employed to calculate the hadronic two-body $B$ decays, for example, see [@du1; @Beneke1; @Beneke2; @Cheng2]. The spectator scattering and annihilation amplitudes (see Fig.\[diag\]) are expressed as the convolution of scattering functions with the light-cone wave functions of the participating mesons [@Beneke1; @Beneke2]. The explicit expressions for the basic building blocks of spectator scattering and annihilation amplitudes have been given by Ref. [@Beneke2], which are also listed in the appendix \[app01\] for convenience. With the asymptotic light-cone distribution amplitudes, the building blocks for annihilation amplitudes of Eq.(\[ai1\]-\[af3\]) could be simplified as [@Beneke2] $$\begin{aligned}
  A_1^i & \simeq & A_2^i \simeq
  2 \pi \alpha_s \Big[ 9\,\Big( X_A - 4 + \frac{\pi^2}{3} \Big)
 + r_\chi^{M_1} r_\chi^{M_2} X_A^2 \Big]
 \label{xai12}, \\
  A_3^i & \simeq &
  6 \pi \alpha_s \big(r_\chi^{M_1} - r_\chi^{M_2} \big)
  \Big( X_A^2 - 2 X_A + \frac{\pi^2}{3} \Big)
 \label{xai3}, \\
  A_3^f & \simeq &
  6 \pi \alpha_s ( r_\chi^{M_1} + r_\chi^{M_2} )
  (2 X_A^2 - X_A)
 \label{xaf3},
 \end{aligned}$$ where the superscripts $i$ (or $f$) refers to gluon emission from the initial (or final) state quarks, respectively (see Fig.\[diag\]). For the $\pi \pi$, $\pi K$ and $K \bar{K}$ final-state, $A_3^i$ is numerically negligible due to $r_\chi^{M_1}$ $\simeq$ $r_\chi^{M_2}$. The model-dependent parameter $X_A$ is used 