\documentclass[12pt,sec]{article}
\usepackage{epsfig}
\usepackage{axodraw}
\setlength{\textwidth}{16.5cm}
  \setlength{\textheight}{23cm}
\setlength{\oddsidemargin}{-0.1cm}
 \setlength{\topmargin}{-1.4cm}
%\renewcommand{\baselinestretch}{2}
\begin{document}

\begin{center}
{\large  Problems of Double Charm Production in $e^+e^-$ Annihilation at $\sqrt{s}=10.6$ GeV}\\[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}

Using the nonrelativistic QCD(NRQCD) factorization formalism, we
calculate the color-singlet cross sections for exclusive
production processes ${e^++e^-\rightarrow J/\psi+\eta_c}$~ and
~$e^++e^-\rightarrow J/\psi + \chi_{cJ}$~$(J=0,1,2)$ at the
center-of-mass energy $\sqrt{s}$=10.6 GeV.  The cross sections are
estimated to be $5.5$fb, $6.7$fb, $1.1$fb, and $1.6$fb for
$\eta_c, \chi_{c0}, \chi_{c1}$ and $\chi_{c2}$ respectively. The
calculated $J/\psi\eta_c$ production rate is smaller than the
recent Belle data by about an order of magnitude, indicating the
failure of perturbative QCD calculation to explain the double
charmonium production data. In addition, we also calculate the
$\chi_{c0}$ inclusive double charm production cross section of
${e^++e^-\rightarrow \chi_{c0}+ c\bar {c} }$ in perturbative QCD
and find that its value, which should be much larger than, is
however saturated by the calculated cross sections of
${e^++e^-\rightarrow J/\psi + \chi_{c0}}$ plus
${e^++e^-\rightarrow \psi(2S) + \chi_{c0}}$. This again indicates
that the perturbative QCD calculation for the charmonium (e.g.
$\chi_{c0}$) inclusive double charm production seriously
underestimates the production rates, and nonperturbative QCD
effects have to be considered at low energies such as
$\sqrt{s}$=10.6 GeV.

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

\end{abstract}

Heavy quarkonium production is interesting in understanding both
perturbative and nonperturbative quantum chromodynamics (QCD). In
recent years the charmonium production has been studied in various
processes, such as in hadron-hadron collision, electron-proton
collission, fixed target experiments, $B$ meson decays, as well as
$Z^0$ decays. Among them, the study of charmonium production in
$e^+e^-$ annihilation is particularly interesting in testing the
quarkonium production mechanisms, the color-singlet model and the
color-octet model in the nonrelativistic QCD (NRQCD)\cite{bbl}
approach. This is not only because of the simpler parton structure
involved in this process, which may be helpful in reducing the
theoretical uncertainty, but also because of the spectacular
experimental prospect opened up by the two $B$ factories with
BaBar and Belle, which will allow a fine data analysis for
charmonium production with more than $10^8$ $e^+e^-$ annihilation
events in the continuum at $\sqrt{s}$=10.6 GeV.

Recently the Belle Collaboration has reported the observation of
prompt $J/\psi$ via double $c\bar{c}$ production from the
${e^+e^-}$ continuum\cite{belle}. For these results, not only the
large fraction of the inclusive $J/\psi$ production due to the
double $c\bar{c}$ production is puzzling\cite{chao}, but also the
exclusive production rate of $J/\psi\eta_c$, $\sigma(e^+e^-
\rightarrow J/\psi \eta_c(\gamma))\times{\cal B}(\eta_c\rightarrow
\geq 4 charged)=(0.033_{-0.006}^{+0.007}\pm0.009)$pb, may not be
consistent with both the previous calculations\cite{kane,brod},
which gave a cross section of a few pb for $J/\psi \eta_c$. In
fact, recent perturbative estimates of the $J/\psi c\bar{c}$ cross
section are only about 0.07 pb\cite{cho,yuan,kiselev}. So the
calculations of exclusive cross sections for $e^+e^-$ annihilation
into $J/\psi \eta_c$ and other double charmonium states such as
$J/\psi \chi_{cJ}(J=0,1,2)$ will be useful to clarify the problem.
Experimentally, aside from $e^+e^-\rightarrow
J/\psi\chi_{c0}$\cite{belle}, Belle \cite{belle2} has also studied
processes $e^+e^-\rightarrow \chi_{c1}X$ and $e^+e^-\rightarrow
\chi_{c2}X$, so we hope that the double charmonia production
involving $\chi_{cJ}(J=0,1,2)$ will be detectable in the near
future. In the following we will calculate the cross sections
$\sigma({e^++e^-\rightarrow J/\psi+\eta_c})$ and
$\sigma({e^++e^-\rightarrow J/\psi+\chi_{cJ}})$ in the leading
order perturbative QCD. To this order ($\sim\alpha_s^2$) the
color-singlet channel is dominant since all color-octet channels
are of high order of $v$, which is the relative velocity of the
charm quark and anti-charm quark in the charmonium, and therefore
suppressed. Furthermore, in order to compare inclusive production
with exclusive production rates associated with the $\chi_{cJ}$
charmonium states, we will calculate the cross section
$\sigma({e^++e^-\rightarrow \chi_{c0}+c\bar{c}})$, and this will
again reveal the failure of perturbative QCD calculation for the
charmonium inclusive production at $\sqrt{s}$=10.6 GeV.

