Resonant contribution of the three-body decay process $\bar{B}_{s}\rightarrow K^{+}K^{-}P$ in perturbation QCD

We present decay diagrams (a)-(i) of the $\bar{B}_{s}\rightarrow\phi$ $(\rho^{0}$, $\omega)$ $P\rightarrow K^{+}K^{-}P$ process in Fig.1, aiming to provide a more comprehensive understanding of the mixing mechanism.

In the above decay diagrams, the decay processes depicted in (a), (d), and (g) represent direct decay modes, where $K^{+}K^{-}$ are produced through $\phi$, $\rho^{0}$, and $\omega$ respectively. The quasi-two-body approach employed in this study is evident from the aforementioned diagrams. Compared to the direct decay processes depicted in diagrams (a), (d), and (g) of Fig.1, the $K^{+}K^{-}$ pair can also be generated through a distinct mixing mechanism. The black dots in the figure represent the resonance effect between these two mesons, denoted by the mixing parameter $\Pi_{V_{i}V_{j}}$. Although the contribution from this mixing mechanism is relatively small compared to other diagrams in Fig.1, it must be taken into consideration.

The amplitude of the $\bar{B}_{s}\rightarrow\phi$ ($\rho^{0},\omega$) $P\rightarrow K^{+}K^{-}P$ decay channel can be characterized in the following manner:

$$A=\left\langle K^{+}K^{-}P\left|H^{T}\right|\bar{B}_{s}\right\rangle+\left\langle K^{+}K^{-}P\left|H^{P}\right|\bar{B}_{s}\right\rangle,$$

(7)

The quantities $\left\langle K^{+}K^{-}P\left|H^{P}\right|\bar{B}_{s}\right\rangle$ and $\left\langle K^{+}K^{-}P\left|H^{T}\right|\bar{B}_{s}\right\rangle$ represent the amplitudes associated with penguin-level and tree-level contributions, respectively. The propagator of the intermediate vector meson can be transformed from the diagonal matrix to the physical state after applying the R matrix transformation. Neglecting higher order terms, the amplitudes can be as demonstrated below:

$\displaystyle\begin{split}\left\langle K^{+}K^{-}P\left|H^{T}\right|\bar{B}_{s}\right\rangle=&\frac{g_{\phi}}{s_{\phi}}t_{\phi}+\frac{g_{\rho}}{s_{\rho}s_{\phi}}\widetilde{\Pi}_{\rho\phi}t_{\phi}+\frac{g_{\omega}}{s_{\omega}s_{\phi}}\widetilde{\Pi}_{\omega\phi}t_{\phi}+\frac{g_{\rho}}{s_{\rho}}t_{\rho}+\frac{g_{\phi}}{s_{\phi}s_{\rho}}\widetilde{\Pi}_{\phi\rho}t_{\rho}\\ &+\frac{g_{\omega}}{s_{\omega}s_{\rho}}\widetilde{\Pi}_{\omega\rho}t_{\rho}+\frac{g_{\omega}}{s_{\omega}}t_{\omega}+\frac{g_{\phi}}{s_{\phi}s_{\omega}}\widetilde{\Pi}_{\phi\omega}t_{\omega}+\frac{g_{\rho}}{s_{\rho}s_{\omega}}\widetilde{\Pi}_{\rho\omega}t_{\omega},\end{split}$

(8)

$\displaystyle\begin{split}\left\langle K^{+}K^{-}P\left|H^{P}\right|\bar{B}_{s}\right\rangle=&\frac{g_{\phi}}{s_{\phi}}p_{\phi}+\frac{g_{\rho}}{s_{\rho}s_{\phi}}\widetilde{\Pi}_{\rho\phi}p_{\phi}+\frac{g_{\omega}}{s_{\omega}s_{\phi}}\widetilde{\Pi}_{\omega\phi}p_{\phi}+\frac{g_{\rho}}{s_{\rho}}p_{\rho}+\frac{g_{\phi}}{s_{\phi}s_{\rho}}\widetilde{\Pi}_{\phi\rho}p_{\rho}\\ &+\frac{g_{\omega}}{s_{\omega s_{\rho}}}\widetilde{\Pi}_{\omega\rho}p_{\rho}+\frac{g_{\omega}}{s_{\omega}}p_{\omega}+\frac{g_{\phi}}{s_{\phi}s_{\omega}}\widetilde{\Pi}_{\phi\omega}p_{\omega}+\frac{g_{\rho}}{s_{\rho}s_{\omega}}\widetilde{\Pi}_{\rho\omega}p_{\omega},\end{split}$

