text:
\section{Introduction}

The inner structure of the light scalars, generally below 2 GeV, has been explored
by the physicists at both experimental and theoretical aspects for several decades.
However, unfortunately, their underlying structure has not yet been well established
and the identification of the considered light scalars is known as a long-standing puzzle
~\cite{Beringer12:pdg}.
But, it is lucky for us that the light scalars could be studied in the decay channels
of heavy flavor $B$ mesons, with the rich data provided by $B$ factories,
the Large Hadron Collider(LHC) experiments ~\cite{Cheng09:bdecays}, and the forthcoming
Super-$B$ factory~\cite{Soni07,SuperB}.
Ever since the $B \to f_0(980) K$ mode was firstly measured by
Belle Collaboration in 2002~\cite{belle02:f98}, then confirmed by BaBar
Collaboration in 2004~\cite{babar04:f98},
more and more channels with $p$-wave light scalars in the final states
of $B$ meson decays have been opened and more precise data have been obtained
~\cite{Beringer12:pdg}.
With the gradually enlarging data samples collected in the  running LHC experiments
and the forthcoming Super-$B$ factory,
it is therefore believed with enough reasons that as a different unique
insight to the nature of the light scalars, the $B$ meson decays
involving $p$-wave light scalars will provide good places in investigating
the physical properties of light scalars.
It is expected that the old puzzles related to the nature of the scalars could receive
new attention through the studies on rare $B$ meson decays involving scalars,
apart from those well-known primary tasks in heavy flavor physics.

Although the underlying structure of the light scalars is still controversial,
the scalar $a_0(1450)$ has been confirmed to be a conventional $q\bar{q}$ meson in lattice
calculations~\cite{Mathur06,Kim97,Bardeen02,Kunihiro04:lattice,Prelovsek04:lattice}
recently. Furthermore, a good SU(3) flavor symmetry is indicated in
the scalar sector through the calculations in lattice QCD~\cite{Mathur06}
on the masses of $a_0(1450)$ and $K_0^*(1430)$. The evaluations on the relevant
$K_0^*(1430)$ (Hereafter, unless otherwise stated,
$K_0^*$ will be adopted to describe the $K_0^*(1430)$ throughout
the paper for the sake of simplicity.) modes therefore draw more attention now.
Recently, the authors in Ref.~\cite{Cheng06:B2SP} proposed two possible
scenarios, namely, Scenario 1(S1) and Scenario 2(S2), to describe the components
of $\ks$ meson in the QCD sum rule method based on the assumption of
conventional two-quark structure:
\begin{itemize}
\item
In S1, the lighter state $\kappa$ near 1 GeV is treated as the lowest lying $q\bar q$
state, while the heavier state $\ks$ above 1 GeV is considered as the corresponding first excited $q \bar q$ state.
\item
In S2, $\ks$ is regarded as ground $q\bar q$ state and the corresponding
first excited state lies between (2.0 $\sim$ 2.3) GeV. Then $\kappa$ is viewed
as the four-quark bound state or hybrid state.
\end{itemize}

The two body charmless hadronic $B$ meson decays to the scalar $\ks$ meson
have been studied intensively, for example,
in Refs.~\cite{Delepine08:b2sp,Cheng06:B2SP,Zhang:scalar,Liu:scalar,Wang:scalar} by employing
different factorization approaches respectively, or in Ref.~\cite{Li:scalar}
even with the inclusion of the new physics contributions from a $Z'$ boson.
This year, the authors of Ref.~\cite{Cheng13:scalar} revisited the $B \to SP, SV$
decays in  the framework of QCD factorization.

On the theory side, it is necessary for us to make all possible
investigations on the decay modes of $B$ meson with the scalar $\ks$ to identify the
favorite one from the proposed S1 and S2 scenarios, which will also be helpful
to obtain the new insights in the properties of the scalar $\ks$;
On the experiment side, however, so far only a preliminary
upper limit at 90\% C.L. on the branching ratio of $B^0 \to {\ks}^0 \overline{K}^{*0}$
decay has been measured by Belle Collaboration~\cite{Chiang10:k0k0b},
\begin{eqnarray}
  Br(B^0 \to {\ks}^0 \overline{K}^{*0}) <  3.3 \times 10^{-6} \;.
  \label{eq:exp1}
\end{eqnarray}
Of course, this measurement would be improved rapidly with the LHC experiments at CERN,
and other relevant channels considered in this work would also be observed
in the near future.

In this work, we will study the branching ratios and CP-violating asymmetries
of $B \to K_0^* K^*$ decays in the standard model (SM) by employing the low energy
effective Hamiltonian~\cite{Buras96:weak} and the
pQCD factorization approach~\cite{Keum01:kpi,Lu01:pipi,Li03:ppnp},
where $B$ stands for $B_{u,d}$ and $B_s$ respectively.
Based on $k_T$ factorization, the pQCD approach is one of the popular factorization
methods for dealing with the B meson exclusive decays.
In the pQCD approach, the parton transverse momentum $k_T$ is kept in order to
eliminate the end-point singularity, while the Sudakov factor
play an important role in suppressing the long-distance contribution \cite{Li03:ppnp}.
We here not only consider the usual factorizable emission diagrams, but also evaluate the nonfactorizable spectator
and the annihilation type contributions simultaneously.
As far as the annihilation contributions are concerned, both the soft-collinear effective theory~\cite{scet:factor}
and the pQCD approach can work, but with rather different viewpoints on the
relevant perturbative calculations~\cite{Stewart08:anni-scet,Chay08:complexanni}.
However, the predictions on the pure annihilation decays based on the pQCD approach
can accommodate the experimental data well, for example,
for the $B_s\to \pi^+\pi^-$ and $B^0\to K^+K^-$ decays as have been done
in Refs.~\cite{Lu03:dsk,Li04:pippim,Lu07:bs2mm,Xiao11:pippim}.
In this work, we will therefore leave the controversies aside and adopt this approach in our analysis.

The paper is organized as follows. Section~\ref{sec:form} is devoted to the ingredients of the basic formalism
in the pQCD approach. The analytic expressions for the decay amplitudes of $B \to \ks K^*$ modes
in the pQCD approach are also collected in this section. The numerical results and phenomenological analysis
for the branching ratios and CP-violating asymmetries of the considered decays
are given in Sec.~\ref{sec:randd}. We summarize and conclude in Sec.~\ref{sec:summary}.

\section{ Formalism}\label{sec:form}

The pQCD approach is one of the popular methods to evaluate the hadronic matrix elements in the heavy $b$-flavor
mesons' decays. The basic idea of the pQCD approach is that it takes into
account the transverse momentum $k_T$ of the valence quarks
in the calculation of the hadronic matrix elements.
The $B$ meson transition form factors, and the spectator and
annihilation contributions are then all calculable in the framework
of the $k_T$ factorization, where three energy scales $m_W,m_B$ and $t\approx \sqrt{m_B \Lambda_{\rm QCD}} $
are involved~\cite{Keum01:kpi,Lu01:pipi,Li95-96:pQCD}.
The running of the Wilson coefficients $C_i(t)$ with $t \geq \sqrt{m_B \Lambda_{\rm QCD}} $ are controlled by
the renormalization group equation (RGE) and can be  calculated perturbatively.
The dynamics below $\sqrt{m_B \Lambda_{\rm QCD}}$ is soft, which is described by the meson wave
functions. The soft dynamics is not perturbative but universal for all channels.
In the pQCD approach, a $B \to M_2 M_3$ decay amplitude is therefore factorized into the convolution of the six-quark
hard kernel($H$), the jet function($J$) and the Sudakov factor($S$) with the
bound-state wave functions($\Phi$) as follows,
\begin{eqnarray}
{\cal A}(B \to M_2 M_3)=\Phi_{B} \otimes H \otimes J \otimes S
\otimes \Phi_{M_2} \otimes \Phi_{M_3}\;, \label{eq:sixquarks}
\end{eqnarray}
The jet function $J$ comes from the threshold resummation, which
exhibits strong suppression effect in the small $x$ (quark momentum fraction) region~\cite{Li02:threshold}.
The Sudakov factor $S$ comes from the $k_T$ resummation, which
provide a strong suppression in the small $k_T$ region~\cite{Sterman89-92:ktfact}.
Therefore, these resummation effects guarantee the removal of the endpoint singularities.

\subsection{Wave Functions and Distribution Amplitudes}\label{ssec:wf}

Throughout this paper, we will use light-cone coordinate $(P^+, P^-, {\bf P_T})$ to describe the meson's momenta with the definitions
$P^{\pm}=(p_0 \pm p_3)/\sqrt{2}$ and ${\bf P_T}=(p_1,p_2)$.
The heavy $B$ meson is usually treated as a heavy-light system and its light-cone wave function
can generally be defined as~\cite{Lu01:pipi,Keum01:kpi,Lu03:form}
\begin{eqnarray}
\Phi_{B,\alpha\beta,ij}&\equiv&
  \langle 0|\bar{b}_{\beta j}(0)q_{\alpha i}(z)|B(P)\rangle \nonumber\\
&=& \frac{i \delta_{ij}}{\sqrt{2N_c}}\int dx d^2 k_T e^{-i (xP^-z^+ - k_T z_T)}
\left\{( P \hspace{-2.8truemm}/  +m_{B})\gamma_5
 \phi_{B}(x, k_T) \right\}_{\alpha\beta}\;;
\label{eq:def-bq}
\end{eqnarray}
where the indices $i,j$ and $\alpha,\beta$ are the Lorentz indices and color indices, respectively,
$P(m)$ is the momentum(mass) of the $B$ meson, $N_c$ is the color factor, and
$k_T$ is the intrinsic transverse momentum
of the light quark in $B$ meson. Note that, in principle, there are two Lorentz
structures of the wave function to be considered in the numerical calculations, however,
the contribution induced by the second Lorentz structure
is numerically small and approximately negligible~\cite{Lu03:form}.

In Eq.~(\ref{eq:def-bq}),
$\phi_{B}(x,k_T)$ is the $B$ meson distribution amplitude
and obeys to the following normalization condition,
\begin{eqnarray}
\int_0^1 dx \phi_{B}(x, b=0) &=& \frac{f_{B}}{2 \sqrt{2N_c}}\;.\label{eq:norm}
\end{eqnarray}
where $b$ is the conjugate space coordinate of transverse momentum $k_T$ and $f_B$
is the decay constant of $B$ meson.
For $B$ meson, the distribution amplitude in the impact $b$
space has been proposed
\begin{eqnarray}
\phi_{B}(x,b)&=& N_Bx^2(1-x)^2
\exp\left[-\frac{1}{2}\left(\frac{xm_B}{\omega_b}\right)^2
-\frac{\omega_b^2 b^2}{2}\right] \;,
\end{eqnarray}
in Refs.~\cite{Keum01:kpi,Lu01:pipi}, where the normalization factor $N_{B}$
is related to the decay constant $f_{B}$ through Eq.~(\ref{eq:norm}).
The shape parameter $\omega_b$ has been fixed at $\omega_b=0.40\pm 0.04$~GeV by using the rich experimental
data on the $B_{u/d}$ mesons with $f_{B_{u/d}}= 0.19$~GeV based on lots of calculations of form factors~\cite{Lu03:form}
and other well-known decay modes of $B_{u/d}$ mesons~\cite{Keum01:kpi,Lu01:pipi}
in the pQCD approach in recent years. By considering the small SU(3) flavor symmetry breaking
effect, the shape parameter $\omega_b$ for $B_s$ meson is taken as $\omega_{B_s}=0.50\pm 0.05$ GeV~\cite{Lu07:bs2mm}.

The light-cone wave function of the light vector meson $K^*$ has been given
in the QCD sum rule method up to twist-3 as~\cite{Ball98:kst}
 \begin{eqnarray}
\Phi^{L}_{K^*,\alpha\beta,ij}&\equiv& \langle K^*(P, \epsilon_L)|\bar q(z)_{\beta j} q(0)_{\alpha i} |0\rangle \nonumber\\
 &=&  \frac{ \delta_{ij}}{\sqrt{2 N_c}} \int^1_0dxe^{ix P\cdot
 z}  \biggl\{ m_{K^*}\, {\epsilon \hspace{-1.8truemm}/\,  }_L \,\phi_{K^*}(x)  +
 {\epsilon \hspace{-1.8truemm}/\,  }_L \,  P \hspace{-2.8truemm}/ \,\phi_{K^*}^t(x)  + m_{K^*}\, \phi_{K^*}^s(x) \biggr\}_{\alpha\beta}\;,
 \end{eqnarray}
for longitudinal polarization,
where $\epsilon_{L}$ denotes the longitudinal polarization
vector of $K^*$, satisfying $P \cdot \epsilon_L=0$,
$x$ denotes the momentum
fraction carried by quark in the meson.

The twist-2 distribution amplitude $\phi_{K^*}$ can be parameterized as:
\begin{eqnarray}
\phi_{K^*}(x)&=&\frac{3f_{K^*}}{\sqrt{2N_c}} x
(1-x)\left[1+3a_{1K^*}^{||}\, (2x-1)+ a_{2K^*}^{||}\, \frac{3}{2} ( 5(2x-1)^2  - 1 )\right]\;.\label{eq:ldav}
\end{eqnarray}
And the asymptotic forms of the twist-3 distribution amplitudes
$\phi^{t}_{K^*}$ and $\phi_{K^*}^{s}$ are adopted~\cite{Li05:kphi}:
\begin{eqnarray}
\phi^t_{K^*}(x) &=& \frac{3f^T_{K^*}}{2\sqrt {2N_c}}(2x-1)^2,\qquad
\phi^s_{K^*}(x)=-\frac{3f_{K^*}^T}{2\sqrt {2N_c}} (2x-1)~.
\end{eqnarray}
Here $f_{K^*}$ and $f_{K^*}^T$ are the decay constants of the $K^*$ meson
with longitudinal and transverse polarization, respectively, whose values are
\begin{eqnarray}
 f_{K^*} &=& 0.217 \pm 0.005 ~~~~~{\rm GeV}\;,  \qquad f_{K^*}^T = 0.185 \pm 0.010 ~~~~~~{\rm GeV}\;.
\end{eqnarray}
The Gegenbauer moments are taken from the recent updates~\cite{Ball05:DAs}:
\begin{eqnarray}
a_{1K^*}^{||}&=&0.03\pm 0.02,\;\;\;\;\;\;\;\;a_{2K^*}^{||}=0.11 \pm 0.09,\;\;
\end{eqnarray}

The light-cone wave function of the light scalar $\ks$ has been analyzed
in the QCD sum rule method ~\cite{Cheng06:B2SP}
 \begin{eqnarray}
\Phi_{\ks,\alpha\beta,ij}&\equiv& \langle \ks(P)|\bar q(z)_{\beta j} q(0)_{\alpha i} |0\rangle \nonumber\\
 &=&  \frac{i \delta_{ij}}{\sqrt{2 N_c}} \int^1_0dxe^{ix P\cdot
 z}  \biggl\{ P \hspace{-2.8truemm}/ \, \phi_{\ks}(x)  +
 m_{\ks}\, \phi_{\ks}^S(x)  + m_{\ks} ( n \hspace{-2.2truemm}/   v \hspace{-2.2truemm}/  - 1)\, \phi_{\ks}^T(x) \biggr\}_{\alpha\beta}\;,
 \end{eqnarray}
where
$n=(1,0,{\bf 0_T})$ and $v=(0,1,{\bf 0_T})$ are the
unit vectors pointing to the plus and minus directions on the light-cone, respectively,
and $x$ denotes the momentum fraction carried by the quark in the $\ks$ meson.

