# Branching Ratio and CP Asymmetry of $B_{s}\to
K^{*}_{0}(1430)\rho(\omega,\phi)$ Decays in the PQCD Approach

Zhi-Qing Zhang 111Electronic address<EMAIL_ADDRESS>Department of
Physics, Henan University of Technology, Zhengzhou, Henan 450052, P.R.China

###### Abstract

In the two-quark model supposition for $K_{0}^{*}(1430)$, which can be viewed
as either the first excited state (scenario I) or the lowest lying state
(scenario II), the branching ratios and the direct CP-violating asymmetries
for decays $\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\phi,K^{*0}_{0}(1430)\omega,K^{*0}_{0}(1430)\rho^{0},K^{*+}_{0}(1430)\rho^{-}$
are studied by employing the perturbative QCD factorization approach. We find
the following results: (a) Enhanced by the color allowed tree amplitude with
large Wilson coefficients $a_{1}=C_{2}+C_{1}/3$, the branching ratio of
$\bar{B}_{s}^{0}\to K^{*+}_{0}(1430)\rho^{-}$ is much larger than those of the
other three decays and arrives at $(3.4^{+0.8}_{-0.7})\times 10^{-5}$ in
scenario I, even $10^{-4}$ order in scenario II, and its direct CP violating
asymmetry is the smallest, around $10\%$, so this channel might be measurable
in the current LHC-b experiments, where a large number (about $10^{12}$) of
$B$ mesons will be produced per year. This high statistics will make the
measurement possible. (b) For the decay modes $\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\omega,K^{*0}_{0}(1430)\rho^{0}$, their direct CP-violating
asymmetries are large, but it might be difficult to measure them, because
their branching ratios are small and less than (or near) $10^{-6}$ in both
scenarios. For example, in scenario I, these values are ${\cal
B}(\bar{B}_{s}^{0}\to K^{*}_{0}(1430)\omega)=(8.2^{+1.8}_{-1.7})\times
10^{-7},{\cal B}(\bar{B}_{s}^{0}\to
K^{*}_{0}(1430)\rho^{0})=(9.9^{+2.1}_{-2.0})\times 10^{-7},{\cal
A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\omega)=-24.1^{+2.8}_{-2.5},{\cal
A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\rho^{0})=26.6^{+2.5}_{-2.5}.$ (c) For the decay
$\bar{B}^{0}_{s}\to K^{*}_{0}(1430)\phi$, the predicted branching ratios are
also small and a few times $10^{-7}$ in both scenarios; there is no tree
contribution at the leading order, so its direct CP-violating asymmetry is
naturally zero.

###### pacs:

13.25.Hw, 12.38.Bx, 14.40.Nd

##  I Introduction

Along with many scalar mesons found in experiments, more and more efforts have
been made to study the scalar meson spectrum theoretically nato ; jaffe ; jwei
; baru ; celenza ; stro ; close1 . Today, it is still a difficult but
interesting topic. Our most important task is to uncover the mysterious
structures of the scalar mesons. There are two typical schemes for their
classification nato ; jaffe . Scenario I: the nonet mesons below 1 GeV,
including $f_{0}(600),f_{0}(980),K^{*}_{0}(800)$, and $a_{0}(980)$, are
usually viewed as the lowest lying $q\bar{q}$ states, while the nonet ones
near 1.5 GeV, including $f_{0}(1370),f_{0}(1500)/f_{0}(1700),K^{*}_{0}(1430)$,
and $a_{0}(1450)$, are suggested as the first excited states. In scenario II,
the nonet mesons near 1.5 GeV are treated as $q\bar{q}$ ground states, while
the nonet mesons below 1 GeV are exotic states beyond the quark model, such as
four-quark bound states.

In order to uncover the inner structures of these scalar mesons, many
factorization approaches are also used to research the $B$ meson decay modes
with a final state scalar meson, such as the generalized factorization
approach GMM , QCD factorization approach CYf0K ; ccysp ; ccysv , and
perturbative QCD (PQCD) approach zqzhang1 ; zqzhang2 ; zqzhang3 ; zqzhang4 ;
zqzhang5 . On the experimental side, along with the running of the Large
Hadron Collider beauty (LHC-b) experiments, some of $B_{s}$ decays with a
scalar meson in the final state might be observed in the current lhc1 ; lhc2 .
In order to make precise measurements of rare decay rates and CP violating
observables in the $B$-meson systems, the LHC-b detector is designed to
exploit the large number of $b$-hadrons produced. LHC-b will produce up to
$10^{12}$ $b\bar{b}$ pairs per year $(10^{7}s)$. Furthermore, it can
reconstruct a $B$-decay vertex with very good resolution, which is essential
for studying the rapidly oscillating $B_{s}$ mesons. In a word, $B_{s}$ decays
with a scalar in the final state can also serve as an ideal platform to probe
the natures of these scalar mesons. So the studies of these decay modes for
$B_{s}$ are necessary in the next a few years.

Here $K^{*}_{0}(1430)$ can be treated as a $q\bar{q}$ state in both scenario I
and scenario II, it is easy to make quantitative predictions in the two-quark
model supposition, so we would like to use the PQCD approach to calculate the
branching ratios and the CP-violating asymmetries for decays
$\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\phi,K^{*0}_{0}(1430)\omega,K^{*0}_{0}(1430)\rho^{0},K^{*+}_{0}(1430)\rho^{-}$
in two scenarios. In the following, $K^{*}_{0}(1430)$ is denoted as
$K^{*}_{0}$ in some places for convenience. The layout of this paper is as
follows. In Sec. II, the decay constants and light-cone distribution
amplitudes of relevant mesons are introduced. In Sec. III, we then analyze
these decay channels using the PQCD approach. The numerical results and the
discussions are given in section IV. The conclusions are presented in the
final part.

##  II decay constants and distribution amplitudes

In general, the $B_{s}$ meson is treated as a heavy-light system, and its
Lorentz structure can be written asgrozin ; kawa

$\displaystyle\Phi_{B_{s}}=\frac{1}{\sqrt{2N_{c}}}(P/_{B_{s}}+M_{B_{s}})\gamma_{5}\phi_{B_{s}}(k_{1}).$
(1)

The contribution of $\bar{\phi}_{B_{s}}$ is numerically small caidianlv  and
has been neglected. For the distribution amplitude $\phi_{B_{s}}(x,b)$ in
Eq.(1), we adopt the following model:

$\displaystyle\phi_{B_{s}}(x,b)=N_{B_{s}}x^{2}(1-x)^{2}\exp[-\frac{M^{2}_{B_{s}}x^{2}}{2\omega^{2}_{b_{s}}}-\frac{1}{2}(\omega_{b_{s}}b)^{2}],$
(2)

where $\omega_{b_{s}}$ is a free parameter, we take $\omega_{b_{s}}=0.5\pm
0.05$ GeV in numerical calculations, and $N_{B_{s}}=63.67$ is the
normalization factor for $\omega_{b_{s}}=0.5$.

