Branching ratios andCPasymmetries ofB→Kη(′)decays in the perturbative QCD approach

Abstract

We calculate the branching ratios and $CP$-violating asymmetries of the four $B\ensuremath{\rightarrow}K{\ensuremath{\eta}}^{(\ensuremath{'})}$ 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 chromomagnetic penguins are also taken into account. The NLO pQCD predictions for the $CP$-averaged branching ratios are $\mathrm{Br}({B}^{+}\ensuremath{\rightarrow}{K}^{+}\ensuremath{\eta})\ensuremath{\approx}3.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, $\mathrm{Br}({B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\eta}}^{\ensuremath{'}})\ensuremath{\approx}51.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, $\mathrm{Br}({B}^{0}\ensuremath{\rightarrow}{K}^{0}\ensuremath{\eta})\ensuremath{\approx}2.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, and $\mathrm{Br}({B}^{0}\ensuremath{\rightarrow}{K}^{0}{\ensuremath{\eta}}^{\ensuremath{'}})\ensuremath{\approx}50.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. The NLO contributions can provide a 70% enhancement to the LO $\mathrm{Br}(B\ensuremath{\rightarrow}K{\ensuremath{\eta}}^{\ensuremath{'}})$, but a 30% reduction to the LO $\mathrm{Br}(B\ensuremath{\rightarrow}K\ensuremath{\eta})$, which play the key role in understanding the observed pattern of branching ratios. The NLO pQCD predictions for the $CP$-violating asymmetries, such as ${\mathcal{A}}_{CP}^{\mathrm{dir}}({K}_{S}^{0}{\ensuremath{\eta}}^{\ensuremath{'}})\ensuremath{\sim}2.3%$ and ${\mathcal{A}}_{CP}^{\mathrm{mix}}({K}_{S}^{0}{\ensuremath{\eta}}^{\ensuremath{'}})\ensuremath{\sim}63%$, agree very well with currently available data. This means that the deviation $\ensuremath{\Delta}S={\mathcal{A}}_{CP}^{\mathrm{mix}}({K}_{S}^{0}{\ensuremath{\eta}}^{\ensuremath{'}})\ensuremath{-}\mathrm{sin}2\ensuremath{\beta}$ in pQCD approach is also very small.