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\begin{document}

\begin{center}
{\LARGE Problems of QCD factorization in exclusive decays of $B$ meson to charmonium}\\[0.8cm]
{\large Zhongzhi Song$~^{(a)}$
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}

\vspace{0.5cm}

\begin{abstract}
We study the exclusive decays of $B$ meson into P-wave charmonium
states $\chi_{cJ}(J=0,1)$ within QCD factorization approach and
find that for $B \rightarrow \chi_{c1} K$ decay, the factorization
breaks down due to logarithmic divergences arising from
nonfactorizable spectator interactions even at twist-2 order, and
the decay rate is too small to accommodate the data, and that for
$B\rightarrow \chi_{c0} K$ decay, there are infrared divergences
arising from nonfactorizable vertex corrections as well as
logarithmic divergences due to spectator interactions even at
leading-order. Our results, together with the problem in $B
\rightarrow J/\psi K$ decay, indicate that QCD factorization can
not be safely applied to the exclusive decays of $B$ meson to
charmonium.

 \vspace{1cm}
 \noindent PACS numbers: 13.25.Hw; 14.40.Gx
\end{abstract}
\vspace{1cm}

 Exclusive nonleptonic $B$-meson decays provide the
opportunity to determine the parameters of the CKM matrix, to
explore CP violation and to observe new physics effects. However,
quantitative understanding of nonleptonic $B$-meson decays is
difficult due to strong-interaction effects. Beneke et al. have
considered two-body nonleptonic $B$-meson decays  extensively
including a light-light as well as a heavy-light final
states\cite{BBNS1,BBNS2,BBNS3}. The general idea is that in the
heavy quark limit $m_b \gg \Lambda_{\mathrm{QCD}}$, the transition
matrix elements of operators in the hadronic decay $B\to M_1M_2$
with $M_1$ being the recoiled meson and $M_2$ being the emitted
meson are given by\cite{BBNS2}
 \bqa \langle M_1M_2|O_i|B\rangle &=& \langle
M_1|j_1|B\rangle\langle M_2|j_2|0\rangle\left[1+\sum
r_n\alpha_s^n+{\cal O}({\Lambda_{\rm QCD}\over m_b})\right]
\nonumber
\\ &=& \sum_jF_j^{BM_1}(m_2^2)\int^1_0 du\,
T_{ij}^I(u)\phi_{M_2}(u) \nonumber
\\ && +\int^1_0 d\xi \,du\,dv
\,T^{II}(\xi,u,v)\phi_B(\xi)\phi_{M_1}(v)\phi_{M_2}(u),
\label{qcdf}
 \eqa
where $M_2$ is a light meson or a quarkonium and $F^{BM_1}$ is the
$B \to M_1$ transition form factor, $\phi_M$ is the light-cone
distribution amplitude, and $T^{I,II}$ are perturbatively
calculable hard scattering kernels. If we neglect strong
interaction corrections, formula (\ref{qcdf}) reproduces the
result of naive factorization. However, hard gluon exchange
between $M_2$ and $BM_1$ system implies a nontrivial convolution
of hard scattering kernels with the distribution amplitude
$\phi_{M_2}$. This is called the QCD-improved factorization. This
method works well for light-light final
states\cite{BBNS1,BBNS3,PP,PV} as well as heavy-light final
states\cite{BBNS2,DL}.

Exclusive decays of $B$ meson to charmonium are important since
those decays e.g. $B \to J/\psi K$ are regarded as the golden
channels for the study of CP violation in $B$ decays. It is
conjectured physically that because the size of the charmonium is
small$(\sim 1/{\as {m_\psi}})$ and its overlap with the (B, K)
system is  negligible\cite{BBNS2}, the same QCD-improved
factorization method as for $B\to\pi\pi$ can be used for $B \to
J/\psi K$ decay. Indeed, the explicit calculations
\cite{chay,cheng} show that the nonfactorizable vertex
contribution is infrared safe and the spectator contribution is
perturbatively calculable at twist-2 order. This small size
argument for the applicability of QCD factorization for the
charmonia is intuitive, but it needs verifying for charmonium
states e.g. the P-wave $\chi_{cJ}$ states other than the $J/\psi$.
In addition, recently BaBar and Belle have measured the exclusive
decays of $B \rightarrow \chi_{cJ} K (J=0,1)$\cite{babar,belle1}.
So, it is instructive to compare the predictions based on the
QCD-improved factorization approach with the experimental data.