In the two-quark picture, the vector decay constant $f_{K^{*}_{0}}$ and the
scalar decay constant $\bar{f}_{K^{*}_{0}}$ for the scalar meson $K^{*}_{0}$
can be defined as

$\displaystyle\langle K^{*}_{0}(p)|\bar{q}_{2}\gamma_{\mu}q_{1}|0\rangle$
$\displaystyle=$ $\displaystyle f_{K^{*}_{0}}p_{\mu},$ (3)
$\displaystyle\langle
K^{*}_{0}(p)|\bar{q}_{2}q_{1}|0\rangle=m_{K^{*}_{0}}\bar{f}_{K^{*}_{0}},$ (4)

where $m_{K^{*}_{0}}(p)$ is the mass (momentum) of the scalar meson
$K^{*}_{0}$. The relation between $f_{K^{*}_{0}}$ and $\bar{f}_{K^{*}_{0}}$ is

$\displaystyle\frac{m_{{K^{*}_{0}}}}{m_{2}(\mu)-m_{1}(\mu)}f_{{K^{*}_{0}}}=\bar{f}_{{K^{*}_{0}}},$
(5)

where $m_{1,2}$ are the running current quark masses. For the scalar meson
$K^{*}_{0}(1430)$, $f_{K^{*}_{0}}$ will get a very small value after the
$SU(3)$ symmetry breaking is considered. The light-cone distribution
amplitudes for the scalar meson $K^{*}_{0}(1430)$ can be written as

$\displaystyle\langle K^{*}_{0}(p)|\bar{q}_{1}(z)_{l}q_{2}(0)_{j}|0\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0}dx\;e^{ixp\cdot z}$ (6)
$\displaystyle\times\\{p\\!\\!\\!/\Phi_{K^{*}_{0}}(x)+m_{K^{*}_{0}}\Phi^{S}_{K^{*}_{0}}(x)+m_{K^{*}_{0}}(n\\!\\!\\!/_{+}n\\!\\!\\!/_{-}-1)\Phi^{T}_{K^{*}_{0}}(x)\\}_{jl}.\quad\quad$

Here $n_{+}$ and $n_{-}$ are lightlike vectors:
$n_{+}=(1,0,0_{T}),n_{-}=(0,1,0_{T})$, and $n_{+}$ is parallel with the moving
direction of the scalar meson. The normalization can be related to the decay
constants:

$\displaystyle\int^{1}_{0}dx\Phi_{K^{*}_{0}}(x)=\int^{1}_{0}dx\Phi^{T}_{K^{*}_{0}}(x)=0,\,\,\,\,\,\,\,\int^{1}_{0}dx\Phi^{S}_{K^{*}_{0}}(x)=\frac{\bar{f}_{K^{*}_{0}}}{2\sqrt{2N_{c}}}\;.$
(7)

The twist-2 light-cone distribution amplitude $\Phi_{K^{*}_{0}}$ can be
expanded in the Gegenbauer polynomials:

$\displaystyle\Phi_{K^{*}_{0}}(x,\mu)$ $\displaystyle=$
$\displaystyle\frac{\bar{f}_{K^{*}_{0}}(\mu)}{2\sqrt{2N_{c}}}6x(1-x)\left[B_{0}(\mu)+\sum_{m=1}^{\infty}B_{m}(\mu)C^{3/2}_{m}(2x-1)\right],$
(8)

where the decay constants and the Gegenbauer moments $B_{1},B_{3}$ of
distribution amplitudes for $K^{*}_{0}(1430)$ have been calculated in the QCD
sum rulesccysp . These values are all scale dependent and specified below:

$\displaystyle{\rm scenarioI:}B_{1}$ $\displaystyle=$ $\displaystyle 0.58\pm
0.07,B_{3}=-1.2\pm 0.08,\bar{f}_{K^{*}_{0}}=-(300\pm 30){\rm MeV},$ (9)
$\displaystyle{\rm scenarioII:}B_{1}$ $\displaystyle=$ $\displaystyle-0.57\pm
0.13,B_{3}=-0.42\pm 0.22,\bar{f}_{K^{*}_{0}}=(445\pm 50){\rm MeV},\quad$ (10)

which are taken by fixing the scale at 1GeV.

As for the twist-3 distribution amplitudes $\Phi_{K^{*}_{0}}^{S}$ and
$\Phi_{K^{*}_{0}}^{T}$, we adopt the asymptotic form:

$\displaystyle\Phi^{S}_{K^{*}_{0}0}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2N_{c}}}\bar{f}_{K^{*}_{0}},\,\,\,\,\,\,\,\Phi_{K^{*}_{0}}^{T}=\frac{1}{2\sqrt{2N_{c}}}\bar{f}_{K^{*}_{0}}(1-2x).$
(11)

The distribution amplitudes up to twist-3 of the vector mesons are

$\displaystyle\langle
V(P,\epsilon^{*}_{L})|\bar{q}_{2\beta}(z)q_{1\alpha}(0)|0\rangle=\frac{1}{2N_{C}}\int^{1}_{0}dxe^{ixP\cdot
z}[M_{V}\epsilon/\,^{*}_{L}\Phi_{V}(x)+\epsilon/\,_{L}^{*}P/\Phi_{V}^{t}(x)+M_{V}\Phi^{s}_{V}(x)]_{\alpha\beta},\quad$
(12)

for longitudinal polarization. The distribution amplitudes can be parametrized
as

$\displaystyle\Phi_{V}(x)$ $\displaystyle=$
$\displaystyle\frac{2f_{V}}{\sqrt{2N_{C}}}[1+a^{\|}_{2}C^{\frac{3}{2}}_{2}(2x-1)],$
(13) $\displaystyle\Phi_{V}^{t}(x)$ $\displaystyle=$
$\displaystyle\frac{3f^{T}_{V}}{2\sqrt{2N_{C}}}(2x-1)^{2},\quad\phi_{V}^{s}(x)=-\frac{3f^{T}_{V}}{2\sqrt{2N_{C}}}(2x-1),$
(14)

where the decay constant $f_{V}$ yao  and the transverse decay constant
$f^{T}_{V}$ pball  are given as the following values:

$\displaystyle f_{\rho}$ $\displaystyle=$ $\displaystyle 209\pm 2{\rm
MeV},f_{\omega}=195\pm 3{\rm MeV},f_{\phi}=231\pm 4{\rm MeV},$ (15)
$\displaystyle f^{T}_{\rho}$ $\displaystyle=$ $\displaystyle 165\pm 9{\rm
MeV},f^{T}_{\omega}=151\pm 9{\rm MeV},f^{T}_{\phi}=186\pm 9{\rm MeV}.$ (16)

Here the Gegenbauer polynomial is defined as
$C^{\frac{3}{2}}_{2}(t)=\frac{3}{2}(5t^{2}-1)$. For the Gegenbauer moments, we
quote the numerical results as pball1 :