For the light scalar meson $\ks$, its leading twist (twist-2) light-cone distribution amplitude
$\phi_{\ks}(x,\mu)$ can be generally expanded as the Gegenbauer polynomials~\cite{Cheng06:B2SP,Li09:B2Sfm}:
\begin{eqnarray}
\phi_{\ks}(x,\mu)&=&\frac{3}{\sqrt{2N_c}}x(1-x)\biggl\{f_{\ks}(\mu)+\bar
f_{\ks}(\mu)\sum_{m=1}^\infty B_m(\mu)C^{3/2}_m(2x-1)\biggr\}, \label{eq:a0-twist-2}
\end{eqnarray}
where $f_{\ks}(\mu)$ and $\bar f_{\ks}(\mu)$, $B_m(\mu)$, and
$C_m^{3/2}(t)$ are the vector and scalar decay constants,
Gegenbauer moments, and Gegenbauer polynomials, respectively.
There is a relation between the vector and scalar decay constants,
\begin{eqnarray}
 \bar f_{\ks} &=& \mu_{\ks} f_{\ks} \;\;\;\;  {\rm and} \;\;\;\;
 \mu_{\ks} = \frac{m_{\ks}}{m_2(\mu)-m_1(\mu)}\;, \label{eq:decay-const}
\end{eqnarray}
where $m_1$ and $m_2$ are the running current quark masses in the scalar $\ks$.
According to Eq.~(\ref{eq:decay-const}), one can clearly find that the vector decay constant
$f_{\ks}$ is proportional to the mass difference between the constituent
$s$ and $u(d)$ quarks, which will result in $f_{\ks}$ being of order $m_s -m_{u(d)}$. Therefore,
contrary to the case of pseudoscalar mesons,
the contribution from the factorizable diagrams with the emission of $\ks$ will be
largely suppressed.

The values for scalar decay constants and Gegenbauer moments in the distribution amplitudes of $\ks$ have been
estimated at scale $\mu=1~ \mbox{GeV}$ in the scenarios S1 and S2~\cite{Cheng06:B2SP}:
\begin{eqnarray}
{\rm S1}: && \quad \bar f_{K_0^*}= -0.300 \pm 0.030~{\rm GeV}, \quad
B_1= 0.58 \pm 0.07, \quad  B_3= -1.20 \pm 0.08,  \nonumber\\
{\rm S2}: && \quad \bar f_{K_0^*}= 0.445 \pm 0.050~{\rm GeV},
\quad B_1= -0.57 \pm 0.13, \quad  B_3= -0.42 \pm 0.22.
\end{eqnarray}

As for the twist-3 distribution amplitudes $\phi_{K_0^*}^S$ and
$\phi_{K_0^*}^T$, we here adopt the asymptotic forms
in our numerical calculations as in Ref.~\cite{Cheng06:B2SP}:
\begin{eqnarray}
\phi^S_{K_0^*}&=& \frac{1}{2\sqrt {2N_c}}\bar f_{K_0^*},\quad
\phi_{K_0^*}^T=
\frac{1}{2\sqrt {2N_c}}\bar f_{K_0^*}(1-2x).
\end{eqnarray}
Here, we should stress that the $k_T$ dependence of the distribution amplitudes in the final states
has been neglected, since its contribution is very small as indicated in Refs.~\cite{Li95-96:pQCD}.
The underlying reason is that the contribution
from $k_T$ correlated with a soft dynamics is strongly suppressed by the Sudakov
effect through resummation for the wave function, which is dominated by a collinear dynamics.
Another reason is just that, unfortunately, up to now, the distribution amplitudes with intrinsic $k_T$-dependence
for the above mentioned light mesons $K^*$ and $\ks$ are not available.

\subsection{Perturbative Calculations} \label{ssec:pcalc}

\begin{figure}[!!htb]
\centering
\begin{tabular}{l}
\includegraphics[width=0.9\textwidth]{fig1.eps}
\end{tabular}
\caption{(Color online) Typical Feynman diagrams contributing to  $B
\to \ks K^*$ decays at leading order. In this figure, $B$ stands for $B_u$,
$B_d$, and $B_s$, respectively. When we exchanged the position of $\ks$ and
 $K^*$, the other eight diagrams
contribute to the considered decay modes will be easily obtained.}
  \label{fig:fig1}
\end{figure}

For the considered $B \to \ks K^*$ decays,
the related weak effective
Hamiltonian $H_{{\rm eff}}$~\cite{Buras96:weak} can be written as
\begin{eqnarray}
H_{\rm eff}\, &=&\, {G_F\over\sqrt{2}}
\biggl\{ V^*_{ub}V_{uq} [ C_1(\mu)O_1^{u}(\mu)
+C_2(\mu)O_2^{u}(\mu) ]
 - V^*_{tb}V_{tq} [ \sum_{i=3}^{10}C_i(\mu)O_i(\mu) ] \biggr\}+ {\rm H.c.}\;,
\label{eq:heff}
\end{eqnarray}
with $q= d$ or $s$, the Fermi constant $G_F=1.16639\times 10^{-5}{\rm
GeV}^{-2}$, Cabibbo-Kobayashi-Maskawa(CKM) matrix elements $V$,
and Wilson coefficients $C_i(\mu)$ at the renormalization scale
$\mu$. The local four-quark
operators $O_i(i=1,\cdots,10)$ are written as
\begin{enumerate}
\item[]{(1) current-current(tree) operators}
\begin{eqnarray}
{\renewcommand\arraystretch{1.5}
\begin{array}{ll}
\displaystyle
O_1^{u}\, =\,
(\bar{q}_\alpha u_\beta)_{V-A}(\bar{u}_\beta b_\alpha)_{V-A}\;,
& \displaystyle
O_2^{u}\, =\, (\bar{q}_\alpha u_\alpha)_{V-A}(\bar{u}_\beta b_\beta)_{V-A}\;;
\end{array}}
\label{eq:operators-1}
\end{eqnarray}

\item[]{(2) QCD penguin operators}
\begin{eqnarray}
{\renewcommand\arraystretch{1.5}
\begin{array}{ll}
\displaystyle
O_3\, =\, (\bar{q}_\alpha b_\alpha)_{V-A}\sum_{q'}(\bar{q}'_\beta q'_\beta)_{V-A}\;,
& \displaystyle
O_4\, =\, (\bar{q}_\alpha b_\beta)_{V-A}\sum_{q'}(\bar{q}'_\beta q'_\alpha)_{V-A}\;,
\\
\displaystyle
O_5\, =\, (\bar{q}_\alpha b_\alpha)_{V-A}\sum_{q'}(\bar{q}'_\beta q'_\beta)_{V+A}\;,
& \displaystyle
O_6\, =\, (\bar{q}_\alpha b_\beta)_{V-A}\sum_{q'}(\bar{q}'_\beta q'_\alpha)_{V+A}\;;
\end{array}}
\label{eq:operators-2}
\end{eqnarray}

\item[]{(3) electroweak penguin operators}
\begin{eqnarray}
{\renewcommand\arraystretch{1.5}
\begin{array}{ll}
\displaystyle
O_7\, =\,
\frac{3}{2}(\bar{q}_\alpha b_\alpha)_{V-A}\sum_{q'}e_{q'}(\bar{q}'_\beta q'_\beta)_{V+A}\;,
& \displaystyle
O_8\, =\,
\frac{3}{2}(\bar{q}_\alpha b_\beta)_{V-A}\sum_{q'}e_{q'}(\bar{q}'_\beta q'_\alpha)_{V+A}\;,
\\
\displaystyle
O_9\, =\,
\frac{3}{2}(\bar{q}_\alpha b_\alpha)_{V-A}\sum_{q'}e_{q'}(\bar{q}'_\beta q'_\beta)_{V-A}\;,
& \displaystyle
O_{10}\, =\,
\frac{3}{2}(\bar{q}_\alpha b_\beta)_{V-A}\sum_{q'}e_{q'}(\bar{q}'_\beta q'_\alpha)_{V-A}\;.
\end{array}}
\label{eq:operators-3}
\end{eqnarray}
\end{enumerate}
with the color indices $\alpha, \ \beta$ and the notations
$(\bar{q}'q')_{V\pm A} = \bar q' \gamma_\mu (1\pm \gamma_5)q'$.
The index $q'$ in the summation of the above operators runs
through $u,\;d,\;s$, $c$, and $b$.
The standard combinations $a_i$ of Wilson coefficients are defined as follows,
  \begin{eqnarray}
a_1&=& C_2 + \frac{C_1}{3}\;, \qquad  a_2 = C_1 + \frac{C_2}{3}\;,\quad
 a_i = C_i + \frac{C_{i \pm 1}}{3}(i=3 - 10) \;.
  \end{eqnarray}
where the upper(lower) sign applies, when $i$ is odd(even).

Similar to $B \to \ks K$ decays~\cite{Liu:scalar}, there are eight types of diagrams
contributing to $B \to \ks K^*$ modes at leading order, as illustrated in Fig.~\ref{fig:fig1}.
They involve two classes of topologies with spectator and annihilation, respectively. Each kind of
topology is classified into factorizable diagrams, in which hard gluon connects the quarks in the
same meson, e.g., Fig.~\ref{fig:fig1} (a) and~\ref{fig:fig1} (b), and nonfactorizable diagrams,
in which hard gluon attaches the quarks in two different mesons, e.g., Fig.~\ref{fig:fig1} (c) and~\ref{fig:fig1} (d).
By calculating these Feynman diagrams, one can get the decay amplitudes of $B \to \ks K^*$ decays.
Because the formulas of $B \to \ks K^*$ are similar to those of $B \to K_0^* K_0^*$~\cite{Liu:scalar},
one can therefore obtain the expressions for all the diagrams just by replacing the corresponding wave functions
and input parameters from $B \to K_0^* K_0^*$. So we do not present the detailed formulas in this paper.

By combining various of contributions from the relevant Feynman diagrams together, the total decay amplitudes
for the considered $B \to \ks K^*$ decays can then read as,
\begin{enumerate}
\item {The total decay amplitudes for charged $B_u$ decays:}
\begin{eqnarray}
{\cal A}(B_u \to K^{*+} {\ksb}^0) &=& \lambda_u \biggl[
M_{nfa} C_1\biggr]
- \lambda_t \biggl[F_{fs} (a_4- \frac{1}{2} a_{10})
+ F_{fs}^{P2} (a_6 -\frac{1}{2} a_8)  \nonumber\\
&&  + M_{nfs} (C_3- \frac{1}{2} C_9) + M_{nfs} (C_5 -\frac{1}{2} C_7)
+ M_{nfa}  \nonumber\\  &&
\times  (C_3 + C_9)+ M_{nfa}^{P1} (C_5 + C_7)
+ f_B F_{fa}^{P2} (a_6 +a_8)\biggr]\;, \label{eq:tda-b2kspk0sb}
\end{eqnarray}
where $\lambda_{u}= V_{ub}^* V_{ud}$ and $\lambda_t = V_{tb}^* V_{td}$. The
decay amplitude of $B_u \to {\ks}^+ \bar{K}^{*0}$ can be obtained directly from
Eq.~(\ref{eq:tda-b2kspk0sb}) with the replacement of $K^{*} \leftrightarrow {\ks}$, but without the
contributions from the term $F_{fs}^{P2}$. The reason is that the emitted vector $K^*$ meson can not
be produced via the scalar or pseudoscalar current.

\item {The total decay amplitudes for neutral $B_d$ decays:}
\begin{eqnarray}
{\cal A}(B_d \to K^{*+} {\ks}^-) &=& \lambda_u \biggl[ M_{nfa} C_2\biggr]-\lambda_t \biggl[
 M_{nfa} (C_4 + C_{10}) + M_{nfa}^{P2} (C_6 + C_8) \nonumber\\  &&
 + M_{nfa}[K^{*+} \leftrightarrow {\ks}^-]
 (C_4 - \frac{1}{2} C_{10})
\nonumber\\  &&
 + M_{nfa}^{P2}[K^{*+} \leftrightarrow {\ks}^-] (C_6 - \frac{1}{2} C_8) \biggr]\;,\label{eq:tda-b2kspk0sm}
\end{eqnarray}

\begin{eqnarray}
{\cal A}(B_d \to K^{*0} {\ksb}^0) &=& -\lambda_t \biggl[ F_{fs}
(a_4 - \frac{1}{2} a_{10}) + F_{fs}^{P2} (a_6 - \frac{1}{2} a_8)
+ (M_{nfs} + M_{nfa} ) \nonumber\\  &&
\times (C_3 -\frac{1}{2} C_9)+(M_{nfs}^{P1}  + M_{nfa}^{P1})(C_5 - \frac{1}{2} C_7)
+ (M_{nfa} \nonumber\\  &&
 + [K^{*0} \leftrightarrow {\ksb}^0] ) (C_4 - \frac{1}{2} C_{10})
 + (M_{nfa}^{P2} + [K^{*0} \leftrightarrow {\ksb}^0] ) \nonumber\\  &&
\times (C_6 - \frac{1}{2} C_8)
 + f_B F_{fa}^{P2} (a_6 - \frac{1}{2}a_8) \biggr]\;.\label{eq:tda-b2ks0k0sb}
\end{eqnarray}
Similarly, the decay amplitudes of $B_d \to {\ks}^+ K^{*-}$ and $B_d \to {\ks}^0 \bar{K}^{*0}$
can also be obtained easily from Eqs.~(\ref{eq:tda-b2kspk0sm}) and (\ref{eq:tda-b2ks0k0sb}) with
the replacements $K^* \leftrightarrow \ks$, respectively, and with the dropping of term $F_{sf}^{P2}$
for the latter mode.