\begin{figure}[t]
\begin{center}
\vspace{-2.8cm}
\includegraphics[width=14cm,height=16cm]{fig1.ps}
\vspace{-4cm}
\end{center}
\caption{ Feynman diagrams for $e^+e^-\rightarrow J/\psi+\eta_c(
\chi_{cJ} )$.} \label{fig1}
\end{figure}
We now write down the scattering amplitude which describes the
creation of two color-singlet $Q\bar{Q}$ pairs in the $e^+e^-$
annihilation process in Fig.~\ref{fig1} as
\begin{eqnarray}
%\hspace{-1.0cm}\hspace{1.0cm}
\label{amp1}   &&\hspace{-1cm}{\cal A}(a+b\rightarrow
Q\bar{Q}({}^{2S_\psi+1}L_{J_\psi})(P_3)+Q\bar{Q}({}^{2S
+1}L_{J})(P_4))= \sum\limits_{L_{\phi z} S_{\psi z}
}\sum\limits_{s_1s_2 }\sum\limits_{jk} \sum\limits_{L_z S_z
}\sum\limits_{s_3 s_4}\sum\limits_{il}\int\frac{d^3
\vec{q}_3}{(2\pi)^3 2q_3^0}\nonumber
\\ & \times&\delta(q_3^0
-\frac{\vec{q_3}^2}{2m_c})Y_{L_\psi L_{\psi
z}}^*(\hat{q_3})<s_1;s_2\mid S_\psi S_{\psi z}> <L_\psi L_{\psi z
};S_\psi S_{\psi z}\mid J_\psi J_{\psi z}><3j;\bar{3}k\mid
1>\nonumber\\&\times&\int\frac{d^3 \vec{q}_4}{(2\pi)^3
2q_4^0}\delta(q_4^0-\frac{\vec{q_4}^2}{2m_c}) Y_{L
L_z}^*(\hat{q_4})<s_3;s_4\mid S S_z><L L_z ;S S_z\mid
J J_z><3l;\bar{3}i\mid 1>\nonumber\\
 &\times&{\cal A}(a+b\rightarrow
 Q_j(\frac{P_3}{2}+q_3)+\bar{Q}_k(\frac{P_3}{2}-q_3)+
 Q_l(\frac{P_4}{2}+q_4)+\bar{Q}_i(\frac{P_4}{2}-q_4)),\nonumber\\
\end{eqnarray}
where $<3j;\bar{3}k\mid 1>=\delta_{jk}/\sqrt{N_c}$ and
$<3l;\bar{3}i\mid 1>=\delta_{li}/\sqrt{N_c}$~,~ $<s_1;s_2\mid
S_\psi S_{\psi z}>$ and $<s_3;s_4\mid S S_z>$~,~ $<L_\psi L_{\psi
z };S_\psi S_{\psi z}\mid J_\psi J_{\psi z}>$ and $ <L L_z ;S
S_z\mid J J_z>$ are respectively the color-SU(3), spin-SU(2), and
angular momentum Clebsch-Gordan coefficients for $Q\bar{Q}$ pairs
projecting out appropriate bound states.

After integrating over the relative momenta $q_3$ and $q_4$, we
get
\begin{eqnarray}
%\hspace{-1.0cm}\hspace{1.0cm}
\label{amp2}   &&\hspace{-2.0cm}{\cal A}(a+b\rightarrow
Q\bar{Q}({}^{2S_\psi+1}L_{J_\psi})(P_3)+Q\bar{Q}({}^{2S
+1}L_{J})(P_4))= \sqrt{C_{L_\psi}}\sqrt{C_L}\sum\limits_{L_{\phi
z} S_{\psi z} }\sum\limits_{s_1s_2 }\sum\limits_{jk}
\sum\limits_{L_z S_z }\sum\limits_{s_3
s_4}\sum\limits_{il}\nonumber\\
&\times&<s_1;s_2\mid S_\psi S_{\psi z}> <L_\psi L_{\psi z };S_\psi
S_{\psi z}\mid J_\psi J_{\psi z}><3j;\bar{3}k\mid
1>\nonumber\\&\times&<s_3;s_4\mid S S_z><L L_z ;S S_z\mid
J J_z><3l;\bar{3}i\mid 1>\nonumber\\
 &\times&\left\{
\begin{array}{ll}
{\cal A}(a+b\rightarrow
 Q_j(\frac{P_3}{2})+\bar{Q}_k(\frac{P_3}{2})+
 Q_l(\frac{P_4}{2})+\bar{Q}_i(\frac{P_4}{2}))&(L=S),\\
\epsilon^*_{\alpha}(L_Z) {\cal A}^\alpha(a+b\rightarrow
 Q_j(\frac{P_3}{2})+\bar{Q}_k(\frac{P_3}{2})+
 Q_l(\frac{P_4}{2})+\bar{Q}_i(\frac{P_4}{2}))
&(L=P),\\
\end{array}
\right.\nonumber\\
\end{eqnarray}
where the coefficients $C_{L_\psi}$ and $C_L$ can be related to
the radial wave function of the bound states $R_{nl}^{(l)} (0)$ as
\begin{equation}
\label{cs} C_S=\frac{1}{4\pi}|R_S (0)|^2,\ \ \
C_P=\frac{3}{4\pi}|R_P'(0)|^2.
\end{equation}
We introduce the spin projection operators $P_{SS_z}(P,q)$ as
\begin{equation}
P_{SS_z}(P,q)\equiv\sum\limits_{s_1s_2
}<s_1;s_2|SS_z>v(\frac{P}{2}+q;s_1)\bar{u}(\frac{P}{2}-q;s_2).
\end{equation}
Expanding $P_{SS_z}(P,q)$ in terms of the relative momentum $q$,
we get the projection operators and their derivatives, which will
be used in our calculation, as follows