$\displaystyle a^{\|}_{2\rho}=a^{\|}_{2\omega}=0.15\pm
0.07,a^{\|}_{2\phi}=0.18\pm 0.08.$ (17)

##  III  the perturbative QCD calculation

Under the two-quark model for the scalar meson $K^{*}_{0}$ supposition, the
decay amplitude for $\bar{B}^{0}_{s}\to VK^{*}_{0}$, where $V$ represents
$\rho,\omega,\phi$, can be conceptually written as the convolution,

$\displaystyle{\cal A}(\bar{B}^{0}_{s}\to
VK^{*}_{0})\sim\int\\!\\!d^{4}k_{1}d^{4}k_{2}d^{4}k_{3}\
\mathrm{Tr}\left[C(t)\Phi_{B_{s}}(k_{1})\Phi_{V}(k_{2})\Phi_{K^{*}_{0}}(k_{3})H(k_{1},k_{2},k_{3},t)\right],$
(18)

where $k_{i}$’s are momenta of the antiquarks included in each meson, and
$\mathrm{Tr}$ denotes the trace over Dirac and color indices. $C(t)$ is the
Wilson coefficient which results from the radiative corrections at short
distance. In the above convolution, $C(t)$ includes the harder dynamics at
larger scale than the $M_{B}$ scale and describes the evolution of local
$4$-Fermi operators from $m_{W}$ (the $W$ boson mass) down to
$t\sim\mathcal{O}(\sqrt{\bar{\Lambda}M_{B_{s}}})$ scale, where
$\bar{\Lambda}\equiv M_{B_{s}}-m_{b}$. The function $H(k_{1},k_{2},k_{3},t)$
describes the four-quark operator and the spectator quark connected by a hard
gluon whose $q^{2}$ is in the order of $\bar{\Lambda}M_{B_{s}}$, and includes
the $\mathcal{O}(\sqrt{\bar{\Lambda}M_{B_{s}}})$ hard dynamics. Therefore,
this hard part $H$ can be perturbatively calculated. The functions
$\Phi_{(V,K^{*}_{0})}$ are the wave functions of the vector meson $V$ and the
scalar meson $K^{*}_{0}$, respectively.

Since the $b$ quark is rather heavy, we consider the $B_{s}$ meson at rest for
simplicity. It is convenient to use the light-cone coordinate
$(p^{+},p^{-},{\bf p}_{T})$ to describe the meson’s momenta,

$\displaystyle p^{\pm}=\frac{1}{\sqrt{2}}(p^{0}\pm p^{3}),\quad{\rm
and}\quad{\bf p}_{T}=(p^{1},p^{2}).$ (19)

Using these coordinates, the $B_{s}$ meson and the two final state meson
momenta can be written as

$\displaystyle P_{B_{s}}=\frac{M_{B_{s}}}{\sqrt{2}}(1,1,{\bf 0}_{T}),\quad
P_{2}=\frac{M_{B_{s}}}{\sqrt{2}}(1-r^{2}_{K^{*}_{0}},r^{2}_{V},{\bf
0}_{T}),\quad
P_{3}=\frac{M_{B_{s}}}{\sqrt{2}}(r^{2}_{K^{*}_{0}},1-r^{2}_{V},{\bf 0}_{T}),$
(20)

respectively, where the ratio $r_{K^{*}_{0}(V)}=m_{K^{*}_{0}(V)}/M_{B_{s}}$,
and $m_{K^{*}_{0}(V)}$ is the scalar meson $K^{*}_{0}$ (the vector meson $V$)
mass. Putting the antiquark momenta in $B_{s}$, $V$, and $K^{*}_{0}$ mesons as
$k_{1}$, $k_{2}$, and $k_{3}$, respectively, we can choose

$\displaystyle k_{1}=(x_{1}P_{1}^{+},0,{\bf k}_{1T}),\quad
k_{2}=(x_{2}P_{2}^{+},0,{\bf k}_{2T}),\quad k_{3}=(0,x_{3}P_{3}^{-},{\bf
k}_{3T}).$ (21)

For these considered decay channels, the integration over $k_{1}^{-}$,
$k_{2}^{-}$, and $k_{3}^{+}$ in Eq.(18) will lead to

$\displaystyle{\cal A}(B_{s}\to VK^{*}_{0})$ $\displaystyle\sim$
$\displaystyle\int\\!\\!dx_{1}dx_{2}dx_{3}b_{1}db_{1}b_{2}db_{2}b_{3}db_{3}$
(22)
$\displaystyle\cdot\mathrm{Tr}\left[C(t)\Phi_{B_{s}}(x_{1},b_{1})\Phi_{V}(x_{2},b_{2})\Phi_{K^{*}_{0}}(x_{3},b_{3})H(x_{i},b_{i},t)S_{t}(x_{i})\,e^{-S(t)}\right],\quad$

where $b_{i}$ is the conjugate space coordinate of $k_{iT}$, and $t$ is the
largest energy scale in function $H(x_{i},b_{i},t)$. In order to smear the
end-point singularity on $x_{i}$, the jet function $S_{t}(x)$ li02 , which
comes from the resummation of the double logarithms $\ln^{2}x_{i}$, is used.
The last term $e^{-S(t)}$ in Eq.(22) is the Sudakov form factor which
suppresses the soft dynamics effectively soft .

For the considered decays, the related weak effective Hamiltonian $H_{eff}$
can be written as buras96

$\displaystyle{\cal
H}_{eff}=\frac{G_{F}}{\sqrt{2}}\,\left[\sum_{p=u,c}V_{pb}V_{pd}^{*}\left(C_{1}(\mu)O_{1}^{p}(\mu)+C_{2}(\mu)O_{2}^{p}(\mu)\right)-V_{tb}V_{td}^{*}\sum_{i=3}^{10}C_{i}(\mu)\,O_{i}(\mu)\right].$
(23)

Here the Fermi constant $G_{F}=1.16639\times 10^{-5}GeV^{-2}$ and the
functions $Q_{i}(i=1,...,10)$ are the local four-quark operators. We specify
below the operators in ${\cal H}_{eff}$ for $b\to d$ transition:

