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\begin{document}


\title{ The QCD factorization in $B \to DKK$ decays}

\author{Zheng-Tao Wei}
 \affiliation{ Institute of Physics, Academia Sinica,
 Taipei, 115 Taiwan, ROC}

\begin{abstract}
A study of hadron pair production mechanism is motivated by the
recent observed decays $\bar B^0\to D^{(*)+}K^-K^0$. We show that
these decays with small invariant mass of kaon pair can be
factorized into a generalized form. The new non-perturbative
quantity is the generalized distribution amplitude which describes
how a quark-antiquark pair transmits into the hadron pair. The
$\bar B^0\to D^{(*)+}K^-K^0$ decays allow for the determination of
the kaon weak form factor in a direct way if the validity of
leading twist predictions of pQCD is tested by the future
experiment.
\end{abstract}

\pacs{13.25.Hw, 12.38.Bx}

\maketitle


Recently, the three body charmful decays $B\to D^{(*)}K^-K^{(*)0}$
were first observed by the BELLE Collaboration \cite{Belle}. They
also found a novel phenomenon that the mass spectra of the
$K^-K^{(*)0}$ pair peaks near their mass threshold. In
\cite{CHST}, a factorization approach which is similar to the
conventional factorization approach in B-meson two body decays
\cite{BSW} is proposed to study $\bar B^0\to D^{(*)+}K^-K^{0}$
decays. The transition matrix element of $\bar B^0\to
D^{(*)+}K^-K^0$ is factorized into the $B\to D$ form factor
multiplied by the $K^-K^0$ weak form factor. The $K^-K^0$ form
factor is constrained from the experimental data for the time-like
electromagnetic (EM) kaon form factors. The branching ratios
obtained by this approach agree well with experiment. The
threshold enhancement is explained by the invariant mass behavior
of the $K^-K^0$ weak form factor. The phenomenological success of
the factorization approach in $\bar B^0\to D^{(*)+}K^-K^{0}$
decays may shed light on some non-trivial QCD dynamics which will
be explored in this study.

The exclusive, non-leptonic three body B meson decays are usually
difficult to understand from the perturbative picture. As we shall
show, the hadronic physics of $\bar B^0\to D^{(*)+}K^-K^0$ decays
with small kaon pair invariant mass is an exception. In this
region, the two kaons are nearly collinear and energetic. It is
possible to apply the production mechanism of two pions in the
process $\gamma^* \gamma\to\pi\pi$ with high virtual photons and
small invariant mass of pion pair \cite{DGPT}. According to this
mechanism, $K^-K^0$ mesons are produced by the hadronization of
$\bar u d$ pair emitted from the W-boson. Since the $\bar u d$
pair moves fast, time dilation effect makes the hadronization
cannot occur until the $\bar u d$ pair moves far away from the
remaining system. The transition $\bar u d\to K^-K^0$ with a small
invariant mass is soft and can be described by a new
non-perturbative, universal matrix element which generalizes the
standard description of a hadron in QCD \cite{BL}. The
contribution from the region with large invariant mass is
suppressed by the time-like kaon form factor. The dominant
contribution comes from the region close to the threshold which
agrees with the observation in experiment \cite{Belle}. For the
$B\to D$ transition, the light spectator does not require a hard
interaction because B and D mesons are both heavy. The energetic
and collinear $\bar u d$ pair in a color-singlet configuration
decouples from the soft gluon interactions. This phenomenon
corresponds to the argument of ``color transparency" in
\cite{Bjorken}.  So, the strong interactions between $K^-K^0$ and
$BD$ systems occur at short distance and can be systematically
calculated in perturbative QCD (pQCD). The above arguments will
lead to a factorization form depicted in Fig. 1. Up to leading
power of $w/m_B$ ($w$ being the kaon pair invariant mass), the
hadronic matrix element of a four-quark operator $Q_i$ in the weak
effective Hamiltonian is expressed as %
\beq %
\langle DKK| Q_i|\bar B \rangle
  =F^{BD}(w^2)\int_0^1dz H(z)\Phi^{KK}(\zeta, z, w^2). %
\eeq %
Here, $F^{BD}(w^2)$ is a $B\to D$ transition form factor at the
momentum transfer $w^2$, and $\Phi^{KK}(z, \zeta, w^2)$ is a
generalized distribution amplitude (GDA) of the kaon pair. $H$
denotes a hard scattering kernel which is perturbatively
calculable. The definition of GDA will be given below. This
factorization formulae is a natural generalization of the QCD
factorization in the $B\to D\pi$ decay given in \cite{BBNS}. Just
as the $B\to D\pi$ decay, the $B\to DKK$ process provides a clean
environment to test the factorization in the three body B-meson
decays.

