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\begin{center}
{\large  Inclusive Charmonium Production via Double $c \bar c$ in
$e^+e^-$  Annihilation}\\[0.8cm]
{ Kui-Yong Liu$~^{(a)}$  Zhi-Guo He$~^{(a)}$ and~Kuang-Ta Chao$~^{(b,a)}$}\\[0.5cm]
{\footnotesize (a)~Department of Physics, Peking University,
 Beijing 100871, People's Republic of China}

{\footnotesize (b)~China Center of Advanced Science and Technology
(World Laboratory), Beijing 100080, People's Republic of China}
\end{center}

\begin{abstract}

Motivated by the recent observation of double charm quark pair
production by Belle, we calculate the complete ${\cal
O}(\alpha_{s}^{2})$ color-singlet inclusive cross sections for
$\eta_{c}$, $J/\psi$ and $\chi_{cJ}$(J=0, 1, 2) via double
$c\bar{c}$ production in $e^+ e^-$ annihilation through a virtual
photon. The complete calculations are compared with the
approximate fragmentation calculations as functions of the
center-of-mass energies. We find that most of the fragmentation
results substantially overestimate the cross sections (e.g. by a
factor of $\sim$4 for $\chi_{c1}$ and $\chi_{c2}$) at the Belle
and BaBar energy $\sqrt{s}=10.6$GeV. For the S-wave charmonium
production, the energy scale at which the fragmentation results
become a good approximation is over 50 GeV, while for the P-wave
charmonium production the corresponding energy scale approaches
100 GeV.

PACS number(s): 12.40.Nn, 13.85.Ni, 14.40.Gx

\end{abstract}

Charnonium is one of the simplest quark-antiquark composite
particles. Charmonium physics has played an important role in the
study of Quantum Chromodynamics (QCD) both perturbatively and
nonperturbatively, since the first charmonium state $J/\psi$ was
discovered in 1974. During the past decade, the study of
charmonium has become more interesting because of the large
discrepancies between the predictions of the color-singlet model
and the observations of $J/\psi$ and $\psi'$ production at several
experimental facilities e.g. at the Fermilab Tevatron \cite{cdf}.

The newly developed nonrelativistic QCD (NRQCD) factorization
formalism \cite{bbl} allows the infrared safe calculation of
inclusive heavy quarkonium production and decay rates. In the
NRQCD production mechanism, a heavy quark-antiquark pair can be
produced at short distances in a conventional color-singlet or a
color-octet state, and then evolves into an observed quarkonium
nonperturbatively. With this color-octet mechanism, one may
explain the Tevatron data on the surplus production of $J/\psi $
and $\psi ^{\prime }$ at large $p_T$, though puzzles about their
polarizations still remain (for a review see \cite{kramer} and
references therein).

To further test the color octet mechanism, it is interesting to
study the charmonium production in $e^+e^-$ annihilation. The
$J/\psi$ inclusive production in $e^+e^-$ annihilation has been
investigated within the color-singlet model \cite{cm1,cm2,cm3} and
the color-octet model \cite{om1,om2,ko}. The angular distribution
and energy distribution of color-singlet $J/\psi$ production at
$\sqrt{s}=10.6GeV$ have been discussed in \cite{cm3}. In
\cite{om1} it is found that a clean signature of the color-octet
mechanism may be observed in the angular distribution of $J/\psi$
production near the end point region. In \cite{om2} contributions
of various color-octet as well as color-singlet channels to the
$J/\psi$ production cross sections are calculated in a wide range
of $e^+e^-$ collider energies. Moreover, the $J/\psi$
polarizations are predicted in \cite{ko}. Recently, BaBar
\cite{babar} and Belle \cite{belle} have measured the direct
$J/\psi$ production in continuum $e^+e^-$ annihilations at
$\sqrt{s}=10.6 GeV$. The total cross section and the angular
distribution seem to favor the NRQCD calculation over the
color-singlet model \cite{babar}, but some issues (e.g. about the
momentum distribution and polarization of $J/\psi$) still remain.

The situation has become even more complicated due to the very
recent observation for the double $c\bar c$ production associated
with $J/\psi$ by Belle\cite{exdou}. The measured exclusive cross
section for $e^+ + e^-\rightarrow J/\psi+\eta_c$ process is an
order of magnitude larger than the theoretical value\cite{double},
and the measured inclusive cross section for $e^+ + e^-\rightarrow
J/\psi+c+\bar c$ ($\sim$0.9 pb)\cite{belle,exdou} is more than
five times larger than NRQCD predictions which are only about
0.1-0.2 pb\cite{cm3,om2,ko} taking into account the differences in
the values of input parameters.

The double $c\bar c$ production associated with $J/\psi$ (both
exclusively and inclusively) is very puzzling and needs a better
understanding for both perturbative and nonperturbative QCD. On
the other hand, experimentally, it is not clear whether the
copious (even dominant) double $c\bar c$ production will also
happen for charmonium states other than $J/\psi$, such as
$\eta_{c}$ and $\chi_{cJ}$(J=0, 1, 2). Among them the $\chi_{c1}$
and $\chi_{c2}$ are more interesting since they have large
branching fractions decaying into $J/\psi+\gamma$ and might be
easier to be detected. In fact, in Ref.~\cite{belle} the inclusive
production for $\chi_{c1}$ and $\chi_{c2}$ was searched for with
the available integrated luminosity of about 30 fb$^{-1}$ at
Belle. As more data are collected in the near future at B
factories we hope that more accurate measurements for the P-wave
and other S-wave charmonium states will be possible. These
measurements will be helpful to clarify the problems associated
with $J/\psi$ double $c\bar{c}$ production.

On the theoretical side, the calculations for S-wave and P-wave
charmonia production via double $c\bar{c}$ are necessary. In this
paper, we will calculate the complete $\cal{O}$$(\alpha_{s}^{2})$
color-singlet inclusive production cross sections for $\eta_{c}$
and $\chi_{cJ}$(J=0,1,2) (along with $J/\psi$) via double
$c\bar{c}$ in $e^+ e^-$ annihilation through a virtual photon.
Then we will compare the complete calculation with the calculation
obtained in the charm quark fragmentation limit, and give their
ratio as functions of the center-of-mass energies and determine
the energy scales at which fragmentation approximations become
reliable.