$\displaystyle\begin{array}[]{llllll}O_{1}^{u}&=&\bar{d}_{\alpha}\gamma^{\mu}Lu_{\beta}\cdot\bar{u}_{\beta}\gamma_{\mu}Lb_{\alpha}\
,&O_{2}^{u}&=&\bar{d}_{\alpha}\gamma^{\mu}Lu_{\alpha}\cdot\bar{u}_{\beta}\gamma_{\mu}Lb_{\beta}\
,\\\
O_{3}&=&\bar{d}_{\alpha}\gamma^{\mu}Lb_{\alpha}\cdot\sum_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Lq_{\beta}^{\prime}\
,&O_{4}&=&\bar{d}_{\alpha}\gamma^{\mu}Lb_{\beta}\cdot\sum_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Lq_{\alpha}^{\prime}\
,\\\
O_{5}&=&\bar{d}_{\alpha}\gamma^{\mu}Lb_{\alpha}\cdot\sum_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Rq_{\beta}^{\prime}\
,&O_{6}&=&\bar{d}_{\alpha}\gamma^{\mu}Lb_{\beta}\cdot\sum_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Rq_{\alpha}^{\prime}\
,\\\
O_{7}&=&\frac{3}{2}\bar{d}_{\alpha}\gamma^{\mu}Lb_{\alpha}\cdot\sum_{q^{\prime}}e_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Rq_{\beta}^{\prime}\
,&O_{8}&=&\frac{3}{2}\bar{d}_{\alpha}\gamma^{\mu}Lb_{\beta}\cdot\sum_{q^{\prime}}e_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Rq_{\alpha}^{\prime}\
,\\\
O_{9}&=&\frac{3}{2}\bar{d}_{\alpha}\gamma^{\mu}Lb_{\alpha}\cdot\sum_{q^{\prime}}e_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Lq_{\beta}^{\prime}\
,&O_{10}&=&\frac{3}{2}\bar{d}_{\alpha}\gamma^{\mu}Lb_{\beta}\cdot\sum_{q^{\prime}}e_{q^{\prime}}\bar{q}_{\beta}^{\prime}\gamma_{\mu}Lq_{\alpha}^{\prime}\
,\end{array}$ (29)

where $\alpha$ and $\beta$ are the $SU(3)$ color indices; $L$ and $R$ are the
left- and right-handed projection operators with $L=(1-\gamma_{5})$,
$R=(1+\gamma_{5})$. The sum over $q^{\prime}$ runs over the quark fields that
are active at the scale $\mu=O(m_{b})$, i.e.,
$(q^{\prime}\epsilon\\{u,d,s,c,b\\})$.

Figure 1:  Diagrams contributing to the decay
$\bar{B}_{s}^{0}\to\rho^{0}K^{*0}_{0}(1430)$ .

In Fig. 1, we give the leading order Feynman diagrams for the channel
$\bar{B}_{s}^{0}\to\rho^{0}K^{*0}_{0}(1430)$ as an example. The Feynman
diagrams for the other decays are similar and not given. The analytic formulas
of each considered decays are similar to those of $B\to f_{0}(980)K^{*}$
zqzhang4  and $B\to K^{*}_{0}(1430)\rho(\omega)$ zqzhang5 . We just need to
replace some corresponding wave functions, Wilson coefficients, and
parameters. Here we do not show these formulas.

Combining the contributions from different diagrams, the total decay
amplitudes for these decays can be written as

$\displaystyle\sqrt{2}{\cal M}(K^{*0}_{0}\rho^{0})$ $\displaystyle=$
$\displaystyle\xi_{u}\left[M_{eK^{*}_{0}}C_{2}+F_{eK^{*}_{0}}a_{2}\right]-\xi_{t}\left[F_{eK^{*}_{0}}\left(-a_{4}+\frac{1}{2}(3C_{7}+C_{8})+\frac{5}{3}C_{9}+C_{10}\right)\right.$
(30)
$\displaystyle\left.+M_{eK^{*}_{0}}(-\frac{C_{3}}{3}+\frac{C_{9}}{6}+\frac{3C_{10}}{2})-(M^{P1}_{eK^{*}_{0}}+M^{P1}_{aK^{*}_{0}})(C_{5}-\frac{C_{7}}{2})+M^{P2}_{eK^{*}_{0}}\frac{3C_{8}}{2}\right.$
$\displaystyle\left.-M_{aK^{*}_{0}}(C_{3}-\frac{1}{2}C_{9})-F_{aK^{*}_{0}}(a_{4}-\frac{1}{2}a_{10})-F^{P2}_{aK^{*}_{0}}(a_{6}-\frac{1}{2}a_{8})\right],$
$\displaystyle\sqrt{2}{\cal M}(K^{*0}_{0}\omega)$ $\displaystyle=$
$\displaystyle\xi_{u}\left[M_{eK^{*}_{0}}C_{2}+F_{eK^{*}_{0}}a_{2}\right]-\xi_{t}\left[F_{eK^{*}_{0}}\left(\frac{7C_{3}}{3}+\frac{5C_{4}}{3}+2a_{5}+\frac{a_{7}}{2}+\frac{C_{9}}{3}-\frac{C_{10}}{3}\right)\right.$
(31)
$\displaystyle\left.+M_{eK^{*}_{0}}(\frac{C_{3}}{3}+2C_{4}-\frac{C_{9}}{6}+\frac{C_{10}}{2})+(M^{P1}_{eK^{*}_{0}}+M^{P1}_{aK^{*}_{0}})(C_{5}-\frac{C_{7}}{2})\right.$
$\displaystyle\left.+M^{P2}_{eK^{*}_{0}}(2C_{6}+\frac{C_{8}}{2})+M_{aK^{*}_{0}}(C_{3}-\frac{1}{2}C_{9})+F_{aK^{*}_{0}}(a_{4}-\frac{1}{2}a_{10})\right.$
$\displaystyle\left.+F^{P2}_{aK^{*}_{0}}(a_{6}-\frac{1}{2}a_{8})\right],$
$\displaystyle{\cal M}(K^{*+}_{0}\rho^{-})$ $\displaystyle=$
$\displaystyle\xi_{u}\left[M_{eK^{*}_{0}}C_{1}+F_{eK^{*}_{0}}a_{1}\right]-\xi_{t}\left[F_{eK^{*}_{0}}\left(a_{4}+a_{10}\right)+M_{eK^{*}_{0}}(C_{3}+C_{9})\right.$
(32)
$\displaystyle\left.+M^{P1}_{eK^{*}_{0}}(C_{5}+C_{7})+M_{aK^{*}_{0}}(C_{3}-\frac{1}{2}C_{9})+M^{P1}_{aK^{*}_{0}}(C_{5}-\frac{1}{2}C_{7})\right.$
$\displaystyle\left.+F_{aK^{*}_{0}}(a_{4}-\frac{1}{2}a_{10})+F^{P2}_{aK^{*}_{0}}(a_{6}-\frac{1}{2}a_{8})\right],$
$\displaystyle{\cal M}(K^{*0}_{0}\phi)$ $\displaystyle=$
$\displaystyle-\xi_{t}\left[F^{P2}_{e\phi}(a_{6}-\frac{a_{8}}{2})+M_{e\phi}(C_{3}-\frac{C_{9}}{2})+(M^{P1}_{e\phi}+M^{P1}_{a\phi})(C_{5}-\frac{C_{7}}{2})\right.$
(33)
$\displaystyle\left.+M_{a\phi}(C_{3}-\frac{1}{2}C_{9})+F_{a\phi}(a_{4}-\frac{1}{2}a_{10})+F_{a\phi}(a_{6}-\frac{1}{2}a_{8})\right.$
$\displaystyle\left.+F_{eK^{*}_{0}}\left(a_{3}+a_{5}-\frac{1}{2}a_{7}-\frac{1}{2}a_{7}\right)+M_{eK^{*}_{0}}(C_{4}-\frac{1}{2}C_{10})\right.$
$\displaystyle\left.+M^{P2}_{eK^{*}_{0}}(C_{6}-\frac{1}{2}C_{8})\right].$

