the hard spectator and annihilation calculations and recent studies within QCDF. In section \[sec03\], focusing on $\pi K$ and $\pi \pi$ puzzles, the effects of spectator scattering and annihilation contributions on $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays are studied in detail in blue[three]{} scenarios. In each scenario, a fitting on relevant parameters are performed. Our conclusions are summarized in section \[sec04\]. Appendix \[app01\] recapitulates the building blocks of annihilation and spectator scattering amplitudes. The input parameters and our fitting approach are given in Appendix \[app02\] and \[app03\], respectively.

Brief Review of Spectator Scattering and Annihilation Amplitudes within QCDF {#sec02}
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The effective Hamiltonian for nonleptonic $B$ weak decays is [@Buchalla:1996vs] $$\begin{aligned}
 {\cal H}_{\rm eff} &=&
  \frac{G_F}{\sqrt{2}}
  \sum\limits_{p,q}
  V_{pb} V_{pq}^{\ast}
  \Big\{
  \sum\limits_{i=1}^{10}
   C_i O_i + C_{7 \gamma} O_{7 \gamma}
   + C_{8g} O_{8g} \Big\}
   + {\rm h.c.}
 \label{eq:eff},
 \end{aligned}$$ where $V_{pb} V_{pq}^{\ast}$ ($p$ $=$ $u$, $c$ and $q$ $=$ $d$, $s$) is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements; $C_{i}$ is the Wilson coefficient corresponding to the local four-quark operator $O_i$; $O_{7 \gamma}$ and $O_{8g}$ are the electromagnetic and chromomagnetic dipole operators.

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With the effective Hamiltonian Eq.(\[eq:eff\]), the QCDF method has been fully developed and extensively 