Following the nonrelativistic QCD (NRQCD) factorization formalism,
the color-singlet scattering amplitude of the process
$e^-(p_1)+e^+(p_2)\rightarrow \gamma^* \rightarrow
c\bar{c}(^{2S+1}L_J)(p)+c(p_c)+\bar{c}(p_{\bar{c}})$ in
Fig.~\ref{feynman} is given by
\begin{eqnarray}
%\hspace{-1.0cm}\hspace{1.0cm}
\label{amp2}   &&\hspace{-1cm}{\cal
A}(e^-(p_1)+e^+(p_2)\rightarrow
c\bar{c}(^{2S+1}L_{J})(p)+c(p_c)+\bar{c}(p_{\bar{c}}))=\sqrt{C_{L}}
\sum\limits_{L_{z} S_{z} }\sum\limits_{s_1s_2 }\sum\limits_{jk}
\nonumber
\\ & \times&\langle s_1;s_2\mid
S S_{z}\rangle \langle L L_{z};S S_{z}\mid J J_{z}\rangle\langle
3j;\bar{3}k\mid 1\rangle\nonumber\\
&\times&\left\{
\begin{array}{ll}
{\cal A}(e^-(p_1)+e^+(p_2)\rightarrow
 c_j(\frac{p}{2};s_1)+\bar{c}_k(\frac{p}{2};s_2)+
 c_l(\frac{p_c}{2};s_3)+\bar{c}_i(\frac{p_{\bar{c}}}{2};s_4))&(L=S),\nonumber\\
\epsilon^*_{\alpha}(L_Z) {\cal
A}^\alpha(e^-(p_1)+e^+(p_2)\rightarrow
 c_j(\frac{p}{2};s_1)+\bar{c}_k(\frac{p}{2};s_2)+
 c_l(\frac{p_c}{2};s_3)+\bar{c}_i(\frac{p_{\bar{c}}}{2};s_4))
&(L=P).
\end{array}
\right.\nonumber\\
\end{eqnarray}
where $c\bar{c}(^{2S+1}L_{J})$ is the $c\bar{c}$ pair produced at
short distances, which subsequently evolve into a specific
charmonium state at long distances, ${\cal A}^\alpha$ is the
derivative of the amplitude with respect to the relative momentum
between the quark and anti-quark in the bound state. In the
nonrelativistic approximation the coefficient $C_{L}$ can be
related to the origin of the radial wave function (or its
derivative) of the bound state as
\begin{equation}
C_{S}=\frac{1}{4\pi}\mid R_{S}(0) \mid^{2},~~~~~
C_{P}=\frac{3}{4\pi}\mid R_{P}'(0) \mid^{2}.
\end{equation}

The spin projection operator can be defined as\cite{pro}
\begin{equation}
P_{SS_z}(p;q)\equiv\sum\limits_{s_1s_2 }\langle
s_1;s_2|SS_z\rangle
v(\frac{p}{2}+q;s_1)\bar{u}(\frac{p}{2}-q;s_2).
\end{equation}

We list the spin projection operators and their derivatives with
respect to the relative momentum, which we will use in the
calculations, as
\begin{equation}
P_{00}(p,0)=\frac{1}{2\sqrt{2}}\gamma_{5}(\not{p}+2m_{c}),
\end{equation}
\begin{equation}
P_{1S_Z}(p,0)=\frac{1}{2\sqrt{2}}\not{\epsilon}(S_z)(\not{p}+2m_{c}),
\end{equation}
\begin{equation}
P_{1S_z}^{\alpha}(p,0)=\frac{1}{4\sqrt{2}m_c}
[\gamma^{\alpha}\not{\epsilon}^*(S_z)(\not{p}+2m_c)-
(\not{p}-2m_c)\not{\epsilon}(S_z)\gamma^{\alpha}].
\end{equation}
For P-wave states we need further relations to reduce the
polarizations
\begin{eqnarray}
\sum\limits_{L_ZS_Z}\epsilon^{*\alpha}(L_Z)\epsilon^{*\beta}(S_Z)\langle
1 L_Z;1 S_Z \mid J=0 J_Z=0
\rangle=\frac{1}{\sqrt{3}}(-g^{\alpha\beta}+\frac{p^{\alpha}p^{\beta}}{M^2})\nonumber\\
\sum\limits_{L_ZS_Z}\epsilon^{*\alpha}(L_Z)\epsilon^{*\beta}(S_Z)\langle
1 L_Z;1 S_Z \mid J=1 J_Z
\rangle=-\frac{i}{\sqrt{2}M}\epsilon^{\alpha\beta\lambda\kappa}p_{\kappa}\epsilon^{*}_{\lambda}(J_z)\\
\sum\limits_{L_ZS_Z}\epsilon^{*\alpha}(L_Z)\epsilon^{*\beta}(S_Z)\langle
1 L_Z;1 S_Z \mid J=1 J_Z
\rangle=\epsilon^{*\alpha\beta}(J_Z)\nonumber
\end{eqnarray}
where M is the mass of the charmonium, which equals to $2m_c$ in
the nonrelativistic approximation.

The calculation of cross sections for $e^{-}+e^{+}\rightarrow
\gamma^*\rightarrow$ charmonium $ + c\bar{c}$ is straightforward
but tedious. As in Ref.~\cite{cm3} we get the differential cross
section as follows
\begin{equation}
\label{cross} \frac{d\sigma(e^{+}+e^{-}\rightarrow \gamma^{*}
\rightarrow {\rm charmonium} +
c\bar{c})}{dz}=\frac{4C_{L}\alpha^{2}\alpha_{s}^{2}}{81m_{c}}(S(z)+\frac{\alpha(z)}{3}).
\end{equation}
where L=S for S-wave, L=P for P-wave, and z is a dimensionless
variable which is the charmonium energy divided by the beam
energy. The functions S(z) and $\alpha(z)$ for different
charmonium states are given in Appendix.

