{33}\\
 H_{T} & = & h_{+-}+h_{-+} \hspace{3.05cm} =\; \frac{1}{2}(H^{22}-H^{11}) \\
 H_{I} & = &  \frac{1}{4}(h_{+0} +h_{0+} -h_{-0}-h_{0-})
              \hspace{.3cm} =\;-\frac{1}{2\sqrt{2}}(H^{13}+H^{31})\label{hi}
\end{eqnarray}
and
\begin{equation}
 C = \frac{8}{3\pi} \left(\frac{\alpha_{s}}{\alpha}\right)^{2} 
     \frac{\Gamma_{ee}}{M_{\psi}}
   = \frac{128\alpha_{s}^{2}}{27\pi} \frac{|\phi(0)|^{2}}{M_{\psi}^{3}}
\label{const}
\end{equation}
%
The structure functions $\sigma_X$ depend on 
$x_1,x_2$ or alternatively on $x_1,y$.

At this point the relative orientation of the electron direction in the
hadron frame has to be specified (Fig. \ref{diag}). The $z$ axis is defined by
the direction of the $\Jpsi$ momentum. The orientation
of the $x$ axis is given by the most energetic gluon.
Thus the momenta of the outgoing hadrons span the $x-z$ plane.
%
\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{file=diag.eps,height=10.cm,width=12.cm,bbllx=0.cm,bblly=0.cm,bburx=20.cm,bbury=15.cm}}
\end{center}
\caption{Definition of the angles $\theta$ and $\phi$\label{diag}}
\end{figure}
%

Analytical expressions for the structure functions
 $\sigma_X$ can be found in appendix \ref{app1}.
Note that
the cross section after integration over the angles is given by
\begin{equation}
\frac{{\rm d}\sigma}{{\rm d}x_1 {\rm d}x_2} =  \sigma_{U+L}
\label{diffwq}
\end{equation}
in agreement\footnote{after corrections ot two misprints:
$6(1-\mu^{2}-y)$ has to be replaced by $6(1+\mu^{2}-y)$ and $(1-\mu^{2})$
by $-2(1-\mu^{2})$.}
with eqs. (8) and (9) in \cite{KuSchT} and (3) in \cite{Keung}.

Integrating the angular distribution in eq. (\ref{winkel})
over the azimuthal angle $\phi$ we obtain the angular distribution of the 
$\Jpsi$
\begin{eqnarray}
\frac{{\rm d}\sigma}{{\rm d}x_1 {\rm d}x_2 {\rm d}\cos\theta }
 & = & 
    \frac{3}{8}  
 \Big[ (1+\cos^{2}\theta) \sigma_{U+L}
          \; +\; (1-3\cos^{2}\theta) \sigma_{L} 
  \Big] \label{winkeltheta} \\
& = &     \frac{3}{8}  (\sigma_{U+L}+\sigma_L)
 \Big[ 1+\alpha \cos^{2}\theta  \Big]  \label{dx1x2al}
\end{eqnarray}
where
\begin{equation}
\label{defalpha}
\alpha=\frac{1-3 \sigma_L/\sigma_{U+L}}{1+\sigma_L/\sigma_{U+L}}
\end{equation}
The corresponding coefficient for the distribution of longitudinally
polarized $\Jpsi$ is denoted by $\alpha^{L}$ and will be calculated below.

Integrating over $\theta$ gives
\begin{equation}
\frac{{\rm d}\sigma}{{\rm d}x_1 {\rm d}x_2 {\rm d}\phi }
 =    \frac{1}{2\pi}  
       \sigma_{U+L}
 \Big[ 1+\beta \cos 2\phi  \Big] 
\label{winkelphi}
\end{equation}
where
\begin{equation}
\beta=\frac{\sigma_T}{\sigma_{U+L}}
\end{equation}

The structure functions are also suited to fully predict the angular
distribution for polarized beams. The cross section, which is still
differential in $\alpha$, is obtained through the replacements
\begin{eqnarray}
(1+\cos^{2}\theta) & \Rightarrow & \frac{1}{2\pi}\left[ 
  (1-s_{z'}^{1}s_{z'}^{2})(1+\cos^{2}\theta) +\sin^{2}\theta 
  (\sin (2\alpha)\; s_{+} + \cos (2 \alpha)\; s_{-})
  \right]  \nonumber \\
(1-3\cos^{2}\theta) & \Rightarrow & \frac{1}{2\pi}\left[ 
  (1-s_{z'}^{1}s_{z'}^{2})(1-3\cos^{2}\theta) -3\sin^{2}\theta 
  (\sin (2\alpha)\; s_{+} + \cos (2 \alpha)\; s_{-})
  \right] \nonumber 
\end{eqnarray}
\begin{eqnarray}
(2\sin^{2}\theta\,\cos(2\phi)) & \Rightarrow & \frac{2}{2\pi} \bigg[
  (1-s_{z'}^{1}s_{z'}^{2})\cos 2\phi\sin^{2}\theta
  \nonumber \\
&&  \mbox{}\hspace{4.mm}+2\cos\theta \;\sin 2\phi 
  (\cos (2 \alpha)\; s_{+} - \sin (2 \alpha)\; s_{-})  \nonumber \\
&&  \mbox{}\hspace{4.mm}
  + (1+\cos^{2}\theta)\cos 2\phi 
  (\sin (2 \alpha)\; s_{+} +\cos (2 \alpha)\; s_{-} ) \bigg] \nonumber \\
(2\sqrt{2}\cos\phi\sin(2\theta)) & \Rightarrow & -\frac{2\sqrt{2}}{2\pi} \bigg[
  (1-s_{z'}^{1}s_{z'}^{2})\sin 2\theta \;\cos\phi
   \nonumber \\
&&  \mbox{}\hspace{12.mm}
  - 2\sin\theta\;\sin \phi 
  (\cos (2 \alpha)\; s_{+} -\sin (2 \alpha)\; s_{-} ) \nonumber \\
&&   \mbox{}\hspace{12.mm}-\sin 2\theta \;\cos\phi 
  (\sin (2 \alpha)\; s_{+} + \cos (2 \alpha)\; s_{-})   \bigg] \nonumber
\end{eqnarray}
with
\begin{eqnarray*}
s_{-} & = & s_{x'}^{1}\,s_{x'}^{2} - s_{y'}^{1}\,s_{y'}^{2}  \\
s_{+} & = & s_{x'}^{1}\,s_{y'}^{2} + s_{y'}^{1}\,s_{x'}^{2}  
\end{eqnarray*}
No new structure functions arise in this case. The spin vectors $\vec{s}_1$
and $\vec{s}_2$ are defined in a coordinate system where the $z'$ axis is
aligned with the electron direction, $\alpha$ denotes the angle between
the plane spanned by $z'$ and $\vec{P}_{\psi}$ and the $x',z'$ plane 
(Fig. \ref{dwal}).
For the special case of natural transverse polarization $\vec{s}$ is 
conveniently aligned in the $x'$ direction and hence $s_{z'}=s_{+}=0$ and
$s_{-}=-1$.
%
\begin{figure}[htb]
\begin{center}
\mbox{ \epsfig{file=dwal.eps,width=10.cm,height=7.cm,bbllx=1.cm,bblly=0.cm,bburx=19.cm,bbury=11.cm} }
\caption{Definition of the angle $\alpha$ \label{dwal}}
\end{center}
\end{figure}



