Effect of resonance for $CP$ asymmetry of the decay process ${\overline{B}}_s\to P{\pi }^{+}{\pi }^{-}$ in perturbative QCD

In the framework of the perturbative QCD (PQCD) approach we study the direct $CP$ asymmetry for the decay channel ${\overline{B}}_s\to P{\pi }^{+}{\pi }^{-}$ around the resonance range via the $\rho -\omega$ mixing mechanism (where $P$ refers to pseudoscalar meson). We find that the $CP$ asymmetry can be enhanced by $\rho -\omega$ mixing when the masses of the ${\pi }^{+}{\pi }^{-}$ pairs are at the area of $\rho -\omega$ resonance, and the maximum $CP$ asymmetry can reach 59% for the relevant decay channels.

Figure 1

Plot of $A_{CP}$ as a function of $\sqrt{s}$ corresponding to central parameter values of CKM matrix elements. The dashed line, dash-dotted, solid line refer to the decay channels of ${\overline{B}}_s^0\to {\rho }^0(\omega )K^0\to {\pi }^{+}{\pi }^{-}{K}^0$, ${\overline{B}}_s^0\to {\rho }^0(\omega ){\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\eta }^{\prime }$, and ${\overline{B}}_s^0\to {\rho }^0(\omega )\eta \to {\pi }^{+}{\pi }^{-}\eta$, respectively.

Figure 2

Plot of $\sin \delta$ as a function of $\sqrt{s}$ corresponding to central parameter values of CKM matrix elements. The dashed line, dash-dotted, solid line refer to the decay channels of ${\overline{B}}_s^0\to {\rho }^0(\omega )K^0\to {\pi }^{+}{\pi }^{-}{K}^0$, ${\overline{B}}_s^0\to {\rho }^0(\omega ){\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\eta }^{\prime }$, and ${\overline{B}}_s^0\to {\rho }^0(\omega )\eta \to {\pi }^{+}{\pi }^{-}\eta$, respectively.)

Figure 3

Plot of r as a function of $\sqrt{s}$ corresponding to central parameter values of CKM matrix elements. The dashed line, dash-dotted, solid line refer to the decay channels of ${\overline{B}}_s^0\to {\rho }^0(\omega )K^0\to {\pi }^{+}{\pi }^{-}{K}^0$, ${\overline{B}}_s^0\to {\rho }^0(\omega ){\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\eta }^{\prime }$ and ${\overline{B}}_s^0\to {\rho }^0(\omega )\eta \to {\pi }^{+}{\pi }^{-}\eta$, respectively.)

Figure 4

Plot of $A_{CP}$ as a function of $\sqrt{s}$ corresponding to central parameter values of CKM matrix elements for ${\overline{B}}_s^0\to {\rho }^0(\omega ){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$.

Figure 5

Plot of $\sin \delta$ as a function of $\sqrt{s}$ corresponding to central parameter values of CKM matrix elements for ${\overline{B}}_s^0\to {\rho }^0(\omega ){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$.

Figure 6

Plot of r as a function of $\sqrt{s}$ corresponding to central parameter values of CKM matrix elements for ${\overline{B}}_s^0\to {\rho }^0(\omega ){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$.