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%\title{ \Large\bf $f_{0}(980)$ production in $B$ decay to charmful mesons  }% Force line breaks with \\

\title{ \large\bf Evidence for two-quark content of $f_{0}(980)$ in exclusive $b\to c$ decays }% Force line breaks with \\

\author{ \bf Chuan-Hung Chen }

\email{chchen@phys.sinica.edu.tw}

\affiliation{  Institute of Physics, Academia Sinica, Taipei,
Taiwan 115, Republic of China }
%

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified


\begin{abstract}
Inspired by large branching ratio (BR) of $B^{+}\to
f_{0}(980)K^{+}$ decay measured by Belle recently, we propose that
a significant evidence of the component of
$n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$ in $f_{0}(980)$ during its
production could be demonstrated by exclusive $b\to c$ processes,
in which
%$f_{0}(980)$ is reconstructed
the reconstructed $f_{0}(980)$ by the decays $\bar{B}\to D^{0(*)}
\pi^{+} \pi^{-}(KK)$ and $\bar{B}\to J/\Psi \pi^{+} \pi^{-}(KK)$
is suggested. We predict the BRs of $\bar{B}\to D^{0(*)} (J/\Psi)
f_{0}(980)$ to be ${\cal {O}}(10^{-4})$ (${\cal {O}}(10^{-5})$)
while the unknown wave functions of $D^{(*)0}$ ($J/\Psi$) are
chosen to fit the observed decays $\bar{B}\to D^{(*)0} \pi^{0}\
(J/\Psi K^{0(*)})$.
\end{abstract}
\pacs{13.25.Hw, 14.40.Cs}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In spite of the successful quark model and QCD theory for strong
interaction, the fundamental questions on the inner structure of
lightest scalar mesons, such as $f_{0}(400-1200)$, $f_{0}(980)$
and $a_{0}(980)$ etc., are still uncertain, although it is over
thirty years since $f_{0}(980)$ was discovered first in phase
shift analysis of elastic $\pi \pi$ scattering \cite{f0}. Besides
the interpretations of $qq\bar{q}\bar{q}$ four-quark states
\cite{4q} or $K\bar{K}$ molecular states \cite{KK} or $q \bar{q}$
states \cite{qq} etc., the possibilities of gluonium states
\cite{MO} and scalar glueballs \cite{Robson} are also proposed. It
might be oversimple to regard them as only one kind of
composition.

It is suggested that in terms of $\gamma \gamma$ \cite{MO} and
radiative $\phi$ \cite{Antonelli} decays, the nature of scalar
mesons can be disentangled. However, with these experiments, the
conclusions such as given by Refs. \cite{De,DP} and Ref. \cite{AG}
are not unique. The former prefers $q \bar{q}$ while the latter is
four-quark content. Nevertheless, according to the data of E791
\cite{E791} and Focus \cite{FOCUS}, the productions of scalar
mesons which are reconstructed from $D$ and $D_{s}$ decaying to
three-pseudoscalar final states and mainly show $q \bar{q}$
contents during their productions, can provide us a further
resolution \cite{Meadows}. In addition, $Z_{0}$ decay data of OPAL
\cite{OPAL} also hint that $f_{0}(980)$, $f_{2}(1270)$ and
$\phi(1020)$ have the same internal structure. Hence, the
compositions of light scalar bosons should be examined further.

% It seems that the
%conclusions on the composition of the mysterious scalar bosons
%depend on how to extrapolate them.
% Probably, the states might be
%different during their productions and decays.

Recently, the decay of $B^{+}\to f_0(980) K^{+}$ with the BR
product of $Br(B^{+}\to f_{0}(980)K^{+})\times Br(f_{0}(980)\to
\pi^+ \pi^-)= (9.6^{+2.5+1.5+3.4}_{-2.3-1.5-0.8})\times 10^{-6}$
has been observed in Belle \cite{Belle1}. The observation not only
displays in the first time $B$ decay to scalar-pseudoscalar final
states but also provides the chance to understand the
characteristics of scalar mesons. Since $B$ meson is much heavier
than $D_{(s)}$ mesons, in the two-body $B$ decays, the outgoing
light mesons will behave as massless particles so that the
perturbative QCD (PQCD) approach \cite{LB,Li}, in which the
corresponding bound states are expanded by Fock states, could
apply. Therefore, as compared to two-parton states, the
contributions of four-parton and gluonium states belong to higher
Fock states. Consequently, we think the effects of $q \bar{q}$
state are more important than those in $D_{s}$ decays. In this
paper, in order to further understand what is the nature of
$f_{0}(980)$ during its production in $B$ decays, we take it to be
composed of $q \bar{q}$ states mainly and use
$|f_{0}(980)>=\cos\phi_{s} |s \bar{s}>+\sin\phi_{s} |n \bar{n}>$
with $n \bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$ to denote its flavor
wave function. We note that so far $\phi_{s}$ could be
$42.14^{+5.8^{0}}_{-7.3}$ \cite{MO} and $138^{0}\pm {6^0}$
\cite{AAN}. With the lowest order criterion, the effects of
four-parton and gluonium states are neglected.