\section{Structure Functions in $\Jpsi$ Decay}

In the previous section we have analyzed the production mechanism
of an {\it unpolarized} $\Jpsi$ by considering angular correlations
between the initial state lepton pair and the $(\Jpsi gg)$-system.
However the angular distribution in eq. (\ref{winkel})
remains valid also for the production of a
{\it polarized} $\Jpsi$:
\begin{eqnarray}
\frac{{\rm d}\sigma^\lambda}{{\rm d}x_1{\rm d}x_2{\rm d}\cos\theta{\rm d}\phi}
 & = &
    \frac{3}{16\pi}  
 \Big[ (1+\cos^{2}\theta) \sigma^\lambda_{U+L}
          \; +\; (1-3\cos^{2}\theta) \sigma^\lambda_{L}  \label{winkelneu} \\
&  & \mbox{}\hspace{4.mm}
 +\; (2\sin^{2}\theta\:\cos (2\phi))\sigma^\lambda_{T}
 \;+\; (2\sqrt{2}\sin (2\theta)\:\cos\phi)\sigma^\lambda_{I} \Big] \nonumber
\end{eqnarray}
Here the superscript $\lambda\in\{U+L,L,T,I\}$ 
refers to the  polarization of the $\Jpsi$.
The structure functions
$\sigma_X^\lambda$ \mbox{}($X\in\{U+L,L,T,I\}$) can be calculated 
by suitable projections
$\cal{P}^{\lambda}_{\delta\delta'}$ instead of
${\cal P}_{\delta\delta'}={\cal P}^{U+L}_{\delta\delta'}=
-g_{\delta\delta'}+P_{\delta}P_{\delta'}/M_{\Psi}^2$ in eq. (\ref{HJpsi}).
The polarization of the $\Jpsi$ determines the angular distribution of
the leptons in the subseqent decay of the $\Jpsi$ for each of the
production cross sections
$\sigma_{U+L}, \sigma_{L}, \sigma_{T}$ and $ \sigma_{I}$.
In analogy to eqs. (\ref{hmdef})-(\ref{hi})
the decay angular distribution can be disentangled by introducing
suitable combinations of {\it density matrix elements of the $\Jpsi$}.
The general form of the decay lepton distribution for each production cross
section  $\sigma_X$ in (\ref{winkel}) is then given by:
\begin{eqnarray}
\frac{{\rm d}\sigma_X}{{\rm d}x_1 {\rm d}x_2 {\rm d}\cos\vartheta {\rm d}\chi}
 & = &
    \frac{3}{16\pi}  
 \Big[ (1+\cos^{2}\vartheta) \sigma_{X}^{U+L}
          \; +\; (1-3\cos^{2}\vartheta) \sigma_{X}^L  \label{winkelpol} \\
&  & \mbox{}\hspace{4.mm}
 +\; (2\sin^{2}\vartheta\:\cos (2\chi))\sigma_{X}^T
 \;+\; (2\sqrt{2}\sin (2\vartheta)\:\cos\chi)\sigma_{X}^I \Big] \nonumber
\end{eqnarray}
Here $\sigma_X^{U+L}$ are the production cross sections for an unpolarized
$\Jpsi$ as given in eq. (\ref{winkel}), i.e. $\sigma_X^{U+L}\equiv\sigma_X$.
Correspondingly
$\sigma_X^L$ are the cross sections for a longitudinally
polarized $\Jpsi$, where the projection
$(-g_{\delta\delta'}+P_{\delta}P_{\delta'}/M_{\Psi}^2)$ in eq.
(\ref{HJpsi}) is replaced by
${\cal{P}}_{\delta\delta'}^L\equiv E_{\delta}^L E_{\delta'}^L$
with
\begin{equation}
E_{\delta}^L=\frac{1}{M_{\Psi}}(|P|,0,0,E_{\Psi})=
\frac{1}{M_{\Psi}}\frac{E_{\Psi}}{|P|}P_\delta-
\frac{M_{\Psi}}{|P|}\frac{1}{\sqrt{s}}q_{\delta}
\end{equation}
the longitudinal polarization vector of the $\Jpsi$ with respect to
its direction of flight in the $\gamma^{\ast}$ rest frame.
Similary projections with
\begin{equation}
{\cal{P}}_{\delta,\delta'}^T\equiv
 E_{\delta}^+ E_{\delta'}^-
+ E_{\delta}^- E_{\delta'}^+
\end{equation}
and
\begin{equation}
{\cal{P}}_{\delta,\delta'}^I\equiv
\frac{1}{4}\left(
 E_{\delta}^+ E_{\delta'}^L
+ E_{\delta}^L E_{\delta'}^+
- E_{\delta}^- E_{\delta'}^L
- E_{\delta}^L E_{\delta'}^-
\right)
\end{equation}
lead to the remaining polarization cross sections in (\ref{winkelpol}).
In this paper we restrict ourselves to the calculation of $\sigma_{X}^{L}$
and $\sigma_{X}^{U+L}$ which are sufficient to describe the 
distribution in the polar angle $\vartheta$.