We first consider $\overline{B}\rightarrow \chi_{c1} K$. The
effective Hamiltonian is written as\cite{BBL}
 \be
H_{\mathrm{eff}} = \frac{G_F}{\sqrt{2}} \Bigl( V_{cb} V_{cs}^*
(C_1 {\cal O}_1 +C_2 {\cal O}_2 ) -V_{tb} V_{ts}^* \sum_{i=3}^{10}
C_i {\cal O}_i \Bigr),
 \ee
where $C_i$ are the Wilson coefficients and the relevant operators
${\cal O}_i$ in $H_{\mathrm{eff}}$ are given by
 \bqa
&& {\cal O}_1=(\overline{s}_{\alpha} b_{\beta})_{V-A} \cdot
(\overline{c}_{\beta} c_{\alpha})_{V-A},\qquad\qquad~ {\cal
O}_2=(\overline{s}_{\alpha} b_{\alpha})_{V-A} \cdot
(\overline{c}_{\beta} c_{\beta})_{V-A},
 \nonumber\\
&& {\cal O}_{3(5)}=(\overline{s}_{\alpha} b_{\alpha})_{V-A} \cdot
\sum_q (\overline{q}_{\beta} q_{\beta})_{V-A(V+A)},~ {\cal
O}_{4(6)}=(\overline{s}_{\alpha} b_{\beta})_{V-A} \cdot \sum_q
(\overline{q}_{\beta} q_{\alpha})_{V-A(V+A)},
\\
&& {\cal O}_{7(9)}={3\over 2}(\overline{s}_{\alpha}
b_{\alpha})_{V-A} \cdot \sum_q e_q (\overline{q}_{\beta}
q_{\beta})_{V+A(V-A)},~ {\cal O}_{8(10)}={3\over
2}(\overline{s}_{\alpha} b_{\beta})_{V-A} \cdot \sum_q e_q
(\overline{q}_{\beta} q_{\alpha})_{V+A(V-A)}.\nonumber
 \eqa

To calculate the decay amplitude, we introduce $\chi_{c1}$ decay
constant as \cite{ball1}
 \be
 \langle \chi_{c1} | \overline{c}
(0) \gamma_{\mu}\gamma_5 c(0) |0\rangle = -if_{\chi_{c1}}
m_{\chi_{c1}}\epsilon^*_{\mu},
\label{vector2}\\
\ee where $m_{\chi_{c1}}$ is the mass of $\chi_{c1}$ and
$\epsilon$ is the $\chi_{c1}$ polarization vector. Here
$f_{\chi_{c1}}$ is the $\chi_{c1}$ decay constant which is
suppressed nonrelativistically as compared to the $J/\psi$ decay
constant, and can be estimated from the potential model. Then the
leading-twist light cone distribution amplitude of $\chi_{c1}$ is
expressed compactly as
 \bqa
  \langle \chi_{c1}
(p,\epsilon)| \overline{c}_{\alpha} (y) c_{\beta} (z) |0\rangle
&=& \frac{i}{4} \int_0^1 du \cdot e^{i(u p\cdot y + (1-u) p \cdot
z)} f_{\chi_{c1}} m_{\chi_{c1}} \bigl( \pslash{\epsilon}^*
\gamma_5 \bigr)_{\beta \alpha} \phi_{\chi_{c1}}(u)
  \label{lcone}
 \eqa
where $u$ and $1-u$ are respectively the momentum fractions of the
$c$ and $\bar c$ quarks inside the $\chi_{c1}$ meson, and the wave
function $\phi_{\chi_{c1}}(u)$ for $\chi_{c1}$ meson is symmetric
under $u\leftrightarrow 1-u$.