Inspired by the large BR of $B^{+}\to f_{0}(980)K^{+}$ decay, we
propose that a significant evidence of the component of $n
\bar{n}$ in $f_{0}(980)$ during its production could be
demonstrated by exclusive $\bar{B}\to D^{(*)0} f_{0}(980)$ and
$\bar{B}\to J/\Psi f_{0}(980)$ processes and $f_{0}(980)$ could be
reconstructed from the decays $\bar{B}\to D^{(*)0} \pi^{+}
\pi^{-}(KK)$ and $\bar{B}\to J/\Psi \pi^{+} \pi^{-}(KK)$. The
results could be as the complement to the three-body decays of
$D_{s}$ that already indicate the existence of $s \bar{s}$
component .

It is known that the tree contributions are dominant effects on
%%$B\to D^{(*)}$ and $B\to J/\Psi$ decays.
exclusive $b\to c$ decays. Only $(V-A)\otimes (V-A)$ four-fermi
interactions need to be considered and the associated dynamical
couplings with the renormalization dependence are simple and
clear. The most difficult thing in our calculations is how to
determine the involving wave functions which are sensitive to the
nonperturbative QCD effects
%absorbed all infrared singularities,
and are universal objects. In $B$ meson case, one can fix it by
$B\to PP$ processes with $P$ corresponding to light pseudoscalars,
which wave functions are defined in the frame of light-cone and
are already derived from QCD sum rule \cite{Ball}. As to the
$D^{(*)0}(J/\Psi)$ wave functions, we can call for the measured
BRs of  color-suppressed $\bar{B}\to D^{0} \pi^{0}$ \cite{BD} and
$\bar{B}\to J/\Psi K^{(*)}$ \cite{BJ} decays. However, it might be
questionable to use the QCD approach for ordinary $ PP$ modes to
deal with $D^{(*)}(J/\Psi)$ decays because they aren't light
mesons anymore. In the heavy $b$ quark limit, fortunately, the
involved scales satisfy $m_{b}>>m_{c}>>\bar{\Lambda}$ with
$m_{b(c)}$ being the mass of $b(c)$-quark and
$\bar{\Lambda}=M_{B}-m_b$ so that the leading power effects in
terms of the expansions of $\bar{\Lambda}/m_{c}$ and $m_{c}/m_{b}$
could be taken as the criterion to estimate the involving
processes. We will see later that not only the obtained BRs of
$\bar{B}\to J/\Psi K^{*}$ but also their helicity components of
decay amplitudes
%, calculated in the framework of PQCD,
are  consistent with current experimental data. It will guarantee
that our predicted results on $f_{0}(980)$ productions of $B$
decays are reliable.