The angles $\vartheta$ and $\chi$ in (\ref{winkelpol}) are
the polar and azimuthal angle of the decay leptons in the $\Jpsi$ rest frame.
The $z$ axis is defined by the direction of the $\Jpsi$ in the
$\gamma^{\ast}$ rest frame.
Integrating the angular distribution in eq. (\ref{winkelpol})
over the azimuthal angle $\chi$ we get
\begin{eqnarray}
\frac{{\rm d}\sigma_{X}}{ {\rm d}x_1{\rm d}x_2{\rm d}\cos\vartheta }
 & = & 
    \frac{3}{8}  
 \Big[ (1+\cos^{2}\vartheta) \sigma_{X}^{U+L}
          \; +\; (1-3\cos^{2}\vartheta) \sigma_{X}^L 
  \Big] \label{winkelthetapol} \\
& = & \mbox{}
    \frac{3}{8}  (\sigma_X^{U+L}+\sigma_X^L)
 \Big[ 1+A_X \cos^{2}\vartheta  \Big] 
\end{eqnarray}
where
\begin{equation}
A_X=\frac{1-3 \sigma_X^L/\sigma_X^{U+L}}{1+\sigma_X^L/\sigma_X^{U+L}}
\hspace{1cm}
\mbox{and}
\hspace{1cm}
A_{U+L}\equiv A
\end{equation}
The angular distribution of longitudinally polarized $\Jpsi$s, which is 
defined in analogy to eqs. (\ref{dx1x2al},\ref{defalpha}) is given by
\begin{equation}
\alpha^{L}=\frac{1-3 \sigma_L^L/\sigma^L_{U+L}}{1+\sigma_L^L/\sigma^L_{U+L}}
\end{equation}
Analytical results for the structure functions $\sigma_{X}^{L}$ are also
listed in the appendix.



\section{Numerical Results}

We used a running coupling constant $\alpha_{s}(Q^{2})$ with
$\Lambda_{\mbox{\footnotesize}}$ = 500 MeV and $n_{f} = 3$.
For the scale $Q^{2}$ either the mass of the $\Jpsi$ ($M_{\psi}=3.097$ GeV) 
or the invariant mass of the two gluon system ($M_{gg}$) is adopted.
$M_{gg}$ falls below 1 GeV in a small fraction of the phase space and in this 
region $\alpha_{s}$ is frozen at the value 1, if the second option
$\alpha_{s}(M_{gg}^{2})$ is adopted \cite{KuSchT}.
\begin{equation}
   \alpha_{s}(Q^{2}) = \left\{  
  \begin{array}{cl}
  \frac{12\pi}{27 \ln[Q^{2}/(0.5GeV)^{2}]} & \mbox{if}\; \sqrt{Q^{2}} >1 GeV \\
     1 & \mbox{if}\; \sqrt{Q^{2}} < 1 GeV 
  \end{array} \right.
  \label{runningalphas}
\end{equation}
%
For the fine structure constant we adopt the running coupling 
\cite{Burkhardt}
\begin{eqnarray}
\alpha(s) & = & \frac{\alpha}{1-\pi_{\gamma\gamma}(s)} \\
\pi_{\gamma\gamma}(s) & = & 0.00165 + 0.00299 \log(1+s/1\mbox{GeV}^{2})
  + \sum_{e,\mu} \frac{\alpha}{3\pi} 
       \left( \log(s/m_{l}^{2}) -\frac{5}{3} \right)
\end{eqnarray}
For the coupling dictated by the wave function at the 
origin\footnote{
In \cite{KuSchT} the experimental value for 
$\Gamma_{ee}$ and the lowest order formula (5)
was used to extract $|\phi(0)|^{2}$. Actually $\Gamma_{ee}$ recieves large 
negative QCD corrections of about -35\%, which are of course included in the 
measurement. This lowered the previous prediction by a factor of about 0.65
compared to the present result.
}
we use the potential $V_{J}$ from \cite{igiono} and obtain
\begin{equation}
\label{neu}
|\phi(0)|^{2}/M_{\psi}^{3} = 2.832 \cdot 10^{-2}.
\end{equation}


\subsection{The total cross section} 

For constant $\alpha_{s}(M_{\psi}^{2})$ an analytical formula for the total 
cross section can be derived ((13) in \cite{KuSchT}):
\begin{eqnarray}
 \sigma_{TOT} & = & \int dx_1 dx_2 \sigma_{U+L} =  \sigma_{\mu^{+}\mu^{-}}  
  \left( \frac{\alpha(s)}{\alpha} \right)^{2} \;  C \; r
  \nonumber \\
&  & \cdot \left[ \frac{2-r-12r^{2}+8r^{3}}{8(1-r)^{2}}\: \ln^{2}
  \left( \frac{1+\sqrt{1-r}}{1-\sqrt{1-r}} \right)
  +  \frac{5-14r+3r^{2}}{2(1-r)^{2}} \ln(r) \right. \nonumber \\
& &\;\; \left. \mbox{}\hspace{4.mm} + \frac{4-9r+8r^{2}}{2(1-r)\sqrt{1-r}} 
  \:\ln \left( \frac{1+\sqrt{1-r}}{1-\sqrt{1-r}} \right)
  \; +   \frac{9}{2} \frac{1-2r+2r^{2}}{(1-r)}  \right] 
 \label{total}\label{TotalWQ} 
\end{eqnarray}
%
\begin{figure}[htb] 
 \begin{center} \mbox{
 \epsfig{file=korr.eps,width=15.cm,height=7.5cm,bbllx=0.cm,bblly=.5cm,bburx=18.cm,bbury=9.5cm} }
 \caption{The total cross section for $\Jpsi$ and $\psi'$. 
  Solid line: $\alpha_{s}(M_{gg})$, dashed line: $\alpha_{s}(M_{\psi})$}
  \label{fig:total}
 \end{center} 
\end{figure}
%
\begin{figure}[htb] 
 \begin{center} \mbox{
 \epsfig{file=korr1.eps,width=12.cm,height=14.cm,bbllx=0.cm,bblly=1.cm,bburx=12.cm,bbury=16.cm} }
 \caption{Influence of initial state radiation on the total cross section
          (Fig. a) and relative size of the corrections (Fig. b) for
         $\alpha_{s}(M_{gg}^2)$ (solid and dotted lines) and      
         $\alpha_{s}(M_{\psi}^2)$ (dashed and dotted lines). }
   \label{fig:total1}
 \end{center} 
\end{figure}
%
\begin{figure}[htb] 
 \begin{center} \mbox{
 \epsfig{file=korr2.eps,width=12.cm,height=11.cm,bbllx=1.cm,bblly=1.cm,bburx=19.cm,bbury=18.cm} }
 \caption{Comparison of the theoretical prediction with the data from
          \protect\cite{PLUTO,CLEO}}
  \label{fig:total2}
 \end{center} 
\end{figure}
For the running $\alpha_{s}(M_{gg}^{2})$ 
the integration has to be performed numerically.
The results for the two choices of $\alpha_{s}$ are shown in Fig. 
\ref{fig:total}. For a comparison with the data the production of the
$\psi'\,\,\,(J^{PC}=1^{--}, M_{\psi'}=3.686\mbox{GeV})$ 
with the subsequent decay $\psi'\rightarrow \Jpsi+X$ has to be added.
This amounts to a relative contribution proportional 
\begin{equation}
\frac{\Gamma(\psi'\rightarrow e^{+}e^{-})}{\Gamma(\Jpsi\rightarrow e^{+}e^{-})}
   \mbox{Br}(\psi'\rightarrow\Jpsi)=0.23.
\end{equation}

