The role and contribution of resonance effect for the decay process of $\bar B^{0}_s \rightarrow π^{+}π^{-}P$

Abstract: The size of the direct CP asymmetry generated during the weak decay of hadrons is attributed to the weak phase and some strong phases. The weak phase comes from the CKM matrix and a strong phase may result from the resonance effect which is produced by the mixing of vector meson $V\left{\rho^{0}(770),\omega(782),\phi(1020)\right}$ to $\pi^+ \pi^-$ meson pairs. $\rho^{0}(770)$ can decay directly into $\pi^+ \pi^-$ meson pairs, both $\omega(782)$ and $\phi(1020)$ can also decay into $\pi^+ \pi^-$ meson pairs with small contribution from isospin symmetry breaking. The main contribution for the middle state vector meson $\rho^{0}(770)-\omega(782)-\phi(1020)$ interference is the mix of $\rho^{0}(770)$, $\omega(782)-\rho^{0}(770)$ and $\phi(1020)-\rho^{0}(770)$. We calculate the CP asymmetry and decay branching ratio for $\bar{B}^0_{s} \rightarrow \pi^+ \pi^- \pi^0 (\bar K ^{0})$ in the framework of QCD factorization and compare them with previous work. We also add the analysis of $\bar{B}^0_{s} \rightarrow \pi^+ \pi^- \eta(\eta^{(')})$ decay process. The results show that the CP asymmetry of these four decay processes are significantly enhanced especially for the $\bar{B}^0_{s} \rightarrow \pi^+ \pi^- \bar K ^{0}$ decay process and the decay branching ratio also changes under resonance effect. These work might provide support for the experimental analysis of the $\bar B^{0}_s$ meson.