The combinations of the Wilson coefficients are defined as usual zjxiao :

$\displaystyle a_{1}(\mu)$ $\displaystyle=$ $\displaystyle
C_{2}(\mu)+\frac{C_{1}(\mu)}{3},\quad
a_{2}(\mu)=C_{1}(\mu)+\frac{C_{2}(\mu)}{3},$ $\displaystyle a_{i}(\mu)$
$\displaystyle=$ $\displaystyle C_{i}(\mu)+\frac{C_{i+1}(\mu)}{3},\quad
i=3,5,7,9,$ $\displaystyle a_{i}(\mu)$ $\displaystyle=$ $\displaystyle
C_{i}(\mu)+\frac{C_{i-1}(\mu)}{3},\quad i=4,6,8,10.$ (34)

##  IV Numerical results and discussions

We use the following input parameters in the numerical calculations pdg08 :

$\displaystyle f_{B_{s}}$ $\displaystyle=$ $\displaystyle
230MeV,M_{B_{s}}=5.37GeV,M_{W}=80.41GeV,$ (35) $\displaystyle V_{ub}$
$\displaystyle=$ $\displaystyle|V_{ub}|e^{-i\gamma}=3.93\times
10^{-3}e^{-i68^{\circ}},V_{ud}=0.974,$ (36) $\displaystyle V_{td}$
$\displaystyle=$ $\displaystyle|V_{td}|e^{-i\beta}=8.1\times
10^{-3}e^{-i21.6^{\circ}},V_{tb}=1.0,$ (37) $\displaystyle\alpha$
$\displaystyle=$ $\displaystyle 100^{\circ}\pm
20^{\circ},\tau_{B_{s}}=1.470\times 10^{-12}s.$ (38)

Using the wave functions and the values of relevant input parameters, we find
the numerical values of the corresponding form factors
$\bar{B}^{0}_{s}\to\phi,K^{*}_{0}(1430)$ at zero momentum transfer

$\displaystyle A^{\bar{B}^{0}_{s}\to\phi}_{0}(q^{2}=0)$ $\displaystyle=$
$\displaystyle 0.29^{+0.05+0.01}_{-0.04-0.01},$ (39) $\displaystyle
F^{\bar{B}^{0}_{s}\to K^{*}_{0}}_{0}(q^{2}=0)$ $\displaystyle=$
$\displaystyle-0.30^{+0.03+0.01+0.01}_{-0.03-0.01-0.01},\quad\mbox{ scenario
I},$ (40) $\displaystyle F^{\bar{B}^{0}_{s}\to K^{*}_{0}}_{0}(q^{2}=0)$
$\displaystyle=$ $\displaystyle
0.56^{+0.05+0.03+0.04}_{-0.07-0.04-0.05},\quad\;\;\;\mbox{ scenario II},$ (41)

where the uncertainties are from $\omega_{b_{s}}=0.5\pm 0.05$ of $B_{s}$ and
the Gegenbauer moment $a_{2\phi}=0.18\pm 0.08$ of the vector meson $\phi$ for
$A^{\bar{B}^{0}_{s}\to\phi}$, and from the decay constant, the Gegenbauer
moments $B_{1}$ and $B_{3}$ of the scalar meson $K^{*}_{0}$ for
$F^{\bar{B}^{0}_{s}\to K^{*}_{0}}$. For the $\bar{B}_{s}\to\phi$ transition
form factor, its value is about $0.30$, which is favored by many model
calculations wuyl ; lucd ; chenghy , while a large value
$A^{\bar{B}_{s}\to\phi}_{0}=0.474$ is obtained by the light-cone sum-rule
method pball1 . The discrepancy can be clarified by the current LHC-b
experiments. As for the form factors $F^{\bar{B}^{0}_{s}\to K^{*}_{0}}_{0}$ in
two scenarios, they are agree well with those given in lirh .

In the $B_{s}$-rest frame, the decay rates of $\bar{B}^{0}_{s}\to
K^{*}_{0}(1430)\rho(\omega,\phi)$ can be written as

$\displaystyle\Gamma=\frac{G_{F}^{2}}{32\pi m_{B_{s}}}|{\cal
M}|^{2}(1-r^{2}_{K^{*}_{0}}),$ (42)

where ${\cal M}$ is the total decay amplitude of each considered decay and
$r_{K^{*}_{0}}$ is the mass ratio, both of which have been given in Sec. III.
The ${\cal M}$ can be rewritten as

$\displaystyle{\cal
M}=V_{ub}V^{*}_{ud}T-V_{tb}V^{*}_{td}P=V_{ub}V^{*}_{ud}\left[1+ze^{i(\alpha+\delta)}\right],$
(43)

where $\alpha$ is the Cabibbo-Kobayashi-Maskawa weak phase angle, and $\delta$
is the relative strong phase between the tree and the penguin amplitudes,
which are denoted as ”T” and ”P,” respectively. The term $z$ describes the
ratio of penguin to tree contributions and is defined as

$\displaystyle
z=\left|\frac{V_{tb}V^{*}_{td}}{V_{ub}V^{*}_{ud}}\right|\left|\frac{P}{T}\right|.$
(44)

From Eq.(43), it is easy to write decay amplitude $\overline{\cal M}$ for the
corresponding conjugated decay mode. So the CP-averaged branching ratio for
each considered decay is defined as

$\displaystyle{\cal B}=(|{\cal M}|^{2}+|\overline{\cal
M}|^{2})/2=|V_{ub}V^{*}_{ud}T|^{2}\left[1+2z\cos\alpha\cos\delta+z^{2}\right].$
(45)

Figure 2: The dependence of the branching ratios for $\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\omega$ (solid curve), $\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\rho^{0}$ (dashed curve) on the Cabibbo-Kobayashi-Maskawa
angle $\alpha$. The left (right) panel is plotted in scenario I (II).