The production cross section 
increases rapidly for smaller $\sqrt{s}$, in particular for the choice with
$\alpha_{s}(M_{gg}^{2})$. Initial state radiation leads,
therefore, to an enhancement up to 12\% in the region of 
interest, if we convolute
the lower order cross section with the radiator function from \cite{Bonneau}.

The influence of initial state radiation as a function of energy is shown in 
Fig.~\ref{fig:total1}. In Fig.~\ref{fig:total2} we compare theoretical 
predictions including initial state 
radiation and the $\psi'$ contribution with the data from the PLUTO and CLEO II
collaborations. Satisfactory agreement is observed. We stress that 
uncalculated higher order corrections induce an uncertainty of the order 50\%.
For example, had we taken $\Gamma_{ee}=5.36$keV and used the Born formula 
(\ref{const}) to derive the wave function instead of eq. (\ref{neu}),
the prediction would have decreased by a factor 0.65.

To reduce or 
remove the aforementioned uncertainties from the choice of $\alpha_{s}$ and 
from the overall normalization, we now concentrate on energy and angular 
distributions which will be normalized to the total cross section.
In section 4.3, we present numerical results for ratios of the structure functions $\sigma_X$,
where the QCD corrections are expected to cancel out to a large extend.
See \cite{mirkes} for an related example.
Therefore,
these quantities are particularly suited to test the validity of the model.


\subsection{Energy distribution}

One of the first distributions that will be measured is the energy 
distribution of the $\Jpsi$. 
It can be calculated analytically \cite{KuSchT}:
\begin{eqnarray}
\frac{{\rm d}\sigma}{{\rm d}y} 
& =&  \sigma_{\mu^{+}\mu^{-}}\: \left(\frac{\alpha(s)}{\alpha}\right)^{2}
 \:C \:r^{2} \nonumber\\
&&\Bigg\{ \frac{\sqrt{y^{2}-4r}}{r}
  \left[
     \frac{(1+2r^{2})}{(1-r)(2r-y)^{2}}
  +  \frac{r(1-4r)}{(1-r)(y-2)^{2}}
  +  \frac{2}{2r-y}+\frac{-2}{y-2}
  \right] \nonumber \\
& & \mbox{} +
  \left[
  -  \frac{8r^{3}-12r^{2}-r+2}{2r(y-2)(r-1)^{2}}
  -  \frac{8r^{3}-12r^{2}-r+2}{2r(2r-y)(r-1)^{2}}
  +  \frac{4r^{3}-4r^{2}-5r+2}{(y-2)^{2}(r-1)r}
  \right. \nonumber \\
& &\mbox{}
  \left. \hspace{5.mm}
  +  \frac{2(4r^{2}-4r+3)}{(r-1)(2r-y)^{2}}
  -  \frac{2(2r^{2}+1)}{(2r-y)^{3}} \right]
  \ln \left(
     \frac{2r-y-\sqrt{y^{2}-4r}}{2r-y+\sqrt{y^{2}-4r}} 
  \right)\Bigg\}
\label{diffen}
\end{eqnarray}
%
\begin{figure}[htb] 
 \begin{center} 
  \mbox{ \epsfig{file=energie.eps,width=9.cm,height=10.cm,bbllx=0.cm,bblly=1.5cm,bburx=15.cm,bbury=18.5cm} } 
 \end{center} 
  \caption{Energy distribution of the $\Jpsi$
for $\alpha_{s}(M_{gg}^2)$  and   $\alpha_{s}(M_{\psi}^2)$.
}
  \label{fig:energy}
\end{figure}
%
and is plotted in Fig.~\ref{fig:energy} for two CMS energies:
4.5 and 10.5 GeV. In contrast to the CEM \cite{KuSchT} the distribution in the 
CSM is nearly flat.
For convenience of the reader we also give the distribution in the scaled
momentum $z = 2|\vec{P}_{\psi}|/\sqrt{s}$ (Fig. \ref{fig:moment}).
%
\begin{figure}[htb] 
 \begin{center} 
  \mbox{ \epsfig{file=moment.eps,width=9.cm,height=11.cm,bbllx=0.cm,bblly=1.cm,bburx=15.cm,bbury=18.cm} }
 \end{center} 
  \caption{Momentum distribution of the $\Jpsi$
for $\alpha_{s}(M_{gg}^2)$  and   $\alpha_{s}(M_{\psi}^2)$.
}
  \label{fig:moment}
\end{figure}


\subsection{Angular distributions}

\begin{figure}[htb]
 \begin{center}
  \mbox{ \epsfig{file=ratios.eps,width=11.cm,height=8.cm,bbllx=2.cm,bblly=2.cm,bburx=18.cm,bbury=16.cm} }
 \end{center}
 \caption{Ratios of structure functions
for $\alpha_{s}(M_{gg}^2)$  and   $\alpha_{s}(M_{\psi}^2)$
}
 \label{fig:diff}
\end{figure}
The results for the ratios
of the structure functions after integration over the 
Dalitz variables as function of $\sqrt{s}$ are shown in figure~\ref{fig:diff}.
For $\sqrt{s}=10.5$GeV the cross sections are given 
in table~\ref{He_cut}, their ratios in table~\ref{QHe_cut}
and the coefficients $\alpha,\alpha^{L},A$ and $A_{L}$ 
calculated from these numbers in table~\ref{aaa}. 
The influence of the scale in $\alpha_{s}$ is still visible but fairly small.
The results differ significantly from
$\alpha=1$, the prediction of the CEM \cite{KuSchT}.

