Branching Ratio and CP Asymmetry of $B\to \pi {\eta }^{(\mathrm{\prime })}$ Decays in the Perturbative QCD Approach

We calculate the branching ratios and CP-violating asymmetries for $B^0 \to \pi^0 \eta^{(\prime)}$and$B^+\to \pi^+ \eta^{(\prime)}$ decays in the perturbative QCD (pQCD) factorization approach here. We not only calculate the usual factorizable contributions, but also evaluate the non-factorizable and annihilation type contributions. Besides the current-current operators, the contributions from the QCD and electroweak penguin operators are also taken into account. The pQCD results for the CP-averaged branching ratios are $Br(B^+ \to \pi^+ \eta) \approx 4.1 \times 10^{-6}$,$Br(B^+ \to \pi^+ \eta^\prime) \approx 2.4 \times 10^{-6}$,and$Br(B^0 \to \pi^0 \eta^{(\prime)}) \approx 0.2 \times 10^{-6}$, which agree very well with the measured values or currently available experimental upper limits. We also predict large CP-violating asymmetries in these decays: $A_{CP}^{dir}(\pi^\pm \etap)\sim A_{CP}^{dir}(\pi^0 \etap)\sim -0.35 $,and$A_{CP}^{mix}(\pi^0 \etap)\sim 0.67$,butwithlargeerrors\mathrm{.}ThepQCDpredictionfor$A_{CP}^{dir}(\pi^\pm \eta)$($A_{CP}^{dir}(\pi^\pm \eta^\prime))$ has the same (opposite) sign with the primary measured values. Further improvements in both theory and experiments are needed to clarify this discrepancy.