Production Rate of Glueball-like $X(2370)$ in $J/\psi$ Radiative Decay

Abstract

$X(2370)$ falls in the mass region of the lowest pseudoscalar glueball predicted by lattice QCD studies and its decay properties are similar to those of $\eta_c$. A previous lattice QCD study finds that the pseudoscalar glueball ($G_{0^-}$) and the lowest pseudoscalar charmonium ($c\bar{c}(0^-)$) can mix with a small mixing angle $\sin\theta$. It is therefore possible that $X(2370)$ and $\eta_c$ are admixtures of $G_{0^-}$ and $c\bar{c}(0^-)$. In this picture, although $\mathrm{Br}(J/\psi\to\gamma\eta_c)$ is insensitive to the small $\sin\theta$, $\mathrm{Br}(J/\psi\to \gamma X(2370))$ can be enlarged drastically by the mixing due to the much larger kinematic factor for $J/\psi\to \gamma X(2370)$ and the much larger transition form factor for $J/\psi\to \gamma (c\bar{c}(0^{-}))$. Depending on the value of $\sin\theta$, $\mathrm{Br}(J/\psi\to \gamma X(2370))$ can be much larger than that of the pure pseudoscalar glueball, namely, $2.3(8)\times 10^{-4}$ that is predicted by a quenched lattice QCD calculation. Present results by BESIII favor a small mixing angle of $\mathcal{O}(1^\circ)$, which can be further constrained by more measurements of $X(2370)$ decays if the mixing picture applies here.