%
\begin{figure}[htb]
 \begin{tabular}{cc}
  \mbox{ \epsfig{file=sigxxc.eps,width=7.cm,height=10.5cm} } &
  \mbox{ \epsfig{file=sigxxr.eps,width=7.cm,height=10.5cm} }
 \end{tabular}
 \caption{Ratios of structure functions for polarized $\Jpsi$
for $\alpha_{s}(M_{gg}^2)$  and   $\alpha_{s}(M_{\psi}^2)$.
}
 \label{fig:sigxx}
\end{figure}
%
To analyze the production mechanism further, it is useful to measure the 
$\Jpsi$ polarization. Here we want to discuss only longitudinal polarization. 
The ratios of the
corresponding structure functions together with those of unpolarized 
$\Jpsi$ are plotted in fig.~\ref{fig:sigxx}. The most striking result is that
the ratio $\sigma^{L}_{L}/\sigma_{L}$ is close to 1. This implies
that longitudinally polarized photons (with respect to the 
$\Jpsi$ momentum) produce dominantly longitudinal $\Jpsi$. Below it will be
demonstrated that this holds true particularly for
energetic $\Jpsi$ (table~\ref{QHe_cut}).

%
\begin{figure}[htb] 
 \begin{center} 
  \mbox{ 
\epsfig{file=alphay.eps,width=13.cm,height=4.5cm,bbllx=.5cm,bblly=1.cm,bburx=16.cm,bbury=7.cm} }
  \mbox{ 
\epsfig{file=ay.eps,width=13.cm,height=4.1cm,bbllx=0.cm,bblly=1.3cm,bburx=15.5cm,bbury=6.8cm} }
 \end{center}
 \caption{ $\alpha(y)$ and $A(y)$ at 4.5 and 10.5 GeV \label{fig:alpha}}
\end{figure}
% 
We now want to concentrate on the coefficients $\alpha,\alpha^{L},A$ and 
$A_{L}$ and their dependence on the $\Jpsi$ energy 
(fig.~\ref{fig:alpha}). The energy dependence of $\alpha$ tells us that the 
angular distributions of slow $\Jpsi$ is fairly flat, the
angular distributions of energetic $\Jpsi$ exhibits a pronounced
$\cos^{2}\theta$ dependence.
Since the coefficient $\alpha$ is negative the $\Jpsi$ are emitted mainly
with momenta perpendicular to the beam axis. 
The dependence of $A$ on the $\Jpsi$ energy
is similar. Energetic $\Jpsi$ decay to less isotropically distributed
leptons than $\Jpsi$ of low momentum. 
Again $A_{L}$ is larger than $A$, which implies 
 that the polarizations of the virtual photon and of the $\Jpsi$
 are strongly correlated.
% that the component of the $\Jpsi$ resulting from $\sigma_{L}$ leads to 
% $\Jpsi$ of more pronounced polarization.

At the $\Upsilon(4S)$ resonance (10.5 GeV) a $\Jpsi$ with high momentum 
cannot originate from $B$ decay. To distinguish between secondary
$\Jpsi$ from $B$ decays and directly produced $\Jpsi$,
a momentum cut at 2 GeV has been employed
which corresponds to a cut 
at $y=$0.702 (to be compared with the phase space limits 
$0.59 \le y \le 1.087$). The structure functions calculated 
with this cut are also listed in table~\ref{He_cut}, their ratios in 
table~\ref{QHe_cut}. These ratios 
are typically larger than those without the cut. 
Since large ratios imply pronounced angular
 distributions and low momentum
$\Jpsi$ are distributed nearly isotropic, this result seems quite plausible.

The data sample expected from the next round of data taking will
presumably not yet allow to measure $\alpha(y)$ and $A(y)$
from the two dimensional distribution in $y$ and in the production or decay
angle. We therefore suggest to split the events into a 
low energy ($0.702\le y\le 0.9$) and high energy
($0.9\le y\le 1.087$) sample, corresponding to a ``cut'' at
$|\vec{P}_{\psi}|$=3.5 GeV (see also fig. 8).

\begin{figure}[htb] 
 \begin{center} 
  \mbox{ 
     \epsfig{file=winkel.eps,width=12.cm,height=15.cm,bbllx=0.cm,bblly=1.cm,bburx=13.cm,bbury=18.cm}}
 \end{center}
 \caption{Differential cross section $d\sigma/d\phi d\cos\theta$ at 10GeV
   for unpolarized (Fig. a) and polarized (Fig. b) $\Jpsi$. \label{fig:winkel}}
\end{figure}
From the two dimensional distribution Fig.~\ref{fig:winkel} it becomes
evident, that the hadronic plane is preferentially aligned with the beam.
This trend is visible for unpolarized $\Jpsi$ already (a) and becomes
even more pronounced for longitudinally polarized $\Jpsi$ (b).
In the experimental analysis the 20\% ``contamination'' from the 
$\psi'\rightarrow\Jpsi+X$ has to be considered, as well as the effect of a 
cut in the $\Jpsi$ energy.


\section{Summary}

Inclusive $\Jpsi$ production in electron positron collisions has been 
evaluated in the nonrelativistic bound state model. Including the leading
corrections from initial state radiation, the running $\alpha_{QED}$
and the $\psi'$ contribution, satisfactory agreement has been obtained. The
prediction of the total cross section is uncertain to about $\pm$50\%
as a consequence of relativistic and higher order QCD corrections
which is in particular reflected in different assumptions on the choice of 
$\alpha_{s}$.
Energy and angular distributions for $\Jpsi$ production and decay 
have also been calculated. They are shown to be less sensitive to these 
uncertainties and may allow to scrutinize the model predictions.
\\