\begin{equation}
\label{pjs} P_{1S_z}(P,0)=\frac{1}{2\sqrt{2}}\ \epsilon\!\!
/^*(S_z)(\not{P}+2m_c),
\end{equation}

\begin{equation}
\label{petc} P_{00}(P,0)=\frac{1}{2\sqrt{2}}\gamma_5(P\!\!\!\!
/+2m_c),
\end{equation}

\begin{equation}
\label{der} P_{1S_z}^{\alpha}(P,0)=\frac{1}{2\sqrt{2}m_c}
[\gamma^{\alpha}\not{\epsilon}^*(S_z)(\not{P}+2m_c)-
(\not{P}-2m_c)\not{\epsilon}\gamma^{\alpha}].
\end{equation}

Then one can calculate the cross sections for the on-shell quarks
in the factorized form of NRQCD\cite{bbl}. The cross section for
${e^++e^-\rightarrow J/\psi+\eta_c}$ process in Fig.~\ref{fig1} is
given by
\begin{eqnarray}
%\hspace{-1.0cm}\hspace{1.0cm}
\label{jsetc} \sigma(a(p_1)+b(p_2)\rightarrow
J/\psi(P_3)+\eta_c(P_4))=\frac{2\pi\alpha^2\alpha_s^2|R_s(0)|^4\sqrt{s-16m_c^2}
}{81m_c^2s^{3/2}}\int^1_{-1}|\bar{M}|^2 d\cos\theta,
\end{eqnarray}
where
\begin{equation}
|\bar{M}|^2=\frac{-16384m_c^2(32m_c^4-t^2-u^2)}{s^5}.
\end{equation}
The Mandelstam variables are defined as
\begin{eqnarray}
s=(p_1+p_2)^2,\\
t=(p_3-p_1)^2,\\
u=(p_3-p_2)^2.
\end{eqnarray}
The cross section for ${e^++e^-\rightarrow J/\psi+\chi_{cJ}}$
process is
\begin{eqnarray}
\label{jskc} &&\hspace{-3.0cm} \sigma(a(p_1)+b(p_2)\rightarrow
J/\psi(P_3)+\chi_{c_J}(P_4))=\nonumber
\\ && \frac{2\pi\alpha^2\alpha_s^2|R_s(0)|^2|R_p'(0)|^2\sqrt{s-16m_c^2}
}{27m_c^2s^{3/2}}\int^1_{-1}|\bar{M}_J|^2 d\cos\theta,
\end{eqnarray}
where  $|\bar{M}_J|^2$ for $\chi_{c0}$, $\chi_{c1}$ and
$\chi_{c2}$ are given by

\begin{eqnarray}
%\hspace{-1.0cm}\hspace{1.0cm}
\label{kc0} &&\hspace{-1.1cm}
|\bar{M}_0|^2=2048(90112m_c^{10}-74752m_c^8t-74752m_c^8u+23360m_c^6t^2
+43136m_c^6tu+23360m_c^6u^2\nonumber\\&&-3152m_c^4t^3-7600m_c^4t^2u-
7600m_c^4tu^2-3152m_c^4u^3+162m_c^2t^4+444m_c^2t^3u\nonumber\\&&+564m_c^2t^2u^2
+444m_c^2tu^3+162m_c^2u^4-t^4u-3t^3u^2-3t^2u^3-tu^4)/(3s^7m_c^2),
\end{eqnarray}
\begin{eqnarray}
%\hspace{-1.0cm}\hspace{1.0cm}
\label{kc1} &&\hspace{-2.5cm}
|\bar{M}_1|^2=32768(1792m_c^8+256m_c^6t+256m_c^6u-56m_c^4t^2
-64m_c^4tu-56m_c^4u^2-4m_c^2t^3\nonumber\\ &&
-20m_c^2t^2u-20m_c^2tu^2-4m_c^2u^3+t^4+2t^3u+2t^2u^2+2tu^3+u^4)/s^7,
\end{eqnarray}
\begin{eqnarray}
%\hspace{-1.0cm}\hspace{1.0cm}
\label{kc2} &&\hspace{-1.5cm}
|\bar{M}_2|^2=4096(145408m_c^{10}-1024m_c^8t-1024m_c^8u-2368m_c^6t^2
-6400m_c^6tu-2368m_c^6u^2\nonumber\\
&&+16m_c^4t^3-208m_c^4t^2u-208m_c^4tu^2
+16m_c^4u^3+24m_c^2t^4+72m_c^2t^3u+96m_c^2t^2u^2\nonumber\\
&&+72m_c^2tu^3 +24m_c^2u^4-t^4u-3t^3u^2-3t^2u^3-tu^4)/(3s^7m_c^2).
\end{eqnarray}

