Branching ratios and CP asymmetries of $B\to K{\eta }^{(\mathrm{\prime })}$ decays in the pQCD approach

We calculate the branching ratios and CP violating asymmetries of the four $B \to K \etap$ decays in the perturbative QCD (pQCD) factorization approach. Besides the full leading order contributions, the partial next-to-leading order (NLO) contributions from the QCD vertex corrections, the quark loops, and the chromo-magnetic penguins are also taken into account. The NLO pQCD predictions for the CP-averaged branching ratios are $Br(B^+ \to K^+ \eta) \approx 3.2 \times 10^{-6}$,$Br(B^\pm \to K^\pm \etar) \approx 51.0 \times 10^{-6}$,$Br(B^0 \to K^0 \eta) \approx 2.1 \times 10^{-6}$,and$Br(B^0 \to K^0 \etar) \approx 50.3 \times 10^{-6}$. The NLO contributions can provide a 70% enhancement to the LO $Br(B\to K\text{etar})$, but a 30% reduction to the LO $Br(B \to K \eta)$, which play the key role in understanding the observed pattern of branching ratios. The NLO pQCD predictions for the CP-violating asymmetries, such as ${\text{acp}}^{dir}(K_S^0\text{etar})\sim 2.3$ and $\acp^{mix}(K^0_S \etar)\sim 63%$, agree very well with currently available data. This means that the deviation $\mathrm{\Delta }S={\text{acp}}^{mix}(K_S^0\text{etar})-\sin 2\beta$ in pQCD approach is also very small.