With Eq.~(\ref{cross}) we can evaluate the inclusive cross
sections for $\eta_{c}$, $J/\psi$ and $\chi_{cJ}$. The input
parameters used in the numerical calculations are\cite{wf}
\begin{equation}
m_e=0, ~~m_c=1.5GeV, ~~\alpha_s(2m_c)=0.26,~~\alpha=1/137,
\end{equation}
\begin{equation}
\mid R_S(0) \mid^2=0.81GeV^3,~~~\mid R_P(0)' \mid^2=0.075GeV^5.
\end{equation}

Now we give the numerical results at the Belle and BaBar energy
$\sqrt{s}=10.6$ GeV.

\begin{equation}
\label{etc}
\sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow\eta_{c}+c\bar{c})=58.7
{\rm fb}
\end{equation}

\begin{equation}
\label{js} \sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow
J/\psi+c\bar{c})=148 {\rm fb}
\end{equation}

\begin{equation}
\label{kc0}
\sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow\chi_{c0}+c\bar{c})=48.8
{\rm fb}
\end{equation}

\begin{equation}
\label{kc1}
\sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow\chi_{c1}+c\bar{c})=13.5
{\rm fb}
\end{equation}

\begin{equation}
\label{kc2}
\sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow\chi_{c2}+c\bar{c})=6.30
{\rm fb}
\end{equation}

The $J/\psi$ production rate is in agreement with other
references\cite{cm3,om2,ko} after taking into account the
differences in the values of the input parameters. In the
z$\gg\delta$ limit, the approximate fragmentation results will be
equivalent to the complete calculations. This is another check for
the validity of the complete calculation. The fragmentation cross
section can be written as

\begin{eqnarray}
\label{frag}
\sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow{\rm
Charmonium}+c\bar{c})=\nonumber\\2\sigma(e^{+}+e^{-}\rightarrow
c\bar{c})\int_{\delta}^{1}{\cal D}_{c\rightarrow{\rm
Charmonium}}(z)dz,
\end{eqnarray}
where $\delta$ is defined as $4m_c/\sqrt{s}$ and ${\cal D}(z)$ is
the heavy quark fragmentation function for S-wave\cite{s} and
P-wave\cite{p}.

The cross sections obtained in the complete calculation and in the
fragmentation approximation as functions of the center-of-mass
energies are plotted in Fig.~\ref{fig1}-\ref{fig5}. All these
cross sections are in units of $\sigma_{cc}=
\sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow c\bar{c})$,
 the cross section for $e^{+}e^{-}$
annihilating into the $c\bar{c}$ quark pair, times $10^{-4}$. One
can find that the cross sections in complete calculations and
fragmentation approximations (all in units of the cross section
for $e^{+}e^{-}$ annihilating to the $c\bar{c}$ pair) are
proportional to the fragmentation probabilities for the charm
quark splitting into charmonia when the z$\gg\delta$ limit is
valid. This is just what the fragmentation approach discribes. The
figures show that except for $\chi_{c0}$, the gap between
fragmentation results and complete calculations is large at low
energies. At the Belle and BaBar energy $\sqrt{s}$=10.6GeV, the
ratios of complete calculations to fragmentation results for
$\eta_{c}$, $J/\psi$, $\chi_{c1}$ and $\chi_{c2}$ are only 0.28,
0.58, 0.25 and 0.25 respectively. As the center-of-mass energy
increases, the ratios of the complete calculations to the
fragmentation results increase and for S-wave charmonia when the
center-of-mass energy is over 50 GeV, the ratios are within
$90\%$; while for P-wave charmonia the corresponding energy scale
is about 100 Gev. The cross sections are rather sensitive to the
input parameters. If we choose $\alpha=1/134, \alpha_{s}=0.28$,
and ${\rm m_{c}}=1.48{\rm GeV}$ at the Belle energy
$\sqrt{s}=10.6$GeV, the cross sections for $\eta_{c}, {\rm
J}/\psi, \chi_{cJ}$(J=0,1,2) become 78.0fb, 194fb, 64.8fb, 18.5fb,
8.57fb respectively.

In summary, We have calculated the complete ${\cal
O}(\alpha_{s}^{2})$ color-singlet inclusive cross sections for
$\eta_{c}$, $J/\psi$ and $\chi_{cJ}$(J=0,1,2) via double
$c\bar{c}$ production in $e^+ e^-$ annihilation through a virtual
photon. The complete calculations are compared to the
fragmentation calculations as functions of the center-of-mass
energies. We find that at the Belle and BaBar energy
$\sqrt{s}$=10.6GeV, except for $\chi_{c0}$, most of the
fragmentation results substantially overestimate the cross
sections (e.g. by a factor of $\sim$4 for $\chi_{c1}$ and
$\chi_{c2}$). For the S-wave charmonium production, the energy
scale at which the approximate fragmentation results tend to be in
agreement with the complete calculations is over 50 GeV, while for
the P-wave charmonium production the corresponding energy scale
approaches 100 GeV.






\section*{Acknowledgments}

The authors thank Z.Z. Song for useful discussions. This work was
supported in part by the National Natural Science Foundation of
China, and the Education Ministry of China.

