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In this paper we perform a comprehensive study of the four $B \to K \etap$decays in the pQCD factorization approach. We calculate the CP-averagedbranching ratios and CP-violating asymmetries of $B \to K\etap$ decays in theordinary $\eta$-$\etar$, the $\eta$-$\etar$-$G$, and the$\eta$-$\etar$-$G$-$\eta_c$ mixing schemes. Besides the full LO contributions,all currently known NLO contributions to $B \to K \etap$ decays in the pQCDapproach are taken into account. From our calculations and phenomenologicalanalysis, we find the following. (a) The NLO contributions in general canprovide significant enhancements to the LO pQCD predictions for the decay ratesof the two $B \to K \etar$ decays, around $70%-89%$ in magnitude, but result inrelatively small changes to $Br(B \to K \eta)$. (b) Although the NLO pQCDpredictions in all three considered mixing schemes agree well with the datawithin one standard deviation, those pQCD predictions in the $\eta$-$\etar$-$G$mixing scheme provide the best interpretation for the measured pattern of $Br(B\to K \etap)$: $Br(B^0 \to K^0 \eta) \approx 1.13\times 10^{-6}$, $Br(B^0 \toK^0 \etar) \approx 66.5 \times 10^{-6}$, $Br(B^\pm \to K^\pm \eta) \approx2.36\times 10^{-6}$, and $Br(B^\pm \to K^\pm \etar) \approx 67.3 \times10^{-6}$, which agree perfectly with the measured values. (c) The NLO pQCDpredictions for the CP-violating asymmetries for the four considered decays arealso in good consistent with the data. (d) The newly known NLO contribution tothe relevant form factors $\calm_{FF}$ can produce about a 20% enhancement tothe branching ratios $Br(B \to K \etar)$, which plays an important role inclosing the gap between the pQCD predictions and the relevant data. (e) Thegeneral expectations about the relative strength of the LO and NLOcontributions from different sources are examined and confirmed by explicitnumerical calculations.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Anatomy of $B \to K \eta^{(\prime)}$ decays in different mixing schemes  and effects of next-to-leading order contributions in the perturbative QCD  approach