(9)

where the tree-level (penguin-level) amplitudes $t_{\rho}\left(p_{\rho}\right)$, $t_{\omega}\left(p_{\omega}\right)$, and $t_{\phi}\left(p_{\phi}\right)$ correspond to the decay processes $\bar{B}_{s}\rightarrow\rho^{0}P$, $\bar{B}_{s}\rightarrow\omega P$ and $\bar{B}_{s}\rightarrow\phi P$, respectively. Here, $s_{V}$ represents the inverse propagator of the vector meson V Chen:1999nxa ; Wolfe:2009ts ; Wolfe:2010gf . Moreover, $g_{V}$ represents the coupling constant derived from the decay process of $V\rightarrow K^{+}K^{-}$ and can be expressed as $\sqrt{2}g_{{\rho}k^{+}k^{-}}=\sqrt{2}g_{\omega k^{+}k^{-}}=-g_{\phi k^{+}k^{-}}=4.54$ Bruch:2004py .

The differential parameter for CP asymmetry can be expressed as follows:

$$A_{CP}=\frac{\left|A\right|^{2}-\left|\overline{A}\right|^{2}}{\left|A\right|^{2}+\left|\overline{A}\right|^{2}}.$$

(10)

In this paper, we perform the integral calculation of A_CP to facilitate future experimental comparisons. For the decay process $\bar{B}_{s}\rightarrow\phi P$, the amplitude is given by $M_{\bar{B}_{s}\rightarrow\phi P}^{\lambda}=\alpha p_{\bar{B}}\cdot\epsilon^{*}(\lambda)$, where $p_{\bar{B}_{s}}$ represents the momenta of the $\bar{B}_{s}$ meson, $\epsilon$ denotes the polarization vector of $\phi$ and $\lambda$ corresponds to its polarization. The parameter $\alpha$ remains independent of $\lambda$. Similarly, in the decay process $\phi\rightarrow K^{+}K^{-}$, we can express $M_{\phi\rightarrow K^{-}K^{+}}^{\lambda}=g_{\phi}\epsilon(\lambda)\left(p_{1}-p_{2}\right)$, where $p_{1}$ and $p_{2}$ denote the momenta of the produced $K^{+}$ and $K^{-}$ particles from $\phi$, respectively. Here, the parameter $g_{\phi}$ represents an effective coupling constant for $\phi\rightarrow K^{+}K^{-}$. Regarding the dynamics of meson decay, it is observed that the polarization vector of a vector meson satisfies $\sum_{\lambda=0,\pm 1}\epsilon^{\lambda}_{\mu}(p)(\epsilon^{\lambda}_{\nu}(p))^{*}=-(g_{\mu\nu}-p_{\mu}p_{\nu}/m_{V}^{2})$. As a result, we obtain the total amplitude for the decay process $\bar{B}_{s}\rightarrow\phi P\rightarrow K^{+}K^{-}P$ Guo:2000uc ; Zhang:2013oqa ; Wang:2015ula :

	$\displaystyle A$	$\displaystyle=\alpha p_{\bar{B}_{s}}^{\mu}\frac{\sum_{\lambda}\epsilon_{\mu}^{*}(\lambda)\epsilon_{\nu}(\lambda)}{s_{\phi}}g_{\phi kk}\left(p_{1}-p_{2}\right)^{\nu}$		(11)
		$\displaystyle=\frac{g_{\phi kk}\alpha}{s_{\phi}}\cdot p_{\bar{B}_{s}}^{\mu}\left[g_{\mu\nu}-\frac{\left(p_{1}+p_{2}\right)_{\mu}\left(p_{1}+p_{2}\right)_{\nu}}{s}\right]\left(p_{1}-p_{2}\right)^{\nu}$
		$\displaystyle=\frac{g_{\phi kk}}{s_{\phi}}\cdot\frac{M_{\bar{B}_{s}\rightarrow\phi\pi^{0}}^{\lambda}}{p_{\bar{B}_{s}}\cdot\epsilon^{*}}\cdot\left(\Sigma-s^{\prime}\right)$
		$\displaystyle=\left(\Sigma-s^{\prime}\right)\cdot\mathcal{A}.$