In naive factorization, we neglect the strong interaction
corrections and the power corrections in
$\Lambda_{\mathrm{QCD}}/m_b$. Then the decay amplitude at leading
order is written as \bqa
 iM_0=2i  f_{\chi_{c1}} m_{\chi_{c1}} \epsilon^*\cdot p_B
F_1 (m_{\chi_{c1}}^2) \frac{G_F}{\sqrt{2}}\Bigl[ V_{cb} V_{cs}^*
(C_2 +\frac{C_1}{N_c} ) -V_{tb} V_{ts}^* (C_3 + \frac{C_4}{N_c}
-C_5 -\frac{C_6}{N_c}) \Bigr], \label{tree2}
 \eqa
where $N_c$ is the number of colors. We do not include the effects
of the electroweak penguin operators since they are numerically
small. The form factors for $\overline{B} \rightarrow K$ are given
as
  \bqa
\langle K(p_K) | \overline{s} \gamma_{\mu} b| B(p_B)\rangle=\Bigl[
(p_B +p_K^{\prime})_{\mu} -\frac{m_B^2-m_K^2}{q^2} q_{\mu} \Bigr]
F_1 (q^2) + \frac{m_B^2-m_K^2}{q^2} q_{\mu} F_0 (q^2),
 \label{vmu2}
  \eqa
where $q= p_B -p_K$ is the momentum of $\chi_{c1}$ with $q^2 =
m_{\chi_{c1}}^2$. We will neglect the kaon mass for simplicity. We
can use the ratio between these two form factors as\cite{chay}
 \be
  \frac{F_0(q^2)}{F_1 (q^2)} = 1-\frac{q^2}{m_B^2}.
 \ee
So we need only one of the two form factors to describe the decay
amplitude.

\begin{figure}[t]
\begin{center}
\vspace{-3.5cm}
\includegraphics[width=14cm,height=18cm]{diagram.eps}
\vspace{-5.5cm}
\end{center}
\caption{ Feynman diagrams for nonfactorizable corrections to
$\overline{B} \rightarrow \chi_{c1} K$.} \label{fvs}
\end{figure}

As we can see easily in Eq.~(\ref{tree2}), this amplitude is
unphysical because the Wilson coefficients depend on the
renormalization scale $\mu$ while the decay constant and the form
factor are independent of $\mu$. However, if we include the order
$\alpha_s$ corrections, it turns out that the $\mu$ dependence of
the Wilson coefficients is cancelled and the overall amplitude is
insensitive to the renormalization scale. Taking nonfactorizable
corrections in Fig.~\ref{fvs}, the full amplitude for $
\overline{B} \rightarrow \chi_{c1} K$ in QCD factorization is
written as
 \be
  iM = 2i f_{\chi_{c1}} m_{\chi_{c1}} \epsilon^*
\cdot p_B F_1 (m_{\chi_{c1}}^2) \frac{G_F}{\sqrt{2}} \Bigl[
V_{cb} V_{cs}^* a_2 -V_{tb} V_{ts}^* (a_3 - a_5) \Bigr],
\label{total2}
 \ee
where the coefficients $a_i$ ($i=2,3,5$) in the naive dimension
regularization(NDR) scheme are given by  \bqa
 && a_2=C_2 +\frac{C_1}{N_c} +\frac{\alpha_s}{4\pi}
\frac{C_F}{N_c} C_1 \Bigl( -18 +12\ln \frac{m_b}{\mu} + f_I +
f_{II}
\Bigr), \nonumber \\
&&a_3=C_3 +\frac{C_4}{N_c} +\frac{\alpha_s}{4\pi} \frac{C_F}{N_c}
C_4 \Bigl( -18 +12 \ln \frac{m_b}{\mu} +f_I + f_{II}
\Bigr), \nonumber \\
&&a_5=C_5 +\frac{C_6}{N_c} -\frac{\alpha_s}{4\pi} \frac{C_F}{N_c}
C_6 \Bigl( -6 +12 \ln \frac{m_b}{\mu} +f_I + f_{II} \Bigr). \eqa

The function $f_I$ is calculated from the four vertex
corrections(a,b,c,d) in Fig.~\ref{fvs} and reads
 \bqa
  f_I&=&\int_0^1 du\  \phi_{\chi_{c1}} (u) \Bigl[ \frac{3(1-2u)}{1-u} \ln
u -3i \pi +3 \ln (1-z)+ \frac{2z (1-u)}{1-z u} \nonumber \\
&&+ \Bigl( \frac{1-u}{(1-z u)^2} -\frac{u}{\bigl(1-z
(1-u)\bigr)^2} \Bigr) z^2 u \ln (z u) + \frac{z^2 (1-u)^2
\Bigl(\ln (1-z) -i\pi\Bigr)}{( 1-z u)^2} \Bigr],
 \eqa
where $z={m_{\chi_{c1}}}^2/{m_B}^2$, and we have already
symmetrized the result with respect to $u\leftrightarrow 1-u$.