\item {The total decay amplitudes for $B_s$ decays:}
\begin{eqnarray}
{\cal A}(B_s \to {\ks}^+ K^{*-}) &=&
\lambda'_u \biggl[ F_{fs} a_1 + M_{nfs} C_1 + M_{nfa} C_2\biggr]
-\lambda'_t \biggl[ F_{fs} (a_4 + a_{10})  \nonumber\\  &&
+ F_{fs}^{P2} (a_6 +a_8)
+ M_{nfs} (C_3 + C_9) + M_{nfs}^{P1} (C_5 + C_7)
  \nonumber\\  &&
 +  M_{nfa}(C_3 - \frac{1}{2}C_9 + C_4 - \frac{1}{2} C_{10})
+ M_{nfa}[{\ks}^+ \leftrightarrow K^{*-}] \nonumber\\  &&
\times (C_4 + C_{10})
+ M_{nfa}^{P1} (C_5 - \frac{1}{2} C_7)
+ M_{nfa}^{P2} (C_6 - \frac{1}{2} C_8)\nonumber\\  &&
+ M_{nfa}^{P2}[{\ks}^+ \leftrightarrow K^{*-}] (C_6 + C_8)
+ f_{B_s} F_{fa}^{P2} (a_6 - \frac{1}{2} a_8) \biggr]\;,\label{eq:tda-bs2k0spksm}
\end{eqnarray}
where $\lambda'_{u}= V_{ub}^* V_{us}$ and $\lambda'_t = V_{tb}^* V_{ts}$, and
\begin{eqnarray}
{\cal A}(B_s \to {\ks}^0 \bar{K}^{*0}) &=& -\lambda'_t \biggl[ F_{fs}
(a_4 - \frac{1}{2} a_{10}) + F_{fs}^{P2} (a_6 - \frac{1}{2} a_8)
+ (M_{nfs} + M_{nfa} )\nonumber\\  &&
\times (C_3 -\frac{1}{2} C_9)
+(M_{nfs}^{P1}  + M_{nfa}^{P1})(C_5 - \frac{1}{2} C_7)
 + (M_{nfa} \nonumber\\  &&
 + [{\ks}^0 \leftrightarrow \bar{K}^{*0}] )
 (C_4 - \frac{1}{2} C_{10})
 + (M_{nfa}^{P2} + [{\ks}^0 \leftrightarrow \bar{K}^{*0}] )\nonumber\\  &&
 \times (C_6 - \frac{1}{2} C_8)
 +f_{B_s} F_{fa}^{P2} (a_6 -\frac{1}{2} a_8) \biggr]\;.\label{eq:tda-bs2k0s0ks0b}
\end{eqnarray}
\end{enumerate}
There are other two $B_s$ decay channels, i.e., $B_s \to K^{*+} {\ks}^-$ and $B_s \to K^{*0} {\ksb}^0$, whose decay amplitudes
can be derived from Eqs.~(\ref{eq:tda-bs2k0spksm})
and (\ref{eq:tda-bs2k0s0ks0b}) by the exchange of $\ks \leftrightarrow K^*$, respectively.
Certainly, the $F_{fs}^{P2}$ term has no contribution to them either.
Note that, based on the discussions of the factorizable annihilation contributions $F_{fa}$
in Ref.~\cite{Liu:scalar}, we here neglected this term in the above decay amplitudes
for the considered $B \to \ks K^*$ decays analytically.

\section{Numerical Results and Discussions} \label{sec:randd}

In this section, we will present the theoretical predictions for the branching ratios and CP-violating asymmetries
for those considered $B \to \ks K^*$ decay modes in the pQCD approach.
In numerical calculations, central values of the input parameters will be
used implicitly unless otherwise stated. The relevant QCD scale~({\rm GeV}), masses~({\rm GeV}),
and $B$ meson lifetime({\rm ps}) are the following ~\cite{Keum01:kpi,Lu01:pipi,Beringer12:pdg}
\begin{eqnarray}
 \Lambda_{\overline{\mathrm{MS}}}^{(f=4)} &=& 0.250\; , \quad m_W = 80.41\;,
 \quad  m_{B}= 5.28\;,  \quad  m_{B_s}= 5.37\;,\quad m_b = 4.8 \;; \nonumber\\
  \tau_{B_u} &=& 1.641\;,  \quad \tau_{B_d}= 1.519\;,
   \quad  \tau_{B_s}= 1.497\;,
\quad m_{K^*}= 0.892\;,
\quad m_{K_0^*(1430)}= 1.425\;.
\label{eq:mass}
\end{eqnarray}
For the CKM matrix elements, we adopt the Wolfenstein
parametrization and the updated parameters $A=0.811$,
 $\lambda=0.22535$, $\bar{\rho}=0.131^{+0.026}_{-0.013}$, and $\bar{\eta}=0.345^{+0.013}_{-0.014}$~\cite{Beringer12:pdg}.

\subsection{Branching Ratios}\label{ssec:cp-brs}

In this subsection, we will analyze the branching ratios of the considered $B \to K_0^* K^*$ decays in
the pQCD approach. For $B \to K_0^* K^*$ decays, the decay rate can be written as
\begin{eqnarray}
\Gamma =\frac{G_{F}^{2}m^{3}_{B}}{32 \pi  } (1-2 r_{K_0^*}^2) |{\cal A}(B
\to K_0^* K^*)|^2\;,\label{eq:bqdr}
\end{eqnarray}
where the corresponding decay amplitudes ${\cal A}$ have been
given explicitly in Eqs.~(\ref{eq:tda-b2kspk0sb}-\ref{eq:tda-bs2k0s0ks0b}).
Using the decay amplitudes obtained in last section, it is straightforward to calculate the
branching ratios with uncertainties as displayed in Table \ref{tab:br-bu} - \ref{tab:br-bs}
for the considered decay modes.
The major errors are induced by the uncertainties of the shape parameters $\omega_b = 0.40 \pm 0.04$~GeV for $B_{u,d}$ decays,
$\omega_{B_s}=0.50 \pm 0.05$~GeV for $B_s$ decays, the scalar decay constant $\bar f_{K_0^*}$ of $\ks$ meson, the decay constants
$f_{K^*}^{(T)}$ of vector $K^*$ meson, the Gegenbauer moments $B_i(i=1, 3)$ for the scalar $K_0^*$,
the Gegenbauer moments $a_{i}(i=1, 2)$ for the vector $K^*$ meson, and CKM matrix elements
$V_i$ ($\bar \rho, \bar \eta$), respectively.

\begin{table*}[htb]
\caption{ The pQCD predictions for the branching ratios of $B_u \to {\ks}^+ \overline{K}^{*0}$ and $K^{*+} {\ksb}^0$ decays
in different scenarios: the first (second)  entry corresponds to S1(S2).}
\label{tab:br-bu}
\begin{center}\vspace{-0.2cm}
{ \begin{tabular}[t]{l|c}
 \hline \hline
 Decay modes     &    Branching ratios                \\
\hline
$B_u \to {\ks}^+ \overline{K}^{*0}$              &$\begin{array}{l}
2.1^{+1.5}_{-0.9}(\omega_{b})
^{+0.4}_{-0.4}(\bar f_{K_0^*})
^{+0.3}_{-0.3}(f^{(T)}_{K^*})
^{+0.1}_{-0.2}(B_{i})
^{+0.2}_{-0.1}(a_{i})
^{+0.1}_{-0.2}(V_i)
^{+0.4}_{-0.2}(a_t)
\times  10^{-7} \\

1.3^{+0.5}_{-0.3}(\omega_{b})
^{+0.3}_{-0.3}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.1}_{-0.1}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.0}_{-0.1}(V_i)
^{+0.4}_{-0.2}(a_t)
\times  10^{-6} \end{array} $
\\
\hline
$B_u \to K^{*+} {\ksb}^0$             &$\begin{array}{l}
6.0^{+1.4}_{-1.0}(\omega_{b})
^{+1.3}_{-1.2}(\bar f_{K_0^*})
^{+0.3}_{-0.3}(f^{(T)}_{K^*})
^{+0.4}_{-0.3}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.5}_{-0.5}(V_i)
^{+0.8}_{-0.5}(a_t)
\times  10^{-7} \\

1.5^{+0.5}_{-0.3}(\omega_{b})
^{+0.4}_{-0.3}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.3}_{-0.1}(B_{i})
^{+0.2}_{-0.1}(a_{i})
^{+0.1}_{-0.1}(V_i)
^{+0.5}_{-0.3}(a_t)
\times  10^{-6}\end{array} $
\\
\hline \hline
\end{tabular}}
\end{center}
\end{table*}
\begin{table*}[htb]
\caption{ Same as Table~\ref{tab:br-bu} but for neutral $B_d \to \ks K^*$ decays in both scenarios.}
\label{tab:br-bd}
\begin{center}\vspace{-0.2cm}
{ \begin{tabular}[t]{l|c}
 \hline \hline
 Decay modes     &    Branching ratios                \\
\hline
$B_d \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0$              &$\begin{array}{l}
6.4^{+0.5}_{-0.6}(\omega_{b})
^{+1.3}_{-1.2}(\bar f_{K_0^*})
^{+0.7}_{-0.7}(f^{(T)}_{K^*})
^{+1.6}_{-1.5}(B_{i})
^{+0.8}_{-0.8}(a_{i})
^{+0.2}_{-0.3}(V_i)
^{+0.2}_{-0.5}(a_t)
\times  10^{-7} \\

5.9^{+0.7}_{-0.6}(\omega_{b})
^{+1.4}_{-1.2}(\bar f_{K_0^*})
^{+0.6}_{-0.4}(f^{(T)}_{K^*})
^{+5.9}_{-3.2}(B_{i})
^{+0.5}_{-0.4}(a_{i})
^{+0.2}_{-0.3}(V_i)
^{+0.7}_{-0.6}(a_t)
\times  10^{-7}
\end{array} $
\\
\hline
$B_d/\bar B_d \to {\ks}^0 \overline{K}^{*0} (K^{*0}  {\ksb}^{0})$              &$\begin{array}{l}
5.0^{+2.2}_{-1.3}(\omega_{b})
^{+1.1}_{-1.0}(\bar f_{K_0^*})
^{+0.6}_{-0.6}(f^{(T)}_{K^*})
^{+0.4}_{-0.3}(B_{i})
^{+0.2}_{-0.1}(a_{i})
^{+0.1}_{-0.3}(V_i)
^{+0.5}_{-0.5}(a_t)
\times  10^{-7} \\

2.3^{+0.8}_{-0.5}(\omega_{b})
^{+0.5}_{-0.5}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.4}_{-0.2}(B_{i})
^{+0.2}_{-0.2}(a_{i})
^{+0.0}_{-0.1}(V_i)
^{+0.7}_{-0.5}(a_t)
\times  10^{-6}
 \end{array} $
\\
\hline \hline
 Decay modes     &    Branching ratios ($10^{-6}$)               \\
\hline
$B_d \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-$             &$\begin{array}{l}
2.8^{+0.2}_{-0.3}(\omega_{b})
^{+0.6}_{-0.6}(\bar f_{K_0^*})
^{+0.1}_{-0.2}(f^{(T)}_{K^*})
^{+0.4}_{-0.4}(B_{i})
^{+0.1}_{-0.0}(a_{i})
^{+0.1}_{-0.2}(V_i)
^{+0.0}_{-0.1}(a_t) \\

1.1^{+0.1}_{-0.1}(\omega_{b})
^{+0.2}_{-0.3}(\bar f_{K_0^*})
^{+0.0}_{-0.1}(f^{(T)}_{K^*})
^{+1.1}_{-0.6}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.0}_{-0.1}(V_i)
^{+0.0}_{-0.1}(a_t) \end{array} $
\\
\hline
$B_d/\bar B_d \to {\ks}^+ K^{*-}$              &$\begin{array}{l}
4.5^{+0.4}_{-0.4}(\omega_{b})
^{+0.9}_{-0.9}(\bar f_{K_0^*})
^{+0.2}_{-0.2}(f^{(T)}_{K^*})
^{+0.6}_{-0.6}(B_{i})
^{+0.3}_{-0.3}(a_{i})
^{+0.2}_{-0.3}(V_i)
^{+0.1}_{-0.1}(a_t)  \\

2.9^{+0.3}_{-0.4}(\omega_{b})
^{+0.6}_{-0.7}(\bar f_{K_0^*})
^{+0.2}_{-0.1}(f^{(T)}_{K^*})
^{+2.0}_{-1.4}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.2}_{-0.2}(V_i)
^{+0.1}_{-0.2}(a_t) \end{array} $
\\
\hline
$B_d/\bar B_d \to K^{*+} {\ks}^{-}$              &$\begin{array}{l}
1.7^{+0.2}_{-0.2}(\omega_{b})
^{+0.3}_{-0.3}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.2}_{-0.2}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.1}_{-0.2}(V_i)
^{+0.1}_{-0.1}(a_t)  \\

1.1^{+0.1}_{-0.2}(\omega_{b})
^{+0.2}_{-0.3}(\bar f_{K_0^*})
^{+0.1}_{-0.0}(f^{(T)}_{K^*})
^{+0.6}_{-0.5}(B_{i})
^{+0.0}_{-0.1}(a_{i})
^{+0.1}_{-0.1}(V_i)
^{+0.0}_{-0.1}(a_t) \end{array} $
\\
\hline \hline
\end{tabular}}
\end{center}
\end{table*}
\begin{table*}[htb]
\caption{ Same as Table~\ref{tab:br-bu} but for strange $B_s \to \ks K^*$ decays in both scenarios.}
\label{tab:br-bs}
\begin{center}\vspace{-0.2cm}
{ \begin{tabular}[t]{l|c}
 \hline \hline
 Decay modes     &    Branching ratios ($10^{-5}$)               \\
\hline
$B_s \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0$              &$\begin{array}{l}
1.3^{+0.2}_{-0.1}(\omega_{bs})
^{+0.3}_{-0.2}(\bar f_{K_0^*})
^{+0.2}_{-0.1}(f^{(T)}_{K^*})
^{+0.4}_{-0.3}(B_{i})
^{+0.2}_{-0.1}(a_{i})
^{+0.0}_{-0.0}(V_i)
^{+0.1}_{-0.1}(a_t)  \\

1.3^{+0.2}_{-0.2}(\omega_{bs})
^{+0.3}_{-0.3}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+1.2}_{-0.7}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.0}_{-0.0}(V_i)
^{+0.1}_{-0.2}(a_t) \end{array} $
\\
\hline
$B_s/\bar B_s \to {\ks}^0 \overline{K}^{*0} (K^{*0} {\ksb}^{0})$              &$\begin{array}{l}
0.9^{+0.3}_{-0.2}(\omega_{bs})
^{+0.2}_{-0.2}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.1}_{-0.1}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.0}_{-0.0}(V_i)
^{+0.1}_{-0.1}(a_t)  \\

5.4^{+1.5}_{-0.9}(\omega_{bs})
^{+1.3}_{-1.1}(\bar f_{K_0^*})
^{+0.2}_{-0.2}(f^{(T)}_{K^*})
^{+0.9}_{-0.5}(B_{i})
^{+0.9}_{-0.6}(a_{i})
^{+0.0}_{-0.0}(V_i)
^{+1.7}_{-1.1}(a_t) \end{array} $
\\
\hline \hline
 Decay modes     &    Branching ratios ($10^{-5}$)               \\
\hline
$B_s \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-$             &$\begin{array}{l}
1.3^{+0.2}_{-0.1}(\omega_{bs})
^{+0.3}_{-0.2}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.4}_{-0.2}(B_{i})
^{+0.1}_{-0.1}(a_{i})
^{+0.1}_{-0.0}(V_i)
^{+0.1}_{-0.1}(a_t)  \\

