Categorizing $SU(3)_f$ representations of scalar mesons by $J/\psi$ decays

The scalar mesons are established for a long time, but their nature is still an open question. In this paper, we investigate the potential of categorizing their $SU(3)_f$ representations via $J/\psi\to SV$ and $\gamma S$, offering a criterion that may illuminate this issue. Here, $S$ ($V$) denotes scalar (vector) mesons. Using the $SU(3)_f$ symmetry with the current data, we find that $f_0(500)$ and $f_0(980)$ are mostly made of singlet and octet $SU(3)_f$ representations, respectively, with the singlet-octet mixing angle of $\theta = (82.9\pm4.4)^{\circ}$. This conclusion is consistent with the caculations of the quark-antiquark ($q\bar{q}$) hypothesis. For the scalar mesons in the range of 1-2 GeV, we discuss the mixings between $q\bar{q}$ and glueballs. Our numerical results suggest that $f_0(1710)$ is likely composed of the scalar glueball. We urge our experimental colleagues to measure $J/\psi \to \rho a_0(980,\ 1450,\ 1710),$ $K^*(892)^{\pm} K^*(700,\ 1430,\ 1950)^{\mp}$ and $\omega f_0(500)$, which provide useful information in the $SU(3)_f$ analysis.