Since the hadronic transition matrix elements of penguin effects
in $\bar{B}\to J/\Psi M$, $M=K,\ K^{*}$, and $f_{0}(980)$, can be
related to tree ones, we  describe the effective Hamiltonian for
the $b\to c\bar{q} d$ transition as
\begin{eqnarray}
H_{{\rm eff}}&=&\frac{G_{F}}{\sqrt{2}}\sum_{q=u,c}V_{q}\left[
C_{1}(\mu ){\cal O}_{1}^{(q)}+C_{2}(\mu ){\cal O}
_{2}^{(q)}\right] \label{eff}
\end{eqnarray}
with ${\cal O}_{1}^{(q)} = \bar{d}_{\alpha} q_{\beta}
\bar{c}_{\beta} b_{\alpha}$ and  ${\cal O}_{2}^{(q)} =
\bar{d}_{\alpha} q_{\alpha} \bar{c}_{\beta} b_{\beta}$, where
$\bar{q}_{\alpha} q_{\beta}=\bar{q}_{\alpha} \gamma_{\mu}
(1-\gamma_{5}) q_{\beta}$, $\alpha(\beta)$ are the color indices,
$V_{q}=V_{qd}^{*}V_{cb}$ are the products of the CKM matrix
elements, $C_{1,2}(\mu )$ are the Wilson coefficients (WCs)
\cite{BBL}. Conventionally, the effective WCs of
$a_{2}=C_{1}+C_{2}/N_{c}$ and $a_{1}=C_{2}+C_{1}/N_{c}$ with
$N_{c}=3$ being color number are more useful.  $q=u$ corresponds
to $\bar{B}\to D^{(*)0} M$ decays while $q=c$ stands for
$\bar{B}\to J/\Psi M$ decays.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%      Topologies          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
According to the effective operators in Eq. (\ref{eff}), we find
that only emission topologies contribute to $\bar{B}\to J/\Psi
f_{0}(980)$ decay, however, the decays $\bar{B}\to D^{(*)0}
f_{0}(980)$ involve both emission and annihilation topologies. To
be more clear, the illustrated diagrams are displayed in Fig.
\ref{topology}. From the figure, we could see obviously that only
$n \bar{n}$ content has the contributions and the factorizable
emission parts, Fig. \ref{topology}(a), are only related to
$\bar{B}\to f_{0}(980)$ form factor. We note that in the
color-suppressed processes the nonfactorizable effects, shown as
Fig. \ref{topology}(b) and (d), are important and should be
considered.
%%%%%%%%%%%%%%Figure %%%%%%%%%%%%%%%%%
\begin{figure}
\includegraphics*[width=1.6
  in]{fig1} % Here is how to import EPS art
\includegraphics*[width=1.6
  in]{fig2}
\caption{Figure (a) and (b) illustrate the factorizable and
non-factorizable emission topologies, respectively, while figure
(c) and (d) correspond to the annihilation topologies.}
\label{topology}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Regarding $f_{0}(980)$ as $q \bar{q}$ contents in $B$ decays, the
immediate question is how to write down the corresponding hadronic
structures and the associated wave functions for this $^{3}P_{0}$
state. One information is that the spin structures of $f_{0}(980)$
should satisfy $\langle 0|\bar{q} \gamma_{\mu} q|
f_{0}(980)\rangle = 0$ and $\langle 0|\bar{q} q |f_{0}(980)\rangle
= m_{f_{0}} \tilde{f}$ in which $m_{f_{0}}(\tilde{f}\approx 0.18)$
\cite{DP} are the mass ( decay constant) of $f_{0}(980)$. In order
to satisfy these local current matrix elements, the light-cone
distribution amplitude for $f_{0}(980)$ should be given by
\begin{eqnarray}
\langle 0|\bar{q}(0)_{j} q(z)_{l}|f_{0}\rangle
&=&\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0} dx e^{-ixP\cdot z} \Big\{
[\, \slash \hspace{-0.2cm} p\, ]_{lj} \Phi_{f_{0}}(x) \nonumber
\\ &&+m_{f}[{\bf 1}]_{lj} \Phi^{p}_{f_{0}}(x)\Big\} \label{daf}
\end{eqnarray}
in which $\Phi^{(p)}_{f_{0}}(x)$ belong to twist-2(3) wave
functions. The charge parity indicates that
$\Phi_{f_{0}}(x)=-\Phi_{f_{0}}(1-x)$ and
$\Phi^{p}_{f_{0}}(x)=\Phi^{p}_{f_{0}}(1-x)$ \cite{CZ} so that
their normalizations are $\int^{1}_{0} dx\Phi_{f_0}(x)=0$ and
$\int^{1}_{0} dx\Phi^{p}_{f_0}(x)=\tilde{f}/2\sqrt{2N_{c}}$. As
usual, we adopt a good approximation that the light-cone wave
functions are expanded in Gegenbauer polynomials. Therefore, we
choose
\begin{eqnarray}
 \Phi^{p}_{f_{0}}(x)&=& \frac{\tilde{f}}{2\sqrt{2N_{c}}} \bigg\{
3(1-2x)^{2}+ G_{1}^{p} (1-2x)^{2} \nonumber
\\
&& \times \left[C^{3/2}_{2}(1-2x)-3 \right]+ G^{p}_{2}
C^{1/2}_4(1-2x)
\bigg\},\nonumber \\
\Phi_{f_{0}}(x)&=& {\tilde{f} \over 2\sqrt{2N_{c}}} G
\left[6x(1-x)C^{3/2}_{1}(1-2x)\right], \label{wavf}
\end{eqnarray}
where $C^{\lambda}_{n}$ are the Gegenbauer polynomials and the
values of coefficients $\{G\}$ haven't been determined yet from
the first principle QCD approach.
%%%%%%%%%%%%%%%%%%%%%%%%