1.5^{+0.4}_{-0.3}(\omega_{bs})
^{+0.4}_{-0.3}(\bar f_{K_0^*})
^{+0.1}_{-0.0}(f^{(T)}_{K^*})
^{+1.0}_{-0.4}(B_{i})
^{+0.1}_{-0.0}(a_{i})
^{+0.1}_{-0.0}(V_i)
^{+0.3}_{-0.2}(a_t) \end{array} $
\\
\hline
$B_s/\bar B_s \to {\ks}^+ K^{*-}$              &$\begin{array}{l}
0.9^{+0.3}_{-0.1}(\omega_{bs})
^{+0.2}_{-0.2}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.2}_{-0.1}(B_{i})
^{+0.0}_{-0.0}(a_{i})
^{+0.0}_{-0.0}(V_i)
^{+0.1}_{+0.1}(a_t)  \\

4.0^{+1.3}_{-0.7}(\omega_{bs})
^{+1.0}_{-0.8}(\bar f_{K_0^*})
^{+0.2}_{-0.1}(f^{(T)}_{K^*})
^{+0.8}_{-0.3}(B_{i})
^{+0.6}_{-0.4}(a_{i})
^{+0.1}_{-0.1}(V_i)
^{+1.4}_{-0.8}(a_t)  \end{array} $
\\
\hline
$B_s/\bar B_s \to K^{*+} {\ks}^{-}$              &$\begin{array}{l}
0.9^{+0.2}_{-0.1}(\omega_{bs})
^{+0.2}_{-0.2}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.1}_{-0.1}(B_{i})
^{+0.2}_{-0.1}(a_{i})
^{+0.0}_{-0.0}(V_i)
^{+0.2}_{-0.1}(a_t)  \\

6.1^{+1.5}_{-0.9}(\omega_{bs})
^{+1.4}_{-1.3}(\bar f_{K_0^*})
^{+0.3}_{-0.2}(f^{(T)}_{K^*})
^{+1.0}_{-0.6}(B_{i})
^{+1.1}_{-0.9}(a_{i})
^{+0.1}_{-0.1}(V_i)
^{+1.9}_{-1.2}(a_t) \end{array} $
\\
\hline \hline
\end{tabular}}
\end{center}
\end{table*}

Based on the above numerical results of the branching ratios given at leading order
in the pQCD approach for the considered decay modes, some remarks are as follows:

\begin{enumerate}
\item[(1)] Generally speaking, the theoretical predictions for the considered decays
in the pQCD approach have relatively large errors arising from the still large uncertainties
of many input parameters. Furthermore, the numerical results for the branching ratios
suffer more from the errors induced by the less constrained hadronic parameters of
the light scalar $\ks$, such as the scalar decay constant $\bar f_{\ks}$ and the Gegenbauer
coefficients $B_i(i=1,3)$. Additionally, in this work, as displayed in the above tables,
the higher order contributions are also simply investigated by exploring the variation
of the hard scale $t_{\rm max}$, i.e., from $0.8t$ to $1.2t$ (not changing $1/b_i, i= 1,2,3$),
in the hard kernel, which have been counted into one of the source of theoretical
uncertainties.

\item[(2)]
The pQCD predictions for the CP-averaged $Br(B_u \to {\ks}^+
\overline{K}^{*0})$ and $Br(B_u \to K^{*+} {\ksb}^0)$ are in the order
of $10^{-6}$ in S2, which are lager than those in S1,
and can be tested by the  future $B$ physics experiments.
Moreover, one can define the ratios of the branching ratios
of the same decay mode but in different scenarios as the following,
 \begin{eqnarray}
 \frac{Br(B_u \to {\ks}^+ \overline{K}^{*0})_{\rm S2}}{Br(B_u \to {\ks}^+ \overline{K}^{*0})_{\rm S1}}
 &=& 6.2\;, \qquad
 \frac{Br(B_u \to K^{*+} {\ksb}^{0})_{\rm S2}}{Br(B_u \to K^{*+} {\ksb}^{0})_{\rm S1}}
 = 2.5\;;
 \end{eqnarray}
where the central values are quoted for clarification.
The above two patterns imply the different QCD dynamics involved in the corresponding decay
channels, which can be tested with the future precision measurements.

\item[(3)]
For the neutral $B_d$ decays, which include the pure penguin contribution modes,
i.e., $B_d \to K^{*0} {\ksb}^0$ and $B_d \to {\ks}^0 \overline{K}^{*0}$, and the pure annihilation
contribution channels, i.e., $B_d \to K^{*+} {\ks}^-$ and $B_d \to {\ks}^+ K^{*-}$, respectively.
The analysis for these four decay modes are a little complicated, which is just because both $B_d$
and $\bar B_d$
can decay into the same final states simultaneously, in other words, the final states
in the considered $B_d$ decays are not the CP-eigensates. Due to the $B_d-\bar B_d$ mixing, it is very difficult for us to distinguish the $B_d$
from the $\bar B_d$. However, fortunately, it is easy to identify the final states in the considered
decays. We therefore sum up $B_d/\bar B_d \to K^{*0} {\ksb}^0$ as one channel, and $B_d/\bar B_d \to
{\ks}^0 \overline{K}^{*0}$ as another. Similarly, we will have $B_d/\bar B_d \to K^{*+} {\ks}^-$ as one mode, and $B_d/\bar B_d \to {\ks}^+ K^{*-}$ as another. Moreover, following the convention by the
experimental measurements~\cite{Beringer12:pdg,Amhis12:hfag}, we also define the averaged quantity of the two channels, i.e., $B_d \to K^{*0} {\ksb}^0 + {\ks}^0 \overline{K}^{*0}$
and $B_d \to K^{*+} {\ks}^{-} + {\ks}^+ K^{*-}$.
The same phenomena will also occur in the decays of the $B_s$ meson.

\item[(4)]
The theoretical predictions on the branching ratios of the $B_d$ meson decays in
the pQCD approach have been presented in Table~\ref{tab:br-bd}.
For the pure penguin $B_d/\bar B_d \to {\ks}^0 \overline{K}^{*0}$, $B_d/\bar B_d \to K^{*0} {\ksb}^0$, and
$B_d \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0$ channels,
the pQCD predictions for the former two decays show that the branching ratios (about $ 2 \times 10^{-6}$)
in S2 are larger than that (about $5 \times 10^{-7}$) in S1, which results in the ratio $Br(B_d/\bar B_d \to {\ks}^0
\overline{K}^{*0}(K^{*0} {\ksb}^0))_{\rm S2}/Br(B_d/\bar B_d \to {\ks}^0 \overline{K}^{*0}(K^{*0} {\ksb}^0))_{\rm S1} \approx 4.6$;
while the pQCD predictions for the latter one in both scenarios are similar, which leads to the ratio
$Br(B_d \to K^{*0} {\ksb}^0 + {\ks}^0 \overline{K}^{*0})_{\rm S1}/
Br(B_d \to K^{*0} {\ksb}^0 + {\ks}^0 \overline{K}^{*0})_{\rm S2} \approx 1.1$. Note that due to the charge conjugation
between the pure penguin channels $B_d \to {\ks}^0 \overline{K}^{*0}$ and $\bar B_d \to {\ksb}^0 K^{*0}$ and the domination
of the real contributions arising from the factorizable emission diagrams, e.g., Fig.~\ref{fig:fig1} (a) and (b),
in the considered channels,
which give the same branching ratios for $B_d/\bar B_d \to {\ks}^0 \overline{K}^{*0}$
and $B_d/\bar B_d \to {\ksb}^0 K^{*0}$ modes,
as presented in Table~\ref{tab:br-bd}. Certainly, the similar phenomena will also appear in the related
$B_s$ meson decays.

As shown in Eq.~(1), only a preliminary upper limit for $B^0 \to {\kst}^0 \overline{K}^{*0}$ decay
is available now. By comparison, one can easily find from Table~\ref{tab:br-bd}
that the pQCD predictions in both scenarios are all consistent with this upper limit.
The branching ratios in the order of $10^{-6}$ and above are expected to be tested in the near future $B$
meson experiments.

\item[(5)]
For the pure annihilation $B_d/\bar B_d \to {\ks}^+ K^{*-}$, $B_d/\bar B_d \to K^{*+} {\ks}^-$,
and $B_d \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-$ channels, as listed in Table~\ref{tab:br-bd}, the pQCD predictions for the
branching ratios in both scenarios are in order of $10^{-6}$.
Furthermore, one can find that the numerical pQCD results
for the branching ratios in S1 are clearly larger than those in S2, which are rather different from the situation
of $B_d \to {\ks}^0 \overline{K}^{*0}$ and $K^{*0} {\ksb}^0$ decays. Although the charge conjugation also exists
in the channels $B_d \to K^{*+} {\ks}^-$ and $\bar B_d \to K^{*-} {\ks}^+$, the interference between tree
and penguin topologies makes the decay amplitudes for the considered modes different from those for
the $B_d$ meson decaying into two neutral final states, which therefore give different branching ratios
for $B_d/\bar B_d \to {\ks}^+ K^{*-}$ and $B_d/\bar B_d \to K^{*+} {\ks}^-$ as exhibited in Table~\ref{tab:br-bd}.

Additionally, one can define the interesting ratios among the same decay modes but in different scenarios,
 \begin{eqnarray}
  \frac{Br(B_d \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-)_{\rm S1}}{Br(B_d \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-)_{\rm S2}} &=& 2.5\;,
  \end{eqnarray}
 \begin{eqnarray}
  \frac{Br(B_d/\bar B_d \to {\ks}^+ K^{*-})_{\rm S1}}{Br(B_d/\bar B_d \to {\ks}^+ K^{*-})_{\rm S2}} &=& 1.6\;, \qquad
  \frac{Br(B_d/\bar B_d \to K^{*+} {\ks}^-)_{\rm S1}}{Br(B_d/\bar B_d \to K^{*+} {\ks}^-)_{\rm S2}} = 1.5\;,
 \end{eqnarray}
where only the central values of the branching ratios are considered for clarification.

Very recently, LHCb~\cite{Powell11:LHCb} and CDF~\cite{Ruffini11:CDF} Collaborations have measured
the pure annihilation modes of charmless hadronic $B$ meson decays, such as $B_d \to K^+ K^-$ and
$B_s \to \pi^+ \pi^-$, respectively. It is therefore believed that such large decay rates (about
$(1 \sim 5) \times 10^{-6}$) for
the considered pure annihilation decays in this paper could be tested by the ongoing LHC experiments
and/or the forthcoming Super-$B$ factory in the near future. If the numerical results of the pure annihilation
decays can be confirmed by the future measurements at the predicted level, on one hand, which will provide
much more evidences to support the successful pQCD approach in calculating the annihilation diagrams;
on the other hand, which will provide more important information on the sizable annihilation contributions
in heavy $B$ meson physics and further shed light on
the underlying mechanism of the annihilated $B$ meson decays.

\item[(6)]
For the considered $B_s$ meson decays, all the predicted branching ratios are in the range
of $(1 \sim 6) \times 10^{-5}$, which can be seen in Table~\ref{tab:br-bs}
and will be tested by the LHC experiments. In terms of the channels
with the neutral final states, which are the pure penguin induced decays, the branching ratios for
the averaged channel $B_s \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0$ are equal to each other
in two scenarios.
The branching ratios for the two summed channels, $B_s/\bar B_s \to {\ks}^0 \overline{K}^{*0}$
and $B_s/\bar B_s \to K^{*0} {\ksb}^0$ in S2, however, are larger than those in S1 with a factor about six.

The pQCD predictions for the branching ratios for the averaged channel $B_s \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-$
are similar in size in both scenarios.
Analogous to the decays with the neutral final states, the pQCD results for
$B_s/\bar B_s \to {\ks}^+ K^{*-}$ and $B_s/\bar B_s \to K^{*+} {\ks}^-$ in S2 are larger than those in S1
with a factor about 4 and 6, respectively.
These results are expected to be examined by the measurements in the future.

\item[(7)]
For the considered pure penguin decays, $B_{d/s} \to {\ks}^0 \overline{K}^{*0}$ and $K^{*0} {\ksb}^0$,
we get the ratios in two scenarios between the branching ratios of $B_d$ and $B_s$ decays in the pQCD approach,
\begin{eqnarray}
\frac{\tau_{B_d}}{\tau_{B_s}} \cdot \frac{Br(B_s \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0)}{Br(B_d \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0)}
 &=& 20.6 \;, \qquad
\frac{\tau_{B_d}}{\tau_{B_s}} \cdot \frac{Br(B_s/\bar B_s \to {\ks}^0 \overline{K}^{*0} (K^{*0} {\ksb}^0))}{Br(B_d/ \bar B_d \to {\ks}^0 \overline{K}^{*0} (K^{*0} {\ksb}^0))}
 = 18.3\;;
\end{eqnarray}
in S1, and
\begin{eqnarray}
\frac{\tau_{B_d}}{\tau_{B_s}} \cdot \frac{Br(B_s \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0)}{Br(B_d \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0)}
 &=& 22.4 \;, \qquad
\frac{\tau_{B_d}}{\tau_{B_s}} \cdot \frac{Br(B_s/\bar B_s \to {\ks}^0 \overline{K}^{*0} (K^{*0} {\ksb}^0))}{Br(B_d/ \bar B_d \to {\ks}^0 \overline{K}^{*0} (K^{*0} {\ksb}^0))}
 = 23.8\;;
\end{eqnarray}
in S2, in which the central values of the branching ratios are quoted.  From the analytical expressions for the decay amplitudes of these $B_d$ and $B_s$ modes, e.g., Eqs.~(\ref{eq:tda-b2ks0k0sb})
and (\ref{eq:tda-bs2k0s0ks0b}), one can easily find that the main difference is just from the involved CKM factors $\lambda_t$ and
$\lambda'_t$ with $|\lambda'_t/\lambda_t|^2 = 22.5$.

\item[(8)]
Frankly speaking, the measurements at the experimental aspect are not yet available
up to now. We therefore can not make any judgements on whether the scenario 1 or scenario 2 of
the scalar $\ks$ is favored by the considered decays.
The pQCD predictions for the branching ratios of the considered $B \to \ks K^*$ decays will be tested by the LHC experiments
and/or forthcoming Super-$B$ facility.