 In the numerical calculations, we choose
$\sqrt{s}=10.6{\rm GeV},~~m_c=1.5{\rm GeV},~ \alpha_s=0.26$,
~$|R_s(0)|^2 =0.810{\rm GeV}^3$ and $|R_p'(0)|^2=0.075 {\rm
GeV^5}$\cite{wf}, and assume that in the non-relativistic
approximation $m_{J/\psi}=m_{\eta_c}=m_{\chi_{cJ}}=2m_c$. The
numerical result for $e^+e^-\rightarrow J/\psi\eta_c$ is
\begin{equation}
\label{jsetac} \sigma(e^+e^-\rightarrow J/\psi\eta_c)=5.5 fb.
\end{equation}

While the numerical result for the cross section of
$e^+e^-\rightarrow J/\psi\eta_c$ is more than a factor of six
smaller than the experimental data\cite{belle} (with uncertainties
due to the unknown decay branching fractions into 4-charged
particles for the $\eta_c$), the calculated ratio of
$\sigma(e^+e^-\rightarrow J/\psi\eta_c)/\sigma(e^+e^-\rightarrow
J/\psi c\bar{c})\approx 0.079$ might be consistent with the
experimental result with the choice of $\sigma(e^+e^-\rightarrow
J/\psi c\bar{c})=70$ fb\cite{cho,yuan,kiselev}.


The cross sections for $J/\psi\chi_{cJ}$ production are given as
\begin{equation}
\label{jskc0} \sigma(e^+e^-\rightarrow J/\psi\chi_{c_0})=6.7 fb,
\end{equation}
\begin{equation}
\label{jskc1} \sigma(e^+e^-\rightarrow J/\psi\chi_{c1})=1.1 fb,
\end{equation}
\begin{equation}
\label{jskc2} \sigma(e^+e^-\rightarrow J/\psi\chi_{c2})=1.6 fb.
\end{equation}

In Fig.~\ref{fig2}, we show the cross sections as functions of the
$e^+e^-$ center-of-mass energy $\sqrt{s}$, and we can see that the
cross sections for $e^+e^-\rightarrow J/\psi\eta_c$,
$e^+e^-\rightarrow J/\psi\chi_{c1}$ and $e^+e^-\rightarrow
J/\psi\chi_{c2}$ decrease rapidly as $\sqrt{s}$ increases.

\begin{figure}[t]
\begin{center}
%\vspace{2cm}
\includegraphics[width=12cm, height=10cm]{fig2.eps}
\vspace{-1cm}
\end{center}
\caption{Cross sections for $\sigma(e^+e^-\rightarrow J/\psi
\eta_c)$(solid line) and $\sigma(e^+e^-\rightarrow J/\psi
\chi_{cJ})$(dashed line for $J=1$, dotted line for $J=2$) plotted
against the $e^+e^-$ center-of -mass energy $\sqrt{s}$ with
z=$\sqrt{s/s_0}$ and $\sqrt{s_0}=10.6 {\rm GeV}$.}
 \label{fig2}
\end{figure}
At $\sqrt{s}=10.6 {\rm GeV}$ if we choose
$\sigma(e^+e^-\rightarrow \chi_{c1}c\bar{c})=18.1 fb$ and
$\sigma(e^+e^-\rightarrow \chi_{c2}c\bar{c})=8.4 fb$ which were
obtained in the fragmentation approximation  \cite{schuler}, then
we have the ratio
\begin{equation}
\sigma(e^+e^-\rightarrow J/\psi\chi_{c1})/\sigma(e^+e^-\rightarrow
\chi_{c1}c\bar{c})=0.061,
\end{equation}
\begin{equation}
\sigma(e^+e^-\rightarrow J/\psi\chi_{c2})/\sigma(e^+e^-\rightarrow
\chi_{c2}c\bar{c})=0.19.
\end{equation}

As for the $\chi_{c0}$ inclusive double charm production rate, it
was given under the fragmentation approximation in
Ref.~\cite{chao} as $\sigma ( e^+e^-\rightarrow
\chi_{c0}c\bar{c})$=5.0 fb. It is puzzling that this inclusive
cross section is even smaller than the exclusive cross section
given in Eq.~(\ref{jskc0}).  In order to clarify this problem here
we recalculate $\sigma ( e^+e^-\rightarrow \chi_{c0}c\bar{c})$ in
a complete form to the $\cal O$$(\alpha_s^2)$ order in
perturbative QCD.