The high ($\sqrt{s^{\prime}}$) and low $\sqrt{s}$ ranges are defined for calculating the invariant mass of $K^{-}K^{+}$. By setting a fixed value for $s$, we can determine an appropriate value for $s^{\prime}$ that fulfills the equation $\Sigma=\frac{1}{2}\left(s_{\max}^{\prime}+s_{\min}^{\prime}\right)$, where ${s}_{\max}^{\prime}({s}_{\min}^{\prime})$ denotes respectively the maximum (minimum) value.

Utilizing the principles of three-body kinematics, we can deduce the local CP asymmetry for the decay $\bar{B}_{s}\rightarrow K^{+}K^{-}P$ within a specific range of invariant mass:

$$A_{CP}^{\Omega}=\frac{\int_{s_{1}}^{s_{2}}\mathrm{~{}d}s\int_{s_{1}^{\prime}}^{s_{2}^{\prime}}\mathrm{d}s^{\prime}\left(\Sigma-s^{\prime}\right)^{2}\left(|\mathcal{A}|^{2}-|\overline{\mathcal{A}}|^{2}\right)}{\int_{s_{1}}^{s_{2}}\mathrm{~{}d}s\int_{s_{1}^{\prime}}^{s_{2}^{\prime}}\mathrm{d}s^{\prime}\left(\Sigma-s^{\prime}\right)^{2}\left(|\mathcal{A}|^{2}+|\overline{\mathcal{A}}|^{2}\right)}.$$

(12)

Our calculation takes into account the dependence of $\Sigma=\frac{1}{2}\left(s_{\max}^{\prime}+s_{\min}^{\prime}\right)$ on $s^{\prime}$. Assuming that $s_{\max}^{\prime}>s^{\prime}>s_{\min}^{\prime}$ represents an integral interval of high invariant mass for the $K^{-}K^{+}$ meson pair, and $\int_{s_{1}^{\prime}}^{s_{2}^{\prime}}\mathrm{d}s^{\prime}(\Sigma-s^{\prime})^{2}$ represents a factor dependent on $s^{\prime}$. The correlation between $\Sigma$ and $s^{\prime}$ can be easily determined through kinematic analysis, as $s^{\prime}$ only varies on a small scale. Therefore, we can consider $\Sigma$ as a constant. This allows us to cancel out the term $\int_{s_{1}^{\prime}}^{s_{2}^{\prime}}\mathrm{d}s^{\prime}(\Sigma-s^{\prime})^{2}$ in both the numerator and denominator, resulting in $A_{CP}^{\Omega}$ no longer depending on the high invariant mass of positive and negative particles.

The three-body decay process is accompanied by intricate and multifaceted dynamical mechanisms. The perturbative QCD (PQCD) method is known for its efficacy in handling perturbation corrections, which has been successfully applied to two-body non-light decay processes and holds promise for quasi-two-body decay processes as well. In the framework of PQCD, within the rest frame of a heavy B meson, the decay process involves the production of two light mesons with significantly large momenta that exhibit rapid motion. The dominance of hard interactions in this decay amplitude arises due to insufficient time for exchanging soft gluons with the final-state mesons. Given the high velocity of these final-state mesons, a hard gluon imparts momentum to the light spectator quark within the B meson, resulting in the formation of a rapidly moving final-state meson. Consequently, this hard interaction is described by six quark operators. The nonperturbative dynamics are encapsulated within the meson wave function, which can be extracted through experimental measurements. On the other hand, employing perturbation theory allows for computation of this aforementioned hard contribution. Quasi-two-body decay can be computed by defining the intermediate state of decay.