It has been shown that by the employ of hierarchy
$M_{B}>>M_{D^{(*)}}>>\bar{\Lambda}$, the $D^{(*)}$ mesons
distribution amplitudes could be described by \cite{Li-P}
\begin{eqnarray}
%\langle 0|\bar{b}(0)_{j} d(z)_{l}|B\rangle
%&=&\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0} dx e^{-ixP\cdot z}
%\nonumber \\
%&&\times \Big\{ [\, \slash \hspace{-0.2cm}
%p\, +M_{B} ]_{lj}\gamma_{5} \Phi_{B}(x)\Big\}, \nonumber \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\langle 0|\bar{d}(0)_{j} c(z)_{l}|D\rangle
&=&\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0} dx e^{-ixP\cdot z}
\nonumber \\
&& \times \Big\{ [\, \slash \hspace{-0.2cm}
p\, +M_{D} ]_{lj} \gamma_{5} \Phi_{D}(x)\Big\}, \nonumber \\
%%%%%%%%%%%%%%%%%%
\langle 0|\bar{d}(0)_{j} c(z)_{l}|D^{*}\rangle
&=&\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0} dx e^{-ixP\cdot z}
\nonumber \\
&& \times \Big\{   [\, \slash \hspace{-0.2cm} p\, +M_{D^{*}}
]_{lj} \, \slash \hspace{-0.18 cm} \varepsilon \,
\Phi_{D^{*}}(x)\Big\},\label{dad}
\end{eqnarray}
where $\varepsilon_{\mu}$ is the polarization vector of $D^{*}$,
the normalizations of wave functions are taken as $\int^{1}_{0}
dx\Phi_{D^{(*)}}(x)=f_{D^{(*)}}/2\sqrt{2N_{c}}$ and $f_{D^{(*)}}$
are the corresponding decay constants. Although the decay
constants and wave functions of $D^{*0}$ meson for longitudinal
and transverse polarizations are different generally, for
simplicity, in our estimations we assume that they are the same.
Since the hadronic structure of $B$ meson was studied before, the
explicit description can be found in Ref. \cite{Chen-Li}. In order
to fit the measured BR of $\bar{B}\to D^{0} \pi^0$ decay, the
involved $D^{(*)}$ wave functions are chosen simply as \cite{Li-P}
\begin{eqnarray}
%\Phi_{B}(x,b)&=&N_{B}x^{2}(1-x)^{2}exp\Big[-\frac{1}{2}\Big(
%\frac{xM_{B}}{\omega_{B}}\Big)-\frac{\omega_{B}^{2}b^{2}}{2}
%\Big],
%\nonumber \\
 \Phi_{D^{(*)}}(x)&=&{3 \over \sqrt{2N_{c}}}
f_{D^{(*)}}x(1-x)[1+0.7 (1-2x)].
\end{eqnarray}
With the same guidance, we also apply the concept to $J/\Psi$
case. For detailed, one can be referred to Ref. \cite{YL}.

