$=$ $X_A^i$ is assumed in our coming numerical evaluation (scenarios I and II, see the next section for detail). Of course, this approximation is neither based on solid ground or from some underlying principle, and should be carefully studied and deserve much research. In fact, our coming phenomenological study (scenarios III) shows that the approximation $X_H$ $=$ $X_A^i$ is allowable with the up-to-date measurement on $B_{u,d}$ ${\to}$ $KK$, ${\pi}K$, ${\pi}{\pi}$ decays. In addition, it can be seen from Eq.(\[hardblock\]) that the spectator scattering corrections depend strongly on the inverse moment parameter ${\lambda}_{B}$ given in Eq.(\[lamdef\]). Recently, the value of ${\lambda}_{B}$ is an increasing concern of theoretical and experimental physicists [@Beneke5; @Beneke4; @Braun; @BaBarBA1; @BaBarBA2; @lambda]. A scrutiny of parameter ${\lambda}_{B}$ becomes imperative. In this paper, we will give some information on ${\lambda}_{B}$ required by present experimental data of $B_{u,d}$ ${\to}$ $K \bar{K}$, ${\pi}K$, ${\pi}{\pi}$ decays.

numerical analysis and discussions {#sec03}
==================================

With the conventions in Ref. [@Beneke2], the decay amplitudes for $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi $ decays within the QCDF framework can be written as $$\begin{aligned}
 %%%%%%%%%%%%%%%%% B- -> pi- k0
 {\cal A}_{ B^- \to \pi^- \bar{K}^0 }
  &=&
 \sum\limits_{p=u,c} V_{pb}V_{ps}^{\ast}
 A_{ \pi K }
 \Big\{ 