Consequently, the nonfactorizable spectator scattering parameters $X_H$ or ($\rho_H$, $\phi_H$) will have great influence on the observable $\Delta A$. Furthermore, a scrutiny of difference between Eq.(\[amp2\]) and Eq.(\[amp3\]), another possible resolution to the $\pi K$ puzzle might be provided by annihilation contributions, such as coefficient $\beta_2$, as suggested in Ref.[@zhu2]. If so, then $\Delta A$ will depend strongly on the nonfactorizable annihilation parameters ($\rho_A^i$, $\phi_A^i$) because $\beta_2$ is proportional to $A_1^i$ in Eq.(\[b12\]). Additionally, it can be seen from Eq.(\[amp2\]) and Eq.(\[amp3\]) that annihilation coefficient $\beta_3^p$ contributes to amplitudes both ${\cal A}_{ B^- \to \pi^0 K^-}$ and ${\cal A}_{ \bar{B}^0 \to \pi^+ K^-}$. If $\beta_3^p$ could offer a large strong phase, then its effect should contribute to the direct $CP$ asymmetries $A_{CP}(B^- \to \pi^0 K^-)$ and $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ rather than $\Delta A$. Due to the fact that the lion’s share of $\beta_3^p$ comes from $N_{c} C_6 A_3^f$ in Eq.(\[b3\]), the direct $CP$ asymmetries $A_{CP}(B^- \to \pi^0 K^-)$ and $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ should vary greatly with the factorizable annihilation parameters $X_A^f$, while $\Delta A$ should be insensitive to variation of parameters ($\rho_A^f$, $\phi_A^f$). The above analysis and speculations are confirmed by Fig.\[cpanni\].

From Eq.(\[amp4\]), it is seen that 