We give the amplitude of the first diagram in Fig.~\ref{fig3} for
$e^+e^-\rightarrow \chi_{c0}c\bar{c}$ as
\begin{eqnarray}
&&\hspace{-2cm}
M=\sum\limits_{L_zS_z}\sigma^*_{\sigma}(L_z)<1L_z;1S_z|J=0,J_z=0>=\frac{ie_ceg_s^2[T^aT^a]_{li}}{\sqrt{3}}
\bar{v}(p_1)\gamma^{\mu}u(p_2)\frac{1}{s}\bar{u}_l(p_c)\nonumber
\\ & \times &[\gamma^{\alpha}P_{1S_z}\gamma_{\alpha}(p,0){\cal O}_{\mu}^{\sigma}+
\gamma^{\alpha}P^{\sigma}_{1S_z}\gamma_{\alpha}{\cal
O}_{\mu}]v_i(p_{\bar{c}}),
\end{eqnarray}
where $e_c=\frac{2}{3}e$ , $T^a$ is the $SU(3)$ color matrix, the
matrix $\cal O$$_{\mu}$ is relevant to the on shell amplitude and
$\cal O$$_{\mu}^{\sigma}$ is its derivative with respect to the
relative momentum between the quarks that form the bound state. We
can also express the contributions of other three diagrams in a
similar way, and our numerical results are obtained with the full
contributions of these four diagrams. Some useful information of
the calculation is given in the Appendix.

\begin{figure}[t]
\begin{center}
\vspace{-2cm}
\includegraphics[width=12cm,height=16cm]{fig3.ps}
\vspace{-5cm}
\end{center}
\caption{ Feynman diagrams for $e^++e^-\rightarrow
\chi_{c0}+c\bar{c}$ process.} \label{fig3}
\end{figure}

We finally get the cross section for this process
\begin{equation}
\sigma(e^+e^-\rightarrow\chi_{c0}c\bar{c})=9.2fb,
\end{equation}
which is larger than that given in Ref.~\cite{chao} under the
fragmentation approximation.

Using $\sigma(e^+e^-\rightarrow\chi_{c0}c\bar{c})=9.2fb$ and
Eq.~(\ref{jskc0}) we get
\begin{equation}
\label{ratio}
\sigma(e^+e^-\rightarrow\chi_{c0}J/\psi)/\sigma(e^+e^-\rightarrow\chi_{c0}c\bar{c})=0.73.
\end{equation}

The ratio in Eq.~(\ref{ratio}) would mean that
$\sigma(e^+e^-\rightarrow\chi_{c0}J/\psi)$ plus
$\sigma(e^+e^-\rightarrow\chi_{c0}\psi(2S))$ will saturate the
$\chi_{c0}$ inclusive production cross section. This is not
reasonable because the $D\bar{D}$ meson pair associated with
$\chi_{c0}$ should have a high production rate~\cite{belle}. This
indicates that the perturbative QCD calculation for inclusive
double charm production seriously underestimates the rate at low
energies such as $\sqrt{s}=$10.6GeV. In Fig.~\ref{fig4}, we show
cross sections for $\sigma(e^+e^-\rightarrow \chi_{c0}
c\bar{c})$(solid line) and $\sigma(e^+e^-\rightarrow J/\psi
\chi_{c_0})$(dotted line) plotted against the $e^+e^-$ center-of
-mass energy $\sqrt{s}$ with z=$\sqrt{s/s_0}$ and $\sqrt{s_0}=10.6
{\rm GeV}$. One can see the ratio Eq.~(\ref{ratio}) decreases
drastically as the center-of-mass energy increases.  The ratio is
$2.4\times 10^{-3}$ at $\sqrt{s}=$32GeV and   $1.5\times 10^{-4}$
at $\sqrt{s}=$53GeV, where the associated $D\bar D$ meson pair
production is expected to dominate. This may imply that the
perturbative QCD calculation for double charm production might
become reliable only when the $e^+e^-$ center-of-mass energy is
much larger than the static energies of the four charm quarks
created in this process. The $\chi_{c0}$ double charm inclusive
production may serve as a good example to show the limited
applicability of perturbative QCD at lower energies such as
$\sqrt{s}=10.6$GeV.

\begin{figure}[t]
\begin{center}
\vspace{-1.0cm}
\includegraphics[width=12cm,height=10cm]{fig4.ps}
\vspace{-1cm}
\end{center}
\caption{cross sections for $\sigma(e^+e^-\rightarrow \chi_{c0}
c\bar{c})$(solid line) and $\sigma(e^+e^-\rightarrow J/\psi
\chi_{c0})$(dotted line) plotted against the $e^+e^-$ center-of
-mass energy $\sqrt{s}$ with z=$\sqrt{s/s_0}$ and $\sqrt{s_0}=10.6
{\rm GeV}$.} \label{fig4}
\end{figure}

In summary, despite of many uncertainties due to the relativistic
corrections, the QCD radiative corrections, the possible
color-octet channel contributions, and the physical parameters
(e.g. the charm quark mass and the strong coupling constant), both
the inclusive and exclusive double charm production cross sections
calculated in perturbative QCD turned out to be seriously
underestimated as compared with data. Therefore we intend to
conclude, as in \cite{chao}, that it is hard to explain the double
charm production data observed by Belle based on perturbative QCD
(including both color-singlet and color-octet channels), and
possible nonperturbative QCD effects have to be considered at
$\sqrt{s}=10.6 {\rm GeV}$.