% =================================================================
% Tabelle 1
\begin{table}
\renewcommand{\baselinestretch}{1.2} 
\begin{center}
\small
\begin{tabular}{|c|c||c|c|c||c|c|}
\hline
scale & SF & 2.5 GeV & 2 GeV & no  & low    & high   \\
      &    & cut     & cut   & cut & energy & energy \\
\hline \hline
$M_{gg}$ & $\sigma_{U+L}$ & $4.88\cdot 10^{-4}$ & $5.39\cdot 10^{-4}$ 
& $6.12\cdot 10^{-4}$ & $2.04\cdot 10^{-4}$ &  $3.33\cdot 10^{-4}$ 
\\ \cline{2-7}
 & $\sigma_{L}$ & $2.21\cdot 10^{-4}$ & $2.38\cdot 10^{-4}$ &
 $2.62\cdot 10^{-4}$ & $6.29\cdot 10^{-5}$ & $1.75\cdot 10^{-4}$
\\ \hline
$M_{\psi}$& $\sigma_{U+L}$ & $5.27\cdot 10^{-4}$  & $6.32\cdot 10^{-4}$ 
& $7.94\cdot 10^{-4}$ & $3.67\cdot 10^{-4}$ & $2.65\cdot 10^{-4}$
\\ \cline{2-7}
 & $\sigma_{L}$ & $2.64\cdot 10^{-4}$ & $2.28\cdot 10^{-4}$
 & $2.79\cdot 10^{-4}$ & $1.15\cdot 10^{-4}$ & $1.15\cdot 10^{-4}$ 
\\ \hline \hline 
$M_{gg}$
 & $\sigma_{U+L}^{L}$ & $3.71\cdot 10^{-4}$ & $4.00\cdot 10^{-4}$ 
 & $4.27\cdot 10^{-4}$ & $1.28\cdot 10^{-4}$ & $2.70\cdot 10^{-4}$ 
\\ \cline{2-7}
 & $\sigma_{L}^{L}$ & $2.16\cdot 10^{-4}$& $2.28\cdot 10^{-4}$
 & $2.43\cdot 10^{-4}$ & $5.47\cdot 10^{-5}$ & $1.73\cdot 10^{-4}$ 
\\ \hline 
$M_{\psi}$
 & $\sigma_{U+L}^{L}$ & $3.81\cdot 10^{-4}$ & $4.35\cdot 10^{-4}$
 & $5.07\cdot 10^{-4}$ & $2.28\cdot 10^{-4}$ & $2.09\cdot 10^{-4}$ 
\\ \cline{2-7}
 & $\sigma_{L}^{L}$ & $1.87\cdot 10^{-4}$ & $2.11\cdot 10^{-4}$
 & $2.43\cdot 10^{-4}$ & $9.78\cdot 10^{-5}$ & $1.14\cdot 10^{-4}$ 
\\ \hline
\end{tabular}
\renewcommand{\baselinestretch}{1.1}
\normalsize
\end{center}
\caption{Structure functions at 10.5 GeV \label{He_cut}}
\end{table}
%
%============================================================
%============================================================
%
\begin{table}
\begin{center}
\renewcommand{\baselinestretch}{1.2}
\small
\begin{tabular}{|c|c||c|c|c||c|c|}
\hline
scale & ratio & 2.5 GeV & 2 GeV & no  & low    & high   \\
      &          & cut     & cut   & cut & energy & energy \\
%\begin{tabular}{|c|c||p{1.5cm}|p{1.5cm}|p{1.5cm}||p{1.5cm}|p{1.5cm}|}
%\hline
%scale & SF & 2.5 GeV cut& 2 GeV cut& no cut & low energy   & high energy  \\
\hline 
\hline
 $M_{gg}$ & $\sigma_{L}/\sigma_{U+L}$ &
  0.455 & 0.441 & 0.427 & 0.31 & 0.52
\\ \hline
 $M_{\psi}$ & $\sigma_{L}/\sigma_{U+L}$ &
  0.372 & 0.361 & 0.351 & 0.30 & 0.43
\\  \hline \hline
 $M_{gg}$ & $\sigma_{L}^{L}/\sigma_{U+L}^{L}$ &
  0.583 & 0.570 & 0.560 & 0.43 & 0.64
\\ \hline
 $M_{\psi}$&$\sigma_{L}^{L}/\sigma_{U+L}^{L}$ &
  0.491 & 0.483 & 0.478 & 0.43 & 0.54
\\ \hline \hline
 $M_{gg}$ & $\sigma_{U+L}^{L}/\sigma_{U+L}$ &
  0.760 & 0.741 & 0.702 & 0.63 & 0.81
\\ \hline
 $M_{\psi}$&$\sigma_{U+L}^{L}/\sigma_{U+L}$ &
  0.724 & 0.689 & 0.638 & 0.62 & 0.79
\\ \hline \hline
 $M_{gg}$ & $\sigma_{L}^{L}/\sigma_{L}$ &
  0.974 & 0.959 & 0.929 & 0.87 & 0.99
\\ \hline
 $M_{\psi}$&$\sigma_{L}^{L}/\sigma_{L}$ &
  0.955 & 0.924 & 0.870 & 0.88 & 0.99
\\ \hline
\end{tabular}
\renewcommand{\baselinestretch}{1.1}
\normalsize
\end{center}
\caption{Ratios of structure functions at 10.5 GeV 
for $\alpha_{s}(M_{gg}^2)$  and   $\alpha_{s}(M_{\psi}^2)$.
\label{QHe_cut}}
\end{table}
%
% =====================================================================
%
\begin{table}
\begin{center}
\renewcommand{\baselinestretch}{1.2}
\small
\begin{tabular}{|c|c||c|c|c||c|c|}
\hline
scale &coeff. & 2.5 GeV & 2 GeV & no  & low    & high   \\
      &       & cut     & cut   & cut & energy & energy \\
%\begin{tabular}{|c|c||p{1.5cm}|p{1.5cm}|p{1.5cm}||p{1.5cm}|p{1.5cm}|}
%\hline
%scale & SF & 2.5 GeV cut& 2 GeV cut& no cut & low energy   & high energy  \\
\hline  
\hline 
$M_{gg}$ & $\alpha$ & -0.25 & -0.22 & -0.20 & 0.05 & -0.37 
\\ \hline
$M_{\psi}$ & $\alpha$ & -0.09 & -0.06 & -0.04 & 0.08 & -0.20
\\ \hline \hline 
$M_{gg}$ & $\alpha^{L}$ & -0.47 & -0.45 & -0.43 & -0.20 & -0.56
\\ \hline
$M_{\psi}$ & $\alpha^{L}$ & -0.31 & -0.30 & -0.29 & -0.20 & -0.40
\\ \hline \hline  
$M_{gg}$ & $A$ & -0.73 & -0.70  & -0.64 & -0.55 & -0.79
\\ \hline
$M_{\psi}$ & $A$ & -0.68 & -0.63 & -0.56 & -0.53 & -0.76
\\ \hline \hline  
$M_{gg}$ & $A_{L}$ & -0.97 & -0.96  & -0.93 & -0.86 & -0.99
\\ \hline
$M_{\psi}$ & $A_{L}$ & -0.95 & -0.92 & -0.86 & -0.87 & -0.99
\\ \hline
\end{tabular}
\renewcommand{\baselinestretch}{1.1}
\normalsize
\end{center}
\caption{Coefficients $\alpha$ and $A$ with different cuts \label{aaa}}
\end{table}
%