\section*{Appendix}

In this Appendix, we give the functions of S and $\alpha$ which
are defined in Eq.~(\ref{cross}).
\begin{eqnarray}
{\rm
S}_{\eta_c}&=&\frac{2\pi}{3s^{2}\delta^{2}z^{3}(z-2)^{6}(z^{2}-\delta^{2})}
\{4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{4+\delta^{2}-4z}}[-96\delta^{6}(2+\delta^{2})(4+\delta^2) \nonumber \\
&&
+96\delta^{6}(64+22\delta^{2}+\delta^{4})z-16\delta^{2}(1920-864\delta^{2}+532\delta^{4}+125\delta^{6}-2\delta^{8})z^{2}
\nonumber \\
&&+8\delta^{2}(9984-5312\delta^{2}+488\delta^{4}+96\delta^{6}-\delta^{8})z^{3} \nonumber \\
&&
+2(6144-47872\delta^{2}+20800\delta^{4}-392\delta^{6}-110\delta^{8}+3\delta^{10})z^{4} \nonumber \\
&&
-4(6144-21376\delta^{2}+4256\delta^{4}+112\delta^{6}+9\delta^{8})z^{5} \nonumber \\
&&
+(14336-51328\delta^{2}+5472\delta^{4}+420\delta^{6}-3\delta^{8})z^{6}\nonumber\\
&&-4(1536-3168\delta^{2}+352\delta^{4}+\delta^{6})z^{7} \nonumber\\
&&
+8(864-36\delta^{2}+13\delta^{4})z^{8}-32(112+11\delta^{2})z^{9}+768z^{10}] \nonumber \\
&&
-3\delta^{2}(z-2)^{4}[8\delta^{6}(2+\delta^{2})-96\delta^{6}z-2\delta^{2}(192-48\delta^{2}+8\delta^{4}-\delta^{6})z^{2}
\nonumber \\
&&
+16\delta^{2}(8+6\delta^{2}-\delta^{4})z^{3}+\delta^{2}(192+40\delta^{2}-\delta^{4})z^{4}
+8(32-4\delta^{2}+\delta^{4})z^{5} \nonumber\\
&&
-8(48+\delta^{2})z^{6}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
\alpha}_{\eta_c}&=&\frac{2\pi}{3s^{2}\delta^{2}z^{3}(z-2)^{6}(z^{2}-\delta^{2})}
\{4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{4+\delta^{2}-4z}}[96\delta^{6}(4+\delta^{2})(6+\delta^{2}) \nonumber \\
&& -96\delta^{6}(64+18\delta^{2}+\delta^{4})z+16\delta^{2}(2688+608\delta^{2}+428\delta^{4}
+43\delta^{6}-2\delta^{8})z^{2} \nonumber \\
&&
-8\delta^{2}(17664+3264\delta^{2}+184\delta^{4}-96\delta^{6}-\delta^{8})z^{3}
\nonumber \\
&&
+2(6144+89344\delta^{2}+7744\delta^{4}-2024\delta^{6}-174\delta^{8}-3\delta^{10})z^{4}
\nonumber \\
&&
-4(6144+22656\delta^{2}-1376\delta^{4}-512\delta^{6}-35\delta^{8})z^{5}
\nonumber \\
&&
+(14336+5504\delta^{2}-5152\delta^{4}-732\delta^{6}-3\delta^{8})z^{6}\nonumber\\
&&-4(1536-1760\delta^{2}-416\delta^{4}+\delta^{6})z^{7}
\nonumber \\
&&
+8(864-196\delta^{2}+13\delta^{4})z^{8}-32(112+11\delta^{2})z^{9}+768z^{10}]
\nonumber \\
&& +3\delta^{2}(z-2)^{4}[8\delta^{6}(6+\delta^{2})-32\delta^{6}z
-2\delta^{2}(64+48\delta^{2}+16\delta^{4}-\delta^{6})z^{2} \nonumber \\
&&
+16\delta^{2}(12-\delta^{2})(2+\delta^{2})z^{3}-(1024+320\delta^{2}-88\delta^{4}-\delta^{4})z^{4}\nonumber \\
&&+8(96-28\delta^{2}-\delta^{4})z^{5}+8(16+\delta^{2})z^{6}]\nonumber\\
&&\times\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
S}_{\psi}&=&\frac{2\pi}{s^{2}\delta^{2}z^{3}(z-2)^{6}(z^{2}-\delta^{2})}
\{4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{4+\delta^{2}-4z}}\nonumber\\
&&\times[-32\delta^{4}(4+\delta^{2})(48+22\delta^{2}+3\delta^{4}) \nonumber \\
&&
+32\delta^{4}(768+400\delta^{2}+66\delta^{4}+3\delta^{6})z\nonumber\\
&&-16\delta^{2}(384+1920\delta^{2}+556\delta^{4}+29\delta^{6}-2\delta^{8})z^{2}
\nonumber \\
&&
+8\delta^{2}(1792+128\delta^{2}-568\delta^{4}-80\delta^{6}-\delta^{8})z^{3} \nonumber \\
&&
+2(2048-11008\delta^{2}+10752\delta^{4}+3176\delta^{6}+98\delta^{8}+3\delta^{10})z^{4}
\nonumber \\
&&
-4(4096-7808\delta^{2}+3424\delta^{4}+600\delta^{6}+17\delta^{8})z^{5}
\nonumber \\
&&
+(38912-20608\delta^{2}+4544\delta^{4}+508\delta^{6}-3\delta^{8})z^{6}
\nonumber \\
&&
-4(13312-800\delta^{2}+120\delta^{4}-3\delta^{6})z^{7}+8(4512-20\delta^{2}-15\delta^{4})z^{8}
\nonumber \\
&& -32(336-\delta^{2})z^{9}+1280z^{10}]
\nonumber \\
&& -\delta^{2}(z-2)^{4}[8\delta^{4}(48+22\delta^{2}+3\delta^{4})-32\delta^{4}(24+5\delta^{2})z \nonumber \\
&& -2\delta^{2}(448+16\delta^{2}+8\delta^{4}-3\delta^{6})z^{2}+16\delta^{2}(56-10\delta^{2}-5\delta^{4})z^{3} \nonumber \\
&& +\delta^{2}(1152+272\delta^{2}-3\delta^{4})z^{4}+8(32-92\delta^{2}+5\delta^{4})z^{5}-56(16+\delta^{2})z^{6} \nonumber \\
&&
+512z^{7}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
\alpha}_{\psi}&=&\frac{2\pi}{s^{2}\delta^{2}z^{3}(z-2)^{6}(z^{2}-\delta^{2})}
\{4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{4+\delta^{2}-4z}}\nonumber\\
&&\times[32\delta^{4}(4+\delta^{2})(16+2\delta^{2}+3\delta^{4})
-32\delta^{4}(256+48\delta^{2}+22\delta^{4}+3\delta^{6})z \nonumber \\
&& +16\delta^{2}(1152+1024\delta^{2}-140\delta^{4}
-53\delta^{6}-2\delta^{8})z^{2} \nonumber \\
&&
-8\delta^{2}(5376+128\delta^{2}-1576\delta^{4}-240\delta^{6}-\delta^{8})z^{3}
\nonumber \\
&&
+2(2048-768\delta^{2}-19968\delta^{4}-6968\delta^{6}-350\delta^{8}-3\delta^{10})z^{4}
\nonumber \\
&&
-4(4096-20096\delta^{2}-11168\delta^{4}-1208\delta^{6}-43\delta^{8})z^{5} \nonumber \\
&&+(38912-75392\delta^{2}-16960\delta^{4}
-996\delta^{6}-3\delta^{8})z^{6}\nonumber \\
&&-4(13312-6304\delta^{2}-872\delta^{4}-3\delta^{6})z^{7}+8(4512-500\delta^{2}-15\delta^{4})z^{8}\nonumber \\
&& -32(336-\delta^{2})+1280z^{10}]
\nonumber \\
&&
+\delta^{2}(z-2)^{4}[8\delta^{4}(16+2\delta^{2}+3\delta^{4})-32\delta^{4}(8-\delta^{2})z\nonumber\\
&&-2\delta^{2}(320-272\delta^{2}+64\delta^{4}-3\delta^{6})z^{2}+16\delta^{2}(40-54\delta^{2}-5\delta^{4})z^{3}\nonumber \\
&& -(1024-720\delta^{4}-3\delta^{6})z^{4}+8(96-36\delta^{2}-
5\delta^{4})z^{5} \nonumber \\
&&
