Nonet symmetry in $\eta$, $\eta\prime$ and $B\rightarrow K\eta,K\eta\prime$ decays
T. N. Pham

The nonet symmetry scheme seems to describe rather well the masses and $\eta-\eta^{\prime}$ mixing angle of the ground state pseudo-scalar mesons. It is expected that nonet symmetry should also be valid for the matrix elements of the pseudo-scalar densitty operators which play an important role in charmless two-body $B$ decays with $\eta$ or $\eta^{\prime}$ in the final state. Starting from the divergences of the $SU(3)$ octet and singlet axial vector currents, we show that nonet symmetry for the pseudo-scalar mass term implies nonet symmetry for the pseudo-scalar density operators. In this nonet symmetry scheme, we find that the branching ratio $B\to PP,PV$, with $\eta$ in the final state agrees well with data, while those with $\eta'$ are underestimated, but by increasing the $B\to \eta'$ form factor by $40-50\%$, one could explain the tree-dominated $B^{-}\to \pi^{-}\eta'$ and $B^{-}\to \rho^{-}\eta'$ measured branching ratios. With this increased form factor and with only a moderate annihilation contribution, we are able to obtain $62\times 10^{-6}$ for the penguin-dominated $B^{-}\to K^{-}\eta'$ branching ratios, quite close to the measured value. This supports the predicted value for the $B\to \eta'$ form factor in PQCD and light-cone sum rules approach. A possible increase by $15\%$ of $\langle 0|\bar{s}\,i\gamma_5 s|s\bar{s}\rangle $ for $\eta_{0} $ would bring the predicted $B^{-}\to K^{-}\eta'$ branching ratio to $69.375\times 10^{-6}$, very close to experiment.

© 2008 The American Physical Society.