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  Formulas
%%%%%%%%%%%%%%%%%%%%%%%%%%
As mentioned before, due to large energy transfer in heavy $B$
meson decays, we can utilize the factorization theorem, in which
decay amplitude can be calculated by the convolution of hard parts
and wave functions \cite{LB,Li}, to stand for the hadronic
effects. Although vector $D^{*0}$ and $J/\Psi$ mesons carry the
spin degrees of freedom, in the $\bar{B}\to D^{*0} (J/\Psi)
f_{0}(980)$ decays only longitudinal polarization is involved.
Expectably, the results should be similar to $D^{0}f_{0}(980)$
one. Hence, we only present the representative formulas for
$\bar{B}\to D^{0} f_{0}(980)$ at amplitude level but give the
predicted BRs for all considered processes. From the Fig.
\ref{topology} and the effective interactions of Eq. (\ref{eff}),
the decay amplitude for $\bar{B}\to D^{0} f_{0}(980)$ is read as
\begin{eqnarray*}
A_{\bar{n} n} &=& {\sin \phi_{s} \over \sqrt{2}} V_{u} \Big[f_{D
}{\cal F}_{e} +{\cal M}_{e}+f_{B} {\cal F}_{a}+{\cal M}_{a} \Big]
\end{eqnarray*}
where ${\cal F}_{e}({\cal M}_{e})$ and ${\cal F}_{a}({\cal
M}_{a})$ are the factorized (non-factorized) emission and
annihilation hard amplitudes, respectively. According to Eqs.
(\ref{daf}) and (\ref{dad}), the typical hard functions are
expressed as
\begin{eqnarray}
{\cal F}_{e}&=&\zeta \int_{0}^{1} dx_{1} dx_{3} \int_{0}^{\infty}
b_{1}db_{1}b_{3}db_{3}\Phi _{B}( x_{1},b_{1})
\nonumber\\
&&  \Big\{ \Big[(1+x_{3})\Phi_{f_{0}}(x_{3})
+r_{f}(1-2x_{3})\Phi^{p}_{f_{0}}(x_{3})\Big]{\cal
E}_{e}(t^{1}_{e}) \nonumber \\
&& +2r_{f}\Phi^{p}_{f_{0}}(x_3){\cal E}_{e}(t^{2}_{e}) \Big\},
\label{fe}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\cal M}_{e}&=&2\zeta \int_{0}^{1} d[x] \int_{0}^{\infty}
b_{1}db_{1} b_{3}db_{3} \Phi _{B}( x_{1},b_{1})\Phi_{D}(x_{2})
\nonumber\\
&& \Big\{ \Big[-(x_{2}+x_{3}) \Phi_{f_{0}}(x_{3})
+r_{f}x_3\Phi^{p}_{f_{0}}(x_{3})\Big] {\cal
E}^{1}_{d}(t^{1}_{d})\nonumber \\
&&+\Big[(1-x_{2})\Phi_{f_{0}}(x_{3})-r_{f}x_{3}\Phi^{p}_{f_{0}}(x_3)\Big]{\cal
E}^{2}_{d}(t^{2}_{d}) \Big\} \label{me}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
with $\zeta=8\pi C_{F} M^{2}_{B}$, $r_{f}=m_{f}/M_{B}$, ${\cal
E}^{i}_{e}(t^{i}_{e})=$ $\alpha_{s}(t^{i}_{e})$ $a_{2}(t^{i}_{e})$
$Su_{B+f_{0}(980)}(t^{i}_{e})$ $h_{e}(\{x\},\{b\})$ and ${\cal
E}^{i}_{d}(t^{i}_{d})=\alpha_{s}(t^{i}_{d})(C_{2}(t^{i}_{d})/N_{c})$
$Su(t^{i}_{d})_{B+D+f_{0}(980)}$ $h_{d}(\{x\},\{b\})$.
$t^{1,2}_{e,d}$, $Su$ and $h_{e,d}$ denote the hard scales of $B$