\newpage
\clearpage
% -----------------------------------------
\begin{appendix}

\section{Analytical expressions of the structure functions \label{app1} }

Remember:
\begin{displaymath}
C\;=\; \left( \frac{\alpha_{s}}{\alpha}\right)^{2}
 \left( \frac{\Gamma_{ee}}{M_{\psi}}\right) \left( \frac{8}{3\pi}\right)
 \; = \; \frac{128\alpha_{s}^{2}}{27\pi} \frac{|\phi(0)|^{2}}{M^{3}_{\psi}}
\end{displaymath}
The structure functions of unpolarized $\Jpsi$ are:
\begin{eqnarray*}
\sigma_{U+L}
& = & C r^{2} \left\{
  \frac{x_{1}(x_{1}+2)}{(x_{2}+r-1)^{2}(x_{1}+x_{2})^{2}}
  +\frac{x_{2}(x_{2}+2)}{(x_{1}+r-1)^{2}(x_{1}+x_{2})^{2}}\right.  \\
 &   & \left.\mbox{}+
  \frac{(x_{1}+x_{2}+r-3)(x_{1}+x_{2}+r-1)}{(x_{1}+r-1)^{2}(x_{2}+r-1)^{2}}
  + \frac{1}{(x_{1}+x_{2})^{2}} \left[\frac{1}{r} \right.\right.  \\
 &   &\left.\left. +
  \frac{6(x_{1}+x_{2}+r-1)^{2}}{(x_{1}+r-1)^{2}(x_{2}+r-1)^{2}}
  +\frac{2(1-r)(x_{1}+x_{2}-1)}{r(x_{1}+r-1)(x_{2}+r-1)}
  \right]\right\}   
\end{eqnarray*} 
\vspace{4.0mm}
\begin{eqnarray*}
\sigma_{L}
& = & 
\frac{Cr^{2}}{(x_{1}+x_{2}-2)^{2}-4r} \cdot  \left[\left( 
  -\frac{2(r-1)(3r-2)(x_{2}-x_{1}+2)}{(x_{1}+r-1)^{2}(x_{2}+r-1)(x_{1}+x_{2})} 
 \right. \right. \\
&& \left. \left.
  - \frac{2(5r-2)}{(x_{1}+r-1)^{2}(x_{2}+r-1)}  
  + \frac{2(4rx_{2}-3x_{2}-7r^{2}+9r-5}{(x_{1}+r-1)(x_{1}+x_{2})^{2}}
  \right.\right.   \\
&& \left. \left.\mbox{}
  + \frac{2(4x_{1}+r-4)}{(x_{1}+r-1)^{2}}
  - \frac{r(2rx_{2}+r^{2} -2r+1)}{(x_{1}+r-1)^{2}(x_{2}+x_{1})^{2}}
  \right) + ( x_{1} \leftrightarrow x_{2}) \right.  \\[2.mm]
&& \left. \mbox{}
  + \frac{r(r^{2}-1)}{(x_{1}+r-1)^{2}(x_{2}+r-1)^{2}}
  + \frac{2(r-1)(9r^{2}-15r+8)}{(x_{1}+r-1)(x_{2}+r-1)(x_{1}+x_{2})^{2}} 
\right.  \\
&& \left. \mbox{}
  - \frac{4(x_{1}+x_{2}-3r+2)}{(x_{2}+x_{1})^{2}}
  - \frac{2(x_{1}+x_{2}+3r)}{(x_{1}+r-1)(x_{2}+r-1)}
  + \frac{1}{r}
  \right] 
\end{eqnarray*}
\vspace{4.0mm}
\begin{eqnarray*}
\sigma_{T}
& = & Cr^{2}
  \frac{(x_{1}+x_{2}+r-1)} {(x_{1}+x_{2})^{2}}
  \left[ \frac{(r+1)^{2}} {r(x_{1}+r-1)^{2}(x_{2}+r-1)^{2}}
  \right.  \\ 
&& \left. \mbox{}
 + \frac{1} {[(x_{1}+x_{2}-2)^{2}-4r]} \cdot 
 \left( \frac{r-1}{r}
   \left( \frac{1}{(x_{1}+r-1)^{2}} + \frac{1}{(x_{2}+r-1)^{2}} \right)
 \right.\right.  \\ 
&& \left.\left. \mbox{}
  + \frac{2[(3r+5)(x_{1}+x_{2})+r^{2}+6r-7]}{(x_{1}+r-1)^{2}(x_{2}+r-1)^{2}}
  \right) \right](x_{1}x_{2}-x_{1}-x_{2}-r+1)
\end{eqnarray*}
\vspace{4.0mm}
\begin{eqnarray*}
\sigma_{I}
&=& Cr^{2}
  \frac{|x_{2}-x_{1}|}{2\sqrt{2}(x_{1}+x_{2})}
  \sqrt{x_{1}+x_{2}+r-1}
  \left[ \frac{3r+1}{r(x_{1}+r-1)^{2}(x_{2}+r-1)^{2}}
  \right.  \\ 
&& \left. \mbox{}
 + \frac{1}{[(x_{1}+x_{2}-2)^{2}-4r]} \cdot \left( \frac{r-1}{r}
 \left( \frac{1}{(x_{1}+r-1)^{2}} + \frac{1}{(x_{2}+r-1)^{2}} \right)
 \right.\right.  \\ 
&& \left.\left. \mbox{}
  + \frac{2[(3r+5)(x_{1}+x_{2})+r^{2}+6r-7]}{(x_{1}+r-1)^{2}(x_{2}+r-1)^{2}}
  \right) \right]  \sqrt{x_{1}x_{2}-x_{1}-x_{2}-r+1} \label{sigi}\\
\sigma_{G} & = & 0
\end{eqnarray*}
%
The following notation is introduced:
\begin{eqnarray*}
z_{1} & = & 1-r-x_{1} \\
z_{2} & = & 1-r-x_{2} \\
z_{3} & = & x_{1}+x_{2} 
\end{eqnarray*}
The structure functions of longitudinally polarized $\Jpsi$ are:
\begin{eqnarray*}