\begin{figure}
\includegraphics*[width=2.5in]{fig1}
\caption{Factorization of process $B\to DKK$ for $w\ll m_B$.}
\label{fig:BDKK}
\end{figure}

For the sake of illustration, our discussion will concentrate on
the decay $\bar B^0\to D^+K^-K^0$. We shall work in the rest frame
of the B meson and use the light-cone variables. Introduce the
total momentum of the kaon pair $P=p_1+p_2$ where $p_{1,2}$ are
momenta of the $K^-,K^0$ respectively, and invariant mass
$P^2=w^2$. The momentum $P$ is chosen to be mainly in the ``$+$"
direction. Define the momentum fraction variable $\zeta\equiv
\frac{p_1^+}{P^+}$. We consider the kinematic region %
\beq %
w^2\ll m_B^2. %
\eeq %
This requirement is necessary to ensure the validity of
factorization. Under the above conventions, we have %
\beq %
&& P_B=(\frac{m_B}{\sqrt 2}, ~\frac{m_B}{\sqrt 2},
  ~\vec 0_\bot), ~~
P_D=(r\frac{m_B}{\sqrt 2}, ~\frac{m_B}{\sqrt 2},
  ~\vec 0_\bot), \non \\
&& P =\bar r(\frac{m_B}{\sqrt 2}, ~\eta\frac{m_B}{\sqrt 2},
   ~\vec 0_\bot), ~~
p_1=\bar r(\zeta \frac{m_B}{\sqrt 2}, ~\bar\zeta\eta\frac{m_B}
  {\sqrt 2}, ~\vec p_{\bot} ), \non \\
&& p_2=\bar r(\bar\zeta \frac{m_B}{\sqrt 2}, ~\zeta\eta\frac{m_B}
  {\sqrt 2}, ~-\vec p_{\bot} ).
\eeq %
where $r=\frac{m_D^2}{m_B^2}$, $\eta=\frac{w^2}{(\bar r m_B)^2}$
and $p_{\bot}^2=(\zeta\bar \zeta w^2-m_K^2)/\bar r ^2$. We have
used the ``bar"-notation for any longitudinal momentum fraction
variable $\bar u=1-u$ throughout this paper. One can obtain the
kinematical constraint on the variable $\zeta$: $\zeta\bar
\zeta\ge \frac{m_K^2}{\bar r^2 w^2}$.