decays, Sudakov factors and hard functions which are arisen from
the propagators of gluon and internal valence quark, respectively.
Their explicit expressions can be found in Ref. \cite{Chen-Li}.
With the same procedure,
the other hard functions can also be derived. %%%%%%%%%%%%%%%%

So far, the still uncertain values are the $\{G\}$ parameters of
the $f_{0}(980)$ wave functions. By the identity of
$<0|\bar{q}\gamma_{\mu}q|V,T>=M_{V}f_{V}\varepsilon_{\mu}(T)$ for
$V$-meson transverse polarization, we find that it is similar to
scalar meson case except the Dirac matrices $\gamma_{\mu}$ and the
associated polarization vector $\varepsilon_{\mu}$. Inspired by
the similarity, we adopt $\Phi^{p}_{f_{0}}(x)$ to be a
$\rho$-meson like wave function and take $G^{p}_{1}\approx 1.5$
and $G^{p}_{2}\approx 1.8$ \cite{Chen}. As to the value of $G$, we
use the corresponding value in $a_{0}(980)$ which is given by the
second reference of \cite{CZ} and get $G\approx 1.11$. By the
chosen values and using Eq. (\ref{fe}) with excluding WC of
$a_{2}$, we immediately get the $\bar{B}\to f_{0}(980)$ form
factor to be 0.38. Is it a reasonable value? In order to
investigate that the obtained value is proper, we employ the
relationship $F^{B\to f_{0}(980)}\sim (M_{D_{s}}/M_{B})^{1/2}
F^{D_{s}\to f_{0}(980)}$, which comes from heavy quark symmetry
limit \cite{HQ}, as a test. According to the calculation of Ref.
\cite{DGNPT}, we know $F^{D_{s}\to f_{0}(980)}\approx 0.6$; and
then, we have $F^{B\to f_{0}(980)}\sim 0.36$. Clearly, it is quite
close to what we obtain. Hence, with the taken values of
parameters, the magnitudes of hard functions are given in Table
\ref{tablehf}. We note that the complex values come from the
on-shell internal quark and all of hard functions are the same in
order of magnitude.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Table for amplitude
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[htb]
\caption{Hard functions (in units of $10^{-2}$) for $\bar{B}\to
D^{0} f_{0}(980)$ decay with %%%$\omega_{B}=0.4$, $f_{B}=0.19$,
$\tilde{f}=0.18$, $f_{D}=0.2$ GeV, $G=1.11$, $G^{p}_{1}=1.5$ and
$G^{p}_{2}=1.8$. }\label{tablehf}
\begin{ruledtabular}
\begin{tabular}{ccccc}
Amp. &  $ \ \ {\cal F}_{e} \  \ $ & $\  \ {\cal M}_{e}\ \ $ & $\ \
{\cal
F}_{a} \ \ $ & $\ \ {\cal M}_{a}\ \ $  \\
\hline $D^{0}f_{0}(980)$ & $-5.95$  &  $-2.66+i1.56$ &
$1.83-i3.60$ & $0.20+i1.12$
\end{tabular}
\end{ruledtabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

