the amplitude ${\cal A}_{ \bar{B}^0 \to K^- K^+}$ depends heavily on coefficients $\beta_1$ and $\beta_4^p$, which are closely associated with the nonfactorizable annihilation parameter $X_A^i$ only. The factorizable annihilation contributions vanish due to the isospin symmetry, which is consistent with the pQCD calculation [@xiao1]. The large branching ratio Eq.(\[HFAGKK\]) would appeal for large nonfactorizable annihilation parameter $X_A^i$ or $\rho_A^i$. The dependence of branching ratio ${\cal B}(\bar{B}^0 \to K^- K^+)$ on the parameters ($\rho_A^i$, $\phi_A^i$) is displayed in Fig.\[branni\].

![The dependence of branching ratio ${\cal B}(\bar{B}^0 \to K^- K^+)$ on nonfactorizable annihilation parameters ($\rho_A^i$, $\phi_A^{i}$). The notes are the same as Fig.\[cpanni\].[]{data-label="branni"}](kkphi.pdf){width="40.00000%"}

            $\rho_H$ $=$ $\rho_A^i$   $\phi_H$ $=$ $\phi_A^i\,[^{\circ}]$         $\rho_A^f$         $\phi_A^f[^{\circ}]$
  -------- ------------------------- ------------------------------------- ------------------------ ----------------------
  Part A    $2.82^{+2.73}_{-1.15}$            $-108^{+44}_{-50}$            $1.07^{+0.30}_{-0.20}$    $-40^{+10}_{-11}$
  Part B    $2.86^{+2.68}_{-1.20}$            $-108^{+42}_{-51}$            $2.72^{+0.30}_{-0.22}$     $166^{+3}_{-4}$

  : Numerical results of annihilation parameters in scenario I.[]{data-label="pikfit"}

                                          Exp. [@HFAG]                    scenario I                          scenario II              S4 [@Beneke2]
  --------------------------------- ------------------------- ----------------------------------- ----------------------------------- ---------------
  $B^- \to \pi^- \bar{K}^0$             $23.79 \pm 0.75$       $20.53^{+1.52+4.28}_{-0.65-3.87}$   $21.54^{+1.60+4.40}_{-0.68-3.99}$      $20.3$
  $B^- \to \pi^0 K^-$                $12.94^{+0.52}_{-0.51}$   $11.29^{+0.88+2.14}_{-0.45-1.96}$   $11.78^{+0.92+2.20}_{-0.47-2.01}$      $11.7$
  $\bar{B}^0 \to \pi^+ K^-$          $19.57^{+0.53}_{-0.52}$   $17.54^{+1.34+3.61}_{-0.65-3.27}$   $18.51^{+1.41+3.73}_{-0.67-3.38}$      $18.4$
  $\bar{B}^0 \to \pi^0 \bar{K}^0$        $9.93 \pm 0.49$       $8.05^{+0.60+1.84}_{-0.27-1.65}$    $8.60^{+0.65+1.90}_{-0.29-1.72}$        $8.0$
  $B^- \to K^- K^0$                      $1.19 \pm 0.18$       $1.45^{+0.13+0.32}_{-0.09-0.29}$    $1.51^{+0.13+0.32}_{-0.09-0.29}$       $1.46$
  $\bar{B}^0 \to K^- K^+$                $0.12 \pm 