The generalized distribution amplitude is a crucial ingredient in
our pQCD approach. Since the kaon meson is spin zero, the axial
part of the weak current does not contribute owing to parity
conservation. Consequently, only the quark vector current is left.
This is different from the case of a single kaon. The leading
twist generalized light-cone distribution amplitude for the
$K^-K^0$ pair is defined by the following matrix element \cite{DGPT}: %
\beq %
\langle K^-(p_1)K^0(p_2)|\bar d(x){\cal P}\gamma_{\mu} u|0\rangle
  ~~~~~~~~~~~~\non \\ %
=P_{\mu}\int_0^1 dz e^{izP^+ x^-}\Phi^{K^-K^0}(z, \zeta,
w^2).
\eeq %
where ${\cal P}$ is the path-ordered exponential of gluon fields
needed to preserve gauge invariance of the matrix element.

The generalized distribution amplitude $\Phi(z, \zeta, w^2)$
provides an important theoretical tool to study the production of
two hadron pair. It is the time-like version of a generalized
parton distribution of hadron. GDA contains much fruitful physical
information. $\Phi(z, \zeta, w^2)$ depends on three variables:
quark fraction $z\equiv \frac{p_{\bar u}^+}{P^+}$, which describes
how the current quark shares the total momentum; hadron fraction
$\zeta=\frac{p_1^+}{P^+}$, which characterizes the momentum
distribution between two hadrons; and the invariant mass $w^2$.
One special feature of GDA is that it is complex in general. The
imaginary part of $\Phi$ is due to rescattering effects or
resonance contributions. The strong phase shift induced by this
soft mechanism is neither power nor perturbarive ($\alpha_s$)
suppressed because the final state interactions between two
hadrons occur at low energies. Thus it gives an origin of a large
strong phase and provides the possibility to observe the large
direct CP violation in three body B decays. In this paper, we will
not explore this point further since only the absolute value of
$\Phi$ is relevant. Another feature of GDA is that it does not
select all the valence quarks of the kaon pair in the hard
scattering at the quark level. The effect of additional quark pair
$\bar s s$ contained in GDA plays a less important role.

The $K^-K^0$ GDA has only one isospin state, i.e. $I=1$. For
iso-vector amplitude $\Phi^{K^-K^0}(z, \zeta, w^2)$, it satisfies %
\beq %
\Phi(z, \zeta, w^2)=\Phi(1-z, \zeta, w^2)=-\Phi(z, 1-\zeta, w^2).
\eeq %
The amplitude is odd under $\zeta \longleftrightarrow 1-\zeta$, so
the skewness of the hadron momentum distribution is described by
the $\zeta$ dependence. The GDA $\Phi^{K^-K^0}(z, \zeta, w^2)$ is
normalized as: %
\beq \label{eq:normal} %
\int_0^1 dz~ \Phi^{K^-K^0}(z, \zeta, w^2)=
  (2\zeta-1)F^{K^-K^0}(w^2),
\eeq %
where $F^{K^-K^0}(w^2)$ is the $K^-K^0$ weak from factor in the
time-like region, which is defined by %
\beq %
\langle K^-(p_1)K^0(p_2)|\bar d(0)\gamma_{\mu} u(0)|0 \rangle
 ~~\non \\=
  (p_1-p_2)_{\mu}F^{K^-K^0}(w^2),
\eeq %
The time-like weak form factor $F^{K^-K^0}(w^2)$ needs to be
determined from experiment. Eq.(\ref{eq:normal}) means that the
time-like form factor can be interpreted from a more general
concept, namely, GDA.

%Is it possible to define the $B\to D$ form factor in another
%concept so that we can understand it from a more general
%foundation of QCD? We will not pursue this problem further but
%point out this similarity.

%We hope to extract it from experiment. Use the well-known isospin
%relation that the matrix element in weak vector transition can be
%determined by the matrix
%element of electromagnetic current, we obtain %
%\beq %
%\langle K^-K^0|\bar d \gamma_{\mu} u|0 \rangle=%
%\langle K^-K^+|j_{\mu}^{EM}|0 \rangle- %
%\langle \bar{K^0}K^0|j_{\mu}^{EM}|0 \rangle
%\eeq %
%where $j_{\mu}^{EM}=\sum\limits_i e_i \bar q_i \gamma_{\mu}q_i$ is
%the electromagnetic current. From the experimental data of
%time-like factors $F^{K^-K^+}$ and $F^{\bar {K^0}K^0}$, the form
%factor $F^{K^-K^0}(w^2)$ can be constrained.

The GDA $\Phi(z, \zeta, w^2)$ will depend on the renormalization
scale in higher order radiative corrections.  Since the scale
dependence is only related to the non-local product of quark
fields, the evolution of $\Phi(z, \zeta, w^2)$ is the same as the
BLER evolution of the pion distribution amplitude \cite{BLER}. In
the limit $\mu\to \infty$, the GDA $\Phi^{K^-K^0}(z, \zeta, w^2)$
has
the asymptotic form \cite{Polyakov} %
\beq %
\Phi^{K^-K^0}(z, \zeta, w^2)=6z(1-z)(2\zeta -1)F^{K^-K^0}(w^2).
\eeq %