\end{enumerate}

Here, based on the numerical calculations of the branching ratios, we also examine the effects
coming from the annihilation diagrams.
In those considered $B \to \ks K^*$ decays, when the annihilation contributions are not taken into account,
the relevant predictions on the branching ratios in the pQCD approach are as follows:
 \begin{eqnarray}
Br(B_u \to {\ks}^+ \overline{K}^{*0}) &=& 7.8 \times 10^{-7}, \qquad
Br(B_u \to K^{*+} {\ksb}^{0}) = 6.1
\times 10^{-7} \;,
\end{eqnarray}

 \begin{eqnarray}
Br(B_d \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0) &=& 1.7 \times 10^{-8}, \quad
Br(B_d/\bar B_d \to {\ks}^0 \overline{K}^{*0}(K^{*0}
{\ksb}^0)) = 1.3 \times 10^{-6}\;,
 \end{eqnarray}

 \begin{eqnarray}
Br(B_s \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0) &=& 3.2 \times 10^{-7}, \quad
Br(B_s/\bar B_s \to {\ks}^0 \overline{K}^{*0}(K^{*0}
{\ksb}^0)) = 2.4 \times 10^{-5}\;,
 \end{eqnarray}

 \begin{eqnarray}
Br(B_s \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-) &=& 0.3 \times 10^{-5},
\end{eqnarray}
\begin{eqnarray}
Br(B_s/\bar B_s \to {\ks}^+ K^{*-}) &=& 2.2 \times 10^{-5}, \quad
Br(B_s/\bar B_s \to K^{*+}
{\ks}^-)=  2.1 \times 10^{-5}\;;
\end{eqnarray}
in scenario 1, and
 \begin{eqnarray}
Br(B_u \to {\ks}^+ \overline{K}^{*0}) &=& 1.8 \times 10^{-6}, \qquad
Br(B_u \to K^{*+} {\ksb}^{0}) = 1.5
\times 10^{-6}\;,
 \end{eqnarray}

 \begin{eqnarray}
Br(B_d \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0) &=& 3.4 \times 10^{-8}, \quad
Br(B_d/\bar B_d \to {\ks}^0 \overline{K}^{*0}(K^{*0}
{\ksb}^0)) = 3.0 \times 10^{-6}\;,
 \end{eqnarray}

 \begin{eqnarray}
Br(B_s \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0) &=& 3.0 \times 10^{-7}, \quad
Br(B_s/\bar B_s \to {\ks}^0 \overline{K}^{*0}(K^{*0}
{\ksb}^0)) = 5.4 \times 10^{-5}\;,
 \end{eqnarray}

 \begin{eqnarray}
Br(B_s \to {\ks}^+ K^{*-} + K^{*+} {\ks}^-) &=& 0.7 \times 10^{-5},
 \end{eqnarray}
 \begin{eqnarray}
Br(B_s/\bar B_s \to {\ks}^+ K^{*-})&=& 5.0 \times 10^{-5}, \quad
Br(B_s/\bar B_s \to K^{*+}
{\ks}^-) =  4.9 \times 10^{-5}\;.
\end{eqnarray}
in scenario 2, in which only the central values are considered for estimating the contributions
arising from the annihilation diagrams in various decay channels. By comparison, one can easily
find the following points:
\begin{itemize}
\item[(1)]
For the charged $B_u$ decays, the weak annihilation contributions play more important roles in the
$B_u \to {\ks}^+ \overline{K}^{*0}$ than that in the $B_u \to K^{*+} {\ksb}^0$.

\item[(2)]
For the pure penguin $B_d$ and $B_s$ decays, apart from the $B_s/\bar B_s \to {\ks}^0 \overline{K}^{*0}(K^{*0} {\ksb}^0)$
in S2, other channels are basically dominated by the weak annihilation contributions, particularly, for
$B_{d/s} \to {\ks}^0 \overline{K}^{*0} + K^{*0} {\ksb}^0$ modes.

\item[(3)]
For other $B_s$ decays, the significant contributions given by the weak annihilation diagrams can also be clearly
observed. Of course, the reliability
of the contributions from the annihilation diagrams to these considered decays calculated in the pQCD approach
will be examined by the relevant experiments in the future.
\end{itemize}

\subsection{CP-violating Asymmetries} \label{ssec:cpv}

Now we turn to the evaluations of the CP-violating asymmetries of $B
\to K_0^* K^*$ decays in the pQCD approach.
For the charged $B_u$ meson decays, the direct CP violation
$A_{CP}^{\rm dir}$ can be defined as,
 \begin{eqnarray}
A_{CP}^{\rm dir} =  \frac{|\overline{\cal A}_f|^2 - |{\cal A}_f|^2}{
 |\overline{\cal A}_f|^2+|{\cal A}_f|^2},
\label{eq:acp1}
\end{eqnarray}
where ${\cal A}_f$ stands for the decay amplitude of $B_u \to {\ks}^+ \overline{K}^{*0}$
and $B_u \to K^{*+} {\ksb}^0$, respectively, while $\overline{{\cal A}}_f$ denotes the
charge conjugation one correspondingly. Using Eq.~(\ref{eq:acp1}), we find the following pQCD predictions
(in unit of $10^{-2}$):
\begin{eqnarray}
A_{CP}^{\rm dir}(B_u \to {\ks}^+ \overline{K}^{*0}) &=& \left\{ \begin{array}{ll}
-32.6^{+10.5}_{-11.3}(\omega_{b})
^{+0.7}_{-0.8}(\bar f_{K_0^*})
^{+3.1}_{-4.0}(f_{K^*})
^{+3.6}_{-4.8}(B_{i})
^{+5.4}_{-1.7}(a_{i})
^{+1.1}_{-2.4}(V_i)
^{+2.1}_{-3.9}(a_t) &
\hspace{0.1cm} ({\rm S1}) \\
-34.9^{+5.0}_{-4.5}(\omega_{b})
^{+0.5}_{-0.4}(\bar f_{K_0^*})
^{+1.6}_{-1.5}(f_{K^*})
^{+6.9}_{-9.0}(B_{i})
^{+1.5}_{-1.6}(a_{i})
^{+1.4}_{-2.2}(V_i)
^{+1.6}_{-0.2}(a_t) &
 \hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.  \label{eq:adir-k0spksb-u}\;, \\
A_{CP}^{\rm dir}(B_u \to {K}^{*+} {\ksb}^0) &=& \left\{ \begin{array}{ll}
\hspace{0.26cm}43.6^{+3.7}_{-2.2}(\omega_{b})
^{+1.1}_{-1.3}(\bar f_{K_0^*})
^{+2.0}_{-1.7}(f_{K^*})
^{+13.0}_{-13.7}(B_{i})
^{+2.8}_{-2.6}(a_{i})
^{+1.7}_{-1.6}(V_i)
^{+2.0}_{-2.7}(a_t) &
\hspace{0.1cm} ({\rm S1}) \\
-67.9^{+4.9}_{-5.2}(\omega_{b})
^{+0.8}_{-1.0}(\bar f_{K_0^*})
^{+1.7}_{-1.6}(f_{K^*})
^{+16.0}_{-14.9}(B_{i})
^{+1.2}_{-0.6}(a_{i})
^{+1.9}_{-3.8}(V_i)
^{+3.3}_{-1.3}(a_t) &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.  \label{eq:adir-kspk0sb-u}\;;
\end{eqnarray}
Note that these two channels exhibit large direct CP-violating asymmetries in both scenarios
in the pQCD approach, which indicates that the contribution of the penguin diagrams is sizable.
Combining the large CP-averaged branching ratios(${\cal O}(10^{-6})$) in S2 with the large
CP violations, which could be clearly detected in $B$ factories and LHC experiments and will
provide important information on further understanding of the QCD dynamics involved in the
considered scalar $\ks$.

As for the CP-violating asymmetries for the neutral $B_{d/s} \to \ks K^*$ decays, the effects of
$B_{d/s}-\bar{B}_{d/s}$ mixing should be considered. Firstly,
for $B_{d/s}/\bar{B}_{d/s} \to {\ks}^0 \overline{K}^{*0}$ and $K^{*0} {\ksb}^0$ decays,
they will not exhibit CP violation in both scenarios 1 and 2,
since they involve the pure penguin
contributions at the leading order in the SM, which can be seen from the decay
amplitudes as given in Eqs.~(\ref{eq:tda-b2ks0k0sb}) and (\ref{eq:tda-bs2k0s0ks0b}).
If the measurements from experiments for the direct CP asymmetries
$A_{CP}^{\rm dir}$ in $B_{d/s} \to {\ks}^0 \overline{K}^{*0}$ and $K^{*0} {\ksb}^0$ decays exhibit obviously nonzero,
which will indicate the existence of new physics beyond the SM and will provide a
very promising place to look for this exotic effect.

However, the study of CP-violation for $B_{d/s}\to {K_0^*}^+ K^{*-}$
and $K^{*+} {K_0^*}^-$ becomes more complicated as
${\ks}^+ \overline{K}^{*-}$ and $K^{*+} {\ksb}^-$
are not CP eigenstates.
The time-dependent CP
asymmetries for $B_{d/s} \to {\ks}^{\pm} K^{*\mp}$ decays are thus given by
\begin{eqnarray}
a_{CP} &\equiv& \frac{\Gamma\left
(\bar{B}_{d/s}(\Delta t) \to {\ks}^{\pm} K^{*\mp}\right) -
\Gamma\left(B_{d/s}(\Delta t) \to {\ks}^{\pm} K^{*\mp}\right )}{ \Gamma\left
(\bar{B}_{d/s}(\Delta t) \to {\ks}^{\pm} K^{*\mp}\right ) + \Gamma\left
(B_{d/s}(\Delta t) \to {\ks}^{\pm} K^{*\mp}\right ) }\nonumber\\
&=& (A_{CP}^{\rm dir} \pm \Delta A_{CP}^{\rm dir})
\cos(\Delta m_{(d/s)}  \Delta t) + (A_{CP}^{\rm mix} \pm \Delta A_{CP}^{\rm mix})
\sin (\Delta m_{(d/s)} \Delta
t), \label{eq:acp-def} \end{eqnarray} where $\Delta m_{(d/s)}$ is the mass
difference between the two neutral $B_{d/s}$ mass eigenstates, $\Delta
t =t_{CP}-t_{tag} $ is the time difference between the tagged
$B_{d/s}$ ($\bar{B}_{d/s}$) and the accompanying
$\bar{B}_{d/s}$ ($B_{d/s}$) with opposite $b$ flavor
decaying to the final CP-eigenstate ${\ks}^{\pm} K^{*\mp}$ at the time $t_{CP}$.
The quantities $A_{CP}^{\rm
dir} ({\cal C}_f)$
and $A_{CP}^{\rm mix} ({\cal S}_f)$ parameterize flavor-dependent
direct CP violation and mixing-induced CP violation, respectively,
and the parameters $\DeltaA_{CP}^{\rm dir}$ and $\DeltaA_{CP}^{\rm mix}$ are
related CP-conserving quantities: $\DeltaA_{CP}^{\rm dir}$ describes the asymmetry
between the rates $\Gamma(B_{d/s} \to {K_0^*}^{+} K^{*-})+
\Gamma(\bar B_{d/s} \to {K_0^*}^- K^{*+} )$ and $\Gamma(B_{d/s} \to {K_0^*}^{-} K^{*+})+
\Gamma(\bar B_{d/s} \to {K_0^*}^+ K^{*-} )$, while $\DeltaA_{CP}^{\rm mix}$ measures the strong
phase difference between the amplitudes contributing to $B_{d/s} \to {\ks}^{\pm} K^{*\mp}$ decays.
Here, we should stress that in the definition
of the above equation, i.e., Eq.~(\ref{eq:acp-def}), the effects arising from
the width difference of $B_s$ meson have been neglected for simplicity.

Following Ref.~\cite{CKMfitter}, we define the transition amplitudes, for $B_d$ decays for example, as follows,
\begin{eqnarray}
{\cal A}_{+-} & \equiv & {\cal A}(B_{d} \to {\ks}^{+} K^{*-})\;, \label{eq:amp-def-1}\qquad
{\cal A}_{-+}  \equiv {\cal A}(B_{d} \to {\ks}^{-} K^{*+})\;, \label{eq:amp-def-2} \nonumber\\
\bar{{\cal A}}_{+-} &\equiv& {\cal A}(\bar B_{d} \to {\ks}^{+} K^{*-})\;,\label{eq:amp-def-3} \qquad
\bar{{\cal A}}_{-+}  \equiv  {\cal A}(\bar B_{d} \to {\ks}^{-} K^{*+})\;; \label{eq:amp-def-4}
\end{eqnarray}
and
\begin{eqnarray}
 \lambda_{+-}={q_{_{B_{d}}}\over p_{_{B_{d}}}}\,{\bar {\cal A}_{+-}\over {\cal A}_{+-}}\;, \qquad
 \lambda_{-+}={q_{_{B_{d}}}\over p_{_{B_{d}}}}\,{\bar {\cal A}_{-+}\over {\cal A}_{-+}}\;;
\end{eqnarray}
where the vector $K^*$ meson is emitted by the $W$ boson in the case of ${\cal A}_{+-}$ and
$\bar {\cal A}_{-+}$, while it contains the spectator quark in the case of ${\cal A}_{-+}$
and $\bar {\cal A}_{+-}$.
Then one can get
\begin{eqnarray}
 A_{CP}^{\rm dir}+\Delta A_{CP}^{\rm dir}&\equiv&{|\lambda_{+-}|^2-1\over |\lambda_{+-}|^2+1}=
{|\bar {\cal A}_{+-}|^2 -|{\cal A}_{+-}|^2\over |\bar {\cal A}_{+-}|^2+|{\cal A}_{+-}|^2}\;, \label{eq:acp-dir-p}
\nonumber\\
 A_{CP}^{\rm dir}-\Delta A_{CP}^{\rm dir}&\equiv& {|\lambda_{-+}|^2-1\over |\lambda_{-+}|^2+1}={|\bar {\cal A}_{-+}|^2-|{\cal A}_{-+}|^2\over |\bar {\cal A}_{-+}|^2+|{\cal A}_{-+}|^2}\;, \label{eq:acp-dir-m}
\end{eqnarray}
and
\begin{eqnarray}
A_{CP}^{\rm mix}+\Delta A_{CP}^{\rm mix} &\equiv& {2\,{\rm Im}\lambda_{+-}\over |\lambda_{+-}|^2+1}={2\,{\rm Im}(e^{-2i\beta_{d}}\bar {\cal A}_{+-} {\cal A}_{+-}^*)\over
 |{\cal A}_{+-}|^2+|\bar {\cal A}_{+-}|^2}, \label{eq:acp-mix-p}\nonumber\\
A_{CP}^{\rm mix}-\Delta A_{CP}^{\rm mix} &\equiv& {2\,{\rm Im}\lambda_{-+}\over |\lambda_{-+}|^2+1}={2\,{\rm Im}(e^{-2i\beta_{d}}\bar {\cal A}_{-+} {\cal A}_{-+}^*)\over
 |{\cal A}_{-+}|^2+|\bar {\cal A}_{-+}|^2}.\label{eq:acp-mix-m}
\end{eqnarray}

