A_{ \pi \pi } \Big\{ \delta_{pu} ( \alpha_2 - \beta_1 )
  - \alpha_{4}^p + \frac{3}{2}\alpha_{3,{\rm EW}}^p
  + \frac{1}{2} \alpha_{4,{\rm  EW}}^p
  \nonumber \\ & &
  - \beta_3^p -2 \beta_4^p
  + \frac{1}{2} \beta_{3,{\rm EW}}^p
  - \frac{1}{2} \beta_{4,{\rm EW}}^p) \Big\}
  \label{amp7}.
  \end{aligned}$$

For the sake for convenient discussion, we reiterate the expressions of the annihilation coefficients [@Beneke2], $$\begin{aligned}
  {\beta}_{i}^{p} &=& b_{i}^{p} B_{M_{1}M_{2}}/A_{M_{1}M_{2}}
   \label{betai}, \\
   b_{1} &=&
   \frac{C_{F}}{N_{c}^{2}}\, C_{1} A_{1}^{i},
   \quad \quad \quad
   b_{2} =
   \frac{C_{F}}{N_{c}^{2}}\, C_{2} A_{1}^{i}
   \label{b12}, \\
   b_{3}^{p} &=&
   \frac{C_{F}}{N_{c}^{2}}\,
   \Big[ C_{3} A_{1}^{i} + C_{5}( A_{3}^{i} + A_{3}^{f} )
   +N_{c} C_{6} A_{3}^{f} \Big]
   \label{b3}, \\
   b_{4}^{p} &=&
   \frac{C_{F}}{N_{c}^{2}}\,
   \Big[ C_{4} A_{1}^{i} + C_6 A_2^i \Big]
   \label{b4}, \\
   b_{3,\rm EW}^p &=&
   \frac{C_{F}}{N_{c}^{2}}\,
   \Big[ C_9 A_1^i + C_7 ( A_3^i + A_3^f )
   + N_c C_8 A_3^f \Big]
   \label{b3ew}, \\
   b_{4,\rm EW}^p &=&
   \frac{C_{F}}{N_{c}^{2}}\,
   \Big[ C_{10} A_1^i + C_8 A_2^i \Big]
   \label{b4ew}.
   \end{aligned}$$

Numerically, coefficients of $b_{3,\rm EW}^p$ and $b_{4,\rm EW}^p$ are negligible compared with the other effective coefficients due to the small electroweak Wilson coefficients, and so their effects would be not discussed in this paper.

In order to illustrate the contributions of annihilation and spectator scattering, we explore three parameter scenarios in which certain parameters are changed freely.

-   Scenario I: 