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

The scalar mesons have been 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 $f_0\left(500\right)$ and $f_0\left(980\right)$ are mostly made of singlet and octet $SU(3{)}_f$ representations, respectively, with the singlet-octet mixing angle of $\theta =(82.9\pm 4.4)°$. This conclusion is consistent with the caculations of the quark-antiquark ($q\overline{q}$) hypothesis. For the scalar mesons in the range of 1–2 GeV, we discuss the mixings between $q\overline{q}$ and glueballs. Our numerical results suggest that $f_0\left(1710\right)$ is likely composed of the scalar glueball. We urge our experimental colleagues to measure $J/\psi \to \rho a_0(980,1450,1710)$, $K^{*}\left(892{)}^{\pm }{K}^{*}\right(700,1430,1950{)}^{\mp }$, and $\omega f_0\left(500\right)$, which provide useful information in the $SU(3{)}_f$ analysis.