By employing the quasi-two-body decay method, the total amplitude of $\bar{B}_{s}\rightarrow\phi$ ($\rho^{0}$, $\omega$) $\pi^{0}$ $\rightarrow K^{+}K^{-}\pi^{0}$ is composed of two components: $\bar{B}_{s}\rightarrow\phi$ ($\rho^{0}$, $\omega$) $\pi^{0}$ and $\phi$ ($\rho^{0}$, $\omega)\rightarrow K^{+}K^{-}$. In this study, we illustrate the methodology of quasi-two-body decay process using the example of $\bar{B}_{s}\rightarrow\phi\pi^{0}\rightarrow K^{+}K^{-}\pi^{0}$, based on the matrix elements involving $V_{tb}$, $V_{ts}^{*}$ and $V_{ub}$,$V_{ub}^{*}$.

	$\displaystyle\sqrt{2}A\left(\bar{B}_{s}\rightarrow\pi^{0}\phi\left(\phi\rightarrow K^{+}K^{-}\right)\right)=$	$\displaystyle\frac{\left.G_{F}p_{\bar{B}_{s}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\phi}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)\right.}{\sqrt{2}s_{\phi}}$		(13)
		$\displaystyle\times\bigg{\{}V_{ub}V_{us}^{*}\left[f_{\pi}F_{\bar{B}_{s}\rightarrow\phi}^{LL}(a_{2})+M_{\bar{B}_{s}\rightarrow\phi}^{LL}(C_{2})\right]\bigg{.}$
		$\displaystyle\left.-V_{tb}V_{ts}^{*}\left[f_{\pi}F_{\bar{B}_{s}\rightarrow\phi}^{LL}\left(\frac{3}{2}a_{9}-\frac{3}{2}a_{7}\right)+M_{\bar{B}_{s}\rightarrow\phi}^{LL}\left(\frac{3}{2}C_{8}+\frac{3}{2}C_{10}\right)\right]\right\},$

where $P_{\bar{B}_{s}}$, $p_{k^{+}}$ and $p_{k^{-}}$ are the momentum of $\bar{B}_{s}$, $K^{+}$ and $K^{-}$, respectively. $C_{i}$ ($a_{i}$) is Wilson coefficient (associated Wilson coefficient), $\epsilon$ is the polarization of vector meson . $G_{F}$ is the Fermi constant. $f_{\pi}$ refers to the decay constants of $\pi$ Li:2006jv . Besides $F_{\bar{B}_{s}\rightarrow\phi}^{LL}$ and $M_{\bar{B}_{s}\rightarrow\phi}^{LL}$ represent emission graphs that are factorable and non-factorable. $F_{ann}^{LL}$ and $M_{ann}^{LL}$ represent annihilation graphs that are factorable and non-factorable. $LL$, $LR$, and $SP$ correspond to three flow structures Ali:2007ff .

The additional representations of the three-body decay amplitudes that necessitate consideration for calculating CP violation through the mixed mechanism in this paper are as follows:

	$\displaystyle 2A\left(\bar{B}_{s}^{0}\rightarrow\right.\left.\rho^{0}\left(\rho^{0}\rightarrow K^{+}K^{-}\right)\pi^{0}\right)=$	$\displaystyle\frac{G_{F}p_{\bar{B}_{s}^{0}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\rho}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)}{\sqrt{2}s_{\rho}}$		(14)
		$\displaystyle\times\left\{V_{ub}V_{us}^{*}\left[f_{B_{s}}F_{ann}^{LL}(a_{2})+M_{ann}^{LL}(C_{2})+f_{B_{s}}F_{ann}^{LL^{\prime}}(a_{2})+M_{ann}^{LL^{\prime}}(C_{2})\right]\right.$
		$\displaystyle-V_{tb}V_{ts}^{*}\left[f_{B_{s}}F_{ann}^{LL}\left(a_{3}+a_{9}\right)\right.\left.-f_{B_{s}}F_{ann}^{LR}\left(a_{5}+a_{7}\right)+M_{ann}^{LL}\left(C_{4}+C_{10}\right)\right.$
		$\displaystyle-M_{ann}^{SP}\left(C_{6}+C_{8}\right)+\left[\pi^{+}\leftrightarrow\rho^{-}\right]+f_{B_{s}}F_{ann}^{LL^{\prime}}\left(a_{3}+a_{9}\right)-f_{B_{s}}F_{ann}^{LR^{\prime}}\left(a_{5}+a_{7}\right)$
		$\displaystyle\left.\left.+M_{ann}^{LL^{\prime}}\left(C_{4}+C_{10}\right)-M_{ann}^{SP^{\prime}}\left(C_{6}+C_{8}\right)+\left[\rho^{+}\leftrightarrow\pi^{-}\right]\right]\right\}.$