Owing to the fact that $B_{d} \to {\ks}^{\pm} K^{*\mp}$ is not a CP eigenstate,
one must also consider the time- and flavor-integrated charge asymmetry,
\begin{eqnarray}
 {\cal A}_{K_0^* K^*} &\equiv& {|{\cal A}_{+-}|^2+|\bar {\cal A}_{+-}|^2
 -|{\cal A}_{-+}|^2-|\bar {\cal A}_{-+}|^2\over |{\cal A}_{+-}|^2
 +|\bar {\cal A}_{+-}|^2+|{\cal A}_{-+}|^2+|\bar {\cal A}_{-+}|^2}\;, \label{eq:acp-tf}
\end{eqnarray}
as another source of possible direct CP-violating asymmetry. Then, by transforming
the experimentally motivated direct CP parameters ${\cal A}_{K_0^*K^*}$ and
$A_{CP}^{\rm dir}$ into the physically motivated choices, one can obtain the direct
CP asymmetries for $B_{d} \to {\ks}^+ K^{*-}$ and ${\ks}^- K^{*+}$ modes as the following,
\begin{eqnarray}
 A_{{\ks}^+ K^{*-}} &\equiv&{\Gamma(\bar{B}_{d} \to {\ks}^{-}K^{*+})-\Gamma(B_{d}\to
 {\ks}^{+}K^{*-})\over  \Gamma(\bar{B}_{d} \to {\ks}^{-}K^{*+})+\Gamma(B_{d}\to
 {\ks}^{+}K^{*-})}= {|\kappa^{+-}|^2-1\over |\kappa^{+-}|^2+1}= - {{\cal A}_{K_0^*K^*}-A_{CP}^{\rm dir}-{\cal A}_{K_0^*K^*}\Delta A_{CP}^{\rm dir}\over 1-\Delta
 A_{CP}^{\rm dir}-{\cal A}_{K_0^*K^*} A_{CP}^{\rm dir}}\;, \label{eq:acp-pm}   \\
 A_{{\ks}^- K^{*+}} &\equiv& {\Gamma( \bar{B}_{d}\to {\ks}^+K^{*-})-\Gamma(B_{d}\to
 {\ks}^{-}K^{*+})\over \Gamma( \bar{B}_{d}\to {\ks}^+K^{*-})-\Gamma(B_{d}\to
 {\ks}^{-}K^{*+})}= {|\kappa^{-+}|^2-1\over |\kappa^{-+}|^2+1}= {{\cal A}_{K_0^*K^*}+A_{CP}^{\rm dir}+{\cal A}_{K_0^*K^*}\Delta A_{CP}^{\rm dir} \over 1+\Delta
 A_{CP}^{\rm dir}+{\cal A}_{K_0^*K^*} A_{CP}^{\rm dir}}\;; \label{eq:acp-mp}
\end{eqnarray}
where
\begin{eqnarray}
 \kappa^{+-}={q_{_{B_{d}}}\over p_{_{B_{d}}}}\,{\bar {\cal A}_{-+}\over {\cal A}_{+-}}\;, \qquad
  \kappa^{-+}={q_{_{B_{d}}}\over p_{_{B_{d}}}}\,{\bar {\cal A}_{+-}\over {\cal A}_{-+}}\;.
\end{eqnarray}
Note that the difference among Eq.~(\ref{eq:acp-pm}) in this paper, Eq.~(161) in Ref.~\cite{CKMfitter}, and
Eq.~(5.20) in Ref.~\cite{Cheng09:b2m2m3}
is just induced from the minus sign in the definition of direct CP-violating asymmetry, i.e.,
$A_{CP}^{\rm dir}$, in
Eq.~(\ref{eq:acp-def}). The CP-violating parameters for the $B_s \to {\ks}^+ K^{*-}$
and $K^{*+} {\ks}^-$ decays can be similarly defined.

Based on the discussions on the CP violations in the above sector,
we can present the numerical results of the considered channels in both
scenarios for the CP-violating asymmetries in the pQCD approach
are as follows,
\begin{eqnarray}
{\cal A}_{{\ks}K^*} &=&  \left\{ \begin{array}{ll}
\hspace{0.28cm}45.4^{+1.3}_{-1.7}(\omega_{b})
^{+0.1}_{-0.2}(\bar f_{K_0^*})
^{+0.0}_{-0.1}(f^{(T)}_{K^*})
^{+0.7}_{-1.1}(B_{i})
^{+0.8}_{-1.3}(a_{i})
^{+1.3}_{-2.2}(V_i)
^{+1.7}_{-2.5}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
\hspace{0.28cm}46.1^{+1.1}_{-0.9}(\omega_{b})
^{+0.4}_{-0.4}(\bar f_{K_0^*})
^{+0.5}_{-0.2}(f^{(T)}_{K^*})
^{+2.6}_{-3.9}(B_{i})
^{+0.7}_{-1.0}(a_{i})
^{+1.4}_{-2.1}(V_i)
^{+4.2}_{-3.8}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-tf-pm-bd}  \;,
\end{eqnarray}
 \begin{eqnarray}
A_{CP}^{\rm dir} &=&  \left\{ \begin{array}{ll}
\hspace{0.18cm}-5.7^{+0.3}_{-0.4}(\omega_{b})
^{+0.1}_{-0.0}(\bar f_{K_0^*})
^{+0.7}_{-0.2}(f^{(T)}_{K^*})
^{+0.5}_{-0.7}(B_{i})
^{+2.9}_{-1.4}(a_{i})
^{+0.3}_{-0.3}(V_i)
^{+0.3}_{-0.3}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
-16.0^{+0.1}_{-0.1}(\omega_{b})
^{+0.3}_{-0.3}(\bar f_{K_0^*})
^{+0.7}_{-0.8}(f^{(T)}_{K^*})
^{+4.2}_{-4.9}(B_{i})
^{+3.4}_{-2.9}(a_{i})
^{+1.0}_{-0.5}(V_i)
^{+1.3}_{-1.8}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dir-pm-bd}  \;,\\
\Delta A_{CP}^{\rm dir} &=&  \left\{ \begin{array}{ll}
\hspace{0.28cm}86.4^{+0.8}_{-0.7}(\omega_{b})
^{+0.1}_{-0.2}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+0.2}_{-0.3}(B_{i})
^{+1.4}_{-1.9}(a_{i})
^{+1.4}_{-1.1}(V_i)
^{+1.7}_{-1.7}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
\hspace{0.28cm}74.0^{+1.1}_{-1.1}(\omega_{b})
^{+0.5}_{-0.8}(\bar f_{K_0^*})
^{+0.0}_{-0.1}(f^{(T)}_{K^*})
^{+1.9}_{-4.8}(B_{i})
^{+0.4}_{-1.0}(a_{i})
^{+0.8}_{-1.2}(V_i)
^{+0.0}_{-0.1}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dird-pm-bd}  \;,
\end{eqnarray}

 \begin{eqnarray}
A_{CP}^{\rm mix} &=&  \left\{ \begin{array}{ll}
-14.2^{+0.3}_{-1.2}(\omega_{b})
^{+0.1}_{-0.0}(\bar f_{K_0^*})
^{+0.2}_{-0.4}(f^{(T)}_{K^*})
^{+1.4}_{-1.7}(B_{i})
^{+0.4}_{-0.4}(a_{i})
^{+2.1}_{-0.9}(V_i)
^{+2.7}_{-4.0}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
\hspace{0.3cm}10.0^{+0.8}_{-0.8}(\omega_{b})
^{+0.6}_{-0.3}(\bar f_{K_0^*})
^{+0.9}_{-0.6}(f^{(T)}_{K^*})
^{+2.2}_{-4.9}(B_{i})
^{+2.8}_{-3.0}(a_{i})
^{+6.2}_{-4.1}(V_i)
^{+0.1}_{-1.1}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-mix-pm-bd}  \;,\\
\DeltaA_{CP}^{\rm mix} &=&  \left\{ \begin{array}{ll}
\hspace{0.18cm}-1.8^{+0.7}_{-0.5}(\omega_{b})
^{+0.2}_{-0.4}(\bar f_{K_0^*})
^{+0.1}_{-0.0}(f^{(T)}_{K^*})
^{+1.6}_{-1.7}(B_{i})
^{+2.0}_{-1.2}(a_{i})
^{+2.1}_{-1.6}(V_i)
^{+1.0}_{-0.3}(a_t)  \% &
\hspace{0.1cm} ({\rm S1}) \\
-28.1^{+1.8}_{-2.0}(\omega_{b})
^{+0.3}_{-0.5}(\bar f_{K_0^*})
^{+1.5}_{-1.5}(f^{(T)}_{K^*})
^{+18.9}_{-7.8}(B_{i})
^{+8.2}_{-7.7}(a_{i})
^{+0.8}_{-0.4}(V_i)
^{+0.5}_{-0.0}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-mixd-pm-bd}  \;,
\end{eqnarray}

 \begin{eqnarray}
A_{{\ks}^+K^{*-}} &=&  \left\{ \begin{array}{ll}
-73.2^{+1.3}_{-2.5}(\omega_{b})
^{+0.5}_{-0.0}(\bar f_{K_0^*})
^{+2.8}_{-1.1}(f^{(T)}_{K^*})
^{+1.9}_{-2.7}(B_{i})
^{+14.6}_{-8.3}(a_{i})
^{+2.1}_{-3.0}(V_i)
^{+2.6}_{-3.2}(a_t)  \% &
\hspace{0.1cm} ({\rm S1}) \\
-83.9^{+0.7}_{-0.7}(\omega_{b})
^{+0.6}_{-0.2}(\bar f_{K_0^*})
^{+1.5}_{-1.4}(f^{(T)}_{K^*})
^{+5.7}_{-2.9}(B_{i})
^{+7.9}_{-5.5}(a_{i})
^{+3.5}_{-1.6}(V_i)
^{+4.0}_{-4.4}(a_t)  \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dir-k0spksm-bd}  \;,\\
A_{{\ks}^-K^{*+}} &=&  \left\{ \begin{array}{ll}
\hspace{0.28cm}43.0^{+1.4}_{-2.0}(\omega_{b})
^{+0.1}_{-0.2}(\bar f_{K_0^*})
^{+0.1}_{-0.4}(f^{(T)}_{K^*})
^{+0.8}_{-1.4}(B_{i})
^{+1.4}_{-1.9}(a_{i})
^{+1.5}_{-2.5}(V_i)
^{+1.8}_{-2.7}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
\hspace{0.28cm}38.5^{+1.1}_{-0.8}(\omega_{b})
^{+0.6}_{-0.6}(\bar f_{K_0^*})
^{+0.2}_{-0.0}(f^{(T)}_{K^*})
^{+4.6}_{-6.7}(B_{i})
^{+1.1}_{-1.1}(a_{i})
^{+1.5}_{-1.9}(V_i)
^{+3.7}_{-3.4}(a_t)  \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dir-k0smksp-bd}  \;,
\end{eqnarray}
for $B_{d} \to {\ks}^+ K^{*-}$ and ${\ks}^- K^{*+}$ decays,
and
\begin{eqnarray}
{\cal A}_{{\ks}K^*} &=&  \left\{ \begin{array}{ll}
\hspace{0.45cm}0.5^{+5.2}_{-5.0}(\omega_{bs})
^{+0.6}_{-0.5}(\bar f_{K_0^*})
^{+2.5}_{-2.3}(f^{(T)}_{K^*})
^{+1.8}_{-2.2}(B_{i})
^{+8.7}_{-7.5}(a_{i})
^{+0.0}_{-0.0}(V_i)
^{+0.8}_{-2.2}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
-20.3^{+2.5}_{-1.7}(\omega_{bs})
^{+0.5}_{-0.4}(\bar f_{K_0^*})
^{+0.1}_{-0.1}(f^{(T)}_{K^*})
^{+1.7}_{-0.4}(B_{i})
^{+3.0}_{-2.0}(a_{i})
^{+0.6}_{-0.8}(V_i)
^{+0.6}_{-0.4}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-tf-pm-bs}  \;,
\end{eqnarray}
 \begin{eqnarray}
A_{CP}^{\rm dir} &=&  \left\{ \begin{array}{ll}
\hspace{0.45cm}3.2^{+4.2}_{-7.4}(\omega_{bs})
^{+0.5}_{-0.5}(\bar f_{K_0^*})
^{+2.8}_{-2.2}(f^{(T)}_{K^*})
^{+2.9}_{-3.6}(B_{i})
^{+12.7}_{-14.2}(a_{i})
^{+0.1}_{-0.1}(V_i)
^{+0.6}_{-2.5}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
-27.2^{+4.1}_{-2.9}(\omega_{bs})
^{+0.2}_{-0.1}(\bar f_{K_0^*})
^{+1.3}_{-1.1}(f^{(T)}_{K^*})
^{+1.8}_{-0.2}(B_{i})
^{+4.6}_{-2.7}(a_{i})
^{+0.9}_{-1.1}(V_i)
^{+0.5}_{-0.0}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dir-pm-bs}  \;,\\
\Delta A_{CP}^{\rm dir} &=&  \left\{ \begin{array}{ll}
-11.5^{+21.7}_{-26.8}(\omega_{bs})
^{+0.6}_{-0.3}(\bar f_{K_0^*})
^{+6.1}_{-6.1}(f^{(T)}_{K^*})
^{+14.8}_{-13.4}(B_{i})
^{+18.1}_{-15.7}(a_{i})
^{+2.4}_{-4.1}(V_i)
^{+12.9}_{-13.6}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
-45.4^{+5.0}_{-4.0}(\omega_{bs})
^{+0.1}_{-0.0}(\bar f_{K_0^*})
^{+3.8}_{-3.5}(f^{(T)}_{K^*})
^{+2.8}_{-5.5}(B_{i})
^{+4.5}_{-2.2}(a_{i})
^{+1.4}_{-2.4}(V_i)
^{+4.2}_{-3.4}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dird-pm-bs}  \;,
\end{eqnarray}

 \begin{eqnarray}
A_{CP}^{\rm mix} &=&  \left\{ \begin{array}{ll}
\hspace{0.28cm}36.5^{+11.5}_{-17.5}(\omega_{bs})
^{+0.5}_{-0.6}(\bar f_{K_0^*})
^{+3.5}_{-3.6}(f^{(T)}_{K^*})
^{+6.9}_{-10.5}(B_{i})
^{+7.1}_{-9.1}(a_{i})
^{+1.4}_{-1.2}(V_i)
^{+1.6}_{-4.3}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
-27.4^{+7.9}_{-5.3}(\omega_{bs})
^{+0.2}_{-0.2}(\bar f_{K_0^*})
^{+0.5}_{-0.3}(f^{(T)}_{K^*})
^{+12.6}_{-5.1}(B_{i})
^{+3.9}_{-3.2}(a_{i})
^{+1.0}_{-0.8}(V_i)
^{+2.9}_{-2.5}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-mix-pm-bs}  \;,\\
\DeltaA_{CP}^{\rm mix} &=&  \left\{ \begin{array}{ll}
-62.3^{+16.6}_{-14.8}(\omega_{bs})
^{+0.6}_{-0.8}(\bar f_{K_0^*})
^{+5.2}_{-4.6}(f^{(T)}_{K^*})
^{+10.3}_{-9.6}(B_{i})
^{+9.0}_{-5.4}(a_{i})
^{+2.0}_{-1.2}(V_i)
^{+0.6}_{-0.2}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
-35.3^{+2.7}_{-1.2}(\omega_{bs})
^{+0.5}_{-0.4}(\bar f_{K_0^*})
^{+5.6}_{-5.4}(f^{(T)}_{K^*})
^{+23.4}_{-14.7}(B_{i})
^{+8.8}_{-9.2}(a_{i})
^{+3.1}_{-1.5}(V_i)
^{+7.4}_{-5.3}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-mixd-pm-bs}  \;,
\end{eqnarray}