+8(80+7\delta^{2})z^{6}-512z^{7}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
S}_{\chi_{c0}}&=&\frac{8\pi}{9s^{3}\delta^{4}z^{5}(z-2)^{8}(z^{2}-\delta^{2})}
\{-4z\sqrt{(1-z)(z^{2}-\delta^{2})(4+\delta^{2}-4z)}\nonumber\\
&&\times[2304\delta^{10}-1152\delta^{8}(26+5\delta^{2})z+192\delta^{6}(640+464\delta^{2}+35\delta^{4})z^{2}\nonumber\\
&& +96\delta^{4}(1152-4816\delta^{2}-1136\delta^{4}-43\delta^{6})z^{3}\nonumber\\
&& +16\delta^{2}(4608-33024\delta^{2}+44752\delta^{4}+4360\delta^{6}+75\delta^{8})z^{4}\nonumber\\
&& -8\delta^{2}(21504-123392\delta^{2}+78448\delta^{4}+2884\delta^{6}-45\delta^{8})z^{5}\nonumber\\
&& -4(12288-156672\delta^{2}+244224\delta^{4}-78128\delta^{6}-512\delta^{8}+21\delta^{10})z^{6}\nonumber\\
&& -2(24576+549888\delta^{2}-356096\delta^{4}+41744\delta^{6}+80\delta^{8}-9\delta^{10})z^{7}\nonumber\\
&& -8(4608-93952\delta^{2}+45728\delta^{4}-1206\delta^{6}+27\delta^{8})z^{8}\nonumber\\
&& +(487424-208384\delta^{2}+119424\delta^{4}+696\delta^{6}-9\delta^{8})z^{9}\nonumber\\
&& -4(155904+4160\delta^{2}+5216\delta^{4}-21\delta^{6})z^{10} \nonumber \\
&&
+16(22976+1480\delta^{2}+85\delta^{4})z^{11}-480(232+11\delta^{2})z^{12}+15104z^{13}]
\nonumber \\
&& +3\delta^{2}(z-2)^{4}[-192\delta^{10}+96\delta^{8}(26+3\delta^{2})z
-64\delta^{6}(160+75\delta^{2}+3\delta^{4})z^{2}\nonumber\\
&& -16\delta^{4}(576-1664\delta^{2}-183\delta^{4}-2\delta^{6})z^{3}\nonumber\\
&& -4\delta^{2}(1536-4608\delta^{2}+4016\delta^{4}+152\delta^{6}+5\delta^{8})z^{4}\nonumber\\
&& +2\delta^{2}(11264-23424\delta^{2}-160\delta^{4}+106\delta^{6}+3\delta^{8})z^{5}\nonumber\\
&& +4(2048-4224\delta^{2}+9952\delta^{4}+248\delta^{6}-27\delta^{8})z^{6}\nonumber\\
&& -(20480-22528\delta^{2}+5312\delta^{4}-368\delta^{6}+3\delta^{8})z^{7}\nonumber\\
&& +4(4096-6496\delta^{2}-600\delta^{4}+17\delta^{6})z^{8}-16(320-472\delta^{2}+7\delta^{4})z^{9}\nonumber \\
&&+32(48+\delta^{2})z^{10}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
\alpha}_{\chi_{c0}}&=&\frac{8\pi}{9s^{3}\delta^{4}z^{5}(z-2)^{8}(z^{2}-\delta^{2})}
\{4z\sqrt{(1-z)(z^{2}-\delta^{2})(4+\delta^{2}-4z)}\nonumber\\
&& \times[2304\delta^{10}-5760\delta^{8}(6+\delta^{2})z+192\delta^{6}(896+424\delta^{2}+35\delta^{4})z^{2}\nonumber\\
&& +96\delta^{4}(384-4528\delta^{2}-904\delta^{4}-43\delta^{6})z^{3}\nonumber\\
&&
+16\delta^{2}(1536+3840\delta^{2}+23536\delta^{4}+2992\delta^{6}+75\delta^{8})z^{4}\nonumber\\
&&
-8\delta^{2}(52224+8704\delta^{2}+3280\delta^{4}+1924\delta^{6}-45\delta^{8})z^{5}\nonumber\\
&&+4(12288+70656\delta^{2}-51200\delta^{4}-34224\delta^{6}-232\delta^{8}-21\delta^{10})z^{6}\nonumber\\
&&+2(24576+336896\delta^{2}+133888\delta^{4}+53904\delta^{6}+376\delta^{8}+9\delta^{10})z^{7}\nonumber\\
&&+16(2304-62720\delta^{2}-11280\delta^{4}-2191\delta^{6}-30\delta^{8})z^{8}\nonumber\\
&&-(487424-605696\delta^{2}-61312\delta^{4}-7016\delta^{6}-9\delta^{8})z^{9}\nonumber\\
&&+4(155904-40768\delta^{2}-2560\delta^{4}-21\delta^{6})z^{10}\nonumber\\
&&-16(22976-504\delta^{2}+85\delta^{4})z^{11}+480(232+11\delta^{2})z^{12}-15104z^{13}]
\nonumber \\
&&
+3\delta^{2}(z-2)^{4}[192\delta^{10}-288\delta^{8}(10+\delta^{2})z
+64\delta^{6}(224+59\delta^{2}+3\delta^{4})z^{2}\nonumber\\
&&+16\delta^{4}(192-1248\delta^{2}-121\delta^{4}-2\delta^{6})z^{3}\nonumber\\
&&+4\delta^{2}(512-12288\delta^{2}+2384\delta^{4}-56\delta^{6}+5\delta^{8})z^{4}\nonumber\\
&&+2\delta^{2}(3072+35968\delta^{2}-160\delta^{4}+50\delta^{6}-3\delta^{8})z^{5}\nonumber\\
&&-4(2048-4992\delta^{2}+8224\delta^{4}-408\delta^{6}-23\delta^{8})z^{6}\nonumber\\
&&+(12288-51200\delta^{2}-3008\delta^{4}-1456\delta^{6}-3\delta^{8})z^{7}\nonumber\\
&&-4(2048-8992\delta^{2}-1224\delta^{4}-17\delta^{6})z^{8}+16(192-616\delta^{2}-7\delta^{4})z^{9}\nonumber\\
&&+32(16+\delta^{2})z^{10}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
S}_{\chi_{c1}}&=&\frac{-8\pi}{3s^{3}\delta^{4}z^{5}(z-2)^{8}(z^{2}-\delta^{2})}
\{4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{(4+\delta^{2}-4z)}}[2304\delta^{8}(3+\delta^{2})(4+\delta^{2})\nonumber\\
&&
-1152\delta^{6}(192+208\delta^{2}+62\delta^{4}+5\delta^{6})z\nonumber\\
&&+192\delta^{4}(3072+6400\delta^{2}+3568\delta^{4}+668\delta^{6}+35\delta^{8})z^{2}\nonumber\\
&&-96\delta^{4}(26624+27808\delta^{2}+9992\delta^{4}+1276\delta^{6}+43\delta^{8})z^{3}\nonumber\\
&&+16\delta^{2}(36864+277248\delta^{2}+195296\delta^{4}+50464\delta^{6}+4406\delta^{8}+75\delta^{10})z^{4}\nonumber\\
&&-8\delta^{2}(258048+521984\delta^{2}+302624\delta^{4}+57800\delta^{6}+2672\delta^{8}-45\delta^{10})z^{5}\nonumber\\
&&-4(98304-753664\delta^{2}-564992\delta^{4}-310048\delta^{6}-37736\delta^{8}-172\delta^{10}+21\delta^{12})z^{6}\nonumber\\
&&+2(983040-659456\delta^{2}+84480\delta^{4}-103008\delta^{6}-9000\delta^{8}-220\delta^{10}+9\delta^{12})z^{7}\nonumber\\
&&-(4784128+2330624\delta^{2}+1528576\delta^{4}+120800\delta^{6}+396\delta^{8}+117\delta^{10})z^{8}\nonumber\\
&&+(6914048+3928064\delta^{2}+1137792\delta^{4}+74544\delta^{6}+1900\delta^{8}-9\delta^{10})z^{9}\nonumber\\
&&-2(3100672+1294336\delta^{2}+200672\delta^{4}+8036\delta^{6}-9\delta^{8})z^{10}\nonumber\\
&&+8(443392+116992\delta^{2}+8048\delta^{4}+35\delta^{6})z^{11}\nonumber\\
&&-64(20288+2808\delta^{2}+51\delta^{4})z^{12}+512(544+33\delta^{2})z^{13}-28672z^{14}]\nonumber \\