Using the input parameters and the wave functions as specified in this section
and Sec. II, we can calculate the branching ratios of the considered modes

$\displaystyle{\cal B}(\bar{B}_{s}^{0}\to K^{*0}_{0}(1430)\phi)$
$\displaystyle=$ $\displaystyle(2.9^{+0.6+0.2+0.6}_{-0.5-0.1-0.5})\times
10^{-7},\mbox{ scenario I},$ (46) $\displaystyle{\cal B}(\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\omega)$ $\displaystyle=$
$\displaystyle(8.2^{+1.7+0.0+0.6}_{-1.6-0.1-0.6})\times 10^{-7},\mbox{
scenario I},$ (47) $\displaystyle{\cal B}(\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\rho^{0})$ $\displaystyle=$
$\displaystyle(9.9^{+2.0+0.0+0.7}_{-1.9-0.1-0.7})\times 10^{-7},\mbox{
scenario I},$ (48) $\displaystyle{\cal B}(\bar{B}_{s}^{0}\to
K^{*+}_{0}(1430)\rho^{-})$ $\displaystyle=$
$\displaystyle(3.4^{+0.7+0.3+0.3}_{-0.6-0.2-0.2})\times 10^{-5},\mbox{
scenario I},$ (49) $\displaystyle{\cal B}(\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\phi)$ $\displaystyle=$
$\displaystyle(9.5^{+2.5+2.8+3.1}_{-1.7-1.9-1.4})\times 10^{-7},\mbox{
scenario II},$ (50) $\displaystyle{\cal B}(\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\omega)$ $\displaystyle=$
$\displaystyle(8.6^{+2.1+0.6+2.2}_{-1.8-0.5-1.5})\times 10^{-7},\mbox{
scenario II},$ (51) $\displaystyle{\cal B}(\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\rho^{0})$ $\displaystyle=$
$\displaystyle(9.6^{+2.2+0.4+2.0}_{-2.0-0.4-2.1})\times 10^{-7},\mbox{
scenario II},$ (52) $\displaystyle{\cal B}(\bar{B}_{s}^{0}\to
K^{*+}_{0}(1430)\rho^{-})$ $\displaystyle=$
$\displaystyle(10.8^{+2.5+1.2+1.9}_{-2.3-1.1-1.7})\times 10^{-5},\mbox{
scenario II},$ (53)

where the uncertainties are mainly from the decay constant, the Gegenbauer
moments $B_{1}$ and $B_{3}$ of the scalar meson $K^{*}_{0}$. From the results,
one can find that the branching ratios of $\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\phi,K^{*+}_{0}(1430)\rho^{-}$ in scenario II are about
$3.2\sim 3.3$ times larger than those in scenario I. While for the decays
$\bar{B}_{s}^{0}\to K^{*0}_{0}(1430)\omega(\rho^{0})$, their branching ratios
for two scenarios are very close to each other, respectively. In these four
decay channels, the branching ratio of $\bar{B}_{s}^{0}\to
K^{*+}_{0}(1430)\rho^{-}$ is the largest one. This is not a surprise: one can
recall that the channel $\bar{B}_{s}^{0}\to K^{+}_{0}\rho^{-}$ also receives a
large branching ratio, about $(2.45^{+1.52}_{-1.29})\times 10^{-5}$ predicted
by the QCD factorization approach Beneke  and about
$(1.78^{+0.78}_{-0.59})\times 10^{-5}$ predicted by the PQCD approach lucd0 .
Certainly, for the other three decays $\bar{B}_{s}^{0}\to
K^{*0}_{0}(1430)\phi,K^{*0}_{0}(1430)\omega(\rho^{0})$, their branch ratios
have the same order with those of the decays $\bar{B}_{s}^{0}\to
K^{0}_{0}\phi,K^{0}_{0}\omega(\rho^{0})$, which are listed in Table I. It is
easy to get the conclusion that the branching ratios of the decays
$\bar{B}_{s}^{0}\to K^{*}_{0}(1430)V$ are not far away from those of
$\bar{B}_{s}^{0}\to KV$, where $V$ represents $\rho,\omega,\phi$. The same
conclusion is also obtained in Ref.zqzhang1 .

Table 1: Comparing the branching ratios of $\bar{B}_{s}^{0}\to K^{0}_{0}\phi,K^{0}_{0}\omega,K^{0}_{0}\rho^{0},K^{+}_{0}\rho^{-}$ predicted in Beneke  and those of $\bar{B}_{s}^{0}\to K^{*0}_{0}(1430)\phi,K^{*0}_{0}(1430)\omega,K^{*0}_{0}(1430)\rho^{0},K^{*+}_{0}(1430)\rho^{-}$ predicted in this work in scenario I .  Mode | Br($\times 10^{-6}$)
---|---
$\bar{B}_{s}^{0}\to K^{*0}_{0}(1430)\phi$ | $0.29^{+0.06+0.02+0.06}_{-0.05-0.01-0.05}$
$\bar{B}_{s}^{0}\to K^{0}_{0}\phi$ | $0.27^{+0.09+0.28+0.09+0.67}_{-0.08-0.14-0.06-0.18}$
$\bar{B}_{s}^{0}\to K^{*0}_{0}(1430)\omega$ | $0.82^{+0.17+0.00+0.06}_{-0.16-0.01-0.06}$
$\bar{B}_{s}^{0}\to K^{0}_{0}\omega$ | $0.51^{+0.20+0.15+0.68+0.40}_{-0.18-0.11-0.23-0.25}$
$\bar{B}_{s}^{0}\to K^{*0}_{0}(1430)\rho^{0}$ | $0.99^{+0.02+0.00+0.07}_{-0.19-0.01-0.07}$
$\bar{B}_{s}^{0}\to K^{0}_{0}\rho^{0}$ | $0.61^{+0.33+0.21+1.06+0.56}_{-0.26-0.15-0.38-0.36}$
$\bar{B}_{s}^{0}\to K^{*+}_{0}(1430)\rho^{-}$ | $34.0^{+0.7+0.3+0.3}_{-0.6-0.2-0.2}$
$\bar{B}_{s}^{0}\to K^{+}_{0}\rho^{-}$ | $24.5^{+11.9+9.2+1.8+1.6}_{-9.7-7.8-3.0-1.6}$

Figure 3: The dependence of the branching ratio for $\bar{B}^{0}_{s}\to K^{*+}_{0}(1430)\rho^{-}$ on the Cabibbo-Kobayashi-Maskawa angle $\alpha$. The left (right) panel is plotted in scenario I (II). Table 2:  Decay amplitudes for decays $\bar{B}^{0}_{s}\to K^{*+}_{0}(1430)\rho^{-},K^{*0}_{0}(1430)\rho^{0}$ ($\times 10^{-2}\mbox{GeV}^{3}$). |  | $F^{T}_{eK^{*}_{0}}$ | $F_{eK^{*}_{0}}$ | $M^{T}_{eK^{*}_{0}}$ | $M_{eK^{*}_{0}}$ | $M_{aK^{*}_{0}}$ | $F_{aK^{*}_{0}}$
---|---|---|---|---|---|---|---
$\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\rho^{0}$ (SI) |  | -22.4 | 4.9 | $-11.7+8.2i$ | $-0.15+0.24i$ | $-0.14+0.11i$ | $-4.1-2.9i$
$\bar{B}^{0}_{s}\to K^{*+}_{0}(1430)\rho^{-}$ (SI) |  | 203 | -8.6 | $6.5-5.2i$ | $0.08-0.28i$ | $0.16-0.09i$ | $5.3+4.4i$
$\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\rho^{0}$ (SII) |  | 27.6 | -7.8 | $0.7+5.9i$ | $0.36-0.20i$ | $0.40+0.20i$ | $3.1+8.1i$
$\bar{B}^{0}_{s}\to K^{*+}_{0}(1430)\rho^{-}$ (SII) |  | -371 | 14.3 | $-0.04-4.6i$ | $0.58+0.41i$ | $-0.58-0.28i$ | $-4.1-11.5i$