While we were about to submit our result, there appeared one paper
which also considered exclusive double-charmonium
production\cite{bl}. Those authors took the QED effects into
account in addition to the QCD effects that we considered.  We
find our result for the exclusive double-charmonium production is
consistent with theirs but we also analyzed the inclusive
processes which were not discussed in Ref.~\cite{bl}.

\section*{Acknowledgments}
The authors thank L.K. Hao and Z.Z. Song for useful discussions.
We also thank C.F. Qiao for useful comments. 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 cross section for the
$e^+e^-\rightarrow \chi_{c0}c\bar{c}$ process shown in
Fig.~\ref{fig3}.
\begin{equation}
d\sigma=\frac{|\bar{M}|^2}{2s(2\pi)^5}\delta^4(p_1+p_2-p_c-p_{\bar{c}}-p)
\frac{d^3p_c}{2E_c}\frac{d^3p_{\bar{c}}}{2E_{\bar{c}}}\frac{d^3p}{2E}.
\end{equation}
It is convenient to rewrite the cross section as
\begin{eqnarray}
&&\hspace{-0.6cm}  d\sigma=
\frac{|\bar{M}|^2}{2s(2\pi)^5}\delta^4(p_1+p_2-\eta-p)\delta^4(\eta-p_c-p_{\bar{c}})
\frac{d^3p_c}{2E_c}\frac{d^3p_{\bar{c}}}{2E_{\bar{c}}}\frac{d^3p}{2E}d^4\eta\nonumber
\\ &&=\frac{|\bar{M}|^2}{2s(2\pi)^5}\delta^4(p_1+p_2-\eta-p)\delta^4(\eta-p_c-p_{\bar{c}})
\frac{d^3p_c}{2E_c}\frac{d^3p_{\bar{c}}}{2E_{\bar{c}}}\frac{d^3p}{2E}\frac{d^3\eta}{2E_\eta}dm_{\eta}^2,\nonumber\\
\end{eqnarray}
where $m_{\eta}^2=\eta^2$.

The integral over the phase-space of $c\bar{c}$ is evaluated in
the corresponding center of mass frame
\begin{eqnarray}
\frac{d^3p_c'}{2E_c'}\frac{d^3p_{\bar{c}}'}{2E_{\bar{c}}'}\delta^4(\eta'-p_c'-p_{\bar{c}}')=
\frac{1}{8m_{\eta}^2}\lambda^{1/2}(m_{\eta}^2,m_c^2,m_{\bar{c}}^2)d\Omega',
\end{eqnarray}
where
$\lambda^{1/2}(m_{\eta}^2,m_c^2,m_{\bar{c}}^2)=\eta^4+m_c^4+m_{\bar{c}}^4-2m_{\eta}^2m_c^2
-2m_{\eta}^2m_{\bar{c}}^2-2m_c^2m_{\bar{c}}^2$.

\noindent The remaining integration are performed in the center of
mass frame of the $e^+e^-$
\begin{eqnarray}
\frac{d^3p}{2E}\frac{d^3\eta}{2E_{\eta}}\delta^4(p_1+p_2-\eta-p)=
\frac{1}{8s}\lambda^{1/2}(s,m_{\eta}^2,m_p^2)d\Omega.
\end{eqnarray}
Finally we have
\begin{equation}
d\sigma=\frac{|\bar{M}|^2C_P}{64s^2(2\pi)^5m_{\eta}m_c}\lambda^{1/2}(m_{\eta}^2,m_c^2,m_{\bar{c}}^2)
\lambda^{1/2}(m_{\eta}^2,m_c^2,m_{\bar{c}}^2)d\Omega'd\Omega
dm_{\eta}.
\end{equation}
The limit of $m_{\eta}$ is
\begin{equation}
m_c+m_{\bar{c}}\leq m_{\eta}\leq \sqrt{s}-m_p.
\end{equation}
To accomplish the integration we use the Lorentz transformation
between the two frames as $L=R_2R_1$, where

\begin{eqnarray}
R_1=\left ( \begin{array}{cccc}
  \sqrt{1+\frac{\vec{p}^2}{m_{\eta}^2}} & 0 & 0 &-\frac{|\vec{p}|}{m_{\eta}}  \\
  0 & 1 & 0 & 0 \\
  0 & 0 & 1 & 0 \\
  -\frac{|\vec{p}|}{m_{\eta}} & 0 & 0 & \sqrt{1+\frac{\vec{p}^2}{m_{\eta}^2}}
\end{array} \right ),
\end{eqnarray}

\begin{eqnarray}
R_2=\left ( \begin{array}{cccc}
  1 & 0 & 0 & 0 \\
  0 & \cos(\theta) & 0 & \sin(\theta) \\
  0 & 0 & 1 & 0 \\
  0 & -\sin(\theta) & 0 & \cos(\theta)
\end{array} \right ).
\end{eqnarray}