One challenging question is that how reliable our results are. In
order to investigate this point, besides the $\bar{B}\to D^{(*)0}
(J/\Psi) f_{0}(980)$ decays, we also calculate $\bar{B}\to D^{0}
\pi,\ J/\Psi (K, K^{*})$ and $\bar{B}\to f_{0}(980)K^{+}$
processes. All of them are already measured in B factories
\cite{BelleBr,BaBarBr}. Due to the calculations and formalisms
being similar to $D^{0}f_{0}(980)$, we directly present the
predicted BRs in Table \ref{tablebr} by taking $\phi_{s}=45^{0}$,
$f_{D^{*}}=0.22$ GeV, $f_{J/\Psi}=0.405$ GeV and the same taken
values of Table \ref{tablehf}. As to the $J/\Psi$ wave functions,
we model it as $\Phi_{J/\Psi}(x)=f_{J/\Psi}
[30x^{2}(1-x)^{2}]/2\sqrt{2N_{c}}$. The BRs of charged $B^{+}\to
J/\Psi M^{+}$ modes can be obtained from neutral modes by using
$Br(\bar{B}^{0}\to J/\Psi M^{0})\tau_{B^+}/\tau_{B^0}$. Hence,
from the Table \ref{tablebr}, we clearly see that our predictions
are consistent with experimental data.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Table for Brs
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
\caption{BRs (in units of $10^{-4}$) with $\phi_{s}=45^{0}$,
$f_{D^{*}}=0.22$, $f_{J/\Psi}=0.405$ GeV and the same taken values
of Table \ref{tablehf}.} \label{tablebr}
%\begin{center}
\begin{ruledtabular}
\begin{tabular}{cccc}
Mode &   Belle \cite{BelleBr} & BaBar \cite{BaBarBr} & This  \vspace{-0.2cm}  \\
 &  &  & work \\
\hline $D^{0}f_{0}(980)$ &   &   &
$2.28$ \\ \hline %%%%%%%%%
       $D^{*0}f_{0}(980)$ &   &  & $2.46$\\ \hline %%%%%%
       $J/\Psi f_{0}(980)$ & &  & $0.10$ \\ \hline %%%%%
       $K^{+} f_{0}(980)$ &  &  &    \\  %%%%%
        $f_{0}\to \pi^{+} \pi^{-}$ & $(9.6^{+2.5+1.5+3.4}_{-2.3-1.5-0.8})\cdot 10^{-2}$ &  &  $0.02$\\ \hline %%%%%
       $D^{0}\pi^{0}$ & $3.1\pm 0.4 \pm 0.5$  &  $2.89\pm 0.29 \pm 0.38$ & $2.60$ \\ \hline  %%%%%
%%       $J/\Psi K^{+}$ & $10.1\pm 0.3\pm 0.8$  &  $10.1\pm 0.3\pm 0.8$ & $8.32$ \\ \hline %%%%%
       $J/\Psi K^{0}$ & $7.9\pm 0.4 \pm 0.9$  & $8.3\pm 0.4 \pm 0.5$ & $8.3$ \\ \hline %%%%
%%%       $J/\Psi \pi^{0}$ & $0.18\pm 0.06 \pm 0.02$  & $0.20\pm 0.06 \pm 0.02$ & $0.15$ \\ \hline %%%%
       $J/\Psi K^{*0}$ & $12.9\pm0.5 \pm 1.3$  & $12.4\pm 0.5 \pm 0.9$&  $13.37$
\end{tabular}
%\end{center}
\end{ruledtabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Moreover, it is worth mentioning that in addition to the $BR$ of
$\bar{B}\to J/\Psi K^*$ decay, its squared helicity amplitudes
$|A_{0}|^{2}$, $|A_{\parallel}|^{2}$ and $|A_{\perp}|^{2}$ with
the normalization of $|A_{0}|^{2}+|A_{\parallel}|^{2}+
|A_{\perp}|^{2}=1$ \cite{CKL} are also given as $0.59$, $0.24$ and
$0.17$, respectively. They are all comparable with the measured
values $0.60\pm0.05 (0.60\pm 0.04)$, $0.21\pm 0.08(0.24\pm0.04)$
and $0.19\pm0.06 (0.16\pm 0.03)$ of Belle (BaBar)
\cite{BelleBr,BaBarBr}. In order to further understand the
dependence of the effects of $n \bar{n}$ content, the BRs as a
function of mixing angle $\phi_{s}$ are shown in Fig. \ref{fig}.
We note that with including twist-2 wave function for
$f_{0}(980)$, our previous result of $B^{+}\to K^{+} f_{0}(980)$
is modified in the small $\phi_{s}$ region \cite{Chen} such that
the BR becomes insensitive to $\phi_{s}$.
%%%%%%%%%%%%%%Figure %%%%%%%%%%%%%%%%%
\begin{figure}
\includegraphics*[width=2.6
  in]{bdf} % Here is how to import EPS art
%\centerline{ \psfig{figure=brdf0.eps,height=1.6 in }
%\psfig{figure=brjf0.eps,height=1.6 in}}
\caption{ BRs as a function of angle $\phi_{s}$. (a) the solid
(dashed) lines are for $\bar{B}\to D^{0}(D^{*0}) f_{0}(980)$
decays while (b) they express $\bar{B}\to J/\Psi f_{0}(980)$ and
$B^{+}\to K^{+} f_{0}(980)$ decays. }\label{fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The subsequent question is
%%%It is interesting to ask
how to search the events for $\bar{B}\to D^{(*)0}f_{0}(980)$ and
$\bar{B}\to J/\Psi f_{0}(980)$ decays. From particle data group of
Ref. \cite{BJ}, we know that $f_{0}(980)$ mainly decays to $\pi
\pi$ and $K K$  and
$R=\Gamma(\pi\pi)/(\Gamma(\pi\pi)+\Gamma(KK))\sim 0.68$.
Therefore, we suggest that the candidates could be found in
$\bar{B}\to D^{(*)0}(J/\Psi)\ \pi \pi\ (KK)$ three-body decay
samples. For illustration, according to the values of Table
\ref{tablebr}, we can estimate that the BR product of
$Br(\bar{B}\to D^{0}f_{0}(980))\times Br(f_{0}(980)\to \pi^+
\pi^-)\approx 1.0\times 10^{-4}$ with $Br(f_{0}(980)\to \pi^+
\pi^-)= 2R/3$. The result is consistent with the measured value of
$(8.0 \pm 0.6\pm 1.5) \times 10^{-4}$ for $\bar{B}\to D^{0}
\pi^{+} \pi^{-} $ decay while $\bar{B}\to D^{0} \rho^{0}$ decay is
determined to have the BR of $(2.9\pm 1.0 \pm 0.4) \times 10^{-4}$
\cite{Belle-conf}.
%%%and $Br(\bar{B}\to D^{0}
%%%\rho^{0})/Br(\bar{B}\to D^{0} \omega)=1.7\pm 1.0$ .