	$\displaystyle 2A\left(\bar{B}_{s}^{0}\rightarrow\pi^{0}\omega\left(\omega\rightarrow K^{+}K^{-}\right)\right)=$	$\displaystyle\frac{G_{F}p_{\bar{B}_{s}^{0}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\omega}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)}{\sqrt{2}s_{\omega}}$		(15)
	$\displaystyle\times$	$\displaystyle\left\{V_{ub}V_{us}^{*}M_{ann}^{LL}\left(c_{2}\right)-V_{tb}V_{ts}^{*}\left[M_{ann}^{LL}\left(\frac{3}{2}c_{10}\right)-M_{ann}^{SP}\left(\frac{3}{2}c_{8}\right)+\left[\pi^{0}\leftrightarrow\omega\right]\right]\right\}.$

	$\displaystyle A\left(\bar{B}_{s}^{0}\rightarrow K^{0}\phi\left(\phi\rightarrow K^{+}K^{-}\right)\right)=$	$\displaystyle\frac{G_{F}p_{\bar{B}_{s}^{0}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\phi}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)}{\sqrt{2}s_{\phi}}$		(16)
		$\displaystyle\times\left\{-V_{tb}V_{td}^{*}\left[f_{\phi}F_{B_{s}\rightarrow K}^{LL}\left(a_{3}+a_{5}-\frac{1}{2}a_{7}-\frac{1}{2}a_{9}\right)+f_{K}F_{B_{s}\rightarrow\phi}^{LL}\left(a_{4}-\frac{1}{2}a_{10}\right)\right.\right.$
		$\displaystyle\left.\left.-f_{K}F_{B_{s}\rightarrow\phi}^{SP}\left(a_{6}-\frac{1}{2}a_{8}\right)+M_{B_{s}\rightarrow K}^{LL}\left(C_{4}-\frac{1}{2}C_{10}\right)+M_{B_{s}\rightarrow\phi}^{LL}\left(C_{3}-\frac{1}{2}C_{9}\right)\right.\right.$
		$\displaystyle\left.\left.-M_{B_{s}\rightarrow K}^{SP}\left(C_{6}-\frac{1}{2}C_{8}\right)-M_{B_{s}\rightarrow\phi}^{LR}\left(C_{5}-\frac{1}{2}C_{7}\right)+f_{B_{s}}F_{ann}^{LL}\left(a_{4}-\frac{1}{2}a_{10}\right)\right.\right.$
		$\displaystyle\left.\left.-f_{B_{s}}F_{ann}^{SP}\left(a_{6}-\frac{1}{2}a_{8}\right)+M_{ann}^{LL}\left(C_{3}-\frac{1}{2}C_{9}\right)-M_{ann}^{LR}\left(C_{5}-\frac{1}{2}C_{7}\right)\right]\right\}.$