&&-3\delta^{2}(z-2)^{4}[-192\delta^{8}(3+\delta^{2})+96\delta^{6}(48+28\delta^{2}+3\delta^{4})z\nonumber\\
&&-16\delta^{4}(768+808\delta^{2}+217\delta^{4}+12\delta^{6})z^{2}\nonumber\\
&&+16\delta^{4}(1600+652\delta^{2}+105\delta^{4}+2\delta^{6})z^{3}\nonumber\\
&&+4\delta^{2}(7168-4352\delta^{2}-360\delta^{4}-59\delta^{6}-5\delta^{8})z^{4}\nonumber\\
&&-2\delta^{2}(24576-3968\delta^{2}+1024\delta^{4}-64\delta^{6}-3\delta^{8})z^{5}\nonumber\\
&&+\delta^{2}(17408-7296\delta^{2}+136\delta^{4}-51\delta^{6})z^{6}\nonumber\\
&&-(8192-12800\delta^{2}-8576\delta^{4}-300\delta^{6}+3\delta^{8})z^{7}\nonumber\\
&&+2(8192-6656\delta^{2}-1328\delta^{4}+17\delta^{6})z^{8}-128(80-10\delta^{2}+\delta^{4})z^{9}\nonumber\\
&&+128(24+5\delta^{2})z^{10}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
\alpha}_{\chi_{c1}}&=&\frac{-8\pi}{3s^{3}\delta^{4}z^{5}(z-2)^{8}(z^{2}-\delta^{2})}
\{-4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{(4+\delta^{2}-4z)}}[2304\delta^{8}(1+\delta^{2})(4+\delta^{2})\nonumber\\
&&-1152\delta^{6}(64+80\delta^{2}+42\delta^{4}+5\delta^{6})z\nonumber\\
&&+192\delta^{4}(1024+2432\delta^{2}+1360\delta^{4}+404\delta^{6}+35\delta^{8})z^{2}\nonumber\\
&&-96\delta^{4}(8192+11872\delta^{2}+3640\delta^{4}+652\delta^{6}+43\delta^{8})z^{3}\nonumber\\
&&+16\delta^{2}(110592+58624\delta^{2}+71328\delta^{4}+12864\delta^{6}+1522\delta^{8}+75\delta^{10})z^{4}\nonumber\\
&&-8\delta^{2}(724992-245504\delta^{2}-21024\delta^{4}-72\delta^{6}+136\delta^{8}-45\delta^{10})z^{5}\nonumber\\
&&+4(98304+1392640\delta^{2}\delta^{4}-349344\delta^{6}
-20648\delta^{8}-1756\delta^{10}-21\delta^{12})z^{6}\nonumber\\
&&-2(983040-856064\delta^{2}\delta^{4}-704352\delta^{6}-37736\delta^{8}-724\delta^{10}-9\delta^{12})z^{7}\nonumber\\
&&+(4784128\delta^{2}\delta^{4}-760736\delta^{6}-20452\delta^{8}-447\delta^{10})z^{8}\nonumber\\
&&-(6914048\delta^{2}\delta^{4}-202576\delta^{6}-4772\delta^{8}-9\delta^{10})z^{9}\nonumber\\
&&+2(3100672\delta^{2}-376864\delta^{4}-15260\delta^{6}-9\delta^{8})z^{10}\nonumber\\
&&-8(443392-47360\delta^{2}-10128\delta^{4}+35\delta^{6})z^{11}
+64(20288+472\delta^{2}+51\delta^{4})z^{12}\nonumber\\
&&-512(544+33\delta^{2})z^{13}+28672z^{14}]\nonumber \\
&&+3\delta^{2}(z-2)^{4}[-192\delta^{8}(1+\delta^{2})+96\delta^{6}(16+12\delta^{2}+3\delta^{4})z\nonumber\\
&&-16\delta^{4}(256+344\delta^{2}+59\delta^{4}+12\delta^{6})z^{2}
+16\delta^{4}(448+404\delta^{2}-9\delta^{4}+2\delta^{6})z^{3}\nonumber\\
&&+4\delta^{2}(5120+1792\delta^{2}-856\delta^{4}+135\delta^{6}-5\delta^{8})z^{4}\nonumber\\
&&-2\delta^{2}(16384+6016\delta^{2}-1088\delta^{4}+24\delta^{6}-3\delta^{8})z^{5}\nonumber\\
&&+(32768-25600\delta^{2}-7040\delta^{4}-1864\delta^{6}-57\delta^{8})z^{6}\nonumber\\
&&-(57344-81408\delta^{2}-11904\delta^{4}-884\delta^{6}-3\delta^{8})z^{7}\nonumber\\
&&+2(16384-27648\delta^{2}-2384\delta^{4}-17\delta^{6})z^{8}-128(48-118\delta^{2}-\delta^{4})z^{9}\nonumber\\
&&-128(8+5\delta^{2})z^{10}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
S}_{\chi_{c2}}&=&\frac{-8\pi}{9s^{3}\delta^{4}z^{5}(z-2)^{8}(z^{2}-\delta^{2})}
\{4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{(4+\delta^{2}-4z)}}\nonumber\\
&&\times[2304\delta^{6}(4+\delta^{2})(144+57\delta^{2}+5\delta^{4})
-1152\delta^{6}(6336+3424\delta^{2}+558\delta^{4}+25\delta^{6})z\nonumber\\
&&-192\delta^{4}(12288-82496\delta^{2}-39168\delta^{4}-5180\delta^{6}-175\delta^{8})z^{2}\nonumber\\
&&+96\delta^{4}(125952-175584\delta^{2}-80408\delta^{4}-8852\delta^{6}-215\delta^{8})z^{3}\nonumber\\
&&+16\delta^{2}(73728\delta^{2}+532640\delta^{4}+310240\delta^{6}+29834\delta^{8}+375\delta^{10})z^{4}\nonumber\\
&&-8\delta^{2}(651264\delta^{2}+224480\delta^{4}+323960\delta^{6}+24824\delta^{8}-225\delta^{10})z^{5}\nonumber\\
&&-4(98304\delta^{2}+3741440\delta^{4}-280928\delta^{6}
-327288\delta^{8}\nonumber\\
&&-15148\delta^{10}+105\delta^{12})z^{6}+2(1179648\delta^{2}\nonumber\\
&&+1172992\delta^{4}-796064\delta^{6}
-273016\delta^{8}-9940\delta^{10}+45\delta^{12})z^{7}\nonumber\\
&&-(7471104\delta^{2}\delta^{4}-864288\delta^{6}-131084\delta^{8}-1377\delta^{10})z^{8}\nonumber\\
&&+(14909440\delta^{2}\delta^{4}-33264\delta^{6}+1164\delta^{8}-45\delta^{10})z^{9}\nonumber\\
&&-2(9654272+318976\delta^{2}+139168\delta^{4}+39524\delta^{6}+447\delta^{8})z^{10}\nonumber\\
&&+8(1980416+242048\delta^{2}+25120\delta^{4}+883\delta^{6})z^{11}\nonumber\\
&&-64(119296+10832\delta^{2}+245\delta^{4})z^{12}+1024(1840+73\delta^{2})z^{13}-188416z^{14}]\nonumber \\
&&-3\delta^{2}(z-2)^{4}[-192\delta^{6}(144+57\delta^{2}+5\delta^{4})
+96\delta^{6}(1008+304\delta^{2}+15\delta^{4})z\nonumber\\
&&+16\delta^{4}(4224-6392\delta^{2}-1731\delta^{4}-60\delta^{6})z^{2}\nonumber\\
&&-16\delta^{4}(13632-916\delta^{2}-705\delta^{4}-10\delta^{6})z^{3}\nonumber\\
&&-4\delta^{2}(15360-56448\delta^{2}-8648\delta^{4}+433\delta^{6}+25\delta^{8})z^{4}\nonumber\\
&&+2\delta^{2}(96256-29568\delta^{2}-13280\delta^{4}-340\delta^{6}+15\delta^{8})z^{5}\nonumber\\
&&+(16384-193536\delta^{2}-28672\delta^{4}+14680\delta^{6}+399\delta^{8})z^{6}\nonumber\\
&&-5(8192-11776\delta^{2}-3712\delta^{4}+604\delta^{6}+3\delta^{8})z^{7}\nonumber\\
&&+2(10240+9728\delta^{2}-2784\delta^{4}-79\delta^{6})z^{8}+512(4-47\delta^{2}+2\delta^{4})z^{9}\nonumber\\
&&+256(12+19\delta^{2})z^{10}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