Thus the shape of the kaon pair invariant mass spectrum in $\bar
B^0\to D^+K^-K^0$ in the heavy quark limit is completely
determined by the time-like weak form factor $F^{K^-K^0}(w^2)$. In
the large invariant mass region, the time-like form factor falls
off by a power of $1/w^2$ from the simple dimensional counting
rule. It chooses the collinear configuration of hadron pair as its
dominant contribution. The threshold enhancement of the mass
spectrum is its a natural prediction of GDA in the processes with
large momentum transfers.

Though it seems that our pQCD approach is substantially different
from the standard pQCD framework in \cite{BL}, their basic ideas
are same. Both are based on the factorization theorem, which
separates short-distance from long-distance physics in a simple
and systematic way. Just as the introduction of $B\to M$ form
factors extends the application of pQCD to B meson two body decays
\cite{BBNS}, the introduction of the generalized distribution
amplitude extends its region of application into more complicated
three body decays. The quark scattering process $b\to c\bar u d$
is same as the case in decay $\bar B^0\to D^+\pi^-$. So the
technical proof of factorization of $\bar B^0\to D^+\pi^-$ in
\cite{BBNS, BPS} can be applied in three body decays $\bar B^0\to
D^+K^-K^0$. The physical principle is color transparency which
states that the soft gluon cannot resolve the structure with a
size much smaller than the typical size of hadron. The only
difference is that the intrinsic transverse momentum in the kaon
pair is of order $w$ rather that $\Lambda_{QCD}$. So the
next-to-leading power refers to the terms proportional to $w/m_B$.
Factorization is broken by the gluon exchange at energy scale of
order $w$.

Now we discuss the phenomenological application of pQCD approach
into $\bar B^0\to D^{(*)+}K^-K^0$ decays. The leading contribution
comes from the collinear region where the momenta of current
quarks and two kaons are replaced by their largest plus variables,
i.e. the invariant mass $w^2$ and transverse momentum are
neglected. The validity of the collinear approximation needs to be
checked by consistency of the perturbative result. To the leading
power of $w/m_B$, the decay amplitudes for
$\bar B^0\to D^{(*)+}K^-K^0$ are %
\beq %
A(\bar B^0\to D^+K^-K^0)=\frac{G_F}{\sqrt 2}V_{ud}^*V_{cb}~
  a_1 F_+(w^2) \non \\
\times F^{K^-K^0}(w^2)(2\zeta-1)(m_B^2-m_D^2),
\eeq %
and %
\beq %
A(\bar B^0\to D^{*+}K^-K^0)=\frac{G_F}{\sqrt 2}
  V_{ud}^*V_{cb}~a_1 A_0(w^2) \non \\
\times F^{K^-K^0}(w^2)(2\zeta-1)
  2m_{D^*}\epsilon^*_{D^*}\cdot P,
\eeq %
where $V_{ij}$ are CKM matrix elements, $F_+, A_0$ are $B\to D(*)$
transition from factors defined in \cite{BBNS}, $a_1$ is the
Wilson coefficient, and $\epsilon^*_{D^*}$ is the polarization
vector of $D^*$. In the above equations, we have chosen the
asymptotic form for $K^-K^0$ generalized distribution amplitude.

The theoretical input parameters are chosen as in \cite{BBNS}:
$a_1=1.05$, $F_+(0)=A_0(0)=0.6$. These parameters provide a well
fit to the decay modes of $\bar B{^0}\to D^{(*)+}\pi$. The only
unknown input is the $K^-K^0$ weak form factor $F^{K^-K^0}(w^2)$.
This from factor has been constrained via an isospin relation from
the experimental data of time-like electromagnetic kaon form
factors in \cite{CHST}. The obtained $F^{K^-K^0}(w^2)$ can be
approximated as a power-law distribution:
$|F^{K^-K^0}(w^2)|\approx \frac{1.4}{w^2}$. The discrepancy
between this simple model and the best fit result in \cite{CHST}
is within 20\% level. It should be noted that this determination
method is not a direct way to extract the $K^-K^0$ weak vector
current form factor. So there are still large theoretical
uncertainties coming from the form factor. Table~\ref{tab:Br} and
Fig.~\ref{fig:spectra} give the numerical results for branching
ratios and kaon pair invariant mass spectra of $\bar B^0\to
D^{(*)+}K^-K^0$ decays.

\begin{table}
\caption{ \label{tab:Br} The branching ratios of $\bar B^0\to
 D^{(*)+}K^-K^0$ in units of $10^{-4}$. ``pQCD" represents  pQCD
 approach.}
\begin{ruledtabular}
\begin{tabular}{lcr}
    & pQCD  &  Experiment       \\ \hline
$\bar B^0\to D^+K^-K^0$
   & 1.99 & $1.6\pm 0.8\pm 0.3$ \\ \hline
$\bar B^0\to D^{*+}K^-K^0$
   & 1.77  & $2.0\pm 1.5\pm 0.4$ \\
\end{tabular}
\end{ruledtabular}
\end{table}

Another test of pQCD comes from the ratio of decay rates
\beq %
r_{K^-K^0} &\equiv& \frac{\Gamma(\bar B^0\to