The momenta in the center mass frame of $e^+e^-$ are

\begin{equation}
p_1=(\sqrt{s}/2, 0, 0, \sqrt{s}/2),
\end{equation}

\begin{equation}
p_2=(\sqrt{s}/2, 0, 0, -\sqrt{s}/2),
\end{equation}

\begin{equation}
p_c=R_2R_1p_c',
\end{equation}

\begin{equation}
p_{\bar{c}}=R_2R_1p_{\bar{c}}',
\end{equation}

\begin{equation}
p=(\sqrt{\vec{p}^2+m_p^2}, |\vec{p}|\sin(\theta),0,
|\vec{p}|\cos(\theta)),
\end{equation}
where $p_c'$ and $p_{\bar{c}}'$ are the momenta of $c$ and $
\bar{c}$ in the $\Omega'$ frame, and they are

\begin{equation}
p_c'=(E_c', |\vec{p_c}'|\sin(\theta')\cos(\theta'),
|\vec{p_c}'|\sin(\theta')\cos(\theta'), \vec{p_c}'|\cos(\theta')),
\end{equation}

\begin{equation}
p_{\bar{c}}'=(E_{\bar{c}}',
-|\vec{p_c}'|\sin(\theta')\cos(\theta'),
-|\vec{p_c}'|\sin(\theta')\cos(\theta'),
-|\vec{p_c}'|\cos(\theta')).
\end{equation}
In Fig.~\ref{fig3} the lower (non-fragmentation) diagrams give
very small contributions, so for simplicity here we only write
down the expressions for the contribution of the upper diagrams
and give

\begin{equation}
|\bar{M}|^2=\frac{2(4\pi)^4\alpha^2\alpha_s^2}{27}(aa+2ab+bb).
\end{equation}
We define $pp_1=p.p_1$, $pp_2=p.p_2$, $pp_3=p.p_c$,
$pp_4=p.p_{\bar{c}}$, $p_{13}=p_1.p_c$, $p_{14}=p_1.p_{\bar{c}}$,
$p_{23}=p_2.p_c$, $p_{24}=p_2.p_{\bar{c}}$. We also notify $aa=bb$
and
\begin{eqnarray}
&& \hspace{-0.5cm}aa = (4(800m^{10}s + 800m^8p_{14}pp_2 +
800m^8p_{24}pp_1+1440m^8pp_3s +160m^6p_{13}p_{24}pp_3 \nonumber
\\ &&+ 160m^6p_{14}p_{23}pp_3 + 1440m^6p_{14}pp_2pp_3 + 1440m^6p_{24}pp_1pp_3 + 900m^6pp_3^2s
\nonumber
\\ &&+ 184m^4p_{13}p_{24}pp_3^2 + 184m^4p_{14}p_{23}pp_3^2 + 856m^4p_{14}pp_2pp_3^2 +
856m^4p_{24}pp_1pp_3^2
 \nonumber
\\ &&+ 216m^4pp_3^3s + 56m^2p_{13}p_{24}pp_3^3 + 56m^2p_{14}p_{23}pp_3^3 + 168m^2p_{14}pp_2pp_3^3
\nonumber
\\ &&+ 168m^2p_{24}pp_1pp_3^3 + 13m^2pp_3^4s + 2p_{13}p_{24}pp_3^4+ 2p_{14}p_{23}pp_3^4))/(3m^2s^2(64m^{12}
\nonumber
\\ && +
192m^{10}pp_3+ 240m^8pp_3^2 + 160m^6pp_3^3 + 60m^4pp_3^4 +
12m^2pp_3^5 + pp_3^6)),
\end{eqnarray}