\sigma_{U+L}^{L} & = & \frac{Cr}{(z_{3}+2)^{2}-4r}
\\&&
\Bigg\{(r-1)\Bigg[
 -4r\frac{r^{3}-r^{2}+4r-2}{z_{2}^{2}z_{3}^{2}}
 -2(r-3)\frac{r^{3}}{z_{1}^{2}z_{2}^{2}}
 -8(r+1)\frac{2r-1}{z_{1}^{2}z_{2}z_{3}}  
\\&& \hspace{1.6cm}
  +4\frac{2r^{3}+11r^{2}-19r+8}{z_{1}z_{2}z_{3}^{2}}
 -2\frac{rz_{2}+rz_{1}+6r^{2}-6r+4}{z_{3}^{2}}
\\&&\hspace{1.6cm}
 +4\frac{r}{z_{3}} 
 -8\frac{r^{3}-r^{2}+3r-2}{z_{2}z_{3}^{2}} 
 +\frac{rz_{2}-3z_{2}-3r^{2}-5r+2}{z_{1}z_{3}} 
\\&&\hspace{1.6cm}
  -4\frac{r^{3}-3r^{2}+1}{z_{1}^{2}z_{3}}
  -4\frac{2r^{2}-1}{z_{1}z_{2}z_{3}}
\Bigg]
\\&&
 +2 
 +3r\frac{z_{2}^{2}}{2z_{1}^{2}}
 -\frac{rz_{1}^{2}+2r^{2}z_{1}-2rz_{1}+4z_{1}+7r^{3}-2r^{2}+3r+2}{z_{2}^{2}} 
\\&&
  -\frac{rz_{1}^{2}+6rz_{1}-8z_{1}+10r^{3}-12r^{2}+8r-4}{2z_{2}^{2}}
 -4\frac{r^{4}-2r^{3}-4r^{2}-r+1}{z_{1}^{2}z_{2}} 
\\&&
  -\frac{5r^{3}-2r^{2}+13r-4}{z_{1}z_{2}}
  -2\frac{r^{4}-5r^{3}-11r^{2}+19r-8}{z_{2}z_{3}^{2}}
 -4(r-2)\frac{r^{3}}{z_{1}z_{2}^{2}} 
\\&&
  +\frac{2r^{2}z_{2}+rz_{2}+r^{2}z_{1}-2rz_{1}+z_{1}+4r^{3}-16r^{2}-r}
    {z_{1}^{2}}
  +\frac{7r^{2}-10r+11}{z_{3}}
\\&&
 -2\frac{r^{4}-5r^{3}-11r^{2}+19r-8}{z_{1}z_{3}^{2}}
 -\frac{2rz_{1}+3z_{1}+18r^{2}+2r+12}{z_{2}}
\\&&
  +4r\frac{3r^{3}-4r^{2}+7r-4}{z_{2}^{2}z_{3}} 
 +\frac{23r^{3}-28r^{2}+49r-36}{z_{2}z_{3}}
 +\frac{2rz_{2}+3z_{2}+8r^{2}}{z_{1}} \Bigg\} \;\;
\\&&
+\Big( z_{1} \leftrightarrow z_{2} \Big)
\end{eqnarray*}
%===================================================================
\newpage
\begin{eqnarray*}
\sigma_{L}^{L} & = &\frac{Cr}{[(z_{3}+2)^{2}-4r]^{2}}
\\&&\Bigg\{
(r-1)^{2}\Bigg[
 +r^{2}\frac{z_{1}^{2}-4z_{1}+8}{z_{2}^{2}z_{3}^{2}}
 +8r\frac{3r-1}{z_{1}^{2}z_{2}z_{3}} 
  +8\frac{r^{2}}{z_{1}z_{2}z_{3}^{2}}
  +3\frac{r^{2}}{z_{3}^{2}}
\\&&\hspace{1.8cm}
 -r^{2}\frac{3z_{1}-4}{z_{2}z_{3}^{2}}
 -r\frac{z_{2}^{2}-2z_{2}+12r-4}{z_{1}^{2}z_{3}} 
  -r^{2}\frac{z_{2}}{z_{1}z_{3}^{2}}
  +4\frac{r^{4}}{z_{1}^{2}z_{2}^{2}}
\Bigg]
\\&&
+(r-1)\Bigg[
 -2r^{2}\frac{z_{1}^{2}-4z_{1}-2r+10}{z_{2}^{2}z_{3}}
 -6\frac{r^{2}}{z_{3}} 
 +4r\frac{2r^{2}-7r+1}{z_{1}z_{2}z_{3}} 
\\&&\hspace{1.6cm}
  +4r\frac{3r^{3}+r^{2}-3r+1}{z_{1}^{2}z_{2}}
 +r\frac{5rz_{2}-3z_{2}-4r+4}{z_{1}z_{3}} 
\\&&\hspace{1.6cm}
  -r\frac{3rz_{2}-3z_{2}-7rz_{1}+z_{1}+6r+2}{z_{2}z_{3}}
 +8\frac{r^{4}}{z_{1}z_{2}^{2}} 
\Bigg]
\\&&
 +\frac{z_{2}^{2}+2z_{1}z_{2}-2rz_{2}+z_{1}^{2}+6rz_{1}+6r^{2}-12r}{2}
\\&&
 +r\frac{rz_{2}^{2}-2r^{2}z_{2}-6rz_{2}+2z_{2}-8z_{1}^{3}
    +15r^{3}+28r^{2}-3r}{2z_{1}^{2}} 
\\&&
  +r\frac{z_{1}^{2}+rz_{1}-z_{1}+14r^{2}+10r-1}{z_{2}}
 +r\frac{19r^{3}-2r^{2}+19r-4}{z_{1}z_{2}} 
\\&&
 +r^{2}\frac{3z_{1}^{2}+8rz_{1}-2z_{1}+15r^{2}-8r+17}{2z_{2}^{2}}
\\&&
  +r\frac{z_{2}^{2}-7rz_{2}-z_{2}+rz_{1}+3r^{2}+10r-4}{z_{1}}
 \Bigg\}
 + \Big( z_{1} \leftrightarrow z_{2} \Big)
\end{eqnarray*}




\end{appendix}
\newpage
% -----------------------------------------



\def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #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\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3}
\def\prc#1#2#3{{\it Phys. Reports }{\bf C #1} (#2) #3}
\def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3}
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\end{document}

%ende

