Determining the effective Wilson coefficient $a_2$ in terms of $\mathrm{BR}(B_s\to {\eta }_c\varphi )$ and evaluating $\mathrm{BR}(B_s\to {\eta }_c{f}_0(980))$

In this work, we investigate decays of $B_s\to {\eta }_c\varphi$ and $B_s\to {\eta }_c{f}_0(980)$ in a theoretical framework. The calculation is based on the postulation that $f_0(980)$ and $f_0(500)$ are mixtures of pure quark states $\frac{1}{\sqrt{2}}(\overline{u}u+\overline{d}d)$ and $\overline{s}s$. The hadronic matrix elements for $B_s\to \varphi$ and $B_s\to f_0(980)$ are calculated in the light-front quark model and the important Wilson coefficient $a_2$ which is closely related to nonperturbative QCD is extracted. However, our numerical results indicate that no matter how the mixing parameter is adjusted to reconcile contributions of $f_0(980)$ and $f_0(500)$, one cannot make the theoretical prediction on $B_s\to {\eta }_c+{\pi }^{+}{\pi }^{-}$ to meet the data. Moreover, the new measurement of $\mathrm{BR}(B_s\to J/\psi +f_0(500))<1.7\times {10}^{-6}$ also negates the mixture scenario. Thus, we conclude that the recent data suggest that $f_0(980)$ is a four-quark state (tetraquark or $K\overline{K}$ molecule), at least the fraction of its pure quark constituents is small.

Figure 1

Feynman diagrams depicting the strong decay.