 \begin{eqnarray}
A_{{\ks}^+K^{*-}} &=&  \left\{ \begin{array}{ll}
\hspace{0.12cm}2.4^{+0.2}_{-0.9}(\omega_{bs})
^{+0.9}_{-1.1}(\bar f_{K_0^*})
^{+0.3}_{-0.2}(f^{(T)}_{K^*})
^{+0.9}_{-1.4}(B_{i})
^{+5.6}_{-4.5}(a_{i})
^{+0.1}_{-0.1}(V_i)
^{+0.1}_{-0.1}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
\hspace{0.12cm}1.7^{+0.3}_{-0.4}(\omega_{bs})
^{+0.6}_{-0.6}(\bar f_{K_0^*})
^{+0.3}_{-0.2}(f^{(T)}_{K^*})
^{+1.2}_{-1.1}(B_{i})
^{+0.4}_{-0.4}(a_{i})
^{+0.1}_{-0.0}(V_i)
^{+0.1}_{-0.1}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dir-k0spksm-bs}  \;,\\
A_{{\ks}^-K^{*+}} &=&  \left\{ \begin{array}{ll}
\hspace{0.45cm}4.1^{+8.2}_{-15.4}(\omega_{bs})
^{+0.1}_{-0.0}(\bar f_{K_0^*})
^{+6.1}_{-4.8}(f^{(T)}_{K^*})
^{+5.9}_{-6.0}(B_{i})
^{+19.6}_{-25.5}(a_{i})
^{+0.3}_{-0.1}(V_i)
^{+1.0}_{-4.8}(a_t)   \% &
\hspace{0.1cm} ({\rm S1}) \\
-63.6^{+10.7}_{-8.4}(\omega_{bs})
^{+0.1}_{-0.0}(\bar f_{K_0^*})
^{+4.4}_{-4.5}(f^{(T)}_{K^*})
^{+4.0}_{-2.8}(B_{i})
^{+11.4}_{-6.7}(a_{i})
^{+1.9}_{-3.2}(V_i)
^{+3.2}_{-2.1}(a_t)   \% &
\hspace{0.1cm} ({\rm S2}) \\ \end{array} \right.   \label{eq:acp-dir-k0smksp-bs}  \;,
\end{eqnarray}
for $B_{s} \to {\ks}^+ K^{*-}$ and ${\ks}^- K^{*+}$ decays.

For the direct CP asymmetries in
the pure annihilation $B_d \to {\ks}^+ K^{*-}$ and $B_d \to K^{*+} {\ks}^-$ decays as defined in Eqs.~(\ref{eq:acp-pm}) and
(\ref{eq:acp-mp})
for example, one
can find from the numerical results shown in Eqs.~(\ref{eq:acp-dir-k0spksm-bd}) and (\ref{eq:acp-dir-k0smksp-bd})
that their signs and magnitudes are rather different in these two modes within theoretical errors. In the former mode,
the direct CP violation is about -73\% in S1 and -84\% in S2; while in the latter one, the direct CP asymmetry
is 43\% in S1 and 39\% in S2. It is clear to find that
the magnitudes of direct CP-violating asymmetries predicted in the pQCD approach for these two modes
in both scenarios are much large, which can be tested at the ongoing LHC and
forthcoming Super-$B$ experiments, by combining the large branching ratios($\sim {\cal O}(10^{-6})$). Furthermore, once the predictions
on the physical quantities in the pQCD approach could be confirmed at the predicted
level by the precision experimental measurements in the future, which can also provide indirect evidences
for the important but
controversial issue on the evaluation of annihilation
contributions at leading power: almost real with tiny strong phase in soft collinear effective theory
or almost imaginary with large strong phase in the pQCD approach?

\section{Summary} \label{sec:summary}

In this work, we studied the charmless hadronic $B_{u/d/s} \to \ks K^*$ decays by employing the pQCD
approach based on the framework of $k_T$ factorization theorem. By regarding the scalar $\ks$ as the
conventional $q\bar q$ meson, then with the help of the light-cone distribution amplitude of $\ks$ up to twist-3
in two scenarios, we explored the physical observables such as branching ratios and CP-violating asymmetries
of the considered channels. It is worth mentioning that, in this paper, as the first estimates to the physical
observables of $B \to \ks K^*$ decays, only the perturbatively short distance contributions at leading order
are investigated. We do not consider the possible long-distance
contributions, such as the rescattering effects, although they should be present, and they
may be large and affect the theoretical predictions. It is beyond
the scope of this work and expected to be studied in the future.

From the numerical evaluations and phenomenological analysis in the pQCD approach, we found the following
results:

\begin{itemize}
\item
The considered $B_u \to {\ks}^+ \overline{K}^{*0}$ and $K^{*+} {\ksb}^0$ decays exhibit large
branching ratios($\sim {\cal O}(10^{-6})$) and large direct CP-violating asymmetries in S2, which
are clearly measurable in B factories and LHC experiments and are helpful to better understand the
QCD behavior of the scalar $\ks$ in turn.

\item
In the considered modes, only the preliminary upper limit on $Br(B_d \to {\ks}^0 \overline{K}^{*0})$ has been
reported by Belle collaboration. The predicted results agree basically with this upper limit and will be
tested by the more precision measurements in the future.

\item
Most of the considered decays are affected significantly by the involved weak annihilation contributions.
The predictions on large
branching ratios and large direct CP violations of the pure annihilation processes $B_d \to {\ks}^+ K^{*-}$ and $K^{*+} {\ks}^-$ can be measured in the ongoing LHC experiments and forthcoming Super-B
factory, which will provide more evidences to help understand the annihilation contributions in
B physics.

\item
The $B_{d/s} \to {\ks}^0 \overline{K}^{*0}$ and $K^{*0} {\ksb}^0$ decays can be viewed as a good platform to test
the exotic new physics beyond the SM if the obviously nonzero direct CP violations could be observed.

\item
Generally speaking, the pQCD predictions for the considered decays still suffer from
large theoretical errors induced by the uncertainties of the input parameters, e.g., mesonic
decay constants, Gegenbauer moments in the universal distribution amplitudes, etc., which
are expected to be constrained by the more and more precision data.

\end{itemize}

\begin{acknowledgments}
X.~Liu thanks Y.M.~Wang for his discussions and comments.
This work is supported by the National Natural Science
Foundation of China under Grants Nos.~11205072 and~11235005,
and by a project funded by the Priority Academic Program Development
of Jiangsu Higher Education Institutions (PAPD),
and by the Research Fund of Jiangsu Normal University under Grant No.~11XLR38.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Form Save returned a 404 error.
On our production server form posting/saving (after clicking on the save button) is now returning a 'The remote server returned an error (404) not found'. We have not configure anything on our prod server, the error just appear out of no where.
Have any of you guys experience this error?
Have you tried creating a sample link and navigate it to the page that you're trying to reach?


text:
Scoot McNairy Boards Two: 'The Legacy Of A Whitetail Deer Hunter' & '478'
By Anthony D'Alessandro
Anthony D'Alessandro
Editorial Director/Box Office Editor
@AwardsTony
More Stories By Anthony
'The Lion King' Feels The Love Thursday Night: Preview Shows Roar $23M – Update
'Smurfs' Director Raja Gosnell Takes On 'Santa: The Adventure Begins'
EXCLUSIVE: Scoot McNairy has joined the Jody Hill comedy The Legacy Of A Whitetail Deer Hunter opposite Josh Brolin and Danny McBride.
Scott Rudin is producing the film along with McBride, Hill and David Gordon Green's Rough House label. Whitetail Deer Hunter follows a sixth-grade boy who is supposed to be bonding with his father on a hunting trip, but is far from interested.
In addition, McNairy has landed the lead opposite Arnold Schwarzenegger in the Darren Aronofsky-produced thriller 478, directed by Elliott Lester (Nightingale) for Lionsgate. 478 centers around a 2002 plane accident and the events that took place 478 days later.
Following his work on those two films, McNairy will return for the third season of the AMC series Halt And Catch Fire. He recently finished work on War Machine, re-teaming for the third time with Brad Pitt, as well as the thriller Sleepless Night, opposite Jamie Foxx.
McNairy can currently be seen in Ross Partridge's Lamb which is going into its third weekend of release from distributor The Orchard. McNairy will re-team with Ben Affleck in Batman v. Superman: Dawn of Justice on March 25. McNairy is repped by The Group Management and William Morris.
The Legacy Of A Whitetail Deer Hunter

redpajama_set_name: RedPajamaCommonCrawl


Did you ever use the 3M kits with the 3 different grades of polish (Green/Yellow/BLue)?
I'm about to buy this kit with the corresponding pads in order to polish my DA40.
I bought my plane almost 1 year ago (he is from 2001, but quite OK for the exterior), I polished it last year with a no name polish only one grade, and the result was quite OK just to have a more shiny aspect.
But I removed the old stripings and decals to make new ones, and I have the ghost traces of the old decals (normal).
Under the old decals it is very shiny and nice looking, and this is what motivates me to polish the complete surface to try to have this aspect back again.
Are you recommending this kit form 3M? Or another one?
What machine are you recommending to apply it? excentric? what RPM? etc?
As I'm in France, I might not be able to buy any US polish that are sold only in the USA.
Check with the local glider pilots, and see what they use. I have used 3m finesse it on my Libelle in the past. Finesse it is the final step ultra fine polish, and it will remove the least amount of gel coat. Remember polish is abrasive. Unless you need to get the oxidation layer off, you may be better off with just a wax based solution.
Gliders has gel coat when da40 are painted?
LP is also used in the marine industry when repainting fiberglass yachts. I know from my decades of boat ownership that LP is one of the most robust finishes one can apply. It weathers very well compared to gel coats.
My 2002 paint finish failed about 2010.
You mean that your painting was dead ?
Was it sleeping outside ?
Further to this discussion -- I'm interested in the safest way to remove an old decal from my DA40. Not one of the decorative decals -- they still look ok -- but an old Connecticut state/town registration sticker. I've tried peeling and (very light) scraping but just can't get the bulk of it off. Is there any tool or chemical remover that won't harm my LP paint finish or the underlying surface?
I've had luck removing these kinds of things by warming the sticker just by rubbing fast and hard with a bare hand and then peeling off.
2002 DA40: MT, PF, 530W/430W, KAP140, ext. baggage, 1090 ES out, 2646 MTOW, 40gal.
I changed my N numbers years ago. Took careful use of a heat gun and a plastic scraper. was a big job.


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As Motherboard reported earlier this week, researchers at Kaspersky discovered that malicious hackers had successfully planted malware posing as an official ASUS security update onto ASUS's servers and signed it with two of the company's legitimate digital certificates.
In its own confirmation of the incident, Symantec revealed that at least 13,000 computers belonging to its customers were infected with the malicious software update pushed from ASUS's Live Update server last year.
Upon installation, the malicious update received by ASUS notebooks launched a scan to determine if it was running on one of the 600 unique devices that the hackers were targeting, all with the intention of downloading further malware.
How was ASUS's infrastructure compromised?
How did the hackers manage to get hold of ASUS's digital certificates in order to sign the code to make it appear as though it really were from ASUS?
Who was behind the attack, and why were they targeting those 600-or-so PCs?
Right now, we don't have answers for any of those questions, though many think the attack's sophisticated and targeted nature could mean that state-sponsored hackers, perhaps with espionage in mind, were responsible for the campaign.
A small number of devices have been implanted with malicious code through a sophisticated attack on our Live Update servers in an attempt to target a very small and specific user group. ASUS customer service has been reaching out to affected users and providing assistance to ensure that the security risks are removed.
ASUS says it has implemented a fix in the latest version (3.6.8) of its Live Update software and introduced additional security measures to prevent similar attacks in the future.
In addition, the company says it has created a security diagnostic tool that users can run on their affected PCs. The tool can be downloaded directly from ASUS's website.
ASUS customers are just the latest in a growing group of victims who have suffered as the result of a supply-chain attack.
Perhaps most infamously, consumer goods manufacturer Reckitt Benckiser and shipping conglomerate Maersk lost $100 million and $300 million in revenue, respectively, as a result of NotPetya ransomware that spread initially via a malicious automatic update for an accounting software package.