In Table II, we list the values of the factorizable and nonfactorizable
amplitudes from the emission and annihilation topology diagrams of the decays
$\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\rho^{0}$ and $\bar{B}^{0}_{s}\to
K^{*+}_{0}(1430)\rho^{-}$. $F_{e(a)K^{*}_{0}}$ and $M_{e(a)K^{*}_{0}}$ are the
$\rho$ meson emission (annihilation) factorizable contributions and
nonfactorizable contributions from penguin operators respectively. The upper
label $T$ denotes the contributions from tree operators. For the decay
$\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\rho^{0}$, there are not diagrams obtained
by exchanging the position of $K^{*0}_{0}$ and $\rho^{0}$ in Fig.1, so there
are not contributions from $F_{e(a)\rho}$ and $M_{e(a)\rho}$. It is same for
the decay $\bar{B}^{0}_{s}\to K^{*+}_{0}(1430)\rho^{-}$. From Table II, one
can find that because of the large Wilson coefficients $a_{1}=C_{2}+C_{1}/3$,
the tree-dominated decay channel $\bar{B}^{0}_{s}\to K^{*+}_{0}(1430)\rho^{-}$
receives a large branching ratio value in both scenarios compared with
$\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\rho^{0}$.

The dependence of the branching ratios for the decays $\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\rho^{0},K^{*0}_{0}(1430)\omega,K^{*+}_{0}(1430)\rho^{-}$ on
the Cabibbo-Kobayashi-Maskawa angle $\alpha$ is displayed in Fig.2 and Fig.3.
The branching ratios of the $K^{*0}_{0}(1430)\rho^{0}$ and
$K^{*+}_{0}(1430)\rho^{-}$ modes increase with $\alpha$, while that of the
$K^{*0}_{0}(1430)\omega$ mode decreases with $\alpha$. The values of
$\cos\delta$ [shown in Eq.(45)] for the decay modes $\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\rho^{0},K^{*+}_{0}(1430)\rho^{-}$ are opposite in sign with
that of $\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\omega$ , and as a result the
behaviors of the branching ratios with the Cabibbo-Kobayashi-Maskawa angle
$\alpha$ for the former are very different with that of the latter. We can
also find that the branching ratio of the decay $\bar{B}^{0}_{s}\to
K^{*+}_{0}(1430)\rho^{-}$ is insensitive to the variation of $\alpha$ in
scenario I. For the decay $\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\phi$, there are
only penguin operator contributions in this channel, so its branching ratio
has no relation with the angle $\alpha$ at the leading order.

Now, we turn to the evaluations of the direct CP-violating asymmetries of the
considered decays in the PQCD approach. The direct CP-violating asymmetry can
be defined as

$\displaystyle{\cal A}_{CP}^{dir}=\frac{|\overline{\cal M}|^{2}-|{\cal
M}|^{2}}{|{\cal M}|^{2}+|\overline{\cal
M}|^{2}}=\frac{2z\sin\alpha\sin\delta}{1+2z\cos\alpha\cos\delta+z^{2}}\;.$
(54)

Here the ratio $z$ and the strong phase $\delta$ are calculable in PQCD
approach, so it is easy to find the numerical values of ${\cal A}_{CP}^{dir}$
(in unit of $10^{-2}$) by using the input parameters listed in the previous
for the considered decays in two scenarios:

$\displaystyle{\cal A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\omega)=-24.1^{+0.0+2.7+0.6}_{-0.0-2.5-0.2},\mbox{ scenario
I},$ (55) $\displaystyle{\cal A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\rho^{0})=26.6^{+0.0+2.5+0.3}_{-0.0-2.5-0.5},\mbox{ scenario
I},$ (56) $\displaystyle{\cal A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*+}_{0}(1430)\rho^{-})=7.7^{+0.0+0.2+0.2}_{-0.0-0.3-0.2},\mbox{ scenario
I},$ (57) $\displaystyle{\cal A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\omega)=-86.7^{+0.1+7.1+1.3}_{-0.1-5.3-2.8},\mbox{ scenario
II},$ (58) $\displaystyle{\cal A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\rho^{0})=84.5^{+0.1+4.9+1.0}_{-0.1-6.3-3.8},\mbox{ scenario
II},$ (59) $\displaystyle{\cal A}_{CP}^{dir}(\bar{B}^{0}_{s}\to
K^{*+}_{0}(1430)\rho^{-})=12.6^{+0.0+0.2+0.8}_{-0.0-0.2-0.6},\mbox{ scenario
II},$ (60)

where the uncertainties are mainly from the decay constant, the Gegenbauer
moments $B_{1}$ and $B_{3}$ of the scalar meson $K^{*}_{0}$. Compared with the
values of the branching ratios, we can find that if the direct CP-violating
asymmetries are sensitive to some parameters, while the branching ratios are
insensitive to them, for example, the decay constant of $K^{*}_{0}$. For the
decays $\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\omega,K^{*0}_{0}(1430)\rho^{0}$,
their direct CP-violating asymmetries in scenario II are more than 3 times
than those in scenario I. In both scenarios, the direct CP-violating
asymmetries of these two decay channels are close to each other in size, while
they are opposite in sign. The reason for this is the following. The mesons
$\rho^{0},\omega$ have very similar mass, decay constant, and distribution
amplitude, only the opposite sign of $d\bar{d}$ in their quark components, and
the difference will appear in penguin operators. From our numerical results,
we can find that the contributions from tree operators for these two channels
(denoted as $T_{K^{*0}_{0}\rho^{0}}$ and $T_{K^{*0}_{0}\omega})$ are really
very close, and those from penguin operators for these two channels (denoted
as $P_{K^{*0}_{0}\rho^{0}}$ and $P_{K^{*0}_{0}\omega}$) are opposite in sign.
Furthermore, the real parts of $P_{K^{*0}_{0}\rho^{0}}$ and
$P_{K^{*0}_{0}\omega}$ in each scenario have large differences in size.