\begin{eqnarray}
{\rm
\alpha}_{\chi_{c2}}&=&\frac{8\pi}{9s^{3}\delta^{4}z^{5}(z-2)^{8}(z^{2}-\delta^{2})}
\{4z\sqrt{\frac{(1-z)(z^{2}-\delta^{2})}{(4+\delta^{2}-4z)}}\nonumber\\
&&\times[2304\delta^{6}(4+\delta^{2})(48+3\delta^{2}+5\delta^{4})
-1152\delta^{6}(2112+576\delta^{2}+170\delta^{4}+25\delta^{6})z\nonumber\\
&&-192\delta^{4}(6144-33088\delta^{2}-7296\delta^{4}-1428\delta^{6}-175\delta^{8})z^{2}\nonumber\\
&&+96\delta^{4}(64512-102688\delta^{2}-16232\delta^{4}-1860\delta^{6}-215\delta^{8})z^{3}\nonumber\\
&&-16\delta^{2}(24576+1006848\delta^{2}-639968\delta^{4}-83584\delta^{6}-5006\delta^{8}-375\delta^{10})z^{4}\nonumber\\
&&+8\delta^{2}(552960+3044608\delta^{2}\delta^{4}-209864\delta^{6}-6608\delta^{8}+225\delta^{10})z^{5}\nonumber\\
&&+4(98304\delta^{2}\delta^{4}+2831904\delta^{6}
+398216\delta^{8}\nonumber\\
&&+2884\delta^{10}-105\delta^{12})z^{6}-2(11796486\delta^{2}\nonumber\\
&&\delta^{4}+4153696\delta^{6}
+362376\delta^{8}+4228\delta^{10}-45\delta^{12})z^{7}\nonumber\\
&&+(74711040\delta^{2}\delta^{4}+3212896\delta^{6}+207588\delta^{8}-285\delta^{10})z^{8}\nonumber\\
&&-(149094402\delta^{2}\delta^{4}+769936\delta^{6}+15836\delta^{8}-45\delta^{10})z^{9}\nonumber\\
&&+2(9654272\delta^{2}-388704\delta^{4}+34020\delta^{6}+447\delta^{8})z^{10}\nonumber\\
&&-8(1980416-193664\delta^{2}-9312\delta^{4}+883\delta^{6})z^{11}\nonumber\\
&&+64(119296+3792\delta^{2}+245\delta^{4})z^{12}-1024(1840+73\delta^{2})z^{13}+188416z^{14}]\nonumber \\
&&-3\delta^{2}(z-2)^{4}[-192\delta^{6}(48+3\delta^{2}+5\delta^{4})
+288\delta^{6}(112+8\delta^{2}+5\delta^{4})z\nonumber\\
&&+16\delta^{4}(1920-3784\delta^{2}+71\delta^{4}-60\delta^{6})z^{2}\nonumber\\
&&-16\delta^{4}(6336-3660\delta^{2}+289\delta^{4}-10\delta^{6})z^{3}\nonumber\\
&&-4\delta^{2}(13312-43392\delta^{2}+9928\delta^{4}-805\delta^{6}+25\delta^{8})z^{4}\nonumber\\
&&+2\delta^{2}(92160-76928\delta^{2}+6944\delta^{4}-676\delta^{6}+15\delta^{8})z^{5}\nonumber\\
&&+(16384-322560\delta^{2}+121856\delta^{4}+10920\delta^{6}+365\delta^{8})z^{6}\nonumber\\
&&-(24576-248320\delta^{2}+96896\delta^{4}+5332\delta^{6}-15\delta^{8})z^{7}\nonumber\\
&&+2(14336-40960\delta^{2}+12768\delta^{4}+79\delta^{6})z^{8}-512(36-55\delta^{2}+2\delta^{4})z^{9}\nonumber\\
&&-256(4+19\delta^{2})z^{10}]\ln\frac{z\sqrt{4+\delta^{2}-4z}+2\sqrt{(1-z)(z^{2}-\delta^{2})}}
{z\sqrt{4+\delta^{2}-4z}-2\sqrt{(1-z)(z^{2}-\delta^{2})}}\}.
\end{eqnarray}