The function $f_{II}$ is calculated from the two spectator
corrections(e,f) in Fig.~\ref{fvs} and it is given by
 \bqa
  f_{II}
= \frac{4\pi^2}{N_c} \frac{f_K f_B}{F_1 (m_{\chi_{c1}}^2) m_B^2}
\frac{1}{1-z} \int_0^1 d\xi \frac{\phi_B (\xi)}{\xi} \int_0^1 dy
\frac{\phi_K (y)}{y} \int_0^1 du
{\phi_{\chi_{c1}}(u)}[\frac{1}{u}+\frac{2z}{y(1-z)}], \label{fII}
\eqa
 where $\phi_B$, $\phi_K$ are the light-cone wave functions
for the $B$ and $K$ mesons respectively. The spectator
contribution depends on the wave function $\phi_B$ through the
integral \be
 \int_0^1 d\xi \frac{\phi_B (\xi)}{\xi} \equiv
\frac{m_B}{\lambda_B}.
 \ee
Since $\phi_B (\xi)$ is appreciable only for $\xi$ of order
$\Lambda_{\mathrm{QCD}}/m_B$, $\lambda_B$ is of order
$\Lambda_{\mathrm{QCD}}$. We will choose $\lambda_B\approx 300$
MeV in the numerical calculation.

\begin{table}[t]
\begin{center}
\begin{tabular}{ c | c c c c c c }
   \hline
    &$C_1$ & $C_2$ & $C_3$ & $C_4$
 & $C_5$ & $C_6$ \\
 \hline
  LO & 1.144 & -0308& 0.014 & -0.030 & 0.009 & -0.038  \\
  NDR & 1.082 & -0.185 & 0.014 & -0.035 & 0.009 & -0.041 \\
 \hline
 \end{tabular}
\caption{ {Leading-order(LO) and Next-to-leading-order(NLO) Wilson
coefficients in NDR scheme(See Ref.\cite{BBL}) with $\mu=4.4$ GeV
and $\Lambda^{(5)}_{\overline{\rm MS}}=225$ MeV.}}
 \label {wilson}
\end{center}
\end{table}

There is an integral in Eq.~(\ref {fII}), which will give
logarithmic divergence. Therefore QCD factorization breaks down
even at leading order. This is different from $B \rightarrow
J/\psi K$ decay which does not have logarithmic divergence at
leading twist\cite{chay,cheng}. The reason is that the logarithmic
divergences arising from the contribution of the vector and tensor
currents are cancelled out in the $B \rightarrow J/\psi K$ decay,
while there is no such cancellation for the $B \rightarrow
\chi_{c1} K$ decay. Following Ref.~\cite {BBNS3}, we treat the
divergent integral as an unknown parameter and write
  \be
\int_0^1 dy \frac{\phi_K (y)}{y^2}=\int_0^1 dy
6(\frac{1}{y}-1)=6(X_H-1),
 \ee
where the asymptotic form $\phi_K (y)=6 y (1-y)$ is used for the
kaon twist-2 light cone distribution amplitude. And we will choose
$X_H=\ln(m_B/\Lambda_{QCD})\approx 2.4$ as a rough estimate in our
calculation.

For numerical analysis, we choose $F_1 (m_{\chi_{c1}}^2) =
0.743$\cite {ball2} and use the following input parameters:
 \bqa
&&m_b=4.8 \ \mbox{GeV}, \ \ m_B=5.28 \ \mbox{GeV}, \ \
m_{\chi_{c1}} =3.5 \ \mbox{GeV}, \nonumber \\ &&f_{\chi_{c1}}=100
\ \mbox{MeV}, \ \ f_B = 180 \ \mbox{MeV}, \ \ f_K = 160 \
\mbox{MeV},
 \eqa
where the decay constant $f_{\chi_{c1}}$ is roughly estimated in
charmonium potential model\footnote{There are uncertainties
associated with the value of $f_{\chi_{c1}}$. However, the
theoretical rates are still smaller than the experimental results
by a factor of two even in the limit $f_{\chi_{c1}} \approx
f_{J/\psi} \approx$ 400 MeV.}.

\begin{table}[tb]
\begin {center}
\begin{tabular}{ c|ccc}
 \hline
 $\phi_{\chi_{c1}}(u)$ &$a_2$&$a_3$&$a_5$ \\   \hline
$6u(1-u)$&0.1255-0.0815i&0.0040+0.0026i&-0.0027-0.0031i \\

$\delta (u-1/2)$&0.1154-0.0814i&0.0043+0.0026i&-0.0031-0.0031i \\
\hline
\end{tabular}
\caption{The coefficients $a_i$ at $\mu=4.4$ GeV with different
choices of $\phi_{\chi_{c1}}(u)$.}
 \label{table1}
\end {center}
\end{table}

