for instance $\Delta A \sim 0.5 \%$ in the S4 scenario of QCDF [@Beneke2].

The so-called ${\pi}{\pi}$ puzzle is reflected by the following two ratios of the $CP$-averaged branching fractions [@pipipuz]: $$R_{+-}^{\pi \pi}
 \equiv 2 \Big[
 \frac{ {\cal B}(B^{-} \to \pi^{-} \pi^0) }
      { {\cal B}(\bar{B}^{0} \to \pi^{+} \pi^{-}) }
 \Big]
 \frac{ \tau_{B^0} }{ \tau_{B^+} },
 \qquad
 R_{00}^{\pi \pi}
 \equiv  2 \Big[
 \frac{ {\cal B}( \bar{B}^{0} \to \pi^0 \pi^0) }
      { {\cal B}( \bar{B}^{0} \to \pi^{+} \pi^{-}) }\Big]
 \label{pipipuzzle}.$$ It is generally expected that branching ratio ${\cal B}(\bar{B}^{0} \to \pi^+ \pi^-) \gtrsim
  {\cal B}(B^{-} \to \pi^{-} \pi^0)$ and ${\cal B}(\bar{B}^{0} \to \pi^+ \pi^-) \gg
  {\cal B}(\bar{B}^{0} \to \pi^0 \pi^0)$ within the SM. To date, the agreement of $R_{+-}^{\pi \pi}$ between the S4 scenario QCDF $R_{+-}^{\pi \pi}(\text{QCDF})$ $=$ $1.83$ [@Beneke2] and the refined experimental data $R_{+-}^{\pi \pi}(\text{Exp.})$ $=$ $1.99 \pm 0.15$ [@HFAG] can be achieved consistently within experimental error, while the discrepancy in $R_{00}^{\pi \pi}$ between the S4 scenario QCDF $R_{00}^{\pi \pi}(\text{QCDF})$ $=$ $0.27$ (where theoretical uncertainties are unenclosed) [@Beneke2] and the progressive experimental data $R_{+-}^{\pi \pi}(\text{Exp.})$ $=$ $1.99 \pm 0.15$ [@HFAG] is unexpectedly large.

It is claimed [@pipipuz] that the so-called ${\pi}{\pi}$ puzzle could be accommodated by the nonfactorizable contributions 