text:
Quick mends	 August 31, 2012 August 31, 2012
Quick mends: Gov. Brown signs river access bill
PADDLE UP: The L.A. Conservation Corps guides a group through still waters and overhanging willows. (Jim Burns
Los Angeles Flood Control, open thy gates!
The dog days of summer have brought us more than just heat and a freakish rain storm: the bill opening the river to recreation and education is now law. I'm surprised by the lack of MSM coverage of this important event.
Filed under: education, Gov. Jerry Brown, Los Angeles River, recreation, SB1201
Political action	 August 18, 2012
Urge Gov. Brown to sign greater L.A. River access into law
From top left, clockwise, the tranquility of carp-filled pools, at the beginning of Glendale Narrows. Once you get past the city locks, you can see self-shadows and nifty bridge architecture. (Jim Burns)
This message from our friends at FOLAR:
"You probably read that SB 1201 is now at the desk of the Governor.
On our website, it shows how citizens can ask him to sign the bill.
Would you mind circulating the contact info to the fishing community?
" See you on the river, Jim Burns
Filed under: FOLAR, Gov. Jerry Brown, Los Angeles River, SB1201
Spots	 August 14, 2012 May 21, 2013
The Golden Trout Wilderness is golden, after all
NO PHOTOSHOP HERE: That's the actual color of Golden Trout … spectacular! (Jim Burns)
Author Thomas McGuane describes fly fishing better than most, and he certainly got it right with his musings about "20-fish days" in "The Longest Silence." Of course, he wrote about stripper bass in Atlantic Ocean boils, yet the sentiment for all fisherman — from stream, to river, to broad-horizon ocean — remains the same: longing to catch lots and lots.
It's a wonderfully greedy obsession and one my son and I tested last week over a couple of days in California's Golden Trout Wilderness. First, topo map in hand, we plied the eastern approach. From Lone Pine off the 395, you take a left at the only stoplight in town, then watch for signs (virtually non-existent) to Horsehoe Meadow Road, drive up the dreaded "Z" (don't slip off the edge …), park and walk. From town to your destination is probably two-to-three hours.
By the way, speaking of signs, you won't see one anywhere in town to announce the GTW, which doesn't open until July. Very strange. And most of the locals seemed bent on driving tourists (many French and Austrians there to hike nearby Mt. Whitney) away. Seriously, Lone Piners, what's up?
Sounds much worse than it actually was, however, because once we arrived at 10,000 feet, our reward was 50 goldens over the day.
"Take one on your first cast," I said to Will, and sure enough his grin as he pulled the first one out of the water said the rest. I was lucky enough to nab No. 50 in late afternoon, exhausted from the day's hiking and catching.
The next day, we approached from the south, bunking in Kernville. This was essentially car fishing, with no topo map required. We quit after a couple of hours with only 27 caught and released. Low water in each spot didn't deter us. After a scant rain year, you can't expect the flows you crave.
Nope, they're not big fish, so if any of you want to laugh, go ahead. The biggest fish was around 12 inches, which is a whopper by golden standards. But, I ask you, isn't this one of the most beautiful species on the planet?
With the right rod in hand, small fish become bigger fish. On a dry, they run, fight, dive and try to get your flouro tied multiple times around that poorly placed log or shock of river vegetation. With the wrong rod, you'll think you're pulling up sardines from the party boat. I used my 2-weight Orvis full flex, matched to a small Battenkill reel, overlined with a 3-weight line.
Any attractor pattern does the trick with these seemingly starving fish, but don't forget your terrestrials. Grasshoppers float for days and were a blast to fish. They also proved a great way to keep the tiniest fish off the hook.
The massive 300,000-acre GTW sits on the Kern Plateau and is accessible from at least three directions. On its eastern edge from Lone Pine, off Hwy. 395; from the south, accessible from the Sequoia National Park around Mineral King, itself a 30-plus mile adventure on a one-lane, dead-end road; or going north from Kernville.
In other words, if you are in reasonably good shape, you can day-trip to some great waters and be home in time for a steak dinner at McNally's Lodge, north of Kernville on M-99.
The Golden Trout is considered a heritage fish, and by catching six different forms of California native trout from their historic drainages and photographing them in situ, you can receive a colorful, personalized certificate featuring the art of fish illustrator Joseph Tomelleri, according to the DFG Web site. The certificate will show six full-color images representing the trout you caught, along with their dates and locations. So far, the DFG has sent out about 150 certificates.
Remarkably, three of the trout native to the state's waters are within the area. Besides the California Golden (technically known as the Golden Trout Creek golden trout), there's also the Little Kern Golden Trout and Kern River Rainbow.
"Common names abound for the golden trout of the Kern River drainable," writes Robert J. Behnke in is authoritative "Trout and Salmon of North America." "This can be confusing because they tend to either pinpoint a fish to a particular stream, such as 'Volcano Creek golden trout,' or encompass a diversity of forms under one name, such as 'California golden trout.' The dozen or so common names for what are really two subspecies (aguabonita and whitei) of rainbow trout reflects the passion that so many have for this pair of jewel-like fish."
Certificate aside, the Goldens we were after end up on many a fly-fisher's bucket list for good reason: their jewel-like beauty. And, although they were once transported to Cottonwood Lakes, then to Arizona and beyond, the only place they naturally occur is right here, where they evolved in isolation from other trout.
My son's a big guy, but it's still true, most are small fish. (Jim Burns)
As I said, they are a feast for the eye, with two red stripes, one on the belly, the other along the lateral line, running to the mouth and under the gill. Also, look for large black spots – up to 10 – that run laterally as well. Put these together with a predominately yellow-gold color and there's little reason for their cousins to enter the beauty contest.
This general description will also come in handy when trying to decided if you've landed a pure golden or a hybrid, created through breeding with hatchery rainbows. Remember, these very distinct markings mean you've got a golden in your net.
David Lentz, who is California Department of Fish and Game's native trout conservation coordinator, said that the small size is because of 140 years of habitat degradation from livestock.
"Continued livestock use results in shallower, wider, warmer water," he said. Waters in their natural state would be both narrower and deeper, which, in turn, would mean fewer goldens that were bigger. The largest section of interconnected meadows for grazing lies in the South Fork of the Kern area. Sections are now being rested for eight-to-nine years at a time to regain this natural habitat. Some environmentalists have argued that the best way to prevent lifestock from grazing in the upper South Fork watershed is to get goldens listed on the Endangered Species Act, according to Behnke.
Filed under: Attractor Patterns, Golden Trout Wilderness, Grasshoppers, Heritage Trout Plaque, Kernville, Lone Pine, Orvis
Spots	 August 11, 2012 August 11, 2012
The DWP needs to do more on the lower Owens
The Lower Owens River Project is all explained on these nifty signs, but that doesn't help your cast through the tules. (Jim Burns)
Stoked by a warm-water fishing article that recently appeared in Cal Fly Fisher mag, my son and I stopped in Lone Pine over the weekend to check out the lower Owens. After all, I'd fished the ponds behind Bishop for bass and panfish, and this piece sang the praises of throwing a bass bug into the river's hot summer waters.
After a two-minute ride from town we found, yes, more water flowed; the weather was unseasonably hot as blazes; and we did spot a good-sized bass near a bank.
But now for that all-important cast … bonk. Only the croak of an insistent bull frog kept us smiling.
The looming LORP problem for the fly fisherman remains terrible access. If you're a tule, you're really a happy camper surrounded by lots of your tule friends, but if you're struggling through them, fly rod in hand, feet in the muck leading to where you might find the river's edge, it's just not so good. Casting? No way. The only casts we got in were right next to the road.
Last summer, reporter Louis Sagahun from the Los Angeles Times penned:
"The largest river restoration ever attempted in the West — intended to support a cornucopia of wildlife and outdoor activities — has left a 62-mile stretch of the Lower Owens so overrun with cattails, cane and bulrushes that it may take decades to bring them under control."
Kind of gives a new meaning to "out in the tules," doesn't it? (Jim Burns)
He was writing about the Lower Owens River Project, LORP for short, that began about six years ago when L.A. Department of Water and Power began putting more water into the river that it had diverted to Los Angeles Aqueduct since 1913.
It's a shame to have the restoration project in full swing, as evidenced by the nifty explanatory signage about the project and a new, shiny access gate, and not be able to fish. Anybody got a lawn mover?
I'd skip this one until there's a solution, possibly like the disabled fishing platform on the ponds outside Bishop.
Filed under: bass, Bishop, Cal Fly Fisher magazine, Lone Pine, Los Angeles Department of Water and Power, Los Angeles Times, Louis Sahagun, Lower Owens River Project, panfish


Remembering VG Tips Cheltenham Festival winners. Briar Hill at 25/1 | Check out VGTips.co.uk!
The Willie Mullins trained, Ruby Walsh ridden, Briar Hill was a winning bet for members of VG Tips at a great SP price of 25-1.
It was the last race of that day in 2013, the bumper, and Briar Hill was one of two Mullins trained horses that I alerted members to because the price was too big to ignore.
As do I. I was right to offer the 25/1 tip to members. Briar Hill went on to win the bumper by 7 lengths.
If you want to join VG Tips click on this link and choose from three payment options.


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Q: U-Sql view to merge duplicates via ranking I have data lying in multiple files with naming convention as {year}/{month}/{date} which have duplicates (every day delta where records may get updated everyday).
I want to create a view that will return the records with the duplicates merged / squashed.
The duplicates will be ranked and only the latest updated records corresponding to each primary key will be returned.
But the use of rowsets in view seems to be not supported. Basically something like this:
CREATE VIEW viewname AS

@sourcedata  = EXTRACT  //schema
from //filenamePattern (regex)
using Extractors.TSV()

@sourceData =  SELECT *,
       ROW_NUMBER() OVER(PARTITION BY primary_Key ORDER BY timestamp DESC) AS RowNumber FROM @SourceData;

SELECT //schema
from @sourceData WHERE RowNumber == 1

So that when I do
select * from viewname
I get the merged data directly from the underlying files.  How to achieve this ?

A: It is possible to have multiple EXTRACT statements in a view stacked together with a UNION statement which would implicitly remove duplicates.  However is there any particular reason you need to use a view?  This will limit your options as you will have to code within the limitations of views (eg they can't be parameterised).  You could also use table-valued function, stored procedure or just a plain old script.  This would give you many more options, especially if your de-duplication logic is complex.  A simple example:
DROP VIEW IF EXISTS vw_removeDupes;

CREATE VIEW vw_removeDupes
    AS

EXTRACT someVal int
FROM "/input/input59a.txt"
USING Extractors.Tsv()

UNION

EXTRACT someVal int
FROM "/input/input59b.txt"
USING Extractors.Tsv();

A: I think it can be solved by table valued function. Have you tried using it?
https://msdn.microsoft.com/en-us/azure/data-lake-analytics/u-sql/u-sql-functions

redpajama_set_name: RedPajamaStackExchange


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VIETNAM: poet, essayist, scholar and Catholic priest Father Nguyen Van Ly conditionally released
Posted on Tuesday June 14th, 2016 by Writers in Prison
26 May 2016 Update #2 to RAN 06/14
PEN International welcomes the early release of poet, essayist, scholar and Catholic priest Father Nguyen Van Ly. Father Nguyen Van Ly had been serving an eight-year prison sentence for 'conducting propaganda against the state' and was due for release in June 2016. His release is thought to be an act of goodwill ahead of United States President Obama's visit to the country. PEN continues to call for the immediate and unconditional release of all those writers and activists imprisoned for the peaceful exercise of their right to freedom of expression in accordance with Article 19 of the International Covenant on Civil and Political Rights (ICCPR), to which Vietnam is a state party.
Please send appeals:
Welcoming the release of poet, essayist, scholar and Catholic priest Father Nguyen Van Ly;
Calling for the immediate and unconditional release of all other writers and activists imprisoned or detained for the peaceful exercise of their right to freedom of expression in accordance with Article 19 of the ICCPR to which Vietnam is state party.
Appeals to:
His Excellency Tran Dai Quang
President of the Socialist Republic of Viet Nam
Hung Vuong street
Ba Dinh district
Mr Nguyen Xuan Phuc
1 Hoang Hoa Tham street
Fax: +84 80 44130/ +84 80 44940
Mr Pham Binh Minh
Minister of Foreign Affairs
1 Ton That Dam street
Email: bc.mfa [at] mofa.gov [dot] vn
Please ask your country's diplomatic representatives in Vietnam to intervene in the case. For some Vietnamese embassies in the world:
'.((strlen('http://www.embassiesabroad.com/embassies-of/Vietnam')>40) ? substr('http://www.embassiesabroad.com/embassies-of/Vietnam',0,40).'…' : 'http://www.embassiesabroad.com/embassies-of/Vietnam').''
Father Nguyen Van Ly, a Catholic priest and co-editor of the underground online magazine Tu do Ngôn luan (Free Speech), was arrested on 19 February 2007 and sentenced to eight years in prison on 30 March 2007 for 'conducting propaganda against the state'. Nguyen Van Ly is a leading member of the pro-democracy movement "Bloc 8406". He was previously detained from 1977-1978, and again from 1983-1992 for his activism in support of freedom of expression and religion. He was sentenced again in October 2001 to 15 years in prison for his online publication of an essay on human rights violations in Vietnam, before being released under amnesty in February 2005. On 30 March 2007, a People's Court in Hue sentenced him to 'conducting propaganda against the Socialist Republic of Viet Nam' under Article 88-1 (c) of the Criminal Code. On 14 November 2009, he reportedly suffered a stroke in prison. Nguyen Van Ly was granted provisional release so that he could seek medical treatment unavailable in prison on 15 March 2010, but was returned to a labour camp in Ha Nam province on 25 July 2011. In September 2010 the United Nations Working Group on Arbitrary Detention called for his immediate and unconditional release.
Nguyen Van Ly was released from the labour camp on 20 May 2016, one month early. He now faces five years of probationary detention, part of the original sentence. Nguyen Van Ly's health is reported to have severely deteriorated in prison.


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Poland, Lithuania and Ukraine want stronger sanctions against Russia
December 21, 2021 9News Nigeria Europe 0
Poland and Lithuania have joined Ukraine in calling for stronger Western sanctions against Moscow, following the alleged Russian troop buildup near the Ukrainian border that has fuelled fears of an invasion.
Polish President Andrzej Duda and Lithuanian President Gitanas Nausėda met with Ukrainian President Volodymyr Zelenskyy for the first presidential trilateral meeting of the "Lublin Triangle" security alliance in Huta in western Ukraine's Carpathian Mountains.
Duda and Nausėda gathered to show their support in the face of what they described as "an ongoing aggression against Ukraine".
US intelligence officials say Russia has amassed 70,000 troops near its border with Ukraine and is preparing for a possible invasion early next year.
Moscow has denied any intention to attack, but demanded that NATO deny membership to Ukraine and other former Soviet countries, and roll back the alliance's military deployments in Central and Eastern Europe.
Nausėda said the buildup is a threat to the entire region and the attempts by Russia to recreate zones of influence will not be tolerated.
"NATO certainly needs to respond not only by the mobilisation and concentration of capabilities in NATO's eastern flank but also by its defence strengthening plans," he added.
In a joint statement issued after the meeting, the three leaders called on other international leaders to step up sanctions against Russia.
"Our common task is to deter the threat posed by Russia and defend Europe from Russia's aggressive policies," Zelenskyy said. "Ukraine, Poland, and Lithuania are the vanguard of that deterrence today."
The Ukrainian leader called for "powerful preventative actions, powerful serious sanctions to exclude any thought about escalation".
READ ALSO COVID-19: UK to offer antibody tests to people who test positive
Russian President Vladimir Putin raised the demand for security guarantees in last week's video call with US President Joe Biden, who warned that Russia would face "severe consequences" if Moscow attacked its neighbour.
Russian diplomats have said that Moscow will have to up the ante if its demands are rejected.
Watch the full interview with Peter Dickinson, Ukraine Editor at the Atlantic Council, in the video player above.


Tangled and True: monster cookies.
I remember loving coming home to fresh baked monster cookies when I was a little girl, as we got older and my mom became increasingly busier she stopped making these little gems, and it was sort of an out of site out of mind type thing, until recently. Right before we left Denver, one of my friends started making these monster cookies. They have quickly become our all time favorite cookie. Not only are they soft, chewy and delicious, the dough freezes amazingly well. So well that we have to foodsaver package them to keep from eating the dough balls straight out of the freezer. They are THAT good.
Cream together the butter and sugars. Beat in eggs, peanut butter and vanilla. Add oat flour, and oats. Once combined add in M&M's. Scoop onto ungreased cookie sheet and bake at 350 for 8-10 minutes.
*These can be "gluten free" if you buy your oats from a specialty store that doesn't process the oats in a facility with other gluten products.