\begin{eqnarray}
&&\hspace{-0cm} ab= (4(400m^{10}s + 400m^8p_{13}pp_2 +
400m^8p_{14}pp_2 + 400m^8p_{23}pp_1+ 400m^8p_{24}pp_1 \nonumber
\\ &&+ 400m^8pp_1pp_2 + 480m^8pp_3s +
400m^8pp_{34}s + 480m^8pp_4s + 80m^6p_{13}p_{24}pp_3 \nonumber
\\ && + 80m^6p_{13}p_{24}pp_4 + 280m^6p_{13}pp_2pp_3 + 440m^6p_{13}pp_2pp_4 + 80m^6p_{14}p_{23}pp_3
 \nonumber
\\ &&+ 80m^6p_{14}p_{23}pp_4 + 440m^6p_{14}pp_2pp_3 + 280m^6p_{14}pp_2pp_4 + 280m^6p_{23}pp_1pp_3
\nonumber
\\ &&+ 440m^6p_{23}pp_1pp_4 + 440m^6p_{24}pp_1pp_3+ 280m^6p_{24}pp_1pp_4 + 240m^6pp_1pp_2pp_3
 \nonumber
\\ &&- 400m^6pp_1pp_2pp_{34} + 240m^6pp_1pp_2pp_4 + 140m^6pp_3^2s + 240m^6pp_3pp_{34}s
 \nonumber
\\ &&+ 476m^6pp_3pp_4s + 240m^6pp_{34}pp_4s + 140m^6pp_4^2s + 40m^4p_{13}p_{24}pp_3^2
+ 104m^4p_{13}p_{24}pp_3pp_4 \nonumber
\\ && + 40m^4p_{13}p_{24}pp_4^2+
20m^4p_{13}pp_2pp_3^2 + 288m^4p_{13}pp_2pp_3pp_4 +
120m^4p_{13}pp_2pp_4^2 \nonumber
\\ &&+ 40m^4p_{14}p_{23}pp_3^2 + 104m^4p_{14}p_{23}pp_3pp_4 + 40m^4p_{14}p_{23}pp_4^2
+ 120m^4p_{14}pp_2pp_3^2\nonumber
\\ && + 288m^4p_{14}pp_2pp_3pp_4 +
20m^4p_{14}pp_2pp_4^2+ 20m^4p_{23}pp_1pp_3^2
 + 288m^4p_{23}pp_1pp_3pp_4  \nonumber
\\ &&+ 120m^4p_{23}pp_1pp_4^2+ 120m^4p_{24}pp_1pp_3^2 + 288m^4p_{24}pp_1pp_3pp_4
 + 20m^4p_{24}pp_1pp_4^2\nonumber
\\ &&  - 240m^4pp_1pp_2pp_3pp_{34} + 144m^4pp_1pp_2pp_3pp_4 - 240m^4pp_1pp_2pp_{34}pp_4
  + 108m^4pp_3^2pp_4s\nonumber
\\ && + 144m^4pp_3pp_{34}pp_4s + 108m^4pp_3pp_4^2s + 28m^2p_{13}p_{24}pp_3^2pp_4
+ 28m^2p_{13}p_{24}pp_3pp_4^2\nonumber
\\ &&+ 12m^2p_{13}pp_2pp_3^2pp_4 +
72m^2p_{13}pp_2pp_3pp_4^2 + 28m^2p_{14}p_{23}pp_3^2pp_4
 + 28m^2p_{14}p_{23}pp_3pp_4^2 \nonumber
\\ &&+ 72m^2p_{14}pp_2pp_3^2pp_4 + 12m^2p_{14}pp_2pp_3pp_4^2
+ 12m^2p_{23}pp_1pp_3^2pp_4 + 72m^2p_{23}pp_1pp_3pp_4^2 \nonumber
\\ &&+ 72m^2p_{24}pp_1pp_3^2pp_4+ 12m^2p_{24}pp_1pp_3pp_4^2
- 144m^2pp_1pp_2pp_3pp_{34}pp_4 + 13m^2pp_3^2pp_4^2s \nonumber
\\ &&+ 2p_{13}p_{24}pp_3^2pp_4^2 + 2p_{14}p_{23}pp_3^2pp_4^2))/(3m^2s^2(64m^{12}
+ 96m^{10}pp_3 + 96m^{10}pp_4+ 48m^8pp_3^2 \nonumber
\\ && + 144m^8pp_3pp_4 +
48m^8pp_4^2 + 8m^6pp_3^3 + 72m^6pp_3^2pp_4 + 72m^6pp_3pp_4^2
+8m^6pp_4^3\nonumber
\\ &&+ 12m^4pp_3^3pp_4+ 36m^4pp_3^2pp_4^2 + 12m^4pp_3pp_4^3
+ 6m^2pp_3^3pp_4^2 + 6m^2pp_3^2pp_4^3 + pp_3^3pp_4^3)).
\end{eqnarray}
\begin{thebibliography}{99}

\bibitem{bbl}  G.T. Bodwin, L. Braaten, and G.P. Lepage, Phys. Rev. {\bf D51}, 1125 (1995).

\bibitem{belle} Belle Collaboration, K. Abe {\it et al.}, .

\bibitem{chao} K.T. Chao and L.K. Hao, .

\bibitem{kane} G.L. Kane, J.P. Leveille and D.M. Scott, Phys.
Lett. {\bf B85}, 115 (1979).

\bibitem{brod} S.J. Brodsky, C.R. Ji, Phys. Rev. Lett. {\bf 55}, 2257 (1985).

\bibitem{cho} P. Cho and A.K. Leibovich, Phys. Rev. {\bf D53},
150 (1996); 53, 6203 (1996). S. Baek, P. Ko, J. Lee and H.S. Song,
J. Kor. Phys. Soc. {\bf 33}, 97 (1998); .

\bibitem{yuan} F. Yuan, C.F. Qiao and K.T. Chao, Phys. Rev. {\bf
D56}, 321 (1997).

\bibitem{kiselev} V.V. Kiselev, A.K. Likhoded and M.V.
Shevlyagin, Phys. Lett. {\bf B332}, 411 (1994).


\bibitem{wf}  E.J. Eichten and C. Quigg, Phys. Rev. {\bf D52}, 1726 (1995).

\bibitem{belle2} Belle Collaboration, K. Abe {\it et al.}, Phys.
Rev. Lett. {\bf 88} 052001 (2002)

\bibitem{schuler} G.A. Schuler and M. V$\ddot{\rm a}$nttinen, Phys.
Rev. {\bf D58}, 017502 (1998).

\bibitem{bl} E. Braaten and J. Lee, .

\end{thebibliography}

\end{document}

