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On Charmless $B\to K_hη^{(')}$ Decays with $K_h=K,K^*,K_0^*(1430),K_2^*(1430)$

We study the charmless decays $B\to K_hη$ and $B\to K_hη'$ within the framework of QCD factorization (QCDF) for $K_h= K,K^*,K_0^*(1430)$ and naive factorization for $K_h=K_2^*(1430)$. There are three distinct types of penguin contributions: (i) $b\to sq\bar q\to sη_{q}$, (ii) $b\to ss\bar s\to sη_s$, and (iii) $b\to s q\bar q\to q \bar K_h$, where $η_q=(u\bar u+d\bar d)/\sqrt{2}$ and $η_s=s\bar s$. $B\to K^{(*)}η^{(')}$ decays are dominated by type-II and type-III penguin contributions. The interference, constructive for $Kη'$ and $K^*η$ and destructive for $Kη$ and $K^*η'$, between type-II and type-III diagrams explains the pattern of $Γ(B\to Kη')\ggΓ(B\to Kη)$ and $Γ(B\to K^*η')\llΓ(B\to K^*η)$. Within QCDF, the observed large rate of the $Kη'$ mode can be naturally explained without invoking flavor-singlet contributions or something exotic. The decay pattern for $B\to K_0^*(1430)η^{(')}$ decays depends on whether the scalar meson $K_0^*(1430)$ is an excited state of $κ$ or a lowest-lying $P$-wave $q\bar q$ state. Hence, the experimental measurements of $B\to K_0^*(1430)η^{(')}$ can be used to explore the quark structure of $K_0^*(1430)$. If $K_0^*(1430)$ is a low-lying $q\bar q$ bound state, we find that $K_0^*η$ has a rate slightly larger than $K_0^*η'$ owing to the fact that the $η$-$η'$ mixing angle in the $η_q,η_s$ flavor basis is less than $45^\circ$, in agreement with experiment. Type-III penguin diagram does not contribute to $B\to K_2^*η^{(')}$ under the factorization hypothesis and type-II diagram dominates.