 D^{*+}K^-K^0)}{\Gamma(\bar B^0\to D^+K^-K^0)}  \\
&\approx& \Bigg ( \frac{A_0(w^2)}{F_+(w^2)}
\frac{(m_B^2-m_{D^*}^2)}{(m_B^2-m_D^2)} \Bigg )^2 =0.95 \non
\eeq %
In the above relation, we have neglected the effects caused by the
phase space difference. The use of this ratio can reduce the
theoretical errors caused by the uncertainties of
$F^{K^-K^0}(w^2)$. In pQCD method, the ratio $r_{K^-K^0}$ is
slightly smaller than 1, while in factorization approach it is
about 2 \cite{CHST}. The difference between the predictions in the
two approaches lies in the collinear approximation adopted in pQCD
approach. Whether this approximation is reasonable or not is
crucial for the validity of applying pQCD at the realistic energy
of $m_B$.

\begin{figure}
\includegraphics*[width=2.6in]{fig2}
\caption{The kaon pair invariant mass spectra of $\bar B^0\to
D^+K^-K^0$ (solid line) and $\bar B^0\to D^{*+}K^-K^0$ (dashed
line). \label{fig:spectra}}
\end{figure}

From Table~\ref{tab:Br}, the theoretical predictions of branching
ratios are consistent with the experimental data. For the ratio
$r_{K^-K^0}$, it needs further tests. The momentum spectra plotted
in Fig.~\ref{fig:spectra} shows a similar momentum distribution
for pseudoscalar and vector D mesons. The fraction comes from the
range $w<1.5\GeV$ is about 35\% which is not sufficient to
guarantee the validity of the collinear approximation. Note that
this numerical result is based on our insufficient information of
the $K^-K^0$  weak vector current form factor. From intuitive
considerations, the $K^-K^0$ form factor in the small invariant
mass region is likely to be enhanced by the resonance contribution
or soft re-scattering effects. The expected momentum spectra
should be more concentrated in the mass region close to the
threshold where the two kaons are collinear. This conjecture is
reinforced by the experimental measurement that the fraction of
$B^-\to D^0 K^- K^0$ signal events in the invariant mass range
$w<1.3\GeV$ is 55\%. The best fit $K^-K^0$ form factor in
\cite{CHST} does not satisfy this criterion. If the future
experiment observes that most of the contribution comes from the
small invariant mass region, such as $w<1.5\GeV$, it will provide
a strong support of our conjecture. Furthermore, we suggest to
extract the $K^-K^0$ weak form factor directly from the momentum
spectrum of $\bar B^0\to D^+K^-K^0$.

Although the $K^-K^0$ form factor is not known accurately at
present, we give a crude estimate about the theoretical errors in
$\bar B^0\to D^{(*)+}K^-K^0$ decays based on power counting. The
next-to-leading power correction is proportional to
$\frac{w}{m_B}$, which is about $20\%$ at the kaon pair threshold.
Considering the interference between the leading and
next-to-leading power contributions, the theoretical accuracy
within $40\%$ is accessible in pQCD method. This accuracy is not
as good as that in $B\to D\pi$, but it is still important in
explaining the experimental data and understanding the hadronic
physics of three body decays.

The principle that the hadron pair produced through
quark-antiquark pair can be applied to other three body B-meson
decays, such as $D\pi\pi$, $\pi\pi K$ etc. For these processes,
more generalized distribution amplitudes are required. The
proposed production mechanism can also be applied to two body
decays to study the weak annihilation contributions in B and D
decays, which had been pointed out long time age \cite{BSW}. The
time-like form factors at the mass scale of heavy mesons can be
determined from experiment. The detailed exploration of this
subject will be given elsewhere.

In conclusion, the exclusive, non-leptonic three body decays $\bar
B^0\to D^{(*)+}K^-K^0$ in the region where kaon pair invariant
mass $w^2\ll m_B^2$ allow a partonic description. The universal,
non-perturbative generalized distribution amplitude builds a new
theoretical bridge from quarks to hadrons and extends pQCD into
more complicated processes. The validity of the leading twist
predictions in this generalized approach requires further
experimental tests in the $\bar B^0\to D^{(*)+}K^-K^0$ decays.
%The generalized distribution amplitudes build up a new bridge from
%quarks to hadrons.


We would like to thank H. Li and C. Chua for many valuable
discussions. This work is partly supported by National Science
Council of R.O.C. under Grant No. NSC 91-2816-M-001-0012-6.


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\end{document}