The asymptotic form of the distribution amplitude
$\phi_{\chi_{c1}}(u)$ is given as $\phi_{\chi_{c1}}(u) = 6u
(1-u)$. In the numerical analysis, we also consider the form
$\phi_{\chi_{c1}}(u) =\delta (u-1/2)$, which comes from the naive
expectation of the distribution amplitude. Although there are
uncertainties associated with the form of the wave function, we
will see shortly that the calculated decay rates are not sensitive
to the choice of the distribution amplitude. The results of
coefficients $a_i$ are listed in Table.~\ref {table1}.

With the help of these coefficients $a_i$, we calculated the decay
branching ratios. For $\phi_{\chi_{c1}} (u) = 6u(1-u)$,
${\mathrm{Br}} (\overline{B} \rightarrow \chi_{c1} K) = 0.18
\times 10^{-4}$. And for $\phi_{\chi_{c1}}(u) = \delta (u-1/2)$,
${\mathrm{Br}} (\overline{B} \rightarrow \chi_{c1} K) = 0.16
\times 10^{-4}$. The measured branching ratio is\cite{babar}
  \be
{\mathrm{Br}} (B^0 \rightarrow \chi_{c1} K^0) = (5.4 \pm 1.4)
\times 10^{-4},
  \ee
which is about thirty times larger than the  theoretical results.

 We now consider the $B \rightarrow \chi_{c0} K$ decay. Unlike the
$\chi_{c1}$ mode that we just analyzed, $\chi_{c0}$ does not have
couplings to $V$ or $A$ currents, so this decay mode is prohibited
in naive factorization. But it could occur if there is an exchange
of an additional gluon\cite{beneke, diehl}. The branching ratio
from experiment is\cite{belle1}
 \be
{\mathrm{Br}} (B^+ \rightarrow \chi_{c0} K^+) = (6.0 \pm 2.1)
\times 10^{-4}.
 \ee

Similar to the calculation performed above, we write the
$\chi_{c0}$ light-cone distribution amplitudes as
 \bqa
  \langle
\chi_{c0} (p)| \overline{c}_{\alpha} (y) c_{\beta} (z) |0\rangle =
\frac{i}{4} \int_0^1 du e^{i(u p\cdot y + (1-u) p \cdot z)}
\Bigl[f_{\chi_{c0}}^s m_{\chi_{c0}}\textsf{1}_{\beta \alpha}
\phi_{\chi_{c0}}^s(u) + f_{\chi_{c0}}^v \bigl( \pslash{p}
\bigr)_{\beta \alpha} \phi_{\chi_{c0}}^v(u)\Bigr],
 \label{lcda0}
 \eqa
where $f_{\chi_{c0}}^s$ and $f_{\chi_{c0}}^v$ denote the
$\chi_{c0}$ decay constants respectively for the scalar and vector
currents. It is easy to see that for the scalar meson $\chi_{c0}$
the decay constants for the vector current and the antisymmetric
tensor current all vanish(i.e., $\langle \chi_{c0}| \overline{c}
\gamma_\mu c|0\rangle=\langle \chi_{c0}| \overline{c}
\sigma_{\mu\nu} c|0\rangle=0$), and only the scalar current decay
constant is nonzero. Therefore on the right hand side of
Eq.(\ref{lcda0}) only the first term makes a real contribution and
it is essentially the leading twist (twist-3) contribution.
However, here we also list the second term (twist-2) in
Eq.(\ref{lcda0}) in order to show that it would also give an
infrared divergence contribution if the vector current decay
constant were not zero.