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\newpage

\begin{figure}
\begin{center}
%\vspace{-2cm}
\includegraphics[width=14cm,height=18cm]{feynman.ps}
%\vspace{-4cm}
\end{center}
\caption{ Feynman diagrams for $e^+ +
e^-\rightarrow\gamma^*\rightarrow$ Charmonium + $c\bar{c}$.}
\label{feynman}
\end{figure}


\newpage

\begin{figure}
\begin{center}
\vspace{5.0cm}
\includegraphics[width=14cm,height=8cm]{etc.eps}
\vspace{-0cm}
\end{center}
\caption{ Cross sections for $e^+ e^-\rightarrow \eta_{c} +
c\bar{c}$ plotted against the center-of-mass energy. Dotted line
illustrates the fragmentation calculation and solid line
illustrates the complete calculation. The cross sections are in
units of $\sigma_{cc}=
\sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow c\bar{c})$
  times $10^{-4}$.
}\label{fig1}
\end{figure}


\newpage

\begin{figure}
\begin{center}
\vspace{5.0cm}
\includegraphics[width=14cm,height=8cm]{js.eps}
\vspace{-0cm}
\end{center}
\caption{ Cross sections for $e^+ e^-\rightarrow J/\psi +
c\bar{c}$ plotted against the center-of-mass energy. Dotted line
illustrates the fragmentation calculation and solid line
illustrates the complete calculation.} \label{fig2}
\end{figure}


\newpage

\begin{figure}
\begin{center}
\vspace{5.0cm}
\includegraphics[width=14cm,height=8cm]{kc0.eps}
\vspace{-0cm}
\end{center}
\caption{ Cross sections for $e^+ e^-\rightarrow \chi_{c0} +
c\bar{c}$ plotted against the center-of-mass energy. Dotted line
illustrates the fragmentation calculation and solid line
illustrates the complete calculation.} \label{fig3}
\end{figure}



\newpage

\begin{figure}
\begin{center}
\vspace{5.0cm}
\includegraphics[width=14cm,height=8cm]{kc1.eps}
\vspace{-0cm}
\end{center}
\caption{ Cross sections for $e^+ e^-\rightarrow \chi_{c1} +
c\bar{c}$ plotted against the center-of-mass energy. Dotted line
illustrates the fragmentation calculation and solid line
illustrates the complete calculation.} \label{fig4}
\end{figure}



\newpage

\begin{figure}
\begin{center}
\vspace{5.0cm}
\includegraphics[width=14cm,height=8cm]{kc2.eps}
\vspace{-0cm}
\end{center}
\caption{ Cross sections for $e^+ e^-\rightarrow \chi_{c2} +
c\bar{c}$ plotted against the center-of-mass energy. Dotted line
illustrates the fragmentation calculation and solid line
illustrates the complete calculation.} \label{fig5}
\end{figure}
\end{document}