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Conclusion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have investigated the possibility to extract the existence of
$n \bar{n}$ component of $f_{0}(980)$ in terms of $\bar{B}\to
D^{(*)0} f_{0}(980)$ and $\bar{B}\to J/\Psi f_{0}(980)$ decays.
Based on the comparable values between the BRs of $\bar{B}\to
D^{0}\pi^{0}$ and $\bar{B}\to J/\Psi M$ decays and current
experimental data, our predictions on the BRs of $\bar{B}\to
D^{(*)0}(J/\Psi) f_{0}(980)$ decays are reliable
and can be tested in the $B$ factories.\\


\noindent {\bf Acknowledgments:}

The author would like to thank H.N. Li and  H.Y. Cheng for their
useful discussions. This work was supported in part by the
National Science Council of the Republic of China under Grant No.
NSC-91-2112-M-001-053. \\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\baselineskip=0.6cm

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\end{document}




\begin{eqnarray}
{\cal F}_{e}&=&\zeta \int_{0}^{1} dx_{1} dx_{3} \int_{0}^{\infty}
b_{1}db_{1}b_{3}db_{3}\Phi _{B}( x_{1},b_{1})
\nonumber\\
&& \Big\{ \Big[(1+x3)\Phi_{f_{0}}(x_{3})
+r_{f}(1-2x_{3})\Phi^{p}_{f_{0}}(x_{3})\Big]{\cal
E}_{e}(t^{1}_{e})
\nonumber\\
&&   +2r_{f}\Phi^{p}_{f_{0}}(x_3){\cal E}_{e}(t^{2}_{e})
\Big\}, \label{fe}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\cal M}_{e}&=&2\zeta \int_{0}^{1} d[x] \int_{0}^{\infty}
b_{1}db_{1} b_{3}db_{3} \Phi _{B}( x_{1},b_{1})\Phi_{D}(x_{2})
\nonumber\\
&& \Big\{ \Big[-(x_{2}+x_{3}) \Phi_{f_{0}}(x_{3})
+r_{f}(x_3)\Phi^{p}_{f_{0}}(x_{3})\Big] {\cal
E}^{1}_{d}(t^{1}_{d})
\nonumber\\
&&+\Big[(1-x_{2})\Phi_{f_{0}}(x_{3})-r_{f}x_{3}\Phi^{p}_{f_{0}}(x_3)\Big]{\cal
E}^{2}_{d}(t^{2}_{d}) \Big\}, \label{me}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
&&{\cal F}_{a}=\zeta \int_{0}^{1} dx_{2} dx_{3} \int_{0}^{\infty}
b_{2}db_{2}b_{3}db_{3}\Phi _{D}( x_{2}) \nonumber\\
 &&\ \ \ \Big\{
\Big[x_{3}\Phi_{f_{0}}(x_{3})
+2r_{f}r_{D}(1+2x_{3})\Phi^{p}_{f_{0}}(x_{3})\Big]{\cal
E}^{1}_{a}(t^{1}_{a})
\nonumber\\
&&\ \ \  - \Big[ x_{2} \Phi_{f_{0}}(x_{3}) +2r_{f}r_{D}
(1+x_{2})\Phi^{p}_{f_{0}}(x_3)\Big]{\cal E}^{2}_{a}(t^{2}_{a})
\Big\}, \label{fa} \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&&{\cal M}_{a}=2\zeta \int_{0}^{1} d[x] \int_{0}^{\infty}
b_{1}db_{1}b_{3}db_{3}\Phi_{B}(x_{1},b_{1})\Phi _{D}( x_{2})
\nonumber\\
&&\ \ \ \Big\{ \Big[x_{2}\Phi_{f_{0}}(x_{3})
+r_{f}r_{D}(2+x_{2}+x_{3})\Phi^{p}_{f_{0}}(x_{3})\Big]{\cal
E}^{1}_{f}(t^{1}_{f})
\nonumber\\
&&\ \ \ - \Big[ x_{3} \Phi_{f_{0}}(x_{3}) +r_{f}r_{D}
(x_{2}+x_{3})\Phi^{p}_{f_{0}}(x_3)\Big]{\cal E}^{2}_{f}(t^{2}_{f})
\Big\}, \label{ma}
\end{eqnarray}

%%%%%%%%%%%%%%Figure %%%%%%%%%%%%%%%%%
\begin{figure}
\includegraphics*[width=1.6
  in]{brdf0} % Here is how to import EPS art
\includegraphics*[width=1.6
  in]{brjf0}
%\centerline{ \psfig{figure=brdf0.eps,height=1.6 in }
%\psfig{figure=brjf0.eps,height=1.6 in}}
\caption{ BRs as a function of angle $\phi_{s}$. (a) the solid
(dashed) lines are for $\bar{B}\to D^{0}(D^{0*}) f_{0}(980)$
decays while (b) they express $\bar{B}\to J/\Psi f_{0}(980)$ and
$B^{+}\to K^{+} f_{0}(980)$ decays. }\label{fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