Because of charge conjugation invariance, $\phi_{\chi_{c0}}^s(u)$
is symmetric while $\phi_{\chi_{c0}}^v(u)$ is anti-symmetric under
$u \leftrightarrow 1-u$. The order $\as$ correction Feynman
diagrams for $B \rightarrow \chi_{c0} K$ are the same as those in
Fig.~\ref{fvs}. In the calculation of $\chi_{c1}$, the
contribution of the four vertex diagrams in Fig.~\ref {fvs} is
infrared safe. However, we find there are infrared divergences
arising from the vertex diagrams in the $B \rightarrow \chi_{c0}
K$ decay. Keeping only the divergent terms, we get
 \bqa
V_{Div}=\frac{\alpha_s C_F}{\pi N_c} i F_1(m_{\chi_{c0}}^2)\cdot
\Bigl\{ m_c m_{\chi_{c0}} f_{\chi_{c0}}^s \int_0^1
\phi_{\chi_{c0}}^s(u) du \Bigl[ -{1\over u}\ln{\lambda} \ln{z}
\Bigr]+m_B^2 f_{\chi_{c0}}^v \int_0^1 \phi_{\chi_{c0}}^v(u) du && \nonumber \\
\times \Bigl[ -\ln{\lambda} \ln{z} +(1-z)[\ln{\lambda}
 +{1\over 2} \ln^2(\frac{\lambda u z}{1-z})
- i \pi \ln(\frac{\lambda u z}{1-z})] \Bigr] \Bigr \},&&
 \eqa
where $z=m_{\chi_{c0}}^2/ m_B^2$ and $\lambda$ corresponds to the
cut of the soft gluon momentum divided by the charm quark mass.
Note that the coefficient of the first divergent term is
proportional to the charm quark mass\footnote{This infrared
divergence will not appear in $B$ decays to light scalar mesons,
since the divergence would vanish if we take the quark mass to be
zero. But for the charm quark the mass can not be treated as
zero.}.

The contribution arising from spectator interactions is given as
 \bqa
  H = \frac{\alpha_s C_F \pi}{ N_c^2} i f_K f_B \Bigl[
  \frac{ m_c m_{\chi_{c0}}}{m_B^2-m_{\chi_{c0}}^2}f_{\chi_{c0}}^s
  \int_0^1 d\xi \frac{\phi_B (\xi)}{\xi} \int_0^1 dy \frac{\phi_K (y)}{{y}^2}
\int_0^1 du \frac{\phi_{\chi_{c0}}^s(u)}{u (1-u)}&& \nonumber \\
+f_{\chi_{c0}}^v\int_0^1 d\xi \frac{\phi_B (\xi)}{\xi} \int_0^1 dy
\frac{\phi_K (y)}{{y}} \int_0^1 du
\frac{\phi_{\chi_{c0}}^v(u)}{1-u}\Bigr].&&
 \label{H}
 \eqa
There are logarithmic divergences even in the leading-twist order.
So, soft gluon exchange dominates $B \rightarrow \chi_{c0} K$
decay, and QCD factorization approach can not be applied to this
decay mode.

In summary, we have studied the exclusive decays of $B$ meson into
P-wave charmonium states $\chi_{cJ}(J=0,1)$ within the QCD
factorization approach and found that for $B \rightarrow \chi_{c1}
K$ decay, the factorization breaks down due to logarithmic
divergences arising from nonfactorizable spectator interactions
even at twist-2 order, and the decay rate is too small to
accommodate the data, and that for $B\rightarrow \chi_{c0} K$
decay, there are infrared divergences arising from nonfactorizable
vertex corrections as well as logarithmic divergences due to
spectator interactions even at the leading-order, indicating that
QCD factorization can not be safely applied to $B\rightarrow
\chi_{cJ} K$ decays. Moreover, we already knew that for
$B\rightarrow J/\psi K$ decay, there are logarithmic divergences
arising from spectator interactions due to kaon twist-3 effects
and the calculated rates are smaller than data by a factor of
8-10\cite{chay,cheng}. This may also be an indication that the QCD
factorization method can not well describe the exclusive decays of
$B$ meson to charmonium. The problems associated with
$B\rightarrow \chi_{cJ} K$ decays are certainly more serious.

Considering the above problems encountered in describing
$B\rightarrow \chi_{cJ}K(J=0,1)$ as well as $B\rightarrow J/\psi
K$ decays, we intend to conclude that in general the small size
argument for charmonia in $B$ decays is questionable, and the QCD
factorization method with its present version can not be safely
applied to exclusive decays of $B$ meson into charmonium, and that
new ingredients or mechanisms such as the effect of the
color-octet $c\bar c$ component in charmonia, other possible
nonperturbative QCD effects, and even some possible new physics
effects beyond the standard model should be introduced to describe
exclusive decays of $B$ meson to charmonium states.


\section*{Acknowledgements}
We are grateful to H.-n. Li for helpful discussions on this work.
We also thank J.P. Ma, C.S. Huang, D.X. Zhang and D.S. Yang for
useful discussions and comments. We would like to thank C. Meng
for checking the calculations. This work was supported in part by
the National Natural Science Foundation of China, and the
Education Ministry of China.
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\end{document}

