On Charmless $B\to K_h{\eta }^{{(}^{\mathrm{\prime }})}$ Decays with $K_h=K,K^{\ast },K_0^{\ast }(1430),K_2^{\ast }(1430)$

We study the charmless decays $B\to K_h\eta$ and $B\to K_h{\eta }^{\mathrm{\prime }}$ within the framework of QCD factorization (QCDF) for $K_h=K,K^{\ast },K_0^{\ast }(1430)$ and naive factorization for $K_h=K_2^{\ast }(1430)$. There are three distinct types of penguin contributions: (i) $b\to sq\bar{q}\to s{\eta }_q$, (ii) $b\to ss\bar s\to s\eta_s$,and(iii)$b\to s q\bar q\to q \bar K_h$,where$\eta_q=(u\bar u+d\bar d)/\sqrt{2}$and$\eta_s=s\bar s$\mathrm{.}$B\to K^{()}\eta^{(')}$ decays are dominated by type-II and type-III penguin contributions. The interference, constructive for $K{\eta }^{\mathrm{\prime }}$ and $K^{\ast }\eta$ and destructive for $K\eta$ and $K^{\ast }{\eta }^{\mathrm{\prime }}$, between type-II and type-III diagrams explains the pattern of $\mathrm{\Gamma }(B\to K{\eta }^{\mathrm{\prime }})\gg \mathrm{\Gamma }(B\to K\eta )$ and $\Gamma(B\to K^\eta’)\ll\Gamma(B\to K^\eta)$\mathrm{.}WithinQCDF,theobservedlargerateofthe$K\eta’$ mode can be naturally explained without invoking flavor-singlet contributions or something exotic. The decay pattern for $B\to K_0^(1430)\eta^{(‘)}$decaysdependsonwhetherthescalarmeson$K_0^(1430)$isanexcitedstateof$\kappa$oralowest-lying$P$-wave$q\bar q$ state. Hence, the experimental measurements of $B\to K_0^{\ast }(1430){\eta }^{{(}^{\mathrm{\prime }})}$ can be used to explore the quark structure of $K_0^{\ast }(1430)$. If $K_0^{\ast }(1430)$ is a low-lying $q\bar{q}$ bound state, we find that $K_0^{\ast }\eta$ has a rate slightly larger than $K_0^{\ast }{\eta }^{\mathrm{\prime }}$ owing to the fact that the $\eta$-${\eta }^{\mathrm{\prime }}$ mixing angle in the ${\eta }_q,{\eta }_s$ flavor basis is less than ${45}^{\circ }$, in agreement with experiment. Type-III penguin diagram does not contribute to $B\to K_2^\eta^{(’)}$ under the factorization hypothesis and type-II diagram dominates.