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

Abstract

We study the charmless decays $B\to K_h\eta$ and $B\to K_h{\eta }^{\prime }$ 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) $b\to sq\bar{q}\to q{\bar{K}}_h$, where ${\eta }_q=(u\bar{u}+d\bar{d})/\sqrt{2}$ and ${\eta }_s=s\bar{s}$. $B\to K^{(\ast )}{\eta }^{{(}^{\prime })}$ decays are dominated by type-II and type-III penguin contributions. The interference, constructive for $K{\eta }^{\prime }$ and $K^{\ast }\eta$ and destructive for $K\eta$ and $K^{\ast }{\eta }^{\prime }$, between type-II and type-III diagrams explains the pattern of $\Gamma (B\to K{\eta }^{\prime })\gg \Gamma (B\to K\eta )$ and $\Gamma (B\to K^{\ast }{\eta }^{\prime })\ll \Gamma (B\to K^{\ast }\eta )$. Within QCDF, the observed large rate of the $K{\eta }^{\prime }$ mode can be naturally explained without invoking flavor-singlet contributions or something exotic. The decay pattern for $B\to K_0^{\ast }(1430){\eta }^{{(}^{\prime })}$ decays depends on whether the scalar meson $K_0^{\ast }(1430)$ is an excited state of $\kappa$ or a lowest-lying $P$-wave $q\bar{q}$ state. Hence, the experimental measurements of $B\to K_0^{\ast }(1430){\eta }^{{(}^{\prime })}$ 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 $K_0^{\ast }{\eta }^{\prime }$ owing to the fact that the $\eta$-${\eta }^{\prime }$ 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 $B\to K_2^{\ast }{\eta }^{{(}^{\prime })}$ under the factorization hypothesis and type-II diagram dominates.

Comments: 18 pages, 1 figure