	$\displaystyle\sqrt{2}A\left(\bar{B}_{s}^{0}\rightarrow K^{0}\rho\left(\rho\rightarrow K^{+}K^{-}\right)\right)=$	$\displaystyle\frac{G_{F}p_{\bar{B}_{s}^{0}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\phi}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)}{\sqrt{2}s_{\rho}}$		(17)
		$\displaystyle\times\left\{V_{ub}V_{ud}^{*}\left[f_{\rho}F_{B_{s}\rightarrow K}^{LL}\left(a_{2}\right)+M_{B_{s}\rightarrow K}^{LL}\left(C_{2}\right)\right]-V_{tb}V_{td}^{*}\left[M_{B_{s}\rightarrow K}^{LR}\left(-C_{5}+\frac{1}{2}C_{7}\right)\right.\right.$
		$\displaystyle\left.+f_{\rho}F_{B_{s}\rightarrow K}^{LL}\left(-a_{4}+\frac{3}{2}a_{7}+\frac{1}{2}a_{10}+\frac{3}{2}a_{9}\right)-M_{B_{s}\rightarrow K}^{SP}\left(\frac{3}{2}C_{8}\right)\right.$
		$\displaystyle\left.\left.+M_{B_{s}\rightarrow K}^{LL}\left(-C_{3}+\frac{1}{2}C_{9}+\frac{3}{2}C_{10}\right)+f_{B_{s}}F_{ann}^{LL}\left(-a_{4}+\frac{1}{2}a_{10}\right)\right.\right.$
		$\displaystyle\left.\left.+f_{B_{s}}F_{ann}^{SP}\left(-a_{6}+\frac{1}{2}a_{8}\right)+M_{ann}^{LL}\left(-C_{3}+\frac{1}{2}C_{9}\right)+M_{ann}^{LR}\left(-C_{5}+\frac{1}{2}C_{7}\right)\right]\right\}.$

	$\displaystyle\sqrt{2}A\left(\bar{B}_{s}^{0}\rightarrow K^{0}\omega\left(\omega\rightarrow K^{+}K^{-}\right)\right)=$	$\displaystyle\frac{G_{F}p_{\bar{B}_{s}^{0}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\omega}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)}{\sqrt{2}s_{\omega}}$		(18)
		$\displaystyle\times\left\{V_{ub}V_{ud}^{*}\left[f_{\omega}F_{B_{s}\rightarrow K}^{LL}\left(a_{2}\right)+M_{B_{s}\rightarrow K}^{LL}\left(C_{2}\right)\right]-V_{tb}V_{td}^{*}\left[M_{B_{s}\rightarrow K}^{LR}\left(C_{5}-\frac{1}{2}C_{7}\right)\right.\right.$
		$\displaystyle\left.+f_{\omega}F_{B_{s}\rightarrow K}^{LL}\left(2a_{3}+a_{4}+2a_{5}+\frac{1}{2}a_{7}+\frac{1}{2}a_{9}-\frac{1}{2}a_{10}\right)\right.$
		$\displaystyle+M_{B_{s}\rightarrow K}^{LL}\left(C_{3}+2C_{4}-\frac{1}{2}C_{9}+\frac{1}{2}C_{10}\right)+M_{ann}^{LL}\left(C_{3}-\frac{1}{2}C_{9}\right)$
		$\displaystyle\left.-M_{B_{s}\rightarrow K}^{SP}\left(2C_{6}+\frac{1}{2}C_{8}\right)+f_{B_{s}}F_{ann}^{LL}\left(a_{4}-\frac{1}{2}a_{10}\right)\right.$
		$\displaystyle\left.\left.+f_{B_{s}}F_{ann}^{SP}\left(a_{6}-\frac{1}{2}a_{8}\right)+M_{ann}^{LR}\left(C_{5}-\frac{1}{2}C_{7}\right)\right]\right\}.$