$\displaystyle T_{K^{*0}_{0}\rho^{0}}$ $\displaystyle=$
$\displaystyle(-34.1+i8.2)\times
10^{-2},P_{K^{*0}_{0}\rho^{0}}=(0.49-i2.5)\times 10^{-2},$ (61) $\displaystyle
T_{K^{*0}_{0}\omega}$ $\displaystyle=$ $\displaystyle(-31.8+i7.6)\times
10^{-2},P_{K^{*0}_{0}\omega}=(-3.7+i2.8)\times 10^{-2},\mbox{ scenario I},$
(62) $\displaystyle T_{K^{*0}_{0}\rho^{0}}$ $\displaystyle=$
$\displaystyle(28.3+i5.9)\times
10^{-2},P_{K^{*0}_{0}\rho^{0}}=(-3.9+i8.1)\times 10^{-2},$ (63) $\displaystyle
T_{K^{*0}_{0}\omega}$ $\displaystyle=$ $\displaystyle(26.4+i5.5)\times
10^{-2},P_{K^{*0}_{0}\omega}=(8.1-i7.2)\times 10^{-2},\mbox{ scenario II}.$
(64)

These values can explain why the two channels have similar CP-violating
asymmetry in size (certainly, their branching ratios are also similar for the
same reason). Using the upper results, we can calculate $\sin\delta$ [shown in
Eq.(54)] in two scenarios:

$\displaystyle\sin\delta_{K^{*0}_{0}\rho^{0}}$ $\displaystyle=$ $\displaystyle
0.91,\sin\delta_{K^{*0}_{0}\omega}=-0.40,\mbox{ scenario I},$ (65)
$\displaystyle\sin\delta_{K^{*0}_{0}\rho^{0}}$ $\displaystyle=$ $\displaystyle
0.97,\sin\delta_{K^{*0}_{0}\omega}=-0.80,\mbox{ scenario II}.$ (66)

These values can explain why the CP-violating asymmetries of these two decays
have opposite signs.

The direct CP-violating asymmetry of $\bar{B}^{0}_{s}\to
K^{*+}_{0}(1430)\rho^{-}$ is the smallest in these decays, about $10\%$, but
its branching ratio is the largest one, about $3.4\times 10^{-5}$ in scenario
I, even at the order of $10^{-4}$ in scenario II. So this channel might be
easily measured at LHC-b experiments.

Figure 4: The dependence of the direct CP asymmetries for $\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\omega$ (solid curve), $\bar{B}^{0}_{s}\to
K^{*+}_{0}(1430)\rho^{-}$ (dotted curve), $\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\rho^{0}$ (dashed curve) on the Cabibbo-Kobayashi-Maskawa
angle $\alpha$. The left (right) panel is plotted in scenario I (II)

From Fig.4(a) and 4(b), one can see that though the direct CP asymmetry values
for each decay in two scenarios are very different in size, they have similar
trends depending on the Cabibbo-Kobayashi-Maskawa angle $\alpha$. As for the
decay $\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\phi$, there is no tree contribution
at the leading order, so the direct CP-violating asymmetry is naturally zero.

##  V Conclusion

In this paper, we calculate the branching ratios and the CP-violating
asymmetries of decays $\bar{B}_{s}^{0}\to K^{*}_{0}(1430)\rho(\omega,\phi)$ in
the PQCD factorization approach. Using the decay constants and light-cone
distribution amplitudes derived from QCD sum-rule method, we find that

  * •

We predict the form factor
$A^{\bar{B}^{0}_{s}\to\phi}_{0}(q^{2}=0)=0.29^{+0.05+0.01}_{-0.04-0.01}$ for
$\omega_{b_{s}}=0.5\pm 0.05$ and the Gegenbauer moment $a_{2\phi}=0.18\pm
0.08$, which agrees well with the values as calculated by many approaches and
disagrees with the value $A^{\bar{B}^{0}_{s}\to\phi}_{0}=0.474$ obtained by
the light-cone sum-rule method. The discrepancy can be clarified by the LHC-b
experiments. The form factors of $\bar{B}^{0}_{s}\to K^{*}_{0}(q^{2}=0)$ in
two scenarios are given as

$\displaystyle F^{\bar{B}^{0}_{s}\to K^{*}_{0}}_{0}(q^{2}=0)$ $\displaystyle=$
$\displaystyle-0.30^{+0.03+0.01+0.01}_{-0.03-0.01-0.01},\quad\mbox{ scenario
I},$ (67) $\displaystyle F^{\bar{B}^{0}_{s}\to K^{*}_{0}}_{0}(q^{2}=0)$
$\displaystyle=$ $\displaystyle
0.56^{+0.05+0.03+0.04}_{-0.07-0.04-0.05},\quad\;\;\;\mbox{ scenario II},$ (68)

where the uncertainties are from the decay constant, the Gegenbauer moments
$B_{1}$ and $B_{3}$ of the scalar meson $K^{*}_{0}$.

  * •

Because of the large Wilson coefficients $a_{1}=C_{2}+C_{1}/3$, the branching
ratios of $\bar{B}^{0}_{s}\to K^{*+}_{0}(1430)\rho^{-}$ are much larger than
those of the other three decays in both scenarios and arrive at a few times
$10^{-5}$ in scenario I, even at the $10^{-4}$ order in scenario II, while its
direct CP-violating asymmetry is the smallest one, around $10\%$. The values
for this channel might be measured by the current LHC-b experiments.

  * •

For the decays $\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\omega,K^{*0}_{0}(1430)\rho^{0}$, their direct CP-violating
asymmetries are large, but it might be difficult to measure them, because
their branching ratios are small and less than (or near) $10^{-6}$ in both
scenarios.

  * •

The values of $\cos\delta$ for the decays $\bar{B}^{0}_{s}\to
K^{*0}_{0}(1430)\rho^{0},K^{*+}_{0}(1430)\rho^{-}$ are opposite in sign with
that for $\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\omega$; as a result, the
behaviors of the branching ratios of the former with Cabibbo-Kobayashi-Maskawa
angle $\alpha$ are very different with that of the latter. Because the values
of $\sin\delta$ are opposite in sign, their direct CP-violating asymmetries of
the former have an opposite sign with that of the latter. Here $\delta$ is the
relative strong phase angle between the tree and the penguin amplitudes.

  * •

Because the mesons $\rho^{0},\omega$ have very similar mass, decay constant,
distribution amplitude, only opposite sign of $d\bar{d}$ in their quark
components, and this difference only appears in the penguin operators; so
these two tree document decays $\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\rho^{0}$
and $\bar{B}^{0}_{s}\to K^{*0}_{0}(1430)\omega$ should have similar branching
ratios and CP-violating asymmetries.

  * •

As for the decay $\bar{B}^{s}_{0}\to K^{*0}_{0}(1430)\phi$, though there exist
large differences between the two scenarios, the predicted branching ratios
are small and a few times $10^{-7}$ in both scenarios. There is no tree
contribution at the leading order, so the direct CP-violating asymmetry is
naturally zero.

## Acknowledgment

This work is partly supported by the National Natural Science Foundation of
China under Grant No. 11047158, and by Foundation of Henan University of
Technology under Grant No.150374. The author would like to thank Cai-Dian Lü
for helpful discussions.

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