	$\displaystyle A\left(\bar{B}_{s}^{0}\rightarrow\eta\phi\left(\phi\rightarrow K^{+}K^{-}\right)\right)=$	$\displaystyle\frac{G_{F}p_{\bar{B}_{s}^{0}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\phi}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)}{\sqrt{2}s_{\phi}}$		(19)
		$\displaystyle\times\left\{\frac{\cos\theta}{\sqrt{2}}\bigg{\{}V_{ub}V_{us}^{*}\left[f_{n}F_{B_{s}\rightarrow\phi}^{LL}\left(a_{2}\right)+M_{B_{s}\rightarrow\phi}^{LL}\left(C_{2}\right)\right]\bigg{.}\right.$
		$\displaystyle-V_{tb}V_{ts}^{*}\left[f_{n}F_{B_{s}\rightarrow\phi}^{LL}\left(2a_{3}-2a_{5}-\frac{1}{2}a_{7}+\frac{1}{2}a_{9}\right)\right.$
		$\displaystyle\left.\left.+M_{B_{s}\rightarrow\phi}^{LL}\left(2C_{4}+\frac{1}{2}C_{10}\right)+M_{B_{s}\rightarrow\phi}^{SP}\left(2C_{6}+\frac{1}{2}C_{8}\right)\right]\right\}$
		$\displaystyle-\sin\theta\left\{-V_{tb}V_{ts}^{*}\left[f_{s}F_{B_{s}\rightarrow\phi}^{LL^{\prime}}\left(a_{3}+a_{4}-a_{5}+\frac{1}{2}a_{7}-\frac{1}{2}a_{9}-\frac{1}{2}a_{10}\right)\right.\right.$
		$\displaystyle+M_{B_{s}\rightarrow\phi}^{SP^{\prime}}\left(C_{6}-\frac{1}{2}C_{8}\right)+f_{B_{s}}F_{ann}^{LL^{\prime}}\left(a_{3}+a_{4}-a_{5}+\frac{1}{2}a_{7}-\frac{1}{2}a_{9}-\frac{1}{2}a_{10}\right)$
		$\displaystyle+M_{ann}^{LL^{\prime}}\left(C_{3}+C_{4}-\frac{1}{2}C_{9}-\frac{1}{2}C_{10}\right)-f_{B_{s}}F_{ann}^{SP^{\prime}}\left(a_{6}-\frac{1}{2}a_{8}\right)$
		$\displaystyle\left.\left.\left.-M_{ann}^{LR^{\prime}}\left(C_{5}-\frac{1}{2}C_{7}\right)-M_{ann}^{SP^{\prime}}\left(C_{6}-\frac{1}{2}C_{8}\right)\right]+\left[\eta_{s}\leftrightarrow\phi\right]\right\}\right\}.$

	$\displaystyle A\left(\bar{B}_{s}^{0}\rightarrow\eta\rho^{0}\left(\rho^{0}\rightarrow K^{+}K^{-}\right)\right)=$	$\displaystyle\frac{G_{F}p_{\bar{B}_{s}^{0}}\cdot\sum_{\lambda=0,\pm 1}\epsilon(\lambda)g_{\rho}\epsilon^{*}(\lambda)\cdot\left(p_{k^{+}}-p_{k^{-}}\right)}{\sqrt{2}s_{\rho}}$		(20)
		$\displaystyle\times\left\{\frac{\cos\theta}{2}\left\{-V_{tb}V_{ts}^{*}\left[f_{B_{s}}F_{ann}^{LL}\left(\frac{3}{2}a_{9}-\frac{3}{2}a_{7}\right)+M_{ann}^{LL}\left(\frac{3}{2}C_{10}\right)-M_{ann}^{SP}\left(\frac{3}{2}C_{8}\right)\right]\right.\right.$
		$\displaystyle\left.+V_{ub}V_{us}^{*}\bigg{[}f_{B_{s}}F_{ann}^{LL}\left(a_{2}\right)+M_{ann}^{LL}\left(C_{2}\right)\bigg{]}+\left[\rho^{0}\leftrightarrow\eta_{n}\right]\right\}$
		$\displaystyle-\frac{\sin\theta}{\sqrt{2}}\left\{V_{ub}V_{us}^{*}\bigg{[}f_{\rho}F_{B_{s}\rightarrow\eta_{s}}^{LL^{\prime}}\left(a_{2}\right)+M_{B_{s}\rightarrow\eta_{s}}^{LL^{\prime}}\left(C_{2}\right)\bigg{]}\right.$
		$\displaystyle\left.\left.-V_{tb}V_{ts}^{*}\left[f_{\rho}F_{B_{s}\rightarrow\eta_{s}}^{LL^{\prime}}\left(\frac{3}{2}a_{7}+\frac{3}{2}a_{9}\right)+M_{B_{s}\rightarrow\eta_{s}}^{LL^{\prime}}\left(\frac{3}{2}C_{10}\right)-M_{B_{s}\rightarrow\eta_{s}}^{SP^{\prime}}\left(\frac{3}{2}C_{8}